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Development of MEMS-Based Piezoelectric Cantilever Arrays for Vibrational Energy Harvesting

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PAGE 1

1 DEVELOPMENT OF MEMS-BASED PIEZO ELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY HARVESTING By ANURAG KASYAP V. S. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 Copyright 2007 by Anurag Kasyap V.S.

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3 To my father

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4 ACKNOWLEDGMENTS Financial support for the research project was provided by NASA. First, I thank my advisor Dr. Louis N. Catta festa for his guidance and support, which was vital for completing my dissertation. I also th ank my co-advisor Dr. Mark Sheplak for advising and guiding me with various aspect s of the project. I would also like to thank Dr. Toshi Nishida for helping me understand the electrical e ngineering aspects of the project. Drs. Khai Ngo and Bhavani Sankar deserve special thanks for finding time to help me out with the project whenever I approached them. I thank all the members of the Interdisciplinary Microsystems group, especially fe llow students Steve Horowitz and Yawei Li for their help with my research. I also thank the University of Florida Depa rtment of Aerospace Engineering, Mechanics, and Engineering Science fo r their financial support. Finally, I want to thank my family and friends for their endless support, particularly my parents whose affection and encouragement has been the driving force for my success as a student and more importantly as a person.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .......10 ABSTRACT....................................................................................................................... ............16 1 INTRODUCTION......................................................................................................... ..........18 Energy Reclamation............................................................................................................. ...18 Energy Resources and Ha rvesting Technologies............................................................19 Self-Powered Sensors......................................................................................................21 Vibration to Electrica l Energy Conversion.............................................................................23 Transduction Mechanisms...............................................................................................24 Electrodynamic transduction....................................................................................29 Electrostatic transduction.........................................................................................31 Piezoelectric transduction........................................................................................33 Microelectromechanical Systems (MEMS)............................................................................40 Piezoelectric MEMS........................................................................................................45 Objectives of Present Work....................................................................................................46 Organization of Dissertation...................................................................................................46 2 PIEZOELECTRIC CANTILEVER BEAM MODELING AND VALIDATION.................. 48 Piezoelectric Composite Beam...............................................................................................49 Analytical Static Model........................................................................................................ ..54 Static Electromechanical Load in the Composite Beam.................................................55 Experimental Verification of the Lumped Element Model....................................................70 3 MEMS PIEZOELECTRIC GENERATOR DESIGN.............................................................84 Power Transfer Analysis........................................................................................................ .84 Nondimensional Analysis.......................................................................................................87 Scaling Theory...............................................................................................................110 Validation of Scaling Theory........................................................................................116 Extension to MEMS.............................................................................................................122 Design of Test Structures..............................................................................................122 Test devices............................................................................................................127 4 DEVICE FABRICATION AND PACKAGING.................................................................. 130 Process Flow................................................................................................................... ......130 Process Traveler............................................................................................................147

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6 Packaging...................................................................................................................... ........148 Vacuum Package...........................................................................................................148 Open Package................................................................................................................150 5 EXPERIMENTAL SETUP................................................................................................. ..153 Ferroelectric Characterization Setup....................................................................................153 Piezoelectric Characterization.......................................................................................155 Electrical Characterization....................................................................................................156 Blocked Electrical Capacitance, ebC and Dielectric Loss, eR......................................156 Mechanical Characterization................................................................................................159 Electromechanical Characterization.....................................................................................163 Open Circuit Voltage Characterization................................................................................164 Voltage and Power Measurements.......................................................................................166 6 EXPERIMENTAL RESULTS AND DISCUSSION............................................................ 168 Ferroelectric Characterization..............................................................................................168 Blocked Electrical Impedance Measurements......................................................................179 Lumped Element Parameter Extraction................................................................................185 Method 1....................................................................................................................... .188 Method 2....................................................................................................................... .197 Method 3....................................................................................................................... .201 Results and Discussion.........................................................................................................206 PZT-EH-09....................................................................................................................206 PZT-EH-07....................................................................................................................212 Summary and Discussion of Results....................................................................................215 7 CONCLUSIONS AND FUTURE WORK........................................................................... 224 Conclusions.................................................................................................................... .......224 Future Work.................................................................................................................... ......230 Second Generation Design Procedure...........................................................................232 Electromechanical Conversion Metrics.........................................................................233 A EULER-BERNOULLI BEAM ANALYSIS: VARIOUS BOUNDARY CONDITIONS .. 237 Euler Bernoulli Beam...........................................................................................................237 Cantilever Beam (Clamped-Free Condition).................................................................237 Clamped-Clamped Beam (Fixed-Fixed Condition)......................................................239 Pin-Pin Beam (Simply Supported)................................................................................243 B DISSIPATION MECHANISMS FOR A VIBRATING CANTILEVER BEAM................ 248 Introduction................................................................................................................... ........248 Overall Mechanical Quality Factor......................................................................................249 Dissipation Mechanisms.......................................................................................................250

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7 Airflow Damping...........................................................................................................251 Intrinsic region :.....................................................................................................252 Molecular region :..................................................................................................252 Viscous region........................................................................................................253 Support Losses...............................................................................................................254 Surface Dissipation........................................................................................................254 Volume Loss..................................................................................................................254 Squeeze Damping Loss.................................................................................................255 Thermoelastic Dissipation.............................................................................................255 Analytical model....................................................................................................257 C TRANSFORMATION OF COORDINATES FOR RELATIVE MOTION........................265 D ELECTRICAL IMPEDANCE FOR A PIEZOELECTRIC MATERIAL............................ 267 E CONJUGATE IMPEDANCE MATCH FOR MAXIMUM POWER TRANSFER............. 270 F UNDESTANDING THE PHYSICS OF THE DEVICE....................................................... 274 G FABRICATION LAYOUTS............................................................................................... ..274 LIST OF REFERENCES.............................................................................................................282 BIOGRAPHICAL SKETCH.......................................................................................................292

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8 LIST OF TABLES Table page 1-1 Conjugate power variables for different energy domains..................................................26 1-2 Vibration based energy harveste rs characterterized for power..........................................40 2-1 Material properties and dimensi ons for a homogenous aluminum beam..........................61 2-2 Material properties and dimensions for a piezoelectric composite aluminum beam.........65 2-3 Material properties and dimensi ons for a homogenous aluminum beam..........................70 2-4 Measured and calculated parame ters for the homogenous beam.......................................71 2-5 Measured and calculated parameters fo r the homogenous beam with a proof mass.........74 2-6 Material properties and dimensions for a piezoelectric composite aluminum beam.........75 2-7 Measured and calculated valu es for a PZT composite beam.............................................76 2-8 Measured and calculated parameters for a PZT composite beam with a proof mass........77 2-9 Comparison between experimental and th eoretical values for power transfer..................82 3-1 List of all device variables that are described in the electromechanical model.................88 3-2 Dimensional representation of all the device variables.....................................................89 3-3 Primary variables used in the dimensional analysis..........................................................90 3-4 List of independent groups............................................................................................93 3-5 Final set of nondimensional groups involving response parameters...............................109 3-6 Material dimensions and properties of composite beam for FEM validation..................117 3-7 Static lumped element parameters fr om FEM and LEM to validate the scaling analysis....................................................................................................................... ......121 3-8 Properties and dimensions used for designing MEMS PZT devices...............................123 3-9 Material properties of pi ezoelectric composite beam......................................................128 3-10 Designed MEMS PZT structures.....................................................................................129 4-1 Residual stress measurements for th e PZT pattern process (source : ARL)....................133

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9 4-2 DRIE recipe conditions for top side etch.........................................................................136 4-3 DRIE recipe conditions for back side etch......................................................................142 4-4 Process traveler for the fabrica tion of micro PZT cantilever arrays................................147 5-1 Reported polarization results (ref: ARL).........................................................................156 5-2 Data acquisiton parameters for mechanical characterization...........................................163 5-3 Data acquisiton parameters for mechanical characterization...........................................164 5-4 Data acquisiton parameters for mechanical characterization...........................................166 6-1 Comparison of ARL's reported hysteresi s parameters with measured values.................178 6-2 Dielectric parameters of all tested design geometries on the device wafer.....................182 6-3 LEM parameters extracted using experimental data.......................................................185 6-4 LEM parameters extracted using Method 1.....................................................................197 6-5 LEM parameters extracted using Method 1.....................................................................201 6-6 LEM parameters extracted using Method 3.....................................................................206 6-7 LEM parameters extracted for PZT-EH-09-01................................................................207 6-8 Extracted LEM parameters for PZT-EH-09-03...............................................................212 6-9 LEM parameters extracted for PZT-EH-07-02................................................................212 6-10 Comparison between theory and experiments for PZT-EH-07.......................................215 6-11 Comparison between theory and ex periments for PZT-EH-09 devices..........................216 6-12 Quality factors for PZT MEMS devices..........................................................................219 A-1 LEM parameters and bending strain for various beams s ubjected to a point load..........246 A-2 LEM parameters and bending strain for various beams subjected to uniform load........247

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10 LIST OF FIGURES Figure page 1-1 Schematic of a typical vibrati on to electrical energy converter.........................................24 1-2 An electromagnetic vibration-powered generator (adapted from Glynne-Jones and White 2001).................................................................................................................... ...30 1-3 Deformation of a piezoceramic material under the influence of an applied electric field.......................................................................................................................... ..........33 1-4 A nonlinear piezoelectric vibration powere d generator (adapted from Umeda et al, 1997).......................................................................................................................... ........38 1-5 Schematic of the proposed cantilever configuration for energy reclamation....................44 2-1 Schematic of a piezoelectric composite beam subject to a base acceleration....................50 2-2 Overall equivalent circuit of composite beam...................................................................52 2-3 Schematic of the piezoelectric cantilever composite beam...............................................55 2-4 Free body diagram of the overall configuration................................................................56 2-5 Free body diagram of the composite beam wh ere the self weights are replaced with equivalent loads............................................................................................................... ..57 2-6 Static model verified with the ideal so lution for a homogenous beam solved for self weight......................................................................................................................... ........61 2-7 Static model verified with the ideal solution for a homogenous beam solved for tip load........................................................................................................................... ..........62 2-8 Deflection modeshape for a composite beam subjected to an input voltage.....................66 2-9 Experimental setup for verifying the el ectro-mechanical lumped element model for meso-scale cantilever beams..............................................................................................72 2-10 Comparison between experiment and theory for tip deflection in a homogenous beam (no tip mass).................................................................................................................. .....73 2-11 Comparison between theory and experime nts for the tip deflection in a homogenous beam with tip mass.............................................................................................................74 2-12 Frequency response of a piezoelectric composite beam (no tip mass)..............................77 2-13 Frequency response for a piezoelectric composite beam (mp=0.476 gm).........................78

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11 2-14 Output voltage for an input acceleration at the clamp.......................................................80 2-15 Output voltage for varying resistive loads.........................................................................81 2-16 Output power across varying resistive loads.....................................................................82 3-1 Thvenin equivalent circuit for the energy reclamation system........................................85 3-2 Schematic of the MEMS PZT device................................................................................88 3-3 Meshed PZT composite cantile ver beam for FE M validation.........................................117 3-4 Short circuit natural freque ncy for a PZT composite beam.............................................118 3-5 Short circuit compliance for a PZT composite beam.......................................................119 3-6 Effective mechanical mass for a PZT composite beam...................................................120 3-7 Effective piezoelectric coefficient for a PZT composite beam........................................121 3-8 Schematic of a single PZT composite beam....................................................................123 4-1 Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer...................................................131 4-2 Sputter deposit Ti/Pt (20 nm /200 nm ) as bottom electrode..............................................131 4-3 Spin coat sol-gel PZT (125/52/48) over th e wafer using a spin-bake-anneal process.....132 4-4 Deposit and pattern Pt for top electrode using liftoff......................................................132 4-5 Pattern opening for access to bottom electrode and wet etch PZT using PZT Etch mask........................................................................................................................... ......133 4-6 Ion milling of PZT and bottom electrode using Ion Milling mask as pattern..................133 4-7 Deposit Au (300 nm ) and pattern bond pads using B ond Pads mask and wet etching....134 4-8 Sidewall profiles on topside of a 4" Si test wafer............................................................136 4-9 Wet etch exposed oxide with BOE and DRIE to BOX from top.....................................137 4-10 Sidewall profiles for backside etching using DRIE.........................................................138 4-11 Curved edges during backside DRIE...............................................................................139 4-12 Onset of silicon grass du ring a backside etch run............................................................140 4-13 Sidewall profiles for a back side etch on a test wafer.......................................................141 4-14 Pattern proof mass on the b ackside and DRIE to BOX...................................................143

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12 4-15 Schematic of final released device...................................................................................144 4-16 SEM pictures of a PZT-EH-07 released device...............................................................145 4-17 SEM pictures of a PZT-EH-09 released device...............................................................145 4-18 Sidewall profiles of released devices...............................................................................146 4-19 Schematic of the bottom of vacu um package for MEMS PZT devices...........................149 4-20 Schematic of glass t op for vacuum package....................................................................149 4-21 An isometric view of the overall vacuum package..........................................................150 4-22 Schematic of open package for MEMS PZT devices......................................................151 4-23 Picture of the open package.............................................................................................152 5-1 Schematic for ferroelectric characterization....................................................................154 5-2 Experimental setup for fe rroelectric characterization......................................................155 5-3 Schematic for blocked elect rical impedance measurement.............................................158 5-4 Experimental setup for electri cal impedance characterization........................................158 5-5 Experimental setup for mechanical and electromechanical characterization..................160 5-6 Experimental setup for vibration a nd velocity measurements with LV..........................161 5-7 Experimental setup for open circuit voltage measurements............................................165 5-8 Experimental setup for open circuit voltage measurements............................................165 5-9 Experimental setup for volta ge and power measurements..............................................166 6-1 A typical P-E hysteresis loop for a piezoel ectric material (adapted from Cady 1964)....169 6-2 A typical -E curve for a piezoelectric material..............................................................170 6-3 Polarization, capacitance and input voltage waveforms for PZT-EH-02-1-1..................171 6-4 Hysteresis plots for PZT-EH-02-1-1................................................................................172 6-5 Pr and Vc for different applied vo ltages for PZT-EH-02-1-1...........................................173 6-6 Normalized Ceb for PZT-EH-02-1-1 during the hysteresis test.......................................174 6-7 Leakage current for PZT-EH02-1-1 subjected to 10V DC.............................................175

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13 6-8 Poling of PZT-EH-02-1-1 at 5V for different times........................................................176 6-9 Poling of PZT-EH-02-1-1 at different temperatures........................................................178 6-10 Variation of Ceb and tan with dc bias and a constant sinusoid, 500 mV at 100 Hz.......179 6-11 Variation of Ceb and tan with source amplitude at 100Hz.............................................180 6-12 Ceb and r for MEMS PZT devices on wafer be fore release for a) PZT-EH-01 (106 geometries) b) PZT-EH-02 (16 geometries ) c) PZT-EH-03 (15 geometries) d) PZTEH-04 (14 geometries) e) PZT-EH-05 (16 geometries)..................................................183 6-13 Ceb and r for MEMS PZT devices on wafer be fore release for a) PZT-EH-06 (150 geometries) b) PZT-EH-07 (12 geometries ) c) PZT-EH-08 (22 geometries) d) PZTEH-09 (108 geometries)...................................................................................................184 6-14 Flowchart for method 1 to extract the LEM parameters from the experimental data......190 6-15 Low frequency electromechanical response data compared with curve fit to extract dm............................................................................................................................... ......194 6-16 Comparison between experiment and LEM based curve fit around resonance for a) electromechanical response b) short-circuit mechanical response..................................195 6-17 Low frequency curve fit compared with experiment to extract Ceb.................................195 6-18 Comparison between experiment and curve fit for low frequency open circuit voltage response to extract Mm.....................................................................................................196 6-19 Experimental data and curve fits for op en circuit voltage response compared around resonance...................................................................................................................... ....196 6-20 Flowchart for parameter extraction using Method 2........................................................199 6-21 Experimental data and curve fits for open circuit voltage response and free electrical impedance compared around resonance..........................................................................201 6-22 Flowchart for LEM parameter extraction implementing Method 3.................................203 6-23 Comparison between experiment and LEM based curve fit for short circuit mechanical and electromechanical response around resonance......................................205 6-24 Experimental data and curve fits for op en circuit voltage response compared around resonance...................................................................................................................... ....205 6-25 Comparison between model and experime nts for PZT-EH-09-01. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance

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14 response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................208 6-26 Comparison between model and experime nts for PZT-EH-09-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................209 6-27 Comparison between model and experime nts for PZT-EH-09-03. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................210 6-28 Comparison between model and experime nts for PZT-EH-09-04. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................211 6-29 Comparison between model and experime nts for PZT-EH-07-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................213 6-30 Comparison between model and experime nts for PZT-EH-07-03. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive load s at resonance...................................................................................214 A-1 Schematic of a cantilever beam.......................................................................................237 A-2 A schematic of clamped-clamped beam..........................................................................240 A-3 Free body iagram of a clamped-clamped beam...............................................................240 A-4 Schematic of a pin-pin beam............................................................................................243 A-5 Free body diagram for a simply supported beam.............................................................243 B-1 A simple schematic of the cantilever beam.....................................................................251 C-1 Vibrating cantilever beam in an accelerating frame of reference....................................265 D-1 Blocked electrical impedance in a parallel network representation.................................267 D-2 Blocked electrical impedance in a series network representation...................................269 E-1 Thevenin equivalent representation conn ected to a external complex impedance..........270 E-2 Thevenin equivalent representati on connected to a resistive load...................................273

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15 F-1 Schematic of the composite beam energy harvester........................................................275 F-2 Free body representation of the device as a two degree of freedom system....................275 F-3 Electromechanical circuit repres entation of the energy harvester...................................276

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16 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEVELOPMENT OF MEMS-BASED PIEZO ELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY HARVESTING By Anurag Kasyap V.S. May 2007 Chair: Louis Cattafesta Cochair: Mark Sheplak Major: Aerospace Engineering In this dissertation, the development of a first generation MEMS-based piezoelectric energy harvester is presented that is designed to convert ambient vibrations into storable electrical energy. The objective of this work wa s to model, design, fabricate and test MEMSbased piezoelectric cantilever ar ray structures to harvest pow er from source vibrations. The proposed device consis ts of a piezoelectric co mposite cantilever beam ( 2SiSiOTiPtPZTPt ) with a proof mass at one end. The pr oof mass essentially translates the input base acceleration to an effective deflecti on at the tip relative to the clamp, thereby generating a voltage in the piezoelectric layer (using 31d mode) due to the induced strain. An analytical electromechanical lumped element mode l (LEM) was formulated to accurately predict the behavior of the piezoelectric composite beam until the first resonance. First, macro-scale PZT composite beams were built and tested to validate the LEM. In addition, a detailed non-dimensi onal analysis was carried out to observe the ov erall device performance with respect to va rious dimensions and properties. Various first generation test structures were designed using a parametric sear ch strategy subject to fi xed vibration inputs and constraints.

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17 The proposed test structures thus designe d using the electromechanical LEM were fabricated using standard sol gel PZT and conventional surface and bulk micro processing techniques. The devices have b een characterized with various frequency response measurements and the lumped element parameters were extracte d from experiments. Finally, they were tested for energy harvesting by measuring the output vol tage and power at resonance for varying resistive loads.

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18 CHAPTER 1 INTRODUCTION This dissertation discusses the modeling, desi gn, fabrication, and ch aracterization of an array of micromachined piezoelectric power ge nerators to harness vibration energy. The reclaimed power is rectified and stored using a power processor (Taylor et al. 2004, Kymissis et al. 1998) for subsequent use by, for example, sensor s. The details of this concept are discussed in subsequent sections. This chapter begins with an introduc tion to energy reclamation, various available resources, and harvestin g technologies. Then, a detail ed description is presented concerning energy reclamation from vibration and its uses in various fields su ch as self-powered sensors, human-wearable electronics and vibrati on control. Finally, it concludes with motivation for microelectromechanical systems (MEMS) and pi ezoelectricity as the tools for this research. An in-depth literature survey is presented to fam iliarize the reader with the previous and current work in these fields. Energy Reclamation Conservation of energy is a fundamental con cept in physics al ong with the conservation of mass and Newtons laws. The law of conservation of energy states that energy can neither be created nor destroyed but only conver ted from one form to another. A useful description of this law in a thermodynamic system is the first law of thermodynamics. It states that the difference between the total rate of inflow of energy into a system minus the total rate of outflow of energy from the system (to the surroundings) equals the tim e rate of change of energy contained within the system. Therefore, energy reclamation, by de finition, relates to converting any form of energy that is otherwise lost to the surroundings into some form of useful power.

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19 Energy Resources and Harvesting Technologies There are two classes of available energy so urces, renewable and non-renewable. Nonrenewable sources, as the name sugge sts, include all that have a li mited supply such as oil, coal, natural gas, etc. These sources take thousands or millions of years to form naturally and cannot be replaced once consumed. They have constitu ted the major part of the United States (U.S.) power supply for a long time. But, with in creasing technology and so cietys ever-growing consumption of energy, these sources could soon be exhausted (National Energy Policy Report, 2001). Hence, it is an ecological and economical necessity to in vestigate alternate sources of energy to meet societal demands. Consequently, re search in the past few decades has focused on using an alternate form, called renewable resources to meet the demand, such as optical, solar, tidal, etc. Jan Krikke, in his editorial article in Per vasive Computing (2005) reviews the current situation in energy harvesting technologies. Ma ny companies in the US, Europe, and Japan are steadily involved in this area as there exists a general fascination with energy scavenging from ambient sources. Many energy harvesting concepts are already available su ch as a self-reliant house (powered by solar energy th at operates all appliances in the house) and a camel fridge, which uses solar energy to operate a refrigerator used to store (below o8 C ) and transport vaccines in African nations. Previous studies have successfully shown th at energy can be recl aimed from renewable sources such as solar and tidal energy (Saraiva 1989). Solar ce lls are an existing technology that is extensively used in self-pow ered watches, calculators, and r ooftop modules for houses. Solar energy has also been harvested on a smaller scale from an array of micro-fabricated photovoltaic cells to produce an overall open circuit voltage of 150 V and a short circuit current of 2.8 A

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20 (Lee et al. 1995). While solar energy has been widely explored and implemented, it becomes difficult to generate power in dark areas. Even though renewable sources can serve as a substitute for the usual power supply resources, energy is still wasted in the form of heat, sound, light and vibrations that can be further reclaimed, at least partially, for future use. For example, thermal energy was generated from a 20.750.9 cm bismuth-telluride thermoel ectric junction to produce 23.5 W for a temperature difference of 20 K (Stark and Stordeur, 1999). Qu et al. in 2001 designed and fabricated a thermoelectric generator, 316200.05 mm consisting of multiple micro Sb-Bi thermocouples embedded in a 50 m epoxy film capable of producing 0.25 V from a temperature difference of 30 K. Kiely et al. (1991; 1994) designed a low cost miniature thermoelectric generator consisting of a silicon on sapphire and silicon on quartz substrate. Another thermoelectric power generator based on silicon technology produced 1.5 W with a temperature difference of 10 C (Glosch et al. 1999). Of all th e renewable sources, optical and thermal energy have been the most popular and widely implemented, even in micro power requirements. However, in applications where li ght and thermal energy are not readily available, alternate sources need to be cons idered such as mechanical energy. In addition, an advantage for mechanical energy conversion over thermal conver sion is that, ideally, it does not require any heat isolation. In addition, scaling ther mal systems to microscale possesses fundamental limitations such as thermal related noise due to thermal fluctuations, temperature based adsorption, etc. (Devoe 2003). In recent years, extensive research ha s been conducted on harvesting undesirable vibrational energy. Although most efforts have b een in the area of meso scale energy harvesting, the focus on microscale has gained importance late ly. The energy thus claimed from vibrational

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21 sources can be stored and later used to power various devices. In the past, efforts in energy reclamation from vibrations have largely focu sed on the available energy in human ambulation (Starner 1996). Reclaiming energy from huma n ambulation has generated immense interest primarily because of its ability to power artificia l organs and human wearable electronic devices. Growing interest in the area of human-wearable electronic devi ces creates a need for portable power sources for these devices. Starner and Pa radiso (2005) describe s various sources from humans for energy harvesting such as body heat, breath, blood-pressure, walking, etc. In addition, heel strike, limb moveme nt, and other gait-related activities are useful sources of strain energy and can be used as alternate methods for powering artificial organs (Antaki et al. 1995). This could replace conventional portable batterie s that are currently restricted by energy limitations, especially for prolonged usage. In addition, batteri es are often bulky and possess a limited shelf life and could be po tentially hazardous due to chem icals. The development of MEMS technology has led to a wide range of app lications for micro actuators and sensors (see, for example, Senturia 2000). It also has enable d implantation of these devices into various host structures, such as medical implants and embedde d sensors in buildings and bridges (Mehregany and Bang 1995). In most of these applications, the devices need to be comp letely isolated from the outside world. These remote devices, alon g with their accompanying circuitry, have their own power supply that is usually powered by batteries. The strides achieved in battery technology have not sufficiently matched the improvements in in tegrated circuit technology. Therefore, developing a microscale self-contained power suppl y offers great potential for applications in remote systems. Self-Powered Sensors The ever-reducing size of CMOS circuitry and correspondingly lowe r power consumption have also provided immense opportunities to desi gn and build micro power generators that can

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22 be ideally integrated with CMOS. Simultaneous research is also being carried out to develop new chip technology to lower the power require ment for electronic equipment (Krikke 2005). The need for self-contained power generators has led to the development of self-powered systems that is an important application fo r energy reclamation, and is currently gaining widespread importance (Shenck and Paradiso 2001) Self-powered systems possess an inherent mechanism to extract power from the ambient environment for their operation. The main objective of self-powered system s is to utilize a generator th at can convert energy from an ambient source to electrical ener gy as long as sufficient energy is available in the ambient source. Consequently, the primary features of self-powered systems in clude power generation, energy extraction, and storage. Ideally, a self -powered device should possess high power density for given size constraints. Attempts to build pe rpetual motion machines date back to as early as the 13th century when the conservation laws had not yet been formulated Glynne-Jones and White (2001) provide a review on available energy resources for self-powered sensors such as vibrations, optical, thermoelectric, etc. Next, some of the relevant work carried ou t in the field of self-powered sensors is examined. As mentioned earlier, heel strike is a resource for strain energy that can be electromechanically transformed into electri cal power. Consequently, shoe-mounted devices have been developed and tested that convert stra in energy induced during heel strike and store it as electrical energy. Kymissis et al (1998) and Shenck and Para diso (2001) designed two novel piezoelectric devices to harness power that were embedded in a shoe. Furthermore, vibrations when available are excellent potential sources for energy harvesting. Meso-scale energy reclamation approaches include rotary generato rs (Lakic 1989), a moving coil electromagnetic generator (Amirtharajah 1998), and a dielectric elastomer w ith compliant electrodes (Pelrine

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23 2001). Single meso-scale piezoelectric cantilevers (Ottman et al. 2002) and stacks (Goldfarb and Jones 1999) have been investigated for energy reclamation but were not operated in a standalone, self-powered mode. Another source for power harvesting is mechanical energy from fluid flow. Taylor et al. (2001) designed an energy harvesting eel that was approximately 1 m long using a piezoelectric polymer to convert fluid flow and vortex-induced strain to generate power. In addition, Allen and Smits (2001) investigated the feasibility of u tilizing a piezoelectric membrane in the wake of a bluff body to indu ce oscillations in the structure generating a capacitance build-up that acts as a voltage source to power a battery in a remote location. Power generation from ocean waves has also been inves tigated involving very la rge-scale piezoelectric generators (Smalser 1997). As a result, there is a clear indication that energy reclamation from strain energy is a promising field in terms of research and applications. The focus of this dissertation is to study the possi bility of using vibrational m echanical energy as a potential source for energy reclamation on a micro scale. Ambient vibration sources, such as household appliances, machinery equipment, and HVAC ducts typically occur at frequencies in the range of 100s of Hz with an acceleration amplitude of 1-10 m/s2 (Roundy et al. 2003). Vibration to Electrical Energy Conversion Continuing the discussion on converting vibrati onal energy to electricity, this can be achieved using a transduction mechanism that eff ectively converts energy from the mechanical domain to the electrical domain. A simple schema tic of a power generator based on vibration is shown in Figure 1-1. The device consists of a sp ring-mass-damper system acting as a single degree of freedom system with an input vibratio n that results in an effective displacement zt.

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24 Figure 1-1: Schematic of a typical vibration to el ectrical energy converter. The following equation is used to represent the behavior of the above system that basically converts the kinetic energy of a vibr ating structure to electrical en ergy by virtue of the relative motion between the base and the inertial mass. M zRzKzMy (1.1) where z is the relative deflection, y is the input displacement, M is the inertial mass, K is the spring constant and R is the effective damping in the sy stem that accounts for mechanical and electrical losses. The above model does not in clude nonlinear effects and is thus valid only under the constraints of linear system theory. It also does not specify the electromechanical transduction mechanism with which the kinetic en ergy is converted to electrical power. These mechanisms are discussed in detail in the following sections. Transduction Mechanisms Vibrational energy reclamation can be achieve d conceptually using different transduction mechanisms. Any transduction mechanism rela tes to energy conversi on from one form to

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25 another. For example, it can involve coupli ng of two or more energy domains such as electrostrictive coupling (Uchino et al 1980), electromagnetic (Hanagan 1997; Kato 1997) and electromechanical coupling (Lee 1990). In his Ph.D. dissertation, Roundy (2003) calculates the theoretical maximum and the practical maximum fo r the energy densities of various transduction mechanisms, namely piezoelectric, electrostatic, and electromagnetic. The expressions were obtained from the basic governing equations of each of the materials and calculated using maximum yield stress for the piezoelectric, the electric field for capacitive, and the maximum magnetic field for electromagnetic materials as the respective upper limits. In his summarized results, he found that piezoel ectric materials possess a practic al maximum energy density of 317.7 mJcm, which is almost four times that of the other transducers. The following paragraphs provide some basic discussion on tran sducer theory and explain electromechanical transduction mechanism in detail. A typical transducer is repres ented using different energy domains associated with power flow from one domain to another. Modeling the energy transfer between domains enables a better representation of the transducer behavior The net power flow between two elements describing the device is repres ented as a product of two te rms called the conjugate power variables (Senturia 2000). Pef (1.2) where e is effort and f is flow. Next, a generalized mome ntum can be defined by integrating the effort over time and is represented as 00.tpetdtp (1.3)

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26 Similarly, a generalized displacement is defined th at is associated with the flow variable, given by 00.tqftdtq (1.4) Here, 0 p and 0 q are the initial momentum and displacement in the element respectively. Consequently, the energy in the element is give n by the product of flow and momentum or effort and displacement as Eqepf (1.5) The ratio between effort and flow results in th e generalized complex impedance of the element. efZ (1.6) Some examples of conjugate power variables for various energy domains (Senturia 2000) are listed in Table 1-1. Table 1-1: Conjugate power variab les for different energy domains. Angular velocity Torque Rotational Mechanical Current Voltage Electrical Flux rate mmf Magnetic Entropy rate Temperature Thermal Volumetric flow Pressure Incompressible flow Velocity Force Translational Mechanical Flow Effort Energy domain Angular velocity Torque Rotational Mechanical Current Voltage Electrical Flux rate mmf Magnetic Entropy rate Temperature Thermal Volumetric flow Pressure Incompressible flow Velocity Force Translational Mechanical Flow Effort Energy domain iIA F N VV TK 1SJKs 31, qQms V A 2PNm 1, uUms N m1 s A transducer is broadly cl assified into energy conser ving and non-energy conserving transducers (Hunt 1982, Fischer 19 55). They can be classified further on factors such as linearity, reciprocity etc. Electromechanical tr ansducers are classified based on force generation

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27 due to the interaction between electric field and charge or magnetic field and current. For electromechanical transduction, there are five major linear energy conserving transducers, namely, electrodynamic, electrostatic, piezo electric, magnetic, and magnetostrictive. All linear conservative transducers are ge nerally represented using simple two-port network theory (Rossi 1988) expressed in im pedance or admittance notation. Here, the impedance form is explained to discuss the various transduction mechanisms. The governing equations for an electromechanical transducer are .ebem memoZT VI TZ FU (1.7) The blocked electrical im pedance is defined as, 0,eb UV Z I (1.8) where 0U indicates that the device is mechanically restricted or blocked from any motion. Alternatively, the freeelectrical impedance, 0 ef FV Z I (1.9) is defined as the electrical impedance when th e device is free or not subjected to any mechanical load. The coupling terms are define d as open circuit electro mechanical transduction impedance and the blocked mechanical-electro transduction impedance, represented as 0 em IV T U (1.10) and 0,me UF T I (1.11)

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28 respectively. The electromechani cal transducer is defined to be reciprocal when the cross diagonal coefficients in Eq. (1.7) are equal, meemTT mo Z is defined as the open-circuit mechanical impedance expressed as the ratio betw een mechanical force and resulting velocity for zero current 0.mo IF Z U (1.12) Alternatively, the ratio between the force and velocity while preventing any voltage from building up defines the short ci rcuit mechanical impedance 0.ms VF Z U (1.13) Both forms of mechanical and electr ical impedances expressed in Eqs. (1.8)-(1.9) and Eqs. (1.12) -(1.13) are related to each other as 21msmoZZ (1.14) and 21,efebZZ (1.15) where 2 is defined as the electrom echanical coupling co efficient that rela tes the amount of energy converted from electric domain to m echanical domain. The coupling coefficient represents the ideal effectiv eness of an electromechanical transducer is defined as 2.emme ebmoTT Z Z (1.16) Two-port network theory can also be represen ted with a corresponding set of coefficients in the admittance form. Fo r reciprocal transducers, emmeTT which implies that the

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29 electromechanical conversion from an applied volta ge to velocity and app lied force to resulting current are equal. Electromechanical transducers are commonly re presented using equivalent circuits with lumped elements and will be explained in detail in Section 2.1. Some of the widely used electromechanical transduction mechanisms fo r energy harvesting involve electromagnetic (specifically electrodynam ic), electrostatic and piezoelectr ic phenomenon that are explained next. Electrodynamic transduction Electrodynamic transduction occurs when ener gy conversion is produced by motion of a current carrying electric conductor subject to a constant magnetic field. This phenomenon is characterized by Laplaces law (Beranek 1986, Tilmans 1996), which defines the force on the electric conductor in terms of the current and the magnetic field through the relation .magFLIB (1.17) Here, magF is termed as Lorentz force, I is the current, B is the magnetic field and L is the length of the conductor. Conversely, the motion of the conductor in the presence of a magnetic field leads to a voltage generation acro ss its terminals, given by Lenzs law VLUB (1.18) In Eq. (1.18), U is the velocity of the conductor and V is the generated voltage. Combining these two laws in a two-port representation yields, 0 0 VBLI FBLU (1.19)

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30 Since eb Z and mo Z for this system are iden tically zero a direct coup ling between electrical and mechanical domains exists. So, an electrodynamic transducer is linear, reciprocal, and direct. Another mechanism called the electroma gnetic transduction is proposed in Figure 1-2 (Glynne-Jones and White 2001, Glynne-Jones et al 2004). This transduction in nonlinear, but can be linearized about its mean state to be re presented as a linear, reci procal transducer. The linearization is valid for small variations in current and magnetic fiel d that are possible by biasing the electrical conductor with an initial current (Tilmans 1997). Figure 1-2: An electromagnetic vibration-powered generator (adapted from Glynne-Jones and White 2001). El-Hami et al. (2001) designed an electromagne tic generator comprised of a magnetic core mounted on the tip of a steel beam. When an i nput vibration is supplied to the structure, the beam vibrates, thereby inducing current in th e coil. They report an output power of 0.53 mW for an input displacement magnitude of 25 m at 322 Hz. The overall volume of the device was 30.24 cm In 2000, Li et al. presented a micr omachined generator that had a permanent

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31 magnet mounted on a spring st ructure and generated 10 W at 2 V DC for an input vibration amplitude of 100 m at 64 Hz from a volume of 31 cm Williams and Yates in 1996 designed an electromagnetic generator 5 5 1 mmmmmm that had a predicted power output of 1 W at 70 Hz and 0.1 mW at 330 Hz for an input vibration amplitude of 50 m Shearwood and Yates in 1997 designed an electromagnetic ge nerator based on a polyimide membrane 2 mm in diameter that could generate 3 W of RMS power at a resonant frequency of 4.4 kHz. Rodriguez et al. (2005) presented their work on the design optimization of an electromagnetic vibrationa l generator to scavenge W s-mWs of power in the frequency range between 10 Hz to 5 kHz. The design proposed in their wo rk consists of a movable magnet mounted on a resonant membrane that induc es a current in a fixed planar coil. Electrostatic transduction Electrostatic transduction is the conversion of energy that is produced by varying the mechanical stress to generate a potential differe nce between two electrodes. An example for this transduction is a simple parallel plate capacitor. If we assume that one plate is moving relative to the other (generally st ationary), due to an external load, the variation in ga p generates a capacitance given by eA Ct x t (1.20) where is the permittivity of the medium separating the plates, A is the area and x t is the distance between the plates that changes about an initial mean distance. The voltage generated between the terminals due to this is eQt Et Ct (1.21)

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32 where Qt is the accumulated charge in the capacitor. From Eq. (1.21), we know that the field has a nonlinear relation with charge and displacement, which implie s that it is nonlinear with current and velocity. In addition, the force gene rated also follows a nonlinear relation with the flow variables. However, the coupled equations can be linearized for small variations about a mean initial condition, generally achieved by appl ying a bias voltage to the plates (Rossi 1988, Tilmans 1997) or by storing a perman ent charge using an electret (Boland et al 2003). The final linearized set of equations are expressed in the two-port form as 1 1o eo o o omV jC VI jx V FU jxjC (1.22) Here, oE and o x are electric field and dist ance between the plates. mC is the mechanical compliance that relates th e force and velocity and eoC is the mean capacitance. Since the effort variables are originally calcula ted using charge and distance, j is the integration factor in the frequency domain to convert them to current and velocity. Although th e cross terms in the matrix are same, diagonal terms do exist, which im plies indirect coupling between the electrical and mechanical domains for an electrostatic tr ansducer. Hence, this system of equations represents a linear, re ciprocal and indirect transduction mechanism. In electrostatic transduction, a relative defl ection induces charge between the electrodes that can be converted to power. For example, at the micro-sc ale, a MEMS variable capacitor has been designed and fabricated to harves t vibrational energy with a chip area of 21.5 1.5 cmand a reported net power output of approximately 8 W (Meninger et al. 2001).

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33 Piezoelectric transduction Piezoelectricity, by definition, is a property of certain material s to physically deform in the presence of an electric field or, conversely, to produce an elec tric charge when mechanically deformed. Piezoelectricity occurs due to the spontaneous separati on of charge within the crystal lattice (Cady 1964). This phenome non, referred to as spontaneous polarization, is caused by a displacement of the electron clouds relative to their individual at oms, as well as a displacement of the positive ions relative to the negative ions within the crystal structure, resulting in an electric dipole. There are a wi de variety of materials that e xhibit this phenomenon, including natural quartz crystals and even human bone. During electrical polari zation, the material becomes permanently elongated in the direct ion of the poling field (polar axis) and correspondingly reduced in the tran sverse direction. Applying a vo ltage in the direction of the poling voltage produces further el ongation along the axis and a co rresponding contraction in the transverse direction subject to its Poissons ratio. This effect is depicted in Figure 1-3, which shows a piezoelectric material under th e influence of an electric field; P is the poling direction and V is the externally applied voltage. Expansion Contraction P P P VV V=0 Figure 1-3: Deformation of a piezoceramic materi al under the influence of an applied electric field. Piezopolymers and piezoceramic materials ar e typically used as transducers for piezoelectric energy harvesting appl ications. Piezoelectric mate rials possess a unique property

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34 that makes them a viable option for electromechanic al transducers. Applying an external electric field across the piezoelectric material induces a mechanical strain in the material, thereby enabling them to function as ac tuators. Conversely, when th e piezoelectric material is mechanically deformed, the resulting strain produ ces a voltage that allows them to operate as a sensor. This strain/electric field characteristic of a piezoelectric material is termed as the piezoelectric effect. Materials with good piezoelectric properties possess high coupling between the mechanical and electrical domains. This effect can be generated using piezopolymers, such as polyvinyledene fluoride (PVDF), or piezoceramics, such as lead zirconium titanate (PZT), Zinc Oxide (ZnO), Aluminum Nitride (AlN) and Barium Titanate. For any linear piezoceramic material (I EEE Standard on Piezoelectricity, 1987), the constitutive governing equati ons can be expressed as T kkjjikiSdE (1.23) and .iiqqijjDdE (1.24) In the above equations, k is the mechanical strain, j is the stress, iD is the electric displacement, iE is the electric field applied to the ceramic, kjS is the proportionality constant between the stress and strain (and is the reciproc al of the elastic modulus of any material), ij is defined as the dielectric permittivity at constant stress, and ikd is the piezoelectric coefficient. The material constants S, ,d and are defined as shown below for a piezoceramic due to its crystal structure (IEEE Standard on Piezoelectricity, 1987)

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35 111213 121113 131333 44 44 66000 000 000 00000 00000 00000 SSS SSS SSS S S S S (1.25) 15 15 31313300000 00000, 000 d dd ddd (1.26) and 11 11 3300 00 00 (1.27) For a typical piezoceramic patch, the electric field is often applied vertically across the ends of the piezoceramic in th e 3-direction, while the stress acts in the 1-direction for the composite beam. Therefore, we extract index 1k from Eq. (1.23) and 3i from Eq. (1.24), since 10, 230 30,E and 120EE Substituting the matrices for the constants and expanding the constitutive equations fo r the one-dimensional case results in 1111313SdE (1.28) and 3311333.DdE (1.29) Rewriting the above equations to expr ess strain in terms of deflection x stress in terms of the force applied F, electric field in terms of an applied voltage V, and the electric displacement in terms of charge q induced in the piezoceramic simplifies them to msm x CFdV (1.30) and

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36 ,mefqdFCV (1.31) where 0ms Vx C F is the short circuit compliance, 0ef Fq C V is the free electrical capacitance, and 0m Fx d V is an effective piezoelectric constant. Equations (1.30) and (1.31) will be used to model the composite cantilever beam in this dissertation. Equations (1.30) and (1.31) when expressed in frequency domain provide the two-port network equations in admittance matrix form as .m ms ef mjd jC UF jC jd I V (1.32) Piezoelectric materials, especi ally PZT, exhibit good strain sensitivity and possess an elastic modulus e.g., 60 GPa that is comparable to many structural materials. This property is essential for effective strain transfer between the layers, which occurs when there is a good impedance match between the piezoceramic and th e shim material. However, PZT is a brittle material and cannot withstand large strains w ithout fracturing unlike PVDF, which is very flexible and easy to handle and shape (Starner 1996). PVDF can sustain higher strains and exhibits higher stability over l ong periods of time. However, the disadvantage of using PVDF instead of PZT is the fact that it has a very low electrical permittivity and, therefore, a much lower coupling factor. Due to this, the electrical response of the device, such as output voltage, power, and overall efficiency ar e significantly lower. Also, th e working frequency range, which can be defined as the difference between the open and short circuit reso nance for the device is greatly decreased due to poor el ectromechanical coupling. A very common application of piezoceramics is that of a bending motor composed of a layer of piezoceramic bonded to a host material. The piezoelectric materi al is assumed to be

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37 firmly attached to the cantilever beam to ensure continuity in strain ac ross the interface (Crawley and deLuis 1987). Thus, when a voltage is applie d to the piezoceramic, an induced moment is concentrated at the ends of the piezoceramic patc h. The maximum induced strain is given by the expression 31pfielddE (1.33) where 31d is the piezoelectric constant, f ield E is the externally appl ied electric field, and p is the strain induced in the piezoceramic. The curvatur e of a bending motor is due to the expansion of one layer and the contraction of the other. This phenomenon occu rs due to an induced moment (Crawley and De Luis 1987) when volta ge is applied to the piezoceramic. Umeda et al. (1996, 1997) performed theoretical and experimental characterizations of a piezoelectric generator based on im pact energy reclamation. In their studies, an oscillating output voltage resulting from an input mechanical impact was rectified and stored in a capacitor. With an initial voltage of over 5 V a maximum efficiency of 35 % was achieved with a prototype generator. The worki ng principle employed in their desi gn is based on a steel ball that freely falls toward the center of a circular me mbrane consisting of bron ze and piezoceramic that vibrates on impact resulting in an alternating cu rrent in the ceramic. A schematic representing their structure is redrawn in Figure 1-4 for reference. Ramsay and Clark (2001) performed a de tailed design study on piezoelectric energy harvesting for bio-MEMS applications. Th eir design employed a simple geometry for harnessing energy from blood flow in the body. The proposed structure consisted of a square PZT-5A plate that is connected to the blood pre ssure on one side and a chamber with constant pressure on the other. Preliminary results reported an output power of 2.3 W from a 1 1 9 cmcmm plate. It was also reported in their work that the device has a mechanical

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38 advantage in converting applied pressure to working stress for piezoelectric conversion, when it functions in the 31-mode than in the 33-mode. Figure 1-4: A nonlinear piezoelectri c vibration powered generator (adapted from Umeda et al, 1997). Glynne-Jones et al. (2001) and White et al. (2001) designed a thic k film piezoelectric composite beam structure that generated 3 W of power at 90 Hz from ambient vibrations. An another paper by the same authors measured 2 W at 80 Hz for a maximum amplitude of 0.9 mm across an optimal resistive load of 333 k. Their device consisted of a macro-scale piezoelectric composite beam that was tapered along its length to ensure constant stress distribution at any point on its length. In 2004, Jame s et al. investigated two appl ications for two self-powered sensors, namely a liquid crystal display and an infra-red link to transmit the data output. The required energy for the prototypes was derived from a 0.17 g 0.23 g vibrating source at 102 Hz In another application of piezoelectric en ergy harvesting, Hausler and Stein (1984) proposed a device that basically consisted of a roll of PVDF material that can be attached between body ribs. They were designed in such a way that regular breathi ng induced a strain in

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39 the material thereby producing power. It wa s tested on a dog by surgically implanting the device, thereby generating micro-watt s of power from the breathing. Roundy and Wright in 2004 designed a piezoel ectric vibration generator consisting of a cantilever bimorph bender with a proof mass at its end. Their design was aimed at generating enough energy from a 1 cm3 to power a 1.9 GHz radio transmitter from the same vibration source. Their design wa s predicted to produce 375 W from a vibration source of 2.5 m/s2 at 120 Hz The lumped element model (LEM) introduced in their work was unconventional and used stress as the effort variable unlike force wh ich is the standard e ffort function for LEM representation. Correspondingly, stra in rate was used as the flow variable in the representation. Sood et al. (2004) developed a piezoelectric mi cro power generator (PMPG) that is based on a piezoelectric layer deposited and patt erned on a membrane consisting of SiO2 and SiNx, followed by a ZrO2 diffusion barrier. The tw o electrodes for the PZT la yer are formed using an inter-digitated top electrode (IDT) w ith Pt/Ti that makes use of the d33 mode (described later in this chapter) to extract power. The premise governing their device was that the d33 coefficient is much higher than 31d of a piezoelectric material. This pot entially results in a higher voltage, but the power density and input acceleration levels ar e not available directly for comparison with other available d31 configurations. The maximum measured power using a direct charging circuit consisting of a full-bridge rect ifier and a capacitor occurred at 5 M of load resistance. The corresponding output voltage and power were 2.4 D CV and 1.01 W respectively (Jeon et al. 2005). Another application for a self-powered piezoelectric device is a Strain Amplitude Minimisation Patch (STAMP) damper that uses pi ezoelectric elements as sensor, actuator and power source. Konak and Powles land (2001) presented their anal ysis on this device that

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40 combined the vibration control aspect of a piezo electric element along with its energy generation characteristic producing a self -powered vibration damper. Table 1-2 compiles all the reported energy ha rvesters discussed in this chapter that generated power from vibration sources using di fferent transduction mechanisms. The columns list the authors, the vibration source (which wa s mostly resonant in nature), the size of the device, and the overall power harvested. Table 1-2: Vibration based energy ha rvesters characterterized for power. Ambient source Size or Mass Power Sood et al. 10 @ 13.9 gkHz 170 260 mm 1.01 W Shearwood et al. 500 @ 4.4 nmkHz 2.5 2.5 700 mmmmm 0.3 W Chandrakasan et al. 500 @ 2.5 nmkHz 500 mg 8 W Li et al. 100 @ 64 mHz 31 cm10 W Roundy et al. 0.25 @ 120 gHz 28 3.6 8.1 mmmmmm 375 W White et al. 0.9 @ 80 mmHz 2.2 W Marzencki et al. 0.5 @ 204 gHz 2 2 0.5 mmmmmm 38 nW El Hami et al. 25 @ 322 mHz 30.24 cm0.53 mW Ching et al. 200 @ 60-110 mHz 31 cm200830 W Stark et al. 20 TK 267 mm20 W Next, a brief introduction to th e application of piezo electric materials in microsystems is presented followed by the proposed PZT based micro energy harvester. Microelectromechanical Systems (MEMS) Some of the earliest ideas about MEMS were initiated by Richard Feynman in his popular speech There is plenty of ro om at the bottom delivered in 1960 (Feynman 1992) followed by Infinitesimal machinery (Feynm an 1993). In the early 1960s, si licon gained a lot of attention as a material for microsystems due to its excel lent properties that su it both electrical and mechanical applications (Peter son 1982). Micromachining is ba sed on fabrication techniques that are used in silicon integrated chips but adds numerous other fabrication techniques as well.

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41 This ability to batch fabricate numerous such de vices in each step is a potentially significant advantage of microfabrication in MEMS. Another major advantage of MEMS is that th eir small size enables suitability for micro applications that were not possible prior to th e advent of MEMS. However, there are some significant considerations, such as packaging for structural robustness, operation in harsh environment, and power requirements that may limit their feasibility in certain applications (Angell et al. 1983). Recently, smart structures that inco rporate MEMS devices were investigated for their importance and use in aer odynamic structures, spacecraft, and vehicles for structural health monitoring (Schoess 1995). The structural configuration adop ted for the device described in this dissertation is that of a piezoelectric composite cantilever beam with an integrated proof mass that functions along the lines of conventional accelerometers. Significant research has been inve sted in understanding a cantilever beam arrangement for en ergy harvesting (Kim et al. 2004). The performance of a piezoelectri c cantilever bimorph in the flexural mode has also been analyzed for scavenging ambient vibration ener gy (Jiang et al. 2005). Their analysis calculates the output voltage, power, and the device efficiency of the composite beam with a concentrated tip mass subjected to a harmonic clamp motion. The analytical dynamic model implemented in their work can be used to design the device appropr iately to tune the frequency and increase the power. However, their work is purely theoretica l and does not provide any experimental data for validation. In addition, model assumes the end mass as a concen trated point load and does not account for its finite stiffness. This dissertation also aims to first develop an analytical model that can be used as a design tool fo r specific energy harvesting applic ations. Furthermore, the validity of the model is investigated for various canoni cal structures both at mesoscale and MEMS. It

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42 uses a different modeling technique called lu mped element modeling that, subject to the assumption that it is valid until the first bending m ode, is analytically simpler. This technique is applicable when the device is small compared to th e characteristic length scale of the distributed physical system. A cantilever configuration is chosen for our energy harvester because it provides the maximum average strain when subjected to a specific load (Appendix A). In addition, a cantilever beam has a lower natural freque ncy compared to beam s with other boundary conditions (Roundy et al 2003). An explanation of these reasons al ong with a proof is provided in Appendix A, where beams subject to different boundary conditions and loads are analyzed to estimate their average strain and na tural frequencies. Therefore, it provides an opportunity to model a slightly different configuration with va riable piezoelectric dimensions from the shim layer. In addition, the proof ma ss, which is generally large (espec ially for MEMS structures), is modeled to account for its mass and its stiffness providing a co mplete accurate model. In addition, the analytical model deve loped can be utilized as a tool to design cantilever based PZT energy harvesters for specific applications. The lumped element modeling technique is investigated in more detail in 2.1. A simple schematic of the propos ed configuration is shown in Figure 1-5. The structure basically consists of a cantilever beam with a proof mass and a thin film of piezoelectric material deposited on the beam. When the device is subj ected to base vibrations, the inertial mass vibrates relative to the base causing bending in th e beam. The strain thus resulting from this relative motion is converted to an effective output voltage by virtue of the piezoceramic transducing element. The piezoce ramic layer converts the mechanic al strain induced due to the vibrations into voltage due to the piezoelectric effect. However, even though the design of the

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43 device is similar to an accelerometer, it is impl emented and operated as a resonant sensor. In other words, the device needs to be tuned to the input vibration frequency so that it operates near its resonance frequency to generate ma ximum power, unlike a conventional accelerometer that operates across a wide bandwidth far removed fr om its resonance. Therefore, the goal of our design is to maximize the performance of the acceler ometer device at its resonance. To provide a brief insight in this area, several investigations have demonstrated the feasibility of fabricating silicon accelerometers. The basi c structure usually consisted of a silicon cantilever with a proof mass made of silicon or is gold plated to increase the sensitiv ity (Seidel and Csepregi 1984). Different transduction mechanisms for accelerometer s such as piezoresistors, piezoelectric films, and electrostatic coupling have been studied in detail, and the advantages and disadvantages of these transducers have been already been published in th e literature (Polla 1995). Piezoelectric accelerometers are of interest to us due to their low power dissipation and high electromechanical coupling (Polla 1995). Howe ver, the major drawback of this design is the difficulty in processing and in tegration with electrical circuitr y. Piezoresisti ve sensors have much higher dissipation and noise floor even tho ugh the processing is relatively straightforward and CMOS compatible. Capacitive accelerometers are favorable in many aspects such as noise, power of dissipation and ease of processing, but are sensitive to dimensional tolerance (Polla 1995; Polla et al 1996). In MEMS, it is difficul t to achieve small and accurate dimensions and a considerable uncertainty exists in the material properties and final dimensions of the device.

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44 Cantilever Beam Proof Mass ao Piezoelectric ElementV MmCms yx Rm Figure 1-5: Schematic of the proposed cantile ver configuration for energy reclamation. Initially accelerometers were fabricated using conventional bulk micromachining techniques. This approach has a clear advantage in the fact that large proof masses can be etched out of a silicon substrate. However, disadvantages with this approach arise during front to back alignment and passivation for integrated circuitr y (DeVoe and Pisano 2001). Additionally, this process consumes a larger die area for bulk etchi ng which is undesirable for batch fabrication. On the other hand, surface micromachining uses standard VLSI techniques and therefore does not pose the above problems. DeVo e and Pisano presented their work on the design, fabrication, and characterization of surface micromachined piezoelectric accelerometers (PiXLs) that consisted of thin film Zinc oxide (ZnO) as the pi ezoelectric material. In addition, they describe some guidelines for robust design based on device sensitivity and resonant frequency. Using a cantilever without a proof mass whose resonant frequency was 3.3 kHz, their results reported a sensitivity of 0.95 f Cg. Addition of a proof mass significantly improved the sensitivity to 13.3 f Cg and 44.7 f Cg, but decreased the correspondi ng resonant frequencies to 2.23 kHz

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45 and 1.02 kHz respectively. The cantile ver accelerometer was modeled using classical EulerBernoulli beam theory similar to the method adopted in our design. However, the model described in their work assumed that the thickness of the piezoelectric layer is negligible compared to the thickness of th e beam. Additionally, it assumes that the ZnO is deposited across the length of the structure and that the elastic moduli of the two materials are comparable in magnitude. This assumption holds true when the active layer is ZnO and the beam is made of silicon. However, in our design where PZT is the piezoelectric layer, the elastic moduli of the two materials are significantly different, and theref ore a detailed static electromechanical model is derived for our cantilever composite beam. Piezoelectric MEMS Silicon is an excellent material for MEMS due to its good mechanical properties such as elastic modulus and density (Peterson 1982). As a re sult, most of the devices that are fabricated in MEMS consist of a silicon substrate. Initia lly, thick film piezoelectric layers were imprinted on micromachined silicon substrates to form the desired structure (Allen et al 1989). Thicknesses in the range of 100 m can be achieved with this process leading to much higher actuation forces compared to conve ntional thin film piezoelectric micro actuators (Barth el al 1988, Terry 1988). Zinc oxide was often used as th e piezoelectric material for most applications until Lead Zirconate Titanate (Pb(Zrx, Ti1-x)O3) gained acceptance. x is the percentage composition of Zr in PZT. It was observed that when x lies between 0.52 and 0.55, the material exhibited high dielectric consta nts and electromechanic al coupling (Wang et al, 1999). PZT has been extensively studied and us ed lately due to its excellen t electromechanical coupling and piezoelectric properties. Piezoelectri c thin films in micro systems are used in a wide variety of applications such as micro actua tors (Lee et al. 1998; Zurn et al. 2001), micro mirrors (Cheng et

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46 al. 2001), micropumps (Nguyen et al 2002), micr ophones (Lee et al. 1996), micro accelerometers (DeVoe and Pisano 2001), fiber bulk wave acous tic resonators (Nguyen et al. 1998) etc. Objectives of Present Work The following chapters describe in detail th e lumped element modeling technique used to represent the composite beam and discuss its use in designing an optimal energy harvesting device to harness maximum power from a vibr ating device. The electromechanical lumped element model thus developed is validated us ing meso-scale experiments. Furthermore, a scaling theory is developed to observe the device behavior as it is reduced in size to a MEMS scale, which is verified using finite element anal ysis. In addition, the fa brication process adopted to build the devices and their char acterization will be presented. The main contributions for this dissertation are as follows: A complete static analytical model of a can tilever composite beam validated using FEM and experiments on candidate devices. Electromechanical lumped element model of a piezoelectric energy harvester, intended to provide a design optimization tool for complete circuit simulation with power processors. A first generation fabrication of a MEMS PZT cantilever array is real ized for vibrational energy harvesting. Design, fabrication and testing of a stand alone MEMS device to demonstrate energy reclamation. Organization of Dissertation The dissertation is organized as follows. Chapter 2 describes in detail the static electromechanical model of the composite beam structure. In particular, lumped element modeling is used to obtain the va rious electromechanical parameters that represent the system. Chapter 3 discusses the detailed non-dimensiona l analysis and the desi gn formulation for the device. Chapter 4 discusses th e fabrication process adopted to build arrays of the MEMS piezoelectric generators. Chapter 5 describes the experimental setup and characterization

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47 procedure for testing energy reclamation devices. Chapter 6 describes the e xperimental results. Chapter 7 concludes the disse rtation with a summary and discussion of future work.

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48 CHAPTER 2 PIEZOELECTRIC CANTILEVER BE AM MODELING AND VALIDATION The objective of this dissertation is to m odel and design MEMS piezoelectric cantilever composite beams with an integrated proof ma ss to reclaim energy from base vibrations. Consequently, these structures will be optimally designed to extract maximum power from the vibrations, subject to some design constraints. Th e ultimate goal is to eventually use an array of such structures to obtain sufficient power to ope rate self-powered sensors. This chapter describes the static electromechanical m odeling of the composite beam using conven tional Euler-Bernoulli beam theory. The shortcomings of this approach are that it does not model nonlinear effects due to large deflections and neglects rotary inertia e ffects. In addition, fa brication-induced inplane residual stresses are neglected in the model. Th ese stresses exist in ME MS structures due to thermal stresses and other sources that mainly occur during layer depositions and other high temperature treatment. For the purpose of this first generation effort, we assume that EulerBernoulli theory is adequate to model, design and characterize the device. However, future models may include the above effects to implement a more complete model. Some of the earlier work in this area was c oncentrated on modeling and testing a canonical cantilever mesoscale composite be am without any proof mass that was excited at its tip with a load (Kasyap 2002). For a known fo rce input, the amount of power ge nerated at the ends of the PZT was used in a flyback converter circuit to reclaim power (Kasyap et al. 2002). However, the previous configuration cannot be di rectly used in real applications because of the nature of its loading condition. In all practi cal applications, the en ergy reclamation device should be directly

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49 attached to the vibrating surface. This alters the complete setup as the device is loaded at the clamp due to the vibrations, instead of the tip. In the new configuration shown in Figure 2-1, the test structure modeled consists of a piezoelectric composite cantilever beam with a proof mass attached to its tip. The composite beam, when directly attached to a vibrating surfa ce, places the whole stru cture in an accelerating frame of reference. The proof mass essentiall y converts the input base acceleration into an effective inertial force at the tip that deflects the beam, there by inducing mechanical strain in the piezoceramic (Yazdi et al. 1998). This strain pr oduces a voltage in the piezoceramic that is converted into usable po wer with the help of an energy reclamation circuit. The motion of the beam depends on the size of the proof mass. If the proof mass is relatively small compared to the effective mass of the beam, it reduces to a cantilever beam subject to an acceleration at the clamp instead of its tip. Alternatively, if the proof mass is very large compared to the effective mass of the beam, it results in large deflections in the beam and consequently, large strains at the clamp. This configuration will be favorable for energy reclamation because a piezoelectric patch, when attached to the beam converts the induced strain into electrical charge. However, if the proof mass is comparable to the actual eff ective mass of the beam, the motion of the beam resembles that of a rigid body and, therefore, mi ght not induce any strain in the beam. These issues are clarified via the model described below. Piezoelectric Composite Beam In this analysis, the test structure consists of a piezoelectric (PZT) composite cantilever beam with a proof mass atta ched to it as shown in Figure 2-1.

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50 Figure 2-1: Schematic of a piezoelectric com posite beam subject to a base acceleration. In Figure 2-1, oa is the input acceleration, ext f is the excitation frequency of the base vibration and V is the resulting voltage from the piezoceramic. The composite beam is modeled using the lumped element modeling technique desc ribed in Hunt (1982) and Rossi (1988). This approach is valid in general when the characte ristic wavelength of the bending waves is very large compared to the geometric length scale a nd, in the case of a cantilever composite beam, is valid up to at least the funda mental bending resonance fre quency (Merhaut, 1979). This approach simplifies the partial di fferential equations governing th e system to coupled ordinary differential equations. In addition, the lumped element modeling tech nique is useful in analyzing and designing coupled energy domain transducer systems. In this approach, we use equivalent circuit elements to effectively represent the coupled electromechan ical behavior of the device. These circuit analogies enable efficient modeling of the inte raction between differen t energy domains in a system. Furthermore, the tools developed for circ uit analysis can be utilized for representing and solving a coupled system with different energy domains.

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51 A piezoelectric composite beam represents an electromechanical system that can be separated primarily into two energy domains c onsisting of electrical and mechanical parts. These two energy domains interact in the equiva lent circuit via a transformer as shown in Figure 2-2. The circuit is obtained by lumping the distributed energy stored and dissipated in the system into simple circuit elements. In this el ectromechanical circuit, force and voltage are the generalized effort variables, while velocity and current are the generalized flow variables (Senturia 2000). An impedance analogy is used to represent the circuit, in which case all elements that share a common effo rt are connected in parallel, and the elements that share a common flow are connected in series. When th e composite beam is subject to a mechanical load, the strain induced in the piezoelectric material generates a voltage, which represents the conversion from the mechanical to the electrical domain. Conversely, the composite beam can be driven with an ac voltage that causes it to vibrate due to the piezoelectric effect. This represents a conversion from the elec trical to the mechanical domain. Figure 2-2 represents the entire equivalent ci rcuit consisting of mech anical and electrical lumped elements representing the composite beam. All elements are labeled and defined in the figure. In the notation shown in Figure 2-2, the first subscript denotes the domain (m for mechanical and e for electric), while the second subscript denotes the condition (s for short circuit and b for blocked). Using the described notation, for example, msC is defined as shortcircuit mechanical compliance, and ebC is the blocked electrical cap acitance of the piezoceramic. F is the effective force applied to the stru cture that is obtained by the product of input acceleration and effective mass lumped at the tip, U is the relative tip velocity with respect to the base, V is the voltage, and I is the current generated at the ends of the piezoceramic. All

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52 the parameters are obtained by lumping the energy at the tip using the relative motion of the tip with respect to the clamp/base. Figure 2-2: Overall equivalent circuit of composite beam. The beam is represented as a spring-mass-damper system by lumping the energy (kinetic and potential) in the beam to an equivalent mass and compliance. The mechanical mass and compliance of the structure can be equated to an equivalent elect rical inductance and capacitance. Similarly, mechanical damping is analogous to electrical resistance. However, mechanical damping cannot be easily estimated fr om first principles a lthough it is a critical parameter for resonant behavior in structures. The same holds true for electrical losses in the device, modeled using e R In principle, the fundamental operation of any power generato r is effectively dependent on the nature of the mechanism by which the energy is extracted. Most microgenerators reported to date can be classified into velocity-damped re sonant generators (VDRGs) or Coulomb-damped resonant generators (CDRGs) as described in Mitcheson et al. (2004). VDRG represents the damping effect as a function of the velocity characterized by a viscous force, while CDRG

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53 represents the same effect using a coulomb fric tional force. Analytical expressions for the dissipated power for these two cases are derived in Mitcheson et al. ( 2004) that provide an estimate for the available power. It should be noted that the aforementioned damping mechanisms are resonant in nature and that wh ile VDRG is widely used and linear, CDRG is a nonlinear representation, although a closed-form solution is ava ilable (Den Hartog 1931; Levitan 1960). An alternate class, namely, Coulombforce parameteric generator (CFPG) is also suggested in their work that ope rates in a non-resonant manner. For the purpose of our analysis, a VDRG implem entation is adopted that represents the damping phenomenon using a viscous effect with an effective damping coefficient. Damping coefficients are typically estimated from experime ntal modal analysis and include effects such as viscous dissipation, boundary cond ition non-ideality, th ermoelastic dissipation, etc. (Srikar and Senturia 2002). A detailed analysis of va rious damping mechanisms is discussed and corresponding empirical re lations are presen ted in Appendix B. The mechanical damping in the system is obtained from the damp ing ratio using the expression 2,m m msM R C (2.1) where m M is the effective mechanical mass of the composite beam (discussed in Section 2-2), and is the mechanical damping ratio. However, th e dielectric loss of th e piezoelectric material can be estimated using an empirical e xpression provided in Jonscher (1999) 1 tan2e nebR fC (2.2) where n f is the natural frequency of the system. tan is the loss tangent de fined as the ratio of resistive and reactive parts of the impedance. Th e theory behind dielectric loss in piezoelectric

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54 materials is described in Mayer goyz and Bertotti (2005) and Jons cher (1999). In this case, the electrical damping is assumed to be in parallel with the capacitor. A discussion about this and an alternate representation for electrical impedance is provided in Appendix D. All the other parameters in the circuit in Figure 2-2 are obtained analytically. The main purpose for modeling the device as a beam is to obtain the lumped parameters such as msC m M and that characterize the circuit. The following sections describe in detail the process of lumped parameter extraction. Analytical Static Model The composite beam clamped at one end is analyzed from first principles using linear Euler-Bernoulli beam theory described in, for example, Beer and Johnston (1992). Therefore, shear and rotary inertia effects ar e neglected. Another assumption is that plane sections remain planar and no geometric nonlinearity exists in th e structure. The following section presents the static analytical model for the composite beam to calculate the lumped element parameters represented in Figure 2-2. In this analysis, the composite beam is solved for its static equilibrium to obtain its transverse deflection for all the sta tic loads acting on it, which permits calculation of the lumped element parameters. When a base acceleration is applied to th e structure, the Euler-Bernoulli governing equations used in our analysis remain valid. However, a Galilean transf ormation of coordinates is carried out, described in detail in Appendix C, to transform to a local coordinate system that treats relative motion of the beam with respect to the clamp. In addition, the effect of the base mass is also analyzed in Appendix F.

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55 Static Electromechanical L oad in the Composite Beam The static electromechanical model is used to calculate the effective short circuit compliance, mass and piezoelectric coefficient of the composite beam. A simple schematic of the composite beam configuration for this case is shown in Figure 2-3. The cantilever composite beam is analyzed for all the static mechanical lo ads acting on it from first principles using the bending beam equation. Figure 2-3: Schematic of the piezo electric cantilever composite beam. In the figure, and psLLl are the lengths of piezoceramic, b eam (i.e., shim) and the proof mass, respectively. The applied voltage will induce a mechanical strain and, hence, a bending moment at the ends of the piezoceramic as described in Cattafesta et al. (2000). The induced moment o M in the composite beam due to an applied voltage to the piezoceramic is given by the expression 3121 2. 2opapppp M EdVbct (2.3)

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56 Here, pE is the elastic modulus of piezoceramic, 31d is the piezoelectric coefficient, appV is the applied voltage to the piezoceramic, 2c is the location of the neutral axis from the bottom of piezoceramic given by the expression 2 222 ,p s sspp ssppt t EttE c EtEt (2.4) where s E and s t represent the elastic m odulus and the thickness of the shim. Similarly, pb and pt are the width and thickness of piezoceramic respectively. The free body diagram for the above configuration, shown in Figure 2-4, essentially replaces th e mass of the composite beam as an equivalent uniform load due to its weight. Figure 2-4: Free body diagram of the overall configuration. where 123, and qqq are the equivalent linear load densities Nm in the composite, shim, and proof mass sections, respectively. Since the configuration represented in Figure 2-4 is assumed to be a linear system, it can be simplified and solved analy tically for the deflection using Euler-Bernoulli beam theory.

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57 Figure 2-5 represents the simplified free body diagram for the composite beam. In Figure 2-5, the clamp is replaced with a reaction bending moment r M and reaction force R Figure 2-5: Free body diagram of the composite beam where the self weight s are replaced with equivalent loads. As indicated in Figure 2-5, the composite beam is uniformly loaded in a piecewise fashion over its total length, sLl. Consequently, each of the uniform loads shown in Figure 2-4 can be replaced with an effective st atic load density defined as ,mqwhg (2.5) 1,ssspppqtbtbg (2.6) and 2.sssqtbg (2.7) Here, and s sbt are the width and th ickness of the shim, and wh are the width and thickness of the proof mass, and p, and s m are the densities of piezoceramic, shim, and proof mass, respectively. Assuming static e quilibrium for the beam in the Figure 2-5, we can obtain expressions for the reaction force an d bending moment at the clamp as 12psp R qLqLLql (2.8)

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58 and 222 12. 222psp rsLLL l MqqqlL (2.9) Let us now divide the composite beam in Figure 2-5 into 3 sections. Section 1 consists of the composite 0p x L the second section consists of the shim psLxL and the third section is the proof mass ss L xLl. The Euler-Bernoulli equation for the beam is then solved using free body diagrams to obtain the bending moment and shear force in each of the sections (Beer and Johnston 1992). Furthermore, the bending moment can be integrated using the Euler-Bernoulli equation to obta in the mode shape. The govern ing equations fo r the sections are 2 2 1 1 2 ; 0, 2rop cwx x EIMRxMqxL x (2.10) 2 2 2 12 2 ; 22p p rpsps sxL L wx EIMRxqLxqLxL x (2.11) 2 2 2 3 12 22 ; 222sp sp p s rps ss mLL LLx L wxxL E IMRxqLxqqLxLl x (2.12) Here, and csmEIEIEI are the flexural rigidi ty in each of the three sections. Further, 123, and wxwxwx are the transverse deflections in each of the sections at a distance x from the clamp. The two clamped boundary conditions and four matching conditions, shown in Eq. (2.13), are obtained from the clamped bounda ry condition and by matching the deflection and slope at each of the interfaces between the sections

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59 1 1 0 12 12 23 2300, , .pp ssx pp xLxL ss xLxLwx w x wLwL wxwx xx wLwL wxwx xx (2.13) The Euler-Bernoulli equations shown in Eqs. (2.10)-(2.12) can now be solved to obtain a piecewise continuous deflection mode shape fo r the beam, that is represented as 222 2 12 3 4 12 1 1222 2462psp so psp cccLLL l qqqlLMx qLqLLqlx qx wx EIEIEI (2.14) 2 2 2 3 4 213 2 24 222 2462s s s sssssL l qqlLx qLqlxCCx qxCC wx EIEIEIEIEI (2.15) and 2 32 4 57 68 3, 2464ss mmmmmqLlxqLlxCCx CC qx wx EIEIEIEIEI (2.16) where the integration constants 1234567, , , CCCCCCC and 8C are given by the following expressions

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60 2 1212 2 2 2122 3 2 4 3 2 51 4 2 62 3 7331, 6 32861, 24 122, 2 1, 2342 6 24 1 62p pssp p psps p spo p os p s s sL CLqqqLLL L CLqqqLLL L CqlLLlM L ML l CLql qL CC qL CC qL CCq 42 841, 1 146. 24ss ssslLLl CCqLqlLLl (2.17) In the above equation, is defined as the ratio between the fl exural rigidity moduli in sections 1 and 2, s c E IEI and is the ratio between the flexural rigidity modulus in sections 2 and 3, msEIEI To check the validity of this general result the deflection modesh ape thus obtained is verified by comparing with a few simple special cases. The first ideal case used to verify the static electromechanical solution is that of a homogenous cantilev er beam subjected to its selfweight. The resulting deflection for this idea l case is given as (Beer and Johnston 1992) 22 2. 4246swss sw sqxLLx x wx EI (2.18) where, s wq is the uniform load acting on the homogenous beam as a result of its own weight. The static short-circuit solution is verified by setting the input vol tage and the proof mass size to zero. The piezoelectric patch is thus absent in this solution and represents a homogenous beam

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61 that deflects due to its self-weight. The two solu tions are plotted for a test case consisting of an Aluminum beam (Al 6061). The properties and dime nsions used in the simu lations are listed in Table 2-1. As indicated in Figure 2-6, the static solutions are identical. Table 2-1: Material proper ties and dimensions for a homogenous aluminum beam. Elastic modulus s E 73 GPa Density s 2718 3kgm Length of the beam s L 127 mm Width of the beam s b 6.35 mm Thickness of the beam s t 1.02 mm Figure 2-6: Static model verified with the ideal solution for a homogenous beam solved for selfweight. Furthermore, the piecewise solution can be ve rified for a homogenous beam subjected to a tip load. From conventional theo ry, for this ideal case, the defl ection modeshape in a beam due to point static tip load is given as (Thomson, 1993)

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62 23, 26tip tiploads sq x x wxL EI (2.19) where tipql is the equivalent tip load. In the piecewis e static solution, the pi ezoelectric patch is absent and the input voltage is se t to zero to generate a similar configuration as before but with a proof mass that contributes an effective tip load. Similar to the previous case, Eq. (2.19) was calculated for the test beam listed in Table 2-1, and the defl ections are plotted in Figure 2-7. Again, the solutions match. Figure 2-7: Static model verified with the ideal solution for a homogenous beam solved for tip load. Now, the complete static mechanical model representing the PZT composite beam has been verified for various test cases. As described in Section 0, the purpose of obtaining a complete electromechanical model is to calculate the lumped element para meters in the circuit

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63 shown in Figure 2-2. The advantage of this soluti on model is that all the parameters are analytical and their scaling dependence on the di mensions can also be obtained which will be further useful in optimizing the structure for maximum power output. The static deflection mode shape can now be used to estimate an equivalent effective mass and compliance that can replace the composite beam as a simple single degree of freedom system. For this configuration, we emphasize that only the mechanical loads are considered, and the piezoceramic is electrically shorted. This c onfiguration effectively eliminates the electrical side from the lumped element circuit represented in Figure 2-2, leading to s hort-circuit electrical condition for the piezoceramic. The potential energy associated with distributed strain energy in the composite beam is given by the expression (Thomson 1993) 2 2 2 0, 2sLlExIxdwx PEdx dx (2.20) where Ex and I x are the local elastic modulus and mome nt of inertia of the section. The above integral equation is determined in each of the three sections and summed to obtain the total potential energy in the beam, 2 22 222 123 222 0. 222p ss psL LLl csm LLEIEIEI dwxdwxdwx PEdxdxdx dxdxdx (2.21) Similarly, the total kinetic energy in the composite beam is given by the integral expression 2 0, 2sLl Lx KEwxdx (2.22)

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64 where L is masslength of the section and wx is the velocity in th e section. We calculate the kinetic energies in the indi vidual sections and add them to obtain the total kinetic energy in the composite beam 222 123 0222p ss psL LLl LcLsLm LLKEwxdxwxdxwxdx. (2.23) Lumping the overall potential strain energy at the tip yields an effective short circuit mechanical compliance for the composite beam 2. 2Ftip msw C PE (2.24) Using the same analogy, an effective mass for the composite beam from its deflection shape is obtained by lumping the kinetic energy of the beam at its tip 22 .Fm tipKE M w (2.25) where, 3FtipswwLl is the resulting tip de flection of the beam due to its self-weight calculated from Eq. (2.16). The natural frequency of the composite beam is calculated from the effective mass and compliance using the expression 11 2n msmf CM (2.26) Next, the electromechanical behavior of the general solution, Eq. (2.14) (2.17), is validated for the case when the piezoceramic composite beam is subjected to an applied voltage. In oder to validate the solution, the proof mass is neglected for this special case. This configuration corresponds to the cantilever piezoelectric actu ator described in Kasyap (2002). The actuator deflection is determined for a test specimen comprised of a piezoceramic patch

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65 attached to an aluminum shim. The dimensions and properties for this beam are listed in Table 2-2. Table 2-2: Material pr operties and dimensions for a piez oelectric composite aluminum beam. Length of the beam s L 101.60 mm Width of the beam s b 6.35 mm Thickness of the beam s t 1.02 mm Elastic modulus of PZT pE 62 GPa Density of PZT p 2500 3kgm Length of the PZT patch pL 25.40 mm Width of PZT pb 6.35 mm Thickness of PZT pt 0.51 mm Piezoelectric coefficient 31d -274 X 10-12 mV Relative permittivity r 3400 Rewriting the static solution from Eq. (2.14) (2.16) for the case when all mechanical loads are neglected in the composite beam yields 2 12o cM x wx EI (2.27) and 23. 2opp cMLL wxwxx EI (2.28) Figure 2-8 compares the deflect ion using the two methods men tioned above. As indicated in the figure, the modeshapes match exactly, whic h indicates that the electromechanical static model accurately represents the struct ure in the absence of a proof mass.

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66 Figure 2-8: Deflection modeshap e for a composite beam subjected to an input voltage. Next, we can use the modeshape for the piezo electric actuator with a proof mass to calculate the effective piezoelectric coefficient, which is defined as the tip deflection resulting from an applied unit voltage. Since we need to obtain the el ectromechanical coupling between the input voltage and the resulting deflection, the static deflection of the composite beam due to all the mechanical loads is subtracted from th e overall deflection. However, we assume the system to be linear, and the solutions can be superimposed. Consequently, the effect of the voltage on the deflection can be decoupled from the overall equations. Th erefore, th e resulting tip deflection due to an input voltage is given as 2Vtip opp ms app cw M LL dL VEI (2.29)

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67 where, Vtipw is the tip deflection due to the applied voltage. After obtaining m M msC and md the rest of the parameters in the circuit, such as , and mebe R CR can be easily obtained as shown below, since they are simple analytical expressions related to these elements. In the electromechanical circuit shown in Figure 2-2, is defined as the turns ratio for the transformation between the electr ical and mechanical domains and is given by the expression .m msd C (2.30) Next, as described earlier ebC is given by 21,m ebef efmsd CC CC (2.31) where efC is the free capacitance of the piezoceramic .p ef pA C t (2.32) Here, is the dielectric permittivity in the piezoceramic and pA is the surface area of the piezoceramic. The resistive elements in the circuit are calculated using Eqs. (2.1) and (2.2). Therefore, all but two of the lu mped element parameters in the circuit have been analytically obtained from the static electromechanical model. Only the mechanical damping and electrical loss are estimated using the empirical relations provided in Eqs. (2.1) and (2.2). It should be noted here that a viscous dampi ng model assumed in this model, represented with an effective damping ratio does not capture all loss mechanisms A more detailed in-depth study of various damping mechanisms is provided in Appendix B. The empirical relations and the estimated

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68 values for the tested MEMS devices are also presented in Table 6-12 along with their experimentally extracted damping. Now, we can represent the composite beam in the lumped element circuit and simulate it for various loading conditions. In the overall c onfiguration, the input to the system is an effective acceleration applied at the clamp, which is replaced with an equivalent inertial force in the circuit (Yazdi et al. 1998). This effective fo rce in the single degree of freedom system is defined as the product of the effective mass and the acceleration of the center of mass of the system. In this analysis, it has been assume d that the input acceleration is equal to the acceleration of the center of mass. As will be demonstrated, th is assumption has proven to be fairly accurate in predicting the dynamic response until the firs t resonance with experimental results. Therefore, the e quivalent force is given as .momFMaMg (2.33) In the above equation, the first term corres ponds to the dynamic input force due to applied acceleration at the clamp. The second term is th e static load on the beam acting due to gravity which indicates the static deflect ion of the beam. This term is, however, not used for dynamic simulations to predict the output voltage and current in the equivalent circu it. Therefore, the input dynamic mechanical power is given as ,inPFU (2.34) where U is the relative instantaneous velocity of the tip with respect to the base. In Figure 2-2, the device is connected to an external load circui t to reclaim power in a real application. Solving for the input velocity in the circuit from UFoverallimpedance we obtain 21 .moebL e ebLmseLMaZR U Z RZZR (2.35)

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69 In the above expression, L R is the external load which is a ssumed to be purely resistive in our analysis (Taylor 2004), eb Z is the blocked electrical impedance, and e ms Z is the short circuit mechanical impedance represented in the electr ical domain that are gi ven by the following expressions 1e eb eb e ebR jC Z R jC (2.36) and 211 ,e msmm msZjMR jC (2.37) where is the frequency of excitation in rads. Therefore, the input power supplied to the composite beam is obtained from Eq. (2.34) as 21 .mo in e ebL ms ebLMa P ZR Z ZR (2.38) The input power to these struct ures when calculated using Eq. (2.38) based on the input base acceleration gives an understanding about th e amount of mechanical energy available for conversion to the electrical domain. The conversi on of mechanical power to electrical energy is related to the coupling factor defined as 2 2.m efmsd CC (2.39) The coupling factor determines the amount of electrical energy available in the piezoceramic that can be reclaimed (Ikeda 1990).

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70 All the analytical expressions for the electromechanical circuit elements have been derived and presented for a composite beam to complete the circuit in Figure 2-2. Since all the lumped parameters excluding the damping ratio in the circ uit are obtained analytically and are dependent on the material dimensions and prop erties, a detailed scaling analys is is carried out in the next chapter to provide a motivation for designing ME MS devices. Furthermore, a simple design strategy is presented to model and design thes e devices for characterizing energy reclamation from vibrations. Experimental Verification of the Lumped Element Model This section summarizes meso-scale experime nts to validate the lumped element model and the corresponding electromechanical circuit for power generation. First, results are presented for experiments carried out to verify the electromechanical lumped element model for its mechanical and electrical behavior. Finall y, overall power transfer estimates are obtained experimentally and compared with the theore tical predictions to validate the model. Initially, experiments were conducted with a clamped aluminum beam that was mounted on a vibrating surface (LDS dynamic shaker mode l V408) to verify the dynamic and static lumped element model. The material dimensions of the test specimen used are listed in Table 23. The properties are listed in Table 2-1 Table 2-3: Material properties and dimensions for a homogenous aluminum beam. Length of the beam 127 mm Width of the beam 6.35 mm Thickness of the beam 1.02 mm Length of the proof mass 3.17 mm Width of proof mass 6.35 mm Thickness of proof mass 8.64 mm

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71 All the lumped element parameters (Figure 2-2) were obtained e xperimentally to validate the model as follows. Static tests were carri ed out by loading the ti p with known masses (that were measured using an OHAUS ma ss balance with resolution of 0.1 mg) and the tip deflection was measured using a Micro-Epsilon laser displacement sensor (OPTONCDT series 2000). An average compliance of the composite beam was obtained by calculating the ratio between the resulting tip deflection and the static load at the tip for all the masses. The difference between the estimated and calcula ted theoretical value using the properties and dimensions is listed in Table 2-4. A simple impact test was carried out to obtain a damped impulse response to estimate the natural frequency of the specimen. The natural frequency thus obtained using the logarithmic decrement method (Craig 1981) was 50.5 Hz. From the measured natural frequency and the effective compliance, the eff ective mass was calculated to be 0.523 gm. Table 2-4: Measured a nd calculated parameters for the homogenous beam. CALCULATED MEASURED UNCERTAINTY Effective mass of the beam, MM 0.540 gm 0.523 gm 3.1 % Effective compliance of the beam, CMS 0.018 m/N 0.019 m/N 5.5 % Natural Frequency, Fn 50.9 Hz 50.5 Hz 0.8 % The structure was then mounted on a vibration shaker as shown in Figure 2-9 that was used to excite the composite beam over a frequency range.

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72 Figure 2-9: Experimental setup for verifying the electro-mechan ical lumped element model for meso-scale cantilever beams. The input acceleration to the structure was me asured using an impedance head (Bruel & Kjaer type 8001). The resulting tip deflection was measured using the displacement sensor. To check mass loading effects in the impedance h ead, the input acceleration measured with the impedance head was initially compared with th e results obtained from a displacement sensor measurement at the same point. It was observed that the resu lts matched very well over the frequency range. The measured re sonant frequency and the complian ce were then used to adjust the mass of the model to match the predicted natural frequency. Figure 2-10 shows a plot of the frequency response function between the tip deflec tion (measured with the displacement sensor) and input acceleration. The magn itude, phase and coherence are in dicated in the plot along with a comparison with the lumped element model pr edictions. The results were found to match well until beyond the first resonance. The parameter plotted in the figure is the transfer function

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73 between the input acceleration and resulting tip de flection. The observed resonant frequency was 50.5 Hz. The frequency response using the LEM is calculated using the expression 1 1mo tip mm msMa w j jMR jC (2.40) The damping ratio was estimated to be 0.005 by matching the response p eaks at the resonant frequency. 0 20 40 60 80 100 10-4 10-2 100 mag (m/s/m/s2) expt LEM 0 20 40 60 80 100 -200 0 200 phase(deg) 0 20 40 60 80 100 0 0.5 1 coherencefrequency (Hz) Figure 2-10: Comparison between experiment a nd theory for tip deflection in a homogenous beam (no tip mass). As indicated in the above plots, the respons e was accurately predicted using the lumped element model. Similar experiments were ca rried out for a homogenous beam with a known proof mass attached at its tip. The addition of th e tip mass to the system leads to an inertial tip load during the vibration. This load will act as shear force along with the static load of the tip

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74 mass due to gravity. The measured tip mass was 0.476 0.1 gmg A similar vibration experiment was carried out for th is device, and the tip deflection was measured as a function of frequency for an input acceleration to the clamp. Figure 2-11 shows a comparison of the frequency response function between theory and e xperiment. As seen in the figure, the plots match well and the resonant frequency was measured to be approximately 37 Hz. The reduction in the resonant frequency is due to the addition of the tip mass. 0 20 40 60 80 100 10-4 10-2 100 mag (m/s/m/s2) expt LEM 0 20 40 60 80 100 -200 0 200 phase(deg) 0 20 40 60 80 100 0 0.5 1 coherencefrequency (Hz) Figure 2-11: Comparison between theory and expe riments for the tip deflection in a homogenous beam with tip mass. Table 2-5: Measured and calcula ted parameters for the homoge nous beam with a proof mass. CALCULATED MEASURED UNCERTAINTY Effective mass of the beam, MM 0.948 gm 0.974 gm 2.7 % Effective compliance of the beam, CMS 0.020 m/N 0.019 m/N 5 % Natural Frequency, Fn 36.4 Hz 37.0 Hz 1.6 %

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75 The estimated and calculated lumped element parameters for the homogenous beam with a proof mass are listed in Table 2-5 along with the estimated uncertainty in the values. After verifying the lumped element model for the two cases mentioned, they were extended to a piezoelectric composite beam. Th e dimensions of the piezoelectric composite beam are listed in Table 2-6. The same tip mass was used for these experiments. Table 2-6: Material pr operties and dimensions for a piez oelectric composite aluminum beam. Length of the beam 103.38 mm Width of the beam 6.35 mm Thickness of the beam 0.51 mm Elastic modulus of PZT 66 GPa Density of PZT 7800 kg/m3 Length of the PZT patch 25.40 mm Width of PZT 6.35 mm Thickness of PZT 0.51 mm Piezoelectric coefficient -190 X 10-12 m/V Relative permittivity 1800 Length of the proof mass 3.17 mm Width of proof mass 6.35 mm Thickness of proof mass 8.64 mm Before performing the vibration experiments, th e static loading test and the impact test were conducted as before to measure the mech anical compliance and the natural frequency. These values are compared with the theoretically calculated parameters using the dimensions and properties in Table 2-7. The relative uncertainties are observed to be higher than the homogenous beam and can be at tributed to the bond layer and the uncertainties in the PZT dimensions (Mathew 2001). A detailed uncerta inty analysis along the lines of what was

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76 described in Kasyap (2002) can be carried out to obtain better estimates. However, for the purpose of this validation, we use the values measured. Table 2-7: Measured and calculated values for a PZT composite beam. CALCULATED MEASURED UNCERTAINTY Effective mass of the beam, MM 0.183 gm 0.176 gm 3.9 % Effective compliance of the beam, CMS 0.038 m/N 0.041 m/N 7.8 % Natural Frequency, Fn 60.55 Hz 59.25 Hz 2.1 % Effective piezoelectric coefficient, deff -1.28e-6 m/V -1.18e-6 m/V 7.8 % Blocked electrical capacitance 5.01 nF (Cef = 5.06 nF ) 4.86 nF (Cef = 4.88 nF ) 3.1 % A vibration experiment was th en carried out using the comp osite beam by mounting it on the vibrating shaker. The clamped base was ha rmonically excited, and the tip deflection was measured using a displacement sensor. Figure 2-12 plots the frequency response and a comparison with the LEM. The response was observed to match well through the first resonance. The mass of the composite beam wa s calculated using the compliance and measured natural frequency. Some of the reasons for higher discrepancy in the composite beam are attributed to the fact that the LEM does not incl ude the epoxy bond layer that was used to attach the PZT with shim. In addition, it does not incor porate the small gap that is provided between the clamp and PZT to prevent any potential shorting dur ing vibration. However, this will not occur in the MEMS device as the clamp will form a part of the substrate itself. This will be understood better in Figures 4.16 and 4.17.

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77 0 20 40 60 80 100 10-4 10-2 100 mag (m/m/s2) expt LEM 0 20 40 60 80 100 -200 0 200 phase(deg) 0 20 40 60 80 100 0 0.5 1 coherencefrequency (Hz) Figure 2-12: Frequency response of a piezo electric composite beam (no tip mass) A similar experiment was c onducted with the composite b eam that has a proof mass attached to its tip. The measured and calcula ted parameters for the specimen are listed along with the uncertainties in Table 2-8. Table 2-8: Measured and calcula ted parameters for a PZT compos ite beam with a proof mass. CALCULATED MEASURED UNCERTAINTY Effective mass of the beam, MM 0.598 gm 0.623 gm 4.1 % Effective compliance of the beam, CMS 0.043 m/N 0.041 m/N 4.6 % Natural Frequency, Fn 31.37 Hz 31.50 Hz 0.5 % Effective piezoelectric coefficient, deff -1.30e-6 m/V -1.18e-6 m/V 7.8 % Blocked electrical capacitance 5.01 nF (Co = 5.06 nF ) 4.86 nF (Co = 4.88 nF ) 3.1 %

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78 Figure 2-13 shows a plot of the frequency res ponse function between the tip deflection and input acceleration and, compares wi th the theoretical predictions us ing static LEM. As is evident from the plot, the response matches well with the predictions and the resonant frequency was found to be at 30 Hz. 0 20 40 60 80 100 10-4 10-2 100 mag (m/m/s2) expt LEM 0 20 40 60 80 100 -200 0 200 phase(deg) 0 20 40 60 80 100 0 0.5 1 coherencefrequency (Hz) Figure 2-13: Frequency response for a piezoelectric composite beam (mp=0.476 gm). Based on the above observations, it can be co ncluded that the lumped mechanical model that was developed is sufficiently accurate in predicting the dynamic behavior of the meso-scale composite beam. To further verify and validate the electr omechanical LEM, the same PZT aluminum composite beam (without the proof mass) was characterized for both its mechanical and electrical response. The dimensions and pr operties of the composite beam are listed in Table 2-6 and therefore not reproduced again.

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79 All the lumped element parameters obtained previously in Table 2-7 are used for subsequent validation. The damp ing ratio for the system was adjusted to match the peaks at resonance in the response obtained both expe rimentally and the lumped element model (Figure 2-12). The resulting damping rati o was estimated to be 0.015 a nd this value was subsequently used in the analysis. The increase in damping ra tio from a homogeneous beam to the composite beam is attributed to the added losses in th e piezoelectric material and the epoxy layer. To measure the effective piezoelectric coeffi cient, an ac voltage was applied to the PZT and the resulting response at the tip was measured using the laser displacement sensor. Ideally, the deflection needs to be measured at dc, but sin ce it is difficult to perform this experimentally, the response was measured at very low frequencie s (~ 20 Hz) where the response is flat. This value was used as the effective piezoelectric coefficient ( dm) for subsequent calculations. The free electrical impedance of the composite beam was measured using a vector impedance meter, and an effective free capacitance was obtained as a result. However, the value for dielectric loss was not measured experimentally and an empirical relation was used (Eq. 2.5) to estimate its value. Therefore, all the lumped elements that can be estimated experimentally were thus obtained and these values were used in the lumped element model to generate the overall response and predict its output characteristics. The resulting values are shown in Table 2-7 and were compared with the theoretical values. Reasonable agreement (better than 8 % ) was obtained between the measured and calculated values. To validate the electrical behavior of th e composite beam, another experiment was conducted wherein the resulting vo ltage across the PZT was measur ed as a function of the input acceleration at the clamp. The frequency res ponse function thus obtained is shown in Figure 2-

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80 14. As indicated in the figure, the model matche d well with the measured response indicating the validity of the complete lumped element model. Figure 2-14: Output voltage for an input acceleration at the clamp. Finally, after verifying the lumped element mo del for the frequency response, a sinusoidal acceleration signal was input at resonance, and th e resulting output voltage was measured across a range of resistive loads to measure the out put power. The results for the measured RMS voltage are shown in the following figure. The plot indicates the voltage generated for unit acceleration input (1 m/s2 RMS) as a function of different resistive loads varying from 10 K to 1 M. As indicated in the figure, the output voltage increases and saturates to a constant value called the open circuit voltage as the load incr eases. In the shown plot, the output voltage is normalized with the input acceleration to compare with the experimental values.

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81 Figure 2-15: Output voltage for varying resistive loads. Figure 2-16 shows the output power 2LLVR generated at the PZT calculated from the measured voltage across the resistive loads for the same input conditions as in Figure 2-15. As indicated in the plot, the power reaches a maximum value at an optimal resistance which occurs when it is equal to the input im pedance of the composite beam. It was observed that the optimal load thus estimated is approximately 404 k which is close to the theoretical value, 450 k. These measured and calculated values for the voltage and power are li sted in the following Table 2-9.

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82 Table 2-9: Comparison between experimental and theoretical values for power transfer. Estimated Measured Optimal load 450 k 404 k Voltage (per unit acceleration) 0.91 V/m/s2 0.85 V/m/s2 Corresponding output power 1.83 W/m2/s4 1.78 W/m2/s4 Figure 2-16: Output power across varying resistive loads. As seen in the above plot s, the lumped element mode l results agre e within 10 % This discrepancy is either within e xperimental uncertainty or accepta bly small for design purposes. It should be noted here that the m echanical and electrical damping are not accurately characterized or known. Using these results a nd conclusions, the LEM is extende d next to the micro-scale by scaling down all the dimensions of the structur e proportionally. The input acceleration is also scaled proportionally to operate the composite beam in the linear region so that the model can be used to predict its behavior. The next chapter describes in detail the scaling analysis for the

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83 composite beam indicating the dependence of a ll the lumped element parameters with the dimensions as they scale down. The next chap ter also describes the motivation for fabricating these structures using MEMS and the inhe rent advantages in their performance.

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84 CHAPTER 3 MEMS PIEZOELECTRIC GENERATOR DESIGN In this chapter, a detailed dimensional analys is is presented for the piezoelectric composite beam. Then, a scaling theory is developed based on the dimensi onal analysis to determine the response of the structure when it is scaled down in size. The objective behind developing a dimensional analysis and scaling theory is to pr ovide a tool that enable s better understanding of the device behavior as a function of dimensions and properties. In addition, it can be used as a tool to optimally design a firs t generation device aimed at specifi c applications. Next, a design strategy is formulated for the composite beam based on a given set of input parameters. In addition, each of the proposed designs is optimized using a parametric search procedure described in this chapter, subject to design and fabrication constr aints, but without any conventional optimi zation techniques. Power Transfer Analysis Recall that all of the lumped parameters cal culated in the previous chapter, with the exception of the empirical damping coefficient, are analytical functions of the material properties and device dimensions. The e quivalent circuit model for th e composite beam can now be attached to an external circuit to harness power. The external ci rcuit has an electrical impedance associated with it, which determines the amo unt of power that can be reclaimed from the composite beam. For the sake of our simulations, we assume that the external circuit is purely resistive (Taylor et al. 2004; Horowitz et al. 2002) and is represented as shown in Figure 3-1. Since most energy harvesters seek to reclaim and store energy (e.g., via a battery) that is later dissipated, a resistive load works best for analys es. In addition, most energy reclamation circuits

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85 present a purely resistive load to the generator. Figure 3-1 is the overall equivalent electromechanical circuit drawn as its Thvenin equivalent. From elementary circuit analysis, it can be proven that maximum power transfer occurs when the complex load impedance L Z is the complex conjugate of the Thvenin impedance *TH Z (derived in Appendix D). In the present case, in which the external circuit presents a purely resistive impedance, the optimal load resistance L R equals the magnitude of the Thvenin impedance in order to maximize power transfer (Appendix D). The Thvenin voltage is defined as the open circuit voltage, and the Thvenin impedance as the short circuit im pedance (Irwin 1996) acr oss the output and is calculated from the original repres entation shown in Figure 2-2. Figure 3-1: Thvenin equivalent circ uit for the energy reclamation system In the circuit, ThV is the equivalent Thv enin voltage, which is 2221 1 1e eb e eb Th e ebmm m e ebR jC F R jC V R j CjMR jC R jC (3.1)

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86 and the Thvenin impedance, Th Z is 222 2221 1 1 1e ebmm m e eb Th e ebmm m e ebR jCjMR jC R jC Z R jCjMR jC R jC (3.2) All the parameters in the above equations are defi ned in the equivalent ci rcuit in Figure 2-2. Consistent with the above discussion, we assume th at the output load is optimal and is therefore equal to the Thvenin impedance as given by the expression (Appendix E) .LTh R Z (3.3) The current across the load can be obtained from Ohms law as .Th L ThLV I Z R (3.4) The rms power across the load is defined as the product of the lo ad and the square of the rms current, given by the expression 2 _1 2LrmsLLPIR (3.5) where LL I R is the voltage across the load. In additi on, the input rms mechanical power to the device was calculated using the expression 11 22inF PUFU (3.6) From Eq. (3.5) and Eq. (3.6), we can calculate the overall electromechanical efficiency of the power transfer across the resistive load

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87 2.LL L in I R P PFU (3.7) Substituting the lumped element expressions for these parameters in terms of material properties and device dimensions provides th e desired scaling dependence of power and efficiency but results in expressions that, because of their algebraic complexity, do not provide any significant physical insight. Instead, dimens ional analysis is used below for the scaling analysis to optimally design an energy reclama tion device and corresponding external circuit that can harness maximum energy from the piezoelectric composite beam. Nondimensional Analysis A list of the all the variables in the el ectromechanical model are listed below in Table 3-1 that describe the dynamic behavior of a piezoelec tric composite beam. Fi rst, a set of primary variables are selected that incorporate the basic dimensions such as length, time, etc. Next, all the other variables used to descri be the composite beam are expr essed as nondimensional groups. These groups are later used to nondimensionali ze the response functions such as modeshape, LEM parameters, etc. providing the dependent groups. A schematic of the device with all dimens ions and properties is shown below in Figure 32. The dimensions and properties have al ready been discussed in Section 2.2.1.

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88 Figure 3-2: Schematic of the MEMS PZT device. Table 3-1: List of all device variables that are described in the electromechanical model. Variable Description s sE Material properties of shim 31,,,pprEd Material properties of piezoelectric layer ,mmE Material properties of proof mass ,, s ssLbt Geometric dimensions of shim ,,pppLbt Geometric dimensions of piezoelectric layer ,,lwh Geometric dimensions of proof mass tan, Dielectric loss tangent and mechanical damping coefficient ,oaf Vibration parameters, ac celeration and frequency Due to the fabrication process that was de signed for the devices, the following conditions hold true, namely, spbbb (3.8) The width of the piezoelectric layer and shim are a ssumed to be same to simplify the analysis. In addition, the shim and proof mass are assumed to be made from the same material. Therefore, and s msmEE (3.9)

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89 For the scaling analysis carried out here, we make the following assumption that simplifies the derivation: namely, pt and h are fixed in the analysis due to fabrication constraints that restrict the thickness of the pi ezoelectric layer and the proof mass. The thickness of the proof mass is formed from the substrate and therefore is equal to the wafer thickness. The thickness of the PZT layer was restricted by ARL process capability, which was 1 m at the time. Listing the remaining variables, we obtain the following tabul ated parameters with their dimensions as indicated in Table 3-2. Table 3-2: Dimensional representati on of all the device variables. Variable Dimensional units s E 12MLT s 3 M L pE 12MLT p 3 M L 31d 112 M LTQ 1322 M LTQ s L L b L s t L pL L l L w L oa 2LT e f 1T For the dimensional analysis, the following in dependent primary variables were defined. These parameters were chosen to include the primary dimensions of length, time, mass, and charge. All the other parameters will be expressed using these primary variables.

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90 Table 3-3: Primary variables used in the dimensional analysis. Independent variable Dimension s L L corresponds to "length" dimension s 3 M L includes the "mass" dimension s E 12MLT includes the "time" dimension 31d 112 M LTQ corresponds to the "charge" dimension The remaining variables are now nondimensiona lized using the 4 rep eating variables to obtain independent dimensionless groups as listed below. These groups will be used to nondimensionalize the piecewise deflection solution ob tained in Eq. 2.9-2.11. Furthermore, the analysis will be extended to nondimensionalize the LEM paramete rs in the equivalent circuit model to finally investigate the devi ce behavior for various topologies. 1p sE E (3.10) 2p s (3.11) We know that p ef pA C t which implies that is dimensionally represented as F m Therefore, 1322 M LTQ (3.12) and the corresponding nondimensional group is 3 2 31sEd (3.13)

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91 Equation (3.13) is a measure of the coupling between the electri cal and mechanical domain and in the ideal case, can be reduced to Eq. 2.33. Furthermore, the device dimensions are scaled as 4sb L (3.14) 5s st L (3.15) 6p sL L (3.16) 7p st L (3.17) From Eq. (3.15) and Eq. (3.17), we obtain 7 7 5pp s ssstt L tLt (3.18) 8sl L (3.19) 9sw L (3.20) From Eq. (3.14) and Eq. (3.20), 9 9 4s sL ww bLb (3.21) 10sh L (3.22) From Eq. (3.15) and Eq. (3.22),

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92 10 10 5s sssL hh tLt (3.23) Similarly, the external vibration paramete rs, such as the acceleration and excitation frequency, are nondimensionalized as 11 o s ssa E L (3.24) 121e s s sf E L (3.25) All the above derived independent primary groups are listed below in Table 3-4. Next, the location of the neutral axis in the composite section 0p x L measured from the bottom can be expressed as (Chapter 2) 22 22 2ssssppp ssppEtEttEt c EtEt (3.26) Dividing the above expression by 2 s sEL and nondimensionalizing with respect to shim thickness yields 22 2 1571572 ,,,, 2ppp ss sssss pp s s ssstEt tt LLLEL c ff Et L t LEL (3.27) Similarly, 1c is denoted as the position of the neutral axis from the top 12 spcttc (3.28) and in nondimensional form becomes

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93 12 1571571,,,,p ssst cc ff LLL (3.29) Table 3-4: List of independent groups. group Dependent variables 1 p s E E 2 p s 3 2 31 sEd 4 s b L 5 s s t L 6 p s L L 7 p s t t 8 s l L 9 w b 10 s h t 11 o s s sa E L 12 1e s s sf E L The bending moments of inertia in each section, 12,, and s spm I III are nondimensionalized as 3 3 1123spb Icct (3.30)

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94 33 12 1 1457 4,,, 3p ssss s st cc b LLLL I f L (3.31) 3 3 223ppb Icct (3.32) 33 22 1457 4,,, 3p ssss p st cc b LLLL I f L (3.33) 3 212 s sbt I (3.34) 3 2 45 4, 12s ss s st b LL I f L (3.35) 312mwh I (3.36) 3 45910 4,,, 12ss m swh LL I f L (3.37) The linear mass densities of each s ection are now nondimensionalized as slensstb (3.38) 45 2,slen s sssst b f LLL (3.39) clenssppttb (3.40) 2457 2,,,slen pp s sssssst t b f LLLL (3.41) mlenmwh (3.42)

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95 45910 2,,,slen m ssssswh f LLL (3.43) Next, the rigidity moduli in each of the b eam sections are nondimensionalized as 2 s s sEIEI (3.44) 2 45 44,ss sssEI I f ELL (3.45) 1 s spp cEIEIEI (3.46) 1 1457 444,,,pp cs sssssEI EI I f ELLEL (3.47) mm mEIEI (3.48) 45910 44,,,mmm ssssEI EI f ELEL (3.49) Let us now define another nondimensional parameter for acceleration due to gravity as s ssg E L (3.50) Nondimensionalizing the uniform line load on the beam due to its weight, we obtain sslenqqg (3.51) 45 2,slen s s ssss ssq g f E ELL L (3.52) cclenqqg (3.53) 2457 2,,,clen c s ssss ssq g f E ELL L (3.54)

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96 mmlenqqg (3.55) 45910 2,,,mlen m s ssss ssq g f E ELL L (3.56) In the static solution for the deflection of th e composite beam described in Chapter 2, we defined two constants 2 1457 2,,,s sss c c ssEI EI EL Cf EI EI EL (3.57) and 2 45910 2,,,m mss s s ssEI EI EL Df EI EI EL (3.58) The Euler-Bernoulli equations that were solv ed earlier to obtain a piecewise continuous deflection modeshape for the beam (in Chapter 2), are rewritten here as 22 2 43 2 124622222pp s ccpsspmcsmso cLL L x xlx qqLqLLqlqqqlLM wx EI (3.59) 2 2 3 4 13 24 222 2462s sms ssm s sssssL l qqlLx qLqlxCCx qx CC wx EIEIEIEIEI (3.60) and 2 32 4 57 68 3. 2464msms m mmmmmqLlxqLlxCCx qxCC wx EIEIEIEIEI (3.61)

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97 where the integration constants 1234567,,,,,, CCCCCCC and 8C are given by the following expressions 2 1 2 2 2 3 2 4 3 51 4 62 3 7331, 6 32861, 24 122, 2 1, 2342 6 24 6p pscsssp p pcsssps p mspo p os pm ss ss msL CLqqqLLL L CLqqqLLL L CqlLLlM L ML l CLql qL CC qL CC qL CC 42 841 1, 2 1 146. 24mss msmssqlLLl CCqLqlLLl (3.62) Removing the effect of moment due to app lied voltage to the piezoelectric layer and nondimensionalizing the deflecti ons with length yields, 123 1245678910,,,,,,,,,ssswww f LLL (3.63) Substituting for 3FtipswwLl and nondimensionalizing, we obtain 1245678910,,,,,,,,Ftip sw f L (3.64) Integrating the deflection across the length of the beam results in the total potential and kinetic energies, represented as

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98 2 22 222 123 222 0222p ss psL LLl csm LLEIEIEI dwxdwxdwx PEdxdxdx dxdxdx (3.65) and 222 123 0222p ss csm psL LLl lenlenlen LLKEwxdxwxdxwxdx (3.66) Nondimensionalizing the energies yields 2 1245678910 35,,,,,,,,ssssPEKE f ELL (3.67) From the energies, the short circuit compliance and mass are extracted and nondimensionalized as 1245678910 3,,,,,,,,m msss ssM CELf L (3.68) The natural frequency shown in Eq. 2.24 is nondimensionalized as 1245678910 3 311 ,,,,,,,, 2 1n ssm s msss s ssss sf f LM E CEL ELL L (3.69) Furthermore, the angular frequency ,n 2n f is a function of the same groups and follows Eq. (3.69). The Rayleigh mechanical damping in the system using its empirical relation listed in Eq. 2.1 is nondimensionalized as 1245678910 3 22,,,,,,,, 1mmn ss sss s s sRM f L LE E L (3.70) It should be noted here that the mechani cal damping model in the system does not accurately represent all loss mech anisms. Some of the general damping losses were investigated

PAGE 99

99 and presented in Appendix B. Although the LEM in this dissertation s till assumes a Raleigh viscous damping effect with an equivalent damp ing ratio, other damping losses are also studied here for their scaling behavior. The loss due to air flow in the viscous region due to device vibration is given by an empirical relation derived in Appendix B as 1 2 2212 61ss nss a eq eqsE kbt Q R RL (3.71) All the variables are defined in Appendix B. For any structure operati ng at fixed conditions, and nk remain constant. So, the nondimensional form simplifies to 2 2 54123 1sss n a eq eq s s sLE k Q R R L L L (3.72) The losses at the support that arise due to the transmitted energy through the clamp during flexural vibration is empirically given as 30.23,s c sL Q t (3.73) and in nondimensional form is 3 51 0.23cQ (3.74) The surface losses are given as 23 s s s s sbtE Q btE (3.75)

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100 where s E is the difference between the adiabati c and isothermal Youngs modulus of the material. It is also known as the dissipation modulus of the surface layer whose thickness is given by The nondimensional form of Eq. (3.75) is 45 4523 s s s sE Q E L (3.76) Similarly, the volume losses are s v s E Q E (3.77) For scaling purposes, we assume that the dissip ation modulus scales proportional to the elastic modulus of the material. In this analysis, the s queeze film damping is negl ected as the vibrations occur in free space without infl uence of walls around the device. Finally the empirical form for thermoelastic loss in a vibrat ing structure is given as 21 80,s t s psnsE k Q CtfE (3.78) where k is thermal conductivity and pC the specific heat capacity at constant pressure for the material. This expression in Eq. (3.78) is a simpler form of the actual expression derived in Eq. B.22. Expanding for the natural freq uency and simplifying leads to 2 380, s s s t spssE L k Q CtE (3.79) The nondimensional form of the quality factor can be expressed as 2 51 80,s t psssE k Q CEt (3.80)

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101 For the mechanical characterization, the fre quency response between the input acceleration and the resulting deflection is already shown in Eq. 2.37. Normalizing the impedance terms, 1mm msjMR jC in the function yields 1245678910 2221 ,,,,,,,,mmms sssssssssjMRjC f LELELE (3.81) Substituting this back into the original equa tion, we obtain the nondimensional form of the response as 124567891011,,,,,,,,.tip sw f L (3.82) Now, for an applied voltage, neglecting all mech anical loads, the moment resulting at the PZT due to the voltage is nondimensionalized as 3 2 31 31 311 2 2pp inss o s ssssEt VEL c b Md L ELLLd d (3.83) 145713 3,,,o ssM f EL (3.84) The effective piezoelectric coefficien t of the composite is a function of 31d and is nondimensionalized as 2Vtip opp ms appapp cw M LL dL VEIV (3.85)

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102 3 14567 31 4 311,,,, 2o p p msss s app c s ssM L L dELL f dL V EI L EL d (3.86) Extending the analysis to the electromechanical parameters, the transduction factor is nondimensionalized as 31 31 m m s s msmsssd dd dEL CCEL (3.87) 1245678910 31,,,,,,,,ssf dEL (3.88) The free and blocked electrical capacitances are similarly expressed in terms of nondimensional groups as 2 2 31 31 p p sss efss p p sL b Lb dELL CdEL t t L (3.89) 12345678910 2 31,,,,,,,,,ef ssC f dEL (3.90) and 2 22 2 31 31 2 31 m ef m ebefss msssmsssd C dd CCdEL CdELCEL (3.91) 12345678910 2 31,,,,,,,,,eb ssC f dEL (3.92) The empirical form of the dielectric loss in the piezoelectric layer is represented as

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103 2 2 31 31111 tan tan 1e eb eb s s s ss s s sR C C E dE dEL E L (3.93) 2 3112345678910,,,,,,,,,s es sE RdEf (3.94) Finally, the electromechanical coupling coefficient is nondimensionalized as 2 22 2 31 12345678910 2 31,,,,,,,,,m m ef efms msss ssd dd f C CC CEL dEL (3.95) The overall short circuit mechanical impedan ce expressed in the el ectrical domain, given in Chapter 2, is rewritten here as 211 .e msmm msZjMR jC (3.96) 2 22 222 31 311 1 .sss e mmms ms ss sssssssss ssLE jMRjC Z dEL LELELE dEL (3.97) Nondimensionalizing the impedance results in 1245678910 2 31,,,,,,,, 1e ms s ssZ f EdE (3.98) Similarly, the blocked electrical impeda nce is also nondimensionalised yielding 1e eb eb e ebR jC Z R jC (3.99)

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104 1 tane ebR C (3.100) 22 3131123456789101 ,,,,,,,,, tanss ess ss ebEE RdEdEf C (3.101) and, 12345678910 2 31,,,,,,,,, 1eb s ssZ f EdE (3.102) In the resulting overall Thvenin equivalent circuit (shown in Figure 3-1), the Thvenin impedance is obtained by combining th e two impedances in parallel as 22 3131 2 31 22 313111 1 11e mseb ss e ssss msebs th e e mseb msebss ss ssssZZ EdEEdE ZZ Z ZZ Z ZEdE EdEEdE (3.103) Therefore, 12345678910 2 31,,,,,,,,,. 1th s ssZ f EdE (3.104) Similarly, the Thvenin voltage is the open ci rcuit voltage across the piezoelectric layer given as

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105 2 3 31 31 31 22 31311 11eb s m ss ebmossos th e e s mseb mseb ss ss ss ssssZ M EdE Z MaLaL V E ZZ Z Zd ELd L EdEEdE (3.105) So, the dimensionless form is 1234567891011 31,,,,,,,,,.th sV f L d (3.106) As evident in the above e quation, the voltage is directly proporti onal to the input acceleration, 11 The optimal load for maximum power transfer occurs when the output resistance equals the absolute value of the Thvenin impedance. 2 31 2 311 1sth Lth ss s ssZ RZ EdE EdE (3.107) and 12345678910 2 31,,,,,,,,,. 1L s ssR f EdE (3.108) The current across the load resistor is given as 31 31 22 313111th s th s Lss th L thLs ss ssssV L VdE ILEd Z R ZR EdEEdE (3.109)

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106 and 1234567891011 31,,,,,,,,,.L s ss sI f E LEd (3.110) The output voltage across the lo ad resistance is given as 31 31 2 311 s LL LLL ss ss sssL IR VIR d E LEd EdE (3.111) and 1234567891011 31,,,,,,,,,.L sV f L d (3.112) The output rms power across the load is obtai ned as the product of the load current and voltage, represented as 2 31 311 2 s LL LLLss s s s ss s sE VI PVIEL L E LEd d (3.113) 2 1234567891011 2,,,,,,,,,.L s ss sP f E EL (3.114) The tip velocity can be obtained by solving for the current in the Thvenin equivalent as 2.ebLmo ee ebmsLmseb Z RMa U ZZRZZ (3.115) Expressing the velocity in di menionless terms simplifies Eq. (3.115) to

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107 1234567891011,,,,,,,,,.s sU f E (3.116) Therefore, the input rms mechanical power is the product of the force and the resulting tip velocity given as 2 311 22mos inss s s ss s ss s M aE U PFUEL E L E L (3.117) and the nondimensional form for th e input power is simplified to 2 1234567891011 2,,,,,,,,,.in s ss sP f E EL (3.118) All the response parameters derived so far are generalized expressions and will simplify near resonance when the reactive portion of the mechanical impedance approaches zero. Hence, at resonance, only the real part of the impedance remains, which is directly related to the dissipative mechanisms in the device. Since th e LEM uses very basic simplified empirical relations for the mechanical damping and electr ical loss, the expressions do not provide any useful physical insight. Therefore, they have not been simplified and/or presented here. The overall electromechanical efficiency expr essed as the ratio between the output and input power is observed to be independent of the base acceler ation (i.e., not a function of 11 ). This holds true subject to the constraints in the model that implies a linear relation between the input acceleration and the tip deflection, which is valid for small deflections. The efficiency is given as 12345678910,,,,,,,,,.L inP f P (3.119)

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108 Another quantity that is gene rally used to parameterize energy harvesters is the power density, usually represented as amount of power generated per unit volume. In some literature, power density is expressed with respect to ma ss and/or volume. The exact usage depends on specific application that poses a restriction on volume or mass of the energy harvester. The power densities with respect to volu me and mass are expressed in Eqs. (3.120) (3.123) 2 3,vL s ss s s s L D s s sP E EL EE P P volume volumeL L (3.120) and in nondimensional form, is represented as 2 1234567891011,,,,,,,,,.vD ss ssP f EE L (3.121) When expressed with respect to mass, the power density is 2 3,mL s ss s s s L D s ss ssP E EL EE P P mass massL L (3.122) And, in nondimensional form, is represented as 2 1234567891011,,,,,,,,,.mD ss sssP f EE L (3.123)

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109 Table 3-5: Final set of nondimensiona l groups involving response parameters. Dependent groups Functional dependence msssCEL 3 m s s M L, 1n s s sf E L 1245678910,,,,,,,, f 31 md d 14567,,,, f 31 s sdEL 1245678910,,,,,,,, f 2 31 ef s sC dEL 2 31 eb s sC dEL 2 31 s es s E RdE 12345678910,,,,,,,,, f 2, 2 311eb s ssZ EdE 2 311L s ssR EdE 12345678910,,,,,,,,, f 2 311e ms s ssZ EdE 1245678910,,,,,,,, f 31 L s ss sI E LEd 31 L s V L d, s s U E 31 th s V L d 1234567891011,,,,,,,,, f 2 L s ss s P E EL 2 in s ss s P E EL 2 1234567891011,,,,,,,,, f vD s s s sP EE L mD s s s ssP EE L 2 1234567891011,,,,,,,,, f 12345678910,,,,,,,,, f

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110 All the above derived variables are summarized in Table 3-5 that list the response functions in terms of the original independent groups. Scaling Theory Let us now use the abov e dimensional analysis to develop a scaling theory for reducing the size of the structure from macro-scal e to micro-scale. This analysis will enable us to investigate the performance metrics of the energy harvester as a function of miniaturization. For this analysis to hold true, it is assumed that the linear Euler-Bernoulli equati on is valid for microscale devices. This is an important assumpti on because the external source acceleration remains constant for an application and th e micro device is subjected to potentially large vibrations. This results in a large deflection and the device can ea sily begin to operate in nonlinear region. We define a variable scale factor called ""s that covers the meso-scale 1 typicallys range to the micro-scale 0.001 typicallys This would, for example, cover the typical dimensions of the structure from a meter down to a millimeter. All the other length scale dimensions, such as the width and thickness of the beam and the size of the piezoceramic, are assumed to be similarly scaled in proportion to the scale factor s. For the purpose of this exercise, the material properties are assumed to be constant over the scale. It should be noted here that the materials used for mesoscale piezoelectric devices are generally Aluminum, Stainless steel, Brass etc while the MEMS devices are mostly made of Silicon. In addition, the properties for bulk piezoelectric material are well known and are generally hi gher than thin film PZT. 31,,,,, = constantssppEEd (3.124) and ,,,,,,, .ssppLbtLtlwhs (3.125)

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111 Using this assumption, the nondimensional groups (Table 3-4) 123, and are constant as they represent the material properties. In addition, 45678910,,,,, and remain constant as both the numerator a nd denominator scale proportionally. If the input acceleration is assumed to be constant, which is true because th e input base vibrations are independent of the harvesting device scale, and the material is fixed, 11 and 12 are proportional to s. Based on these trends, the dependence of all the nondime nsional parameters described earlier can be evaluated as a function of the scale factor. Th e effective short-circuit mechanical compliance expressed in Eq. (3.68) remains constant across the scale as all the dependant groups are constant. 1 constant, which implies .msssmsCELC s (3.126) Similarly, the nondimensional effective mass varies as 3 3 3constant, which implies .m ms ssM M Ls L (3.127) The mechanical damping in the system can be expressed as 22 2constant, implies .m ms sssR R Ls LE (3.128) Again, it is noted that the mechanical damping model is not perfectly valid, but is calculated nevertheless to complete the overall scaling analys is. In reality, overall damping is dependent on various mechanisms described in Eq. (3.71) to Eq. (3.80) that have to be studied for their scaling behavior. The air flow loss qua lity factor is scaled as 2 2 5constant 23amn ssQkPk E (3.129)

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112 which will remain constant for no change in pr essure and length to thickness ratio. This holds true for a specific material operating at the same conditions such as temperature, molar mass, etc. Next, the support dissipation is given as 3 51 0.23constantcQ (3.130) The support losses remain constant as long as the length to thickness ratio is maintained. However, it is inversely proportional to th e thickness and could become significant if the thickness and length are of same order. The surface dissipation mechanism is shown in Eq. (3.131) and is related to the thickness of the thin film layer on the substrate. If all the dimensions are proportionally scaled, they bear no effect on surface dissipation. 45 4523 s s s sE Q E L (3.131) Next, the volume loss is studied, which is only depe ndent on material propert ies, specifically its elastic modulus and therefore is independent of scaling here. ,s v s E Q E (3.132) Finally, thermoelastic di ssipation shown in Eq. (3.133) has a significant on the size of the device. It is related to the material properties such as modulus and density. In addition, it depends on the length and thickness of the devi ce. But when all dimensions are scaled prop ortionally, it increases with a decrease in thickness. Therefore, the thermoelastic quality factor increases at reduced scale. This observati on is particularly important he re as the MEMS PZT energy harvesters are shown in Chap ter 6 to posses a much higher Q as opposed to the meso-scale

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113 devices. This can be attributed to the fact th at thermoelastic damping becomes significant at micro-scale and dominates most other loss mechanisms. Hence, a higher Q was measured. 2 51 80,s t psssE k Q CEt (3.133) Next, the natural frequency of the device is represented as 11 constant, implies 1n n s s s sf f Ls E L (3.134) From Eq. (3.69), the angular frequency n is proportional to s. The normalized frequency response between the in put acceleration and th e resulting deflecti on, represented in Eq. (3.82)is 11tip sw s L Therefore, 2tipswLss (3.135) Therefore, the tip deflection scales down as the square of the dimensions thereby increasing the operating range for the device in it s linear region. The linear behavi or limit is usually expressed as a function of the ratio betw een the tip deflection and length for a cantilever beam (Section 3.3.1). Now, the nondimensional effective piezoele ctric coefficient of the composite (shown in Eq. (3.86)) is a function of 31d and scales as 31constant, implies constant.m md d d (3.136) Similarly, the transduction factor that rela tes the electrical domain and the mechanical domain is nondimensionalized in Eq. (3.88) and scales as

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114 31constant, implies .s ssLs dEL (3.137) The free and blocked electrical capac itances are similarly scaled with s as 22 3131;constant, implies ;ef eb efebs ssssC C CCLs dELdEL (3.138) According to this scaling theory here, the dielec tric permittivity and piezoelectric coefficients are assumed to remain constant. These properties ar e highly dependent on the fabrication process and are therefore subject to large variations at the micro-sc ale. The nondimensional dielectric loss in the piezoelectric layer, shown in Eq. (3.94) is 2 31constant, implies constant.s ese sE RdER (3.139) In addition, the electromechanical coupling coefficient 2 also remains constant versus scale. Furthermore, the power tr ansfer parameters can also be reduced to expressions that show their dependence on the scale factor. A similar anal ysis is carried out to reduce these parameters to obtain a simple relation with the scale fact or. The short-circuit mechanical impedance, blocked electrical impedance and the Thvenin equivalent impedance remain constant with s as 222 313131,,constant. 111e msebth sss ssssssZZZ EdEEdEEdE (3.140) Similarly, the nondimensional Th venin voltage and the output voltage across the load are represented as 2 11 3131,, implies, ,.th L thLs ssV V sVVsLs LL dd (3.141)

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115 The optimal load for maximum power transfer which is the magnitude of the Thvenin impedance also remains constant with s. Again, the optimal load is a direct measure of the overall damping in the system and therefore follows the damping assumptions made in the model. For a better estimate, each of the losses needs to be included into the model and their scaling behavior studied. The current across the lo ad resistor given by the ratio of output voltage and load scales as 2 11 31, implies, .L Ls s ss sI sIsLs E LEd (3.142) The output rms power across the load and input rms mechanical power sc ale with respect to s as 22224 11 22,, implies, ,.in L Lins ss ssss ssP P sPPsLs EE ELEL (3.143) The overall electromechanical efficiency rema ins constant with the scale factor and is independent if all the device dimensions are sc aled proportionally. The power densities with respect to volume and mass are expressed in Eqs. (3.120) (3.123) scale as 22 11,, which implies ,vm vmDD DD ssss sssssPP sPPs EEEE LL (3.144) To summarize, a comprehensive nondimensiona l tool was developed using the analytical LEM that was investigated for its scaling behavi or for miniaturization. This tool will enable variation of various individual properties and dimensions to study the device response for specific conditions and applications.

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116 Validation of Scaling Theory The developed scaling theory needs to be va lidated, and the predicte d results and response of various parameters need to be corroborated. To verify the scaling theory experimentally would mean designing test specimens over an apprec iable dimensional range followed by detailed characterization to extract the lumped elem ent parameters, which is difficult and time consuming. In addition, obtai ning custom made commercial PZT patches over the required dimensional scale is fairly expensive. Instead an FEM validation is easier to carry out by changing the device dimensions and performing simulations. A simple validation was carried out by simulating a sample PZT-Aluminum composite beam using ABAQUS to calculate the static lumped element parameters such as natural frequency, short-circuit mechanical complian ce and mass and the effective piezoelectric coefficient (Table 3-7). A PZT cantilever composite beam clamped at one end with a proof mass at the other end was assumed with arbitrary dime nsions. The material di mensions and properties of the base test specimen simulated are listed in Table 3-6. The composite beam was modeled using 3-D solid elements that consisted of 20-no de quadratic hexahedran elements for shim and proof mass and 20-node quadratic he xahedran piezoelectric elements for the PZT layer. The final meshed structure is shown in Figure 3-3. The computed values compared favorably with the analytical values. The sample was then reduced proportionally in all it dimensions to micro-scale, and the dependence of these parameters was observed and compar ed with the predicted trend.

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117 Table 3-6: Material dimensi ons and properties of composite beam for FEM validation. Material Dimensions (s=1) Length of beam 1 m Width of beam 0.2 m Thickness of beam 0.1 m Length of PZT 0.5 m Width of PZT 0.2 m Thickness of PZT 0.05 m Length of proof mass 0.2 m Width of proof mass 0.05 m Thickness of proof mass 0.05 m Material Properties Elastic modulus of shim 73.263 GPa Density of shim 2718.33 kg/m3 Elastic modulus of PZT 66 GPa Density of PZT 7800 kg/m3 Piezoelectric coefficient -190e-12 m/V Dielectric constant 1800 Figure 3-3: Meshed PZT composite can tilever beam for FEM validation.

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118 Simulations were performed for the short-circ uit case to obtain the eigenvalue solution for the beam. From the modeshape, the deflections along the center line were extracted and integrated to estimate the kinetic and potential energies. Lumping these energies at the tip yielded the short circuit mechanical compliance a nd mass. The eigenvalue analysis directly gives the natural frequency of the beam. Another si mulation was carried out to obtain the static deflection due to an applied vol tage to the PZT. The tip defl ection corresponding to unit applied voltage is the effective piezoelectric coefficient md of the composite beam. Assuming that 1 s represents the base structure, the dimensions of the beam we re proportionate ly scaled down 1000 times in regular intervals and simulations were performed to estimate the lumped element parameters as described before. Figure 3-4: Short circuit natural fr equency for a PZT composite beam. Figure 3-4 shows the short circuit mechanical natural frequencies obtained directly from the eigenvalue solution using FEM. The scale vari ed from the base dimensions down to the micro-scale, for example, the length of the beam varied from 1 m to 1 mm. As predicted from the

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119 scaling theory, the natural frequencies incr eased with decreasing dimensions given by 1nf s verified by the results in Figure 3-4. The difference between the natural frequencies from FEM and those estimated from theory can be attributed to the fact that the model employs a static modeshape while the FEM is based on the frequency dependent modeshape. Since 3-D stress elements were used, the defl ection across the width of the beam is not constant and therefore, the deflections along the center line are extr acted from the eigen modeshape. The 2-D deflection sh ape thus obtained is integrated to obtain the stored potential energy, which is lumped at the tip to estimate an effective compliance of the beam. This was repeated for all the test cases and the results are presented in Figure 3-5. As indicated in the figure, the short circuit compliance in creases proportionally with decreasing s, given by 1msC s as predicted with the scaling theory. Figure 3-5: Short circuit compliance for a PZT composite beam.

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120 In Figure 3-5, the compliance predicted from FEM is lower than the model because the FEM predicts higher natura l frequencies, shown in Figure 3-4. Higher natural frequency implies lower compliance. Next, the mechanical mass is calculated from the natural frequency and compliance using 21 2m nmsM f C (3.145) The results obtained for all the test cases are plotted in Figure 3-6. As indicated in the plots, the mass scales down as 3m M s since all the three dime nsions vary proportionally decreasing the overall volume cubically. Howeve r, the masses from both methods are nearly equal, primarily because the differences in natural frequency and compliance compensate inversely to minimize the difference in mass. Figure 3-6: Effective mechanical mass for a PZT composite beam. Finally, from the static voltage simulation where the deflection was calculated for a unit applied voltage to the PZT, th e tip deflection was directly ex tracted and compared with the model. The results are shown in Table 3-7. The effective piezoel ectric coefficient remained

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121 constant over the scaling range validati ng the trend predicted from the model (Figure 3-7). In this case, both models being static, th e values match very well within 2%. Figure 3-7: Effective piezoelectric co efficient for a PZT composite beam. The results represented by the a bove plots are summarized in Table 3-7. The columns indicate the LEM parameters for various scales represented with s. The static LEM parameters thus obtained matched with the predicted trend and th erefore serve as a good validation for our design methodology. Table 3-7: Static lumped element parameters from FEM and LEM to validate the scaling analysis. F n (Hz) C ms (m/N) M m (kg) d m (m/V) s FEM Theory FEM Theory FEM Theory FEM Theory 1 72 67 2.17e-7 2.42e-7 22.20 23.07 -1.60e-8 -1.57e-8 0.5 145 126 4.30e-7 5.73e-7 2.80 2.76 -1.60e-8 -1.57e-8 0.1 724 616 2.17e-6 3.09e-6 0.022 0.022 -1.60e-8 -1.57e-8 0.05 1452 1231 4.29e-6 6.20e-6 2.80e-3 2.70e-3 -1.60e-8 -1.57e-8 0.01 7260 6154 2.13e-5 3.10e-5 2.26e-5 2.16e-5 -1.60e-8 -1.57e-8 0.005 14521 12310 4.29e-5 6.20e-5 2.80e-6 2.69e-6 -1.60e-8 -1.57e-8 0.001 72604 61540 2.14e-4 3.10e-4 2.24e-8 2.20e-8 -1.60e-8 -1.57e-8

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122 Now that we have verified the scaling anal ysis using ABAQUS, we can design the test specimens and make reasonably accurate predictions for its response characteristics, subject to the validity of the model assu mptions and boundary conditions. Extension to MEMS After completing the scaling analysis for the lumped element and power transfer parameters, we now explore this concept on a micro-scale. Some of the features observed in MEMS structures that were sugge sted by the scaling theory described earlier are as follows. First, some parameters that control the power ha rvesting aspects, such as device efficiency and coupling factor, remained constant even though the device was reduced in size. This holds true in the ideal case and is not valid if the piezoelectric properties vary at micro-scale. However, parameters such as overall power delivered to a load and even power density decrease due to scaling and miniaturization. On th e other hand, an advantage of mi crofabrication is the feasibility to batch-fabricate many devices simultaneously af ter standardizing the pr ocess. Furthermore, fabricating arrays of such devi ces enables parallel or series connection of the electrodes to enhance the output voltage or current to m eet the micro power processor requirements. Design of Test Structures In order to design the test specimens at the micro-scale, the target natural frequencies used in this design analysis are 100 Hz, 1 kHz and 10 kHz. These frequencies were selected to broadly cover the widely occurring vibration fre quencies (Chapter 1). The dimensions of the composite beam are designed to match this frequenc y. The properties of the materials are fixed. The composite beam is comprised of a PZT thin film deposited using so l gel process on bulk silicon shim. The material properties used in this analysis were obtained from existing literature (Horowitz 2005) and ARL and are listed in Table 3-8.

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123 Table 3-8: Properties and dimensions used for designing MEMS PZT devices. Elastic modulus of Silicon 169 GPa Density of Silicon 2330 kg/m3 Elastic modulus of PZT 60 GPa Density of PZT 7500 kg/m3 Piezoelectric coefficient -45 X 10-12 m/N Relative permittivity 900 Damping ratio 0.01 tan 0.02 The objective of this design formulation is to build test structures to match the target natural frequencies. In addition, th ey are designed for specific acceleration levels0.1, 1, 10 ggg that are indicative of ambient acceler ation levels in many applications. The acceleration levels act as the criterion to enfo rce the linear elastic limit. The LEM is valid for small deflections and will fail to predict the behavior if the resonant deflection for a specific acceleration is beyond the elastic limit. Furthe rmore, after estimating the amount of power available for reclamation in these beams, th e dimensions of each are optimized to obtain maximum power. The schematic (elevation and pl an) of a typical cantil ever composite beam that is designed is shown in Figure 3-8. Figure 3-8: Schematic of a single PZT composite beam.

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124 The design constraints that ar e considered for the design fo rmulation are listed below. Our goal is to fit as many such can tilever structures as possible in 1 1 500 cmcmm which is the overall dye size. Therefore, we divide the dye further into smaller cells, typically of the size, 3 3 mmmm or 5 5 mmmm squares according to the size of the composite beam that consist of multiple cantilevers connected to a single proof mass. Due to the constraint on the dye size, th e height of the proof mass is fixed as 500 m which is the thickness of the wafer. For simplicity, the width of the piezoelectric layer and the shim are consider ed equal in the desi gn formulation. In addition, the thickness of the piezoelectric layer is fixed at 0.5 m due to fabrication constraints. The resulting natural frequency of the stru cture should match the external vibration frequency for maximum power generation. Clearance needs to be provided to the stru cture on the bottom and side wall for it to vibrate. For simplicity, we assume that th e clearance at the bo ttom needs to be less than the tip deflection at resonance. In addition, the mechanical damping predominantly controls the amplitude of th e tip deflection at resonance. We have assumed a damping ratio of 0.01 in our design analysis which held true for most meso-scale cantilever structures verified experimentally. However, as discussed earlier in this chapter, the damping phenom ena are different at microscale and could vary, depending on how the loss mechanisms scale with device size. The main constraint of the analysis is the li near elastic limit. In order to stay below 2 % of the linear elastic limit, the tip defl ection of a cantilever beam must be less

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125 than 0.4 times the actual length (Appendix D). Therefore to incorporate this in our design, the dynamic response from the lumped element model is used to predict the tip deflection at resonance. This tip deflection, which reaches a maximum at resonance, is ensured to stay w ithin the linearly elastic range. In addition, the stress fracture limit for S ilicon was considered as a constraint. The yield strength for Silicon is 7 GPa (Petersen 1982) and the stress limit for the devices was designed to be less than 10 % of the yield stress. Since no yield strength data was readily available for thin film P ZT, only Silicon stress state was analyzed to meet the constraint. Finally, the design has to meet the energy reclamation circuit requirements. The minimum overall open circuit voltage should be 2 V and the output current should be at least 20 A The blocked electrical capac itance should be less than 4eb mC R (Shengwen 2005). A simple optimization technique was adopted to design the test structures. However, for the second generation devices designed in the fu ture, a detailed and thorough optimization needs to be carried out. The techni que adopted here is rudimentar y and is based on a parametric analysis, by varying the dimens ions and observing the overall res ponse subject to constraints. The process that was adopted is listed in the following steps. The overall available area is 1 1 cmcm As described earlier, this total area is divided into various cells. The first step involves deci ding the area of each cell. From the scaling analysis, we observed that as the dimensi ons become smaller, the natural frequency increases. So for the lowest na tural frequency, the w hole dye is used as a single cell. For

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126 1 kHz cantilevers, the cell size was typically assumed to be 3 5 mmmm that leads to 6 cells or3 3 mmmm which leads to 9 such cells in the overall dye. The number of structures in each cell has to be decided. For convenience, it was assumed that there were 3 cantilever beams attached to a single proof mass at their tips. Consequently, this number n varied anywhere from 1-4 fo r the designs to generate enough power from the device. We can choose the width of the proof mass, w after providing e nough clearance at the edges. Similarly, we can also choose the overall length of the structure s L lclamp The size of the clamp in all cases was assumed to be anywhere between 0.5 mm and 1 mm, depending on the overall length of the composite beam. Once the number of beamsn and w have been established, the width of the beam can be calculated s pbandb Using the dimensions of accelerometers liste d in DeVoe and Pisano (2001) and Vizvary (2001), some typical ratios for s sLb (5, 10) and s sLt (50, 100, 200) were obtained to act as base dimensions. For some cases, th e dimensions of existing meso-scale aluminum beams (Table 2-6, Table 2-7) were reduced to micro scale to estimate these typical ratios. Using these typical ratios and width of the beam, basic length and thickness was calculated. To obtain the required natura l frequency, the lengths and thicknesses of the beam and proof mass were varied by tria l and error. Some knowledge about how these dimensions affect the mass and compliance of the beam (from the scaling analysis) was useful in

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127 arriving at the final dimensions. For exampl e, if we wish to decrease the natural frequency, we need to either increase the compliance or the mass. We can increase the length of the beam to increase its compliance, which would restrict th e length of the proof mass as the overall length is fixed. In addi tion, increasing the thic kness of the proof mass will also increase the mass which will in turn increase the natural frequency. Figure 3-9 shows the top view of a typical cell that has a 3-element array of such composite beam structures. In the figure, n is the number of individua l cantilevers present in one cell structure, which is genera lly 3 for most of the designs. Figure 3-9: Layout fo r a single cell comprising of an array of composite beams. Test devices Using the above design method ology, various test struct ures were designed. The following table lists all the properties of the test specimens designed to achieve the target natural frequencies.

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128 Table 3-9: Material pr operties of piezoelectric composite beam. Elastic modulus of Silicon s E 169 GPa Density of Silicon s 2330 3kgm Elastic modulus of PZT pE 60 GPa Density of PZT p 7500 3kgm Piezoelectric coefficient 31d -100 X 10-12 mV Relative permittivity r 1000 Table 3-10 shows the dimensions of the test stru ctures designed to achieve the target natural frequencies for a specific input ac celerations. The predicted natu ral frequency is obtained from the calculated compliance and mass using the di mensions and properties selected. It was difficult to design an optimal structure for 0.1 g, due to the restrictions in area and natural frequency. However, the same st ructures that were designed for 1 g and 10 g can be used for 0.1 g, but would generate less power compared to th e other designs. In the table, the first row lists the targeted frequency and the correspon ding input acceleration. Since, the design and optimization procedure adopted here is slightly crude, the resulting de signs possess different natural frequency. While perf orming the experiments, the input vibrations applied to the structure will be altered to match the frequencies. All the other values listed in the table are selfexplanatory. The last row indicates the number of cells n in 1 1 cmcm and number of cantilevers per cellm. The nomenclature used to denote the 9 different designs are indicated in the first row of Table 3-10.

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129 Table 3-10: Designed MEMS PZT structures. PZT-EH1 2 3 4 5 6 7 8 9 Ls (mm) 1 4 6 6 6 0.6 1 7 0.5 bs (mm) 0.2 1 2 2 1 0.3 1 1 0.5 ts (um) 12 12 12 12 12 12 12 12 12 Lp (mm) 1 3 6 6 4 0.4 1 5 0.5 bp (mm) 0.2 1 2 2 1 0.3 1 1 0.5 tp (um) 1 1 1 1 1 1 1 1 1 l (mm) 1 4 3 2 0.5 0.2 2.5 0.5 1.8 w (mm) 0.8 3 3 3 3 0.8 4 2 0.8 h (um) 500 500 500 500 500 500 500 500 500 # of cantilevers/chip 3 3 3 3 3 3 1 4 3 Fn (Hz) 354 25 30 40 68 3055 127 66 501 normalized P_load (nW/g) per cantilever 3 540 480 250 49 5.75E-05 88 35 3

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130 CHAPTER 4 DEVICE FABRICATION AND PACKAGING This chapter begins with a detailed descri ption of the fabrication process of micro piezoelectric structures and the associated micr ofabrication challenges. A final process flow consisting of 6 Photosciences 5090 CRSL masks and the recipes for device release are presented. The device is based on a Silicon-on-Insulator (SOI) wafer consisting of Si/SiO2/Ti/Pt/PZT/Pt as the layered structure. The fabr ication involved uses standard sol gel PZT and conventional surface micromachining techniques. The thin films were deposited using sputtering or evaporation techniques, and sol ge l PZT is deposited using a 3-step process involving spin, bake and anneal steps. The final step involved patterning the beams fr om the top and proof mass from the backside and deep reactive ion etching to re lease the devices. Furthermore, a discussion of packaging used in the experimental characteriza tion of the test devices is provided followed by the mounting mechanism employed fo r the devices in the package. Process Flow The proposed test structures designed using the electromechanical lumped element model were fabricated using conventional surface a nd bulk micro processing techniques (Madou 1998). The first three major steps in the fabricati on, which involved the PZT layer deposition and patterning, were carried out at the Army Research laboratory (ARL) through MEMS Exchange. ARL has a standard PZT process (Piekarski et al 1999) available, from which the process flow was devised and the corresponding mask-sets were generated. Their process involves three masks for depositing and patterning the PZT layer along with its electrodes. The fabrication was carried out on a 4 Silicon On Insulator (SOI) wafer that had a 0.4 m buried oxide layer

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131 separating the overlayer from the bulk silicon substrate. The active overlayer thickness was 12 m of Silicon. First, the base SOI wafer was RCA cleaned to remove surface particles and impurities. Silicon dioxide 20.1 mSiO was deposited on the wafer using the Plasma Enhanced Chemical Vapor Deposition (PEC VD) technique as shown in Figure 4-1. The 2SiO layer is necessary to prevent possible lead diffusion into the silicon overlayer during the PZT anneal step. Any lead diffusion will im mensely affect the performance of the PZT and could result in the loss of its piezoelectric property. SiO2 (100 nm) Si (12 um) SiO2(BOx) Figure 4-1: Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer. Next, Titanium 20 Tinm and Platinum 200 Ptnm are sputter deposited on the whole wafer to form the bottom electrode, as shown in Figure 4-2. A Titanium seed layer is provided to ensure good adhesion be tween Pt and the substrate. SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Figure 4-2: Sputter deposit Ti/Pt (20 nm /200 nm ) as bottom electrode. Next, sol gel PZT Lead,/Zirconium,/Titanium, PbZrTi of composition 125/52/48 is deposited using a 3-step process involving spin, bake and anneal steps as shown in Figure 4-3. The sol gel is initially spun at a certain rotationa l speed (typically at 3000 rpm for 30 s) in a spinner, after which the wafer is pyrolized on a hot plate at 350 C for 5060 s For higher

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132 thicknesses of PZT, multiple spin and bake steps ar e carried out before the final anneal step is done at a much higher temperature (typically o700 C for 1 min in the presence of 2O ) in a rapid thermal anneal (RTA) furnace. These multiple sp in and bake steps are adopted to maintain consistent PZT properties even at higher thickness. PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Figure 4-3: Spin coat sol-gel P ZT (125/52/48) over the wafer usi ng a spin-bake-anneal process. Next, the top electrode for the PZT 200 Ptnm is patterned using a lift-off photolithography technique with the TopElectrode mask as shown in Figure 4-4. Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Top_electrode mask Figure 4-4: Deposit and pattern Pt for top electrode using liftoff. In the next step, openings are patterned on the device areas to provide access to the bottom electrode. The exposed PZT in those areas is wet etched revealing the bottom Pt underneath as shown in Figure 4-5. A PZTEtch mask was used to pattern the access holes on the electrode. The PZT in open areas is etch ed using a combination of hydrofluoric acid/ hydrochloric acid/de-ionized wate r that had an etch rate of 23 nms. The residues left behind in the previous etch were removed using d ilute nitric acid and hydrogen peroxide.

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133 Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) PZT_mask Figure 4-5: Pattern opening for access to bottom electrode and wet etch PZT using PZT Etch mask. The final step in the PZT process at ARL i nvolved patterning the el ectrode area using an I onMilling mask. The exposed PZT and bottom / PtTi are physically etched using Ion milling process resulting in the PZT feat ures along with top and bottom electrodes with provisions for bond pads. The profile for this step is shown in Figure 4-6 Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Ion_Milling mask Figure 4-6: Ion milling of PZT and bottom electrode using Ion Milling mask as pattern. After the PZT process was completed, the wafe rs were returned for device release steps, which were carried out at UF. The residual stre sses in each layer patterned as provided by ARL are listed in Table 4-1. Table 4-1: Residual stress measurements fo r the PZT pattern pro cess (source : ARL). Film As Deposited Stress PECVD Oxide (0.1 m film) -45.7 MPa Annealed Bottom Ti/Pt (20/200 nm film) 558.1 MPa Annealed

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134 PZT (1000 nm film) 109.7 MPa Oxide/Ti/Pt/PZT Stack 175 MPa* Top Pt (200 nm film) -15 0 MPa* as provided for a general process. The rest were measured specifically for this process. Before performing the release, Gold A u was deposited on the wafer using e-beam deposition technique. Alternativel y, gold can also be deposited using the sputtering technique, but an e-beam was chosen for our process due to its availability. Then, patterns were transferred to the wafer to protect the bond pads as shown in Figure 4-7. The remaining exposed A u areas were removed using Transene Gold (Au) Etchant (Potassium Iodide solution) protecting the bond pads that were patterned using the B ondPads mask. Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Bond_Pads mask Figure 4-7: Deposit Au (300 nm ) and pattern bond pads using B ond Pads mask and wet etching. In the next step, the beams are released from the top. A positive photoresist (PR), AZ 1512 was first spun at 4000 rpm for 40 s to approximately deposit 1.5 m of PR. This was followed by soft baking the wafer for 60 s on a hotplate at o95 C. The B eamEtch mask is used to pattern the release trenches for the beams with Karl Suss MA-6 Mask Aligner, following which, the wafer was hard baked for 60 min at o90 C Subsequently, the pad oxide deposited in the first step is wet etched using BOE 2:6:1 HFHO Removal of this oxide is critical as any residual oxide in the exposed ope n areas will prevent silicon et ching during topside release. Consequently, 12 m of overlayer is removed using Deep Reactive Ion Etching (DRIE) with the

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135 STS-MESC Multiplex ICP etcher. DRIE is a highl y anisotropic etching process that involves number of etchpassivation cycles to achieve the vertically directional profile (also called Bosch process). During the et ch cycle, highly reactive 6SF gas is used along with 2O to perform a nearly isotropic etch of Si. In the deposition cycle, 48CF is used as a passivation layer to protect the etched area from further etching. Th en, the process switches to an etch cycle where the energetic plasma ions are collimated and bo mbarded to remove the passivation layer from the bottom of the previously etched trench. This is repeated for a prescribed number of cycles to achieve a specific etch depth. The process paramete rs can be chosen such that aspect ratios of greater than 20:1 can be obtained. Even side walls with fairly negative or positive slope angles can be achieved depending on the application. Initially, a stan dard DRIE recipe recommended by STS was used to observe the etch profiles on 4 test wafers. The process parameters were varied to eventually obtain a standard recipe for our layout. Some i ssues encountered during this iterative process are shown here. In Figure 4-8, an SEM picture of a sidewall profile for PZTEH-06 is shown that was taken at an angle of o40 C. It also shows the co rresponding profile for PZT-EH-09 on the right. The scalloped lines on the sidewall can be seen, which indicate the 15 etchpassivation cycles.

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136 PZT-EH-09 PZT-EH-06 @ 40 deg Figure 4-8: Side wall profiles on topside of a 4" Si test wafer. In our process, the etching from the top stops at the buried oxide la yer (BOX) as shown in Figure 4-9. The etch selectivity in DRIE with Si and 2SiO is approximately 1:100 and therefore, the BOX layer acts as an etch stop. Th e process conditions used for top side release are listed in Table 4-2. Table 4-2: DRIE recipe c onditions for top side etch. Gas Flow Rates 486285 130 13 CFsccmSFsccmOsccm Platen power 12 W Pressure 83.6%manualcontrol Etch : Passivation 12:7 Number of cycles 13 DRIE was carried out for 12 cycles before th e buried oxide layer was revealed. An extra etchpassivation cycle was performed to over etch and remove any residues.

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137 Photoresist(5 um) Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Beam_Etch mask Figure 4-9: Wet etch expos ed oxide with BOE and DRIE to BOX from top. After the beam release step, the remain ing PR was removed using Acetone and subsequently rinsed in methanol and de-ioni zed (DI) water. First, AZ 1529 was spun on the patterned wafer 4000 rpm for 40 s to approximately deposit 2.9 m which protects the topside. The wafer was soft baked on a hotplate for 60 s at o95 C. Next, AZ 9260 positive photoresist was spun on the backside of patterned wafer at 2000 rpm for 40 s to approximately deposit 9.8 m of PR. After soft baking for 30 min at o90 C, the proof masses were patterned on back side using EVG-620 mask aligner with the ProofMass mask, followed by a hard bake for 60 min at o90 C. The EVG-620 mask aligner enable s back to front alignment to ensure that the beams and the proof masses were pe rfectly aligned with each other. The patterned wafer was then etched from the backside using DRIE technique. First, as carried out previously this process was performed on 4 SOI test wafers to optimize the process and recipe conditions. So me of the phenomenon observed during many iterations carried out to optimize the pro cess are explained below with pictures. For the backside release, the patterned wafe r is typically mounted on a carrier wafer, ( Siorpyrex) and loaded in the DRIE machine. To prevent any thermal loading on the wafer that could potentially produce uneven etching, AIT cool grease 7016 was used to attach the

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138 patterned wafer to the carrier wafe r. The carrier wafer is required for long and/or through etches to provide mechanical integrity dur ing clamping of process wafer. Figure 4-10: Sidewall profiles for backside etching using DRIE. In Figure 4-10, on the left, shown etched area corresponds to PZT-EH-06 on a test wafer. Two regions are zoomed in and shown on the right to indicate the sidewall profiles for the proof mass. As is evident from the figure, vertical stria tions are observed. In this test, the process wafer was attached to a carrier wafer using AiT Cool-Grease 7016 that was applied only along the edges of the wafer preventing any cool grease from contaminating the features on the topside.

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139 Figure 4-11: Curved edges during backside DRIE. Figure 4-11 shows another etch area correspondi ng to a larger device that clearly indicates curved edges near the corners of the proof mass. Initial efforts with this test run, shown in Figure 4-10 and Figure 4-11 produced highly non-uniform etchi ng across the wafer with the features in the center of the wafer etching mu ch slower than the areas furt her away. The process conditions for this run remained the same as listed in Table 4-2. It was hypothesi zed at this point that thermal effects on the wafer were causing the highl y non-uniform etching. In the next test run, a blank 4 bulk Si wafer was patterned with the ProofMass mask. Since the Si wafer was not patterned on the topside, cool grease was app lied evenly over the entire wafer. This time, however, the cool grease was spread evenly at an elevated temperature o50 C to ensure uniform contact across the wafer and provide good thermal dissipation. The process conditions

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140 remained the same as Table 4-2 except that the etchpassivation ratio was changed to 13:7 with an overrun of 1:0.5. The overrun allows compensation for the ramp in flow rate from zero to specified rate for the gases during etch and passivation cycles. Figure 4-12: Onset of silicon gras s during a backside etch run. SEM pictures were obtained agai n by cleaving the pr ocess wafer at different locations to observe the etch results. Figure 4-12 shows an etch area that indicated the onset of Si grass. A small area from the etch bottom is magni fied on the right that clearly shows 10 m of grass. A further magnified image is provided to show the scallops in the grass from etchpassivation cycles. In addition, it also indicates incr easing width of grass with pointed tops.

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141 Figure 4-13: Sidewall profiles for a backside etch on a test wafer. In Figure 4-13, the sidewall profile s of two etch areas are provi ded that show a significant negative profile with cu rved corners and concave bottom. The non-uniform etching across the wafer persisted even in this test run, with measured etch lags as large as 6080 m for 350 etchpassivation cycles. In the layout, the etch areas in the center are several orders higher than those further away. The la rgest etch feature was 270 mm and the smallest was 20.5 mm that corresponded to dimensions of 10 mm and 200 m respectively. In addition, the exposed Si area was calculated to be approximately 38.8 % of the overall wafer area. Based on our layout design and di scussions with STS, it was unders tood that to evenly etch through 500 m of the wafer with the current pattern would be difficult. A more detailed process optimization needed to be performed to carefully investigate the effects of etching by varying each of the recipe parameters individually. In view of the time constraint for fabrication, it was decided to dice the patterned wafer into individual dyes and etch them separately to release the devices. The dicing process involves attaching the process wafer to a carrier wafer using hot melt glue and sawing. The diced wafer pieces can be removed from the carrier wafer by heating and melting the glue that binds the two. An added advant age with prior dicing is that if the devices are diced after release, any damage to the suspended cantilevers during the mechanical sawing process will lower the overall yield of the structures.

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142 Another test run was carried out on a 4 bulk Si wafer that was patterned using the ProofMass mask. The wafer was later diced into small dyes consisting of a few devices of each design. These dyes were mounted on a carrier wa fer using cool grease and separately etched using DRIE. The non-uniformity in etching was mini mal with etch lags of less than 10 cycles. For this process, the recipe pa rameters remained the same as Table 4-2, except that the etchpassivation ratio was 13:7 without any overrun. The final process conditions are listed in Table 4-3. Table 4-3: DRIE recipe cond itions for back side etch. Gas Flow 486285 130 13 CFsccmSFsccmOsccm Platen 12 W Pressure 83.6%manualcontrol Etch : 12:7 Number of 350 After ensuring that the above methodology of rel ease worked for all device sizes, the same process was employed on the original PZT process wafer that had been released from the top. The wafer was first patterned on the backside using the photolithogra phy process described previously with ProofMass mask. It was diced into individua l dyes that consisted of 1 or 2 devices using the dicing saw. The dyes were mounted on a p yrex carrier wafer using cool grease to ensure good thermal dissipation. DRIE was th en performed until the BOX layer was revealed to release the structures from the backside as shown in Figure 4-14. The number of cycles to release varied from 350 to 400 depending on the designs.

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143 Photoresist(10 um) Photoresist(5 um) Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Proof_Mass mask Figure 4-14: Pattern proof mass on th e backside and DRIE to BOX. DRIE was followed by immersing carrier wafer and the chip in Acet one for a few hours to separate the chip and stri p the photoresist. The chip was then attached to a 2 Si wafer using AZ 9260 photoresist to protect the PZT on the t opside. The wafer was then etched in 6:1 BOE to remove the 0.4 m BOX layer. The PR binding the chip to the carrier wafer was stripped using acetone, followed by chip rinse in DI water and dr y to obtain the final PZT devices. A schematic of the devices when released is shown in Figure 4-15. In the figure, 3 cantilevers are shown attached to a single proof mass. The number of cantilevers per proof mass typically varies between 14 depending on the designs listed in Table 310. It should be noted that extra care was taken to ensure that BOE never came in cont act with the PZT, as initially, some released devices appear shorted and HF is known to etch PZT.

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144 Pt (200 nm) PZT(1 um) SiO2 (100 nm) Si (12 um) SiO2(BOx) Ti/Pt (20 nm/200 nm) Figure 4-15: Schematic of final released device. Shown below in Figure 4-16 and Figure 4-17 are SEM pictures of two designs that were released using DRIE. Several devices of these de signs were released, packaged and characterized for mechanical, electrical and electromechanical response. The experimental setup and results are presented in later chapters (Chapter 5 and Chapter 6). Figure 4-16 shows two views of PZTEH-07 that consists of a PZT composite beam, 1 1 mmmm attached to a single proof mass, 2.5 4 mmmm Figure 4-17 shows PZT-EH-09 comp rising of three beams each 0.5 0.5 mmmm attached to one proof mass, 1.8 2.4 mmmm

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145 Figure 4-16: SEM pictures of a PZT-EH-07 released device. Figure 4-17: SEM pictures of a PZT-EH-09 released device. Although the devices releas ed completely, the reci pe used for DRIE is still not optimized and issues such as negative profile, slightly concave bottom and excessive undercut were not addressed in detail. From discussions with STS, it was found that a negative profile cannot be avoided with large etch areas due to reentry of gases during the etch step. In addition, it was suggested that increasi ng the platen power by 23 W will ensure a more directional and flatter etch bottoms. Furthermore, reducing the pressure duri ng etch step to 4030 mT and 22 mT during the passivation step may help the etch prof iles. However, since the modified DRIE recipe seemed to release the devices al beit imperfect sidewalls (see Figure 4-18) current process was continued for release of subsequent devices. The suggestions for increased power and lowered pressure could be explored in future to optimize the process to obtain straighter walls.

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146 Furthermore, the layout and device locations on the wafer need to be rearranged to obtain a more symmetric pattern to ensure uniform etching. Figure 4-18 shows SEM pictures of sidewall for PZT-EH-07. As is evident in the figure, the sidewalls exhibit vertical st riations that have been magnifi ed on the right. The scallops indicate etchpassivation cycles that are approximately 1.11.2 m each. In addition, a significant undercut is observed as shown in the figure below. Furthermore, insufficient passivation occurs in some ar eas of the sidewall that is shown on the right bottom of Figure 4-18. Figure 4-18: Sidewall profile s of released devices.

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147 Process Traveler A complete process flow that was finally us ed to fabricate the MEMS PZT devices is summarized in Table 4-4. Table 4-4: Process traveler for the fabr ication of micro PZT cantilever arrays 1 Start with SOI wafer with 4000 A oxide (12 m of silicon overlayer) 2 Deposit 100 nm of SiO2 using PECVD 3 Spin deposit bottom electrode. Deposit 20 200 nmnm of Ti/Pt using sputtering technique. 4 Spin coat PZT (125/52/48) solution at 3000 RPM for 30 s. Pyrolize at 350 C for 1 min. Repeat spin/pyrolize 3 times to achieve 1 m thick PZT. Furnace anneal at 700 C for 1 min (O2). 5 Spin photoresist on front surface and patte rn top electrode using the same thickness for the top electrode as th e bottom electrode. Deposit 200 nm Pt using sputtering technique. Strip resist to liftoff Pt using Acetone 6 Etch PZT in 3:1:1 ammonium biflouride/hyd rochloric acid/DI water using etch mask. Etch PZT residues left behind by previous etch with dilute nitric acid/hydrogen peroxide etchant 7 Spin photoresist on topside and pattern to etch the bottom electrode using Ion milling. 8 Deposit Au (300 nm) and pattern bond pads using B ond Pads mask and wet etching. 9 Spin photoresist on topside and pattern to etch the beams. Plasma etch (DRIE) to BOX from the top to release the beams 10 Deposit thick photoresist on the bottom al ong with protecting frontside. Pattern bottom for etching the proof mass. Plasma et ch (DRIE) from backside that stops on BOX layer 11 Etch oxide (BOE) on backside to remove oxide. Finally, strip photoresist on top and bottom using Acetone to release the device

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148 Packaging The individual chips after release, were then packaged for experimental characterization. Two packages were designed for this purpose, namely, a vacuum package and an open package. The chips were mounted in a Lucite package, described next, and the bond pads were connected to copper leads that allow for external elect rical access during experimental measurements. Vacuum Package A two-part vacuum package was devised to mi nimize any fluid loading on the device that will result in a higher mechanical damping at resonance. Figure 4-19 shows a schematic of the bottom piece of the vacuum package th at basically consisted of a 221 Lucite base and a 110.5 center cavity with 5 mm wide support to mount the chip s. The base has a threaded hole at the bottom to facilitate rigid mounting to a vibrating surface (for eg., Bruel and Kjaer mini shaker 4810) and a 14 vacuum port. The vacuum package is significantly larger than the device itself, primarily to provide considerable co ntact between the Lucite base and the top cover during pump down. A rubber gasket is also provided between th e base and cover to hold the vacuum for a long time. Copper posts go through the Lucite base and extend on the other side to serve as external leads.

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149 1 in 1 in 5 mm 2 in 2 in 2 in 2 in A A B B Bottom View Top View Sectional view at BB Sectional view at AA 1 mm 10-32 threaded hole (0.25 in) Pressure tap for vacuum 1 in 0.5 in LUCITE BOTTOM Figure 4-19: Schematic of the bottom of vacuum package for MEMS PZT devices. Figure 4-20 shows the top part of the package comprising of a glass/plexiglass cover to essentially act as a transparent surface for la ser based measurements during characterization. 2 in 0.5 in GLASS TOP Top View Bottom View Figure 4-20: Schematic of glass top for vacuum package. A 3-D schematic of the completely assembled vacuum package is shown in Figure 4-21. The devices are mounted on the center support in the cavity using a two-part silver epoxy

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150 (Epotek H2OE). The silver e poxy was cured in an oven at o90 C for 90 min to provide a rigid support to the device. Although cu ring time can be reduced at higher temperatures, the above temperature was chosen to stay below th e glass transition temperature for Lucite o100 C. The electrode bond pads of the chips were connected to the copper posts with gold wires using a wire-bonder. In areas where wi re-bonding was difficult, silver epoxy was used to create the contacts. The package as shown in the Figure 4-21 has a threaded hole in the bottom to be mounted on a vibrating shak er for characterization. Glass Lucite bottom External leads (Cu) Device Wire bonds (Au) Figure 4-21: An isometric view of the overall vacuum package. Open Package The open package shown in Figure 4-22 is similar to the vacuum package without the transparent top part and the vacuum port. Since this package will not be pumped down, a smaller package was devised comprising of a 111 Lucite base and a 0.50.50.5 center cavity

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151 with 3 mm wide support to mount the chips. The pack age also has a threaded hole in the bottom to be mounted on a vibrating shaker for characterization. 0.5 in 0.5 in 3 mm 1 in 1 in 1 in 1 in A A B B Bottom View Top View Sectional view at BB Sectional view at AA 1 mm 10-32 threaded hole (0.25 in) 1 in 0.5 in LUCITE BOTTOM Figure 4-22: Schematic of open package for MEMS PZT devices. An optical photograph of the open package is provided in Figure 4-23. In the figure, 2 devices of PZT-EH-09 are mounted on the packag e which is attached to vibrating surface (a shaker) on the bottom. Gold wi res are used to provide cont act with the bond pads on the individual devices. The wires we re connected to the copper posts with Silver epoxy using the curing process described earlier.

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152 Cu posts MEMS PZT device B & K shaker External copper leads Lucite base Figure 4-23: Picture of the open package. Following fabrication and packaging of the ME MS PZT cantilever devices, experimental setups were designed to completely characterize them for lumped element parameter extraction. Furthermore, the devices were characterized fo r their frequency respons e and power generation. The next chapter describes these experimental setups and the data acquisition conditions employed for the experiments.

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153 CHAPTER 5 EXPERIMENTAL SETUP Prior to nondimensionalizing th e lumped element model and developing the scaling theory, the model was validated experimentally on ca ndidate aluminum-PZT composite beams at a macro-scale as described in Chapter 2. This chapter continues from there and describes the experimental setup and procedur es used to characterize the ME MS PZT cantilever generators. First, the devices were tested for their dielect ric and piezoelectric prope rties. Next, various characterization experiments were designed to extr act all of the lumped element parameters to complete the equivalent circuit. The values extr acted are ultimately compared (Section 6.5) with the predictions from the model, providing clear insight and direction fo r further analysis and validation. Detailed descriptions of the mechanical, electrical, and electromechanical characterization setups are prov ided in this chapte r along with the data acquisition methods employed. Finally, methods to excite the devices at resonance are described, in which the output voltage and power with respect to input accele ration were measured across varying resistive loads. Ferroelectric Characterization Setup Before releasing devices from the processe d wafer that was returned from the Army Research Laboratory (ARL), it was characterized initially for the dielec tric and ferroelectric properties of the PZT layer. The wafer was m ounted in a probe station and the electrical measurements were taken using a Radiant Techno logies Precision LC Ferroelectric Tester. Two probes were used to access the top and bottom el ectrodes of the PZT layer on the wafer. A bipolar triangular voltage wavefo rm is applied to the PZT layer for a specified time, typically

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154 10 ms 100 Hz, and the current during that period is measured. The triangular waveform was generated by cycling increments of voltage de termined by the number of points for one whole time period. Then, the current is integrated across the area (i.e., using a valu e input by the user) to obtain the accumulated charge in the area to give an effective polarization. The voltage signal was divided into either 101 or 201 points to cover the entire time period and the polarization for each value was obtained and plotted. This pro cess was carried out for peak voltages ranging from 1 60 VV. Experiments were not extended for higher voltages beyond the saturation voltage of the PZT thin film. The pol arization saturated at approximately 60 V for all tested PZT layers. A simple schematic of the setup used is shown in Figure 5-1 and the results of these experiments are presented in the next chapter. A photograph of the actual setup used is shown here in Figure 5-2. The area and the thickness of the dielectric were specified to calculate the remnant polarization and coercive fields. current voltage Radiant Ferroelectric Tester Base PZT wafervoltage current probes microscope Figure 5-1: Schematic for ferroelectric characterization.

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155 Figure 5-2: Experiment al setup for ferroelect ric characterization. Piezoelectric Characterization The piezoelectric material on the processed wafer is not poled. Hence, we need to polarize the PZT to the recommended voltage for a reco mmended duration (at a specified temperature if applicable). The literature (K holkin et al. 1998, Roeder et al. 1998) suggests a polarizing temperature of slightly below the Curie temperat ure and using a strong electric field. Although ARL recommends applying 5-10 V for 5-10 minutes to polarize the PZT at room temperature, it reports that the PZT is usually partially poled during fabrication and ma y not need polarization for most of their applications. Therefore, using the Radiant (Preci sion LS) Ferroelectric Materials Analyzer, hysteresis te sts were conducted to obtain th e hysteresis for the PZT layer and estimate its remnant polarization and coercive fields. The hysteresis curve is plotted with polarization as a function of the applied voltage on the x axis The applied voltage profile in this experiment is a triangular waveform describe d earlier that cycles from a negative peak to a positive value in increments over time. The remnant polarization is the intercept on the y axis and the coercive field is obtained from the intercept on the x-axis where the polarization changes

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156 orientation. These values were then compar ed with the values provided by ARL for the fabricated wafers that are listed in Table 5-1. If the remnant polarization is considerably larger, the devices need not be polarized. These experi ments that capture the hysteresis behavior of PZT were repeated on known geometries on the wafer after poling at different voltages for different times and elevated temperatures to obser ve their effect. The hysteresis behavior along with results for the experiments conducted to extr act the relevant paramete rs is further explained in detail in chapter 6. Table 5-1: Reported polari zation results (ref: ARL) Thickness of PZT 1.016 10% m Remnant Polarization (Pr) 22.5 C/cm2 Coercive Field (Vc) 80 kV/cm Electrical Characterization Next, electrical impedance measurements we re carried out to estimate the electrical lumped element parameters in the circuit. The first two parameters that are measured are the blocked electrical capacitance, ebC, and the dielectric loss, e R in the PZT layer. Furthermore, the released devices after packaging are charac terized for their free electrical impedance ef Z The packaged devices are generally connected in se ries or parallel if they contain an array of PZT cantilevers. So, the overall free electrical impedance for the connection is measured. Blocked Electrical Capacitance, ebC and Dielectric Loss, e R Blocked electrical impedance measurements we re obtained from the process wafer before device release. These measurements give an es timate of the blocked capacitance and dielectric

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157 loss of the device geometries. An effective relative permittivity value for the piezoelectric material can be extracted from the measured cap acitance using the dimensions of the devices. ebC, defined in Eq. 2.25, is the capacitance in the piezoelectric laye r when the device is blocked (no resulting motion). So, measuring the capacitance be fore releasing the structures from the wafer directly yields ebC. A HP 4294A Precision Vector Impedance Analyzer was used to measure the blocked electrical impedance. This response is curve-fitted using an assumed circuit topology (e.g., consisting of a capacitor and resistor in para llel). An effective value for ebC and e R is thus extracted from the response. In the probe station (shown in Figure 5-3), the fixture compensation was initially performed by calibrating for short and open circu its to eliminate any residual impedance of the cables and probes. Then the two probes were co nnected to the top and bottom electrode and the blocked electrical impedance was measured using the Impedance An alyzer. A known input voltage, 500 ebmVforZ was applied to the PZT, and the corresponding current was simultaneously measured across a frequency range First, an experiment was conducted to measure the electrical impeda nce as a function of the ap plied voltage varying from 10 1 mVV and it was observed that the capacitance varies with the input amplitude. Therefore, an intermediate value of 500 mV was chosen to characterize all measurements. A schematic of the experimental setup is shown in Figure 5-3.

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158 Figure 5-3: Schematic for blocked electrical impedance measurement. A photograph of the experimental setup used for the impedan ce measurements is shown in Figure 5-4. The setup basically co nsists of a probe station on which the process wafer is mounted and probes are used to provide contacts to the electrodes. Figure 5-4: Experimental setup for electrical impedance characterization. The start frequency for the impedance an alyzer is fixed by the instrument at 40 Hz. The impedance was thus measured across a frequency range of 40 1000 HzHz The real and

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159 imaginary parts of the impedance were extracted to estimate an effective dielectric loss factor, tan which is given as Re tan Im1e ebZ R R Z XjC (5.1) In the LEM, the dielectric loss in the PZT layer is calculated using an empirical relation given as 1 tan2e ebR f C, where tan was assumed to be 0.04 (from existing literature) for our predictions. The actual values for tan will be extracted for each of the devices from experimental measurements. From the blocked im pedance experiments, an effective value for tan can be estimated using Eq. (5.1) and compared with the assumed tan The new value will be used henceforth to estimate other lu mped element parameters in the circuit. It should also be noted that the top and bo ttom electrode extend into the clamp (shown in the process flow in Figure 4-16). Therefore, the measured capaci tance includes the PZT material on the clamp that acts like a pa rasitic capacitance for the electrom echanical characterization. Mechanical Characterization Overall, six total devices that consisted of two devices of type PZT-EH-07 and four devices of PZT-EH-09 were characterized. PZ T-EH-09-1 and PZT-EH09-2 belong to Design-9 that consists of 3 cantilever beams attached to a single proof mass. In these two devices, all the PZT electrodes are connected in parallel for characterization. PZT-EH -09-3 and PZT-EH-09-4 are also of the same design type, but only one P ZT electrode is used as external electrical connection. This will compare the performance of the device for one PZT versus the PZT layers connected in parallel. Furthermore, PZT-EH-071 and PZT-EH-07-2 are of design type 7. The device dimensions for the two de signs are listed for reference in Table 3-10. A detailed

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160 schematic of the experimental setup used to ch aracterize the PZT energy harvesters is shown in Figure 5-5. Figure 5-5: Experimental setup for mechanic al and electromechanical characterization. In the setup shown in a picture in Figure 5-6, the packaged device is mounted on the Bruel & Kjaer mini shaker (type 4810). The whol e setup is mounted underneath the microscope using a lab jack that provides th e flexibility to move the shaker both horizontally and vertically. This arrangement is necessary to get the device in focus and also to traverse to different spots on the device.

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161 Figure 5-6: Experimental se tup for vibration and veloci ty measurements with LV. The MEMS PZT device was rigidly m ounted on the mini shaker using a 1" 10-32 threaded screw with a locking nut. The sc hematic of the package and the mounted devices were explained in Section 4.2. The whole setup is mounted underneath an Olympus BX 60 optical microscope with a 5 x objective lens to perform velocity m easurements. A Polytec scanning laser vibrometer (LV) as shown in Figure 5-6 is connected to the mi croscope to obtain velocity measurements. The laser vibrometer consists of a microscope adapte r (Model OFV 074) that enables scanning very small areas with the laser. The laser spot can be as small as 10 m with a 100 x magnification lens. The laser also consists of a fiber interferometer (Model OFV 511), a scanner controller (Model MS V-Z-040) and a vibrometer c ontroller (Model OFV 3001s) that effectively enables velocity measurements across a scanning range. A signal is generated using the waveform generator inside the LV that is used to drive the shak er through a B&K power amplifier (type 2718). The resul ting acceleration to the device is measured by focusing the laser on the clamp of the MEMS device th at is rigidly mounted to the package. This gives an estimate of the input acceleration to the device mimicking an input base vibration. Sample records are obtained for a periodic chirp input and its corresponding Fourier Transform (FFT) was

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162 calculated. A uniform window function was applied for the data with 1600 FFT lines and averaged 50-100 times to minimize any random errors during data acquisition. The data acquisition parameters are summarized in Table 52. Next, the device was horizontally traversed and the laser was focused on the tip of the can tilever beam and the co rresponding tip velocity was measured for the same input conditions. A relative tip motion was cal culated by subtracting out the clamp motion. Consequently, a freque ncy response was obtained between the input acceleration and resultin g tip deflection. The resonant frequency is extracted from the peak in the frequency response spectrum. A nominal mechanical damping, m R in the system is estimated by matching the frequency response peaks at resonance between the model a nd experiment. Since damping affects only the resonance, we can obtain the value di rectly by altering the value of so that the resonance peaks match. For all PZT-EH-09 devices, the frequency response was measured over 01 kHz that was divided with 1600 FFT lines resulting in a frequency resolution of 0.625 Hz. The number of averages for the frequency response of PZTEH-09-1 and PZT-EH-09-2, was 50, while 100 averages were measured for PZT-EH-09-3 and P ZT-EH-09-4. In the case of PZT-EH-07 devices, the frequency response was obtained for 00.5 kHz using 800 FFT lines with a frequency resolution of 0.625 Hz. 100 averages were obtained for all measurements with PZT-EH-07-2 and PZT-EH-07-3. Since a periodic chirp sign al was used as an input, a uniform window function was used for all tested devices

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163 Table 5-2: Data acquisiton parameters for mechanical characterization. Waveform Perodic chirp Frequency Range 0-1 kHz Number of FFT lines 1600 Window Uniform Resolution 0.625 Hz Number of Averages 50 or 100 During the experiment, the laser sensitivity an d the measurement range were adjusted to ensure good signal quality and resolution. Measurements we re carried out with the PZT electrodes shorted to avoid any electromechanical effect A corresponding open circuit measurement was also taken, but the coupling between the mechanic al and electrical domains is minimal resulting in minimal changes in the response or resonance. Electromechanical Characterization The effective piezoelectric coefficient, defined as the static tip deflection for an applied voltage to the PZT, is obtained from the electromechanical charac terization. However, it is difficult to apply a static voltage and measure a static deflec tion because the voltage dissipates across the PZT in the form of dielectric loss Alternatively, the effective piezoelectric coefficient, md, can be experimentally determined from the low frequency ~1050 Hz asymptotic dc response of the device when excited using an ac voltage. For the electromechanical characteri zation, the same setup shown in Figure 5-5 was used. Instead of having the PZT el ectrodes shorted, they act as the input to apply a voltage that results in a tip velocity measuring using the laser vibr ometer. An input periodic chirp signal across prescribed frequency range was used to electri cally excite the structure and the frequency response between the input voltage and the result ing tip deflection was extracted. The data acquisition parameters for this expe riment are listed in Table 5-3.

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164 Table 5-3: Data acquisiton parameters for mechanical characterization. Waveform Periodic chirp Frequency Range 0-1 kHz Number of FFT lines 1600 Window Uniform Resolution 0.625 Hz Number of Averages 50 or 100 The low frequency response of the experiment that measures the tip deflection per unit voltage directly provides an eff ective piezoelectric coefficient md. Open Circuit Voltage Characterization All of the LEM parameters are extracted using a least squares fit to the experimentally obtained frequency response with the LEM equations described in Chapter 2. The details about this extraction are summarized in Chapter 6. Each of the frequency response obtained from the mechanical, electrical, and elect romechanical characterization provides the corresponding LEM parameters in each energy domain. Next, the open circuit voltage is measured across a prescribed frequency range for a known input acceleration. A periodic chirp signal is generated using the SRS spectrum analyzer that is applied to the device mechanically th rough the LDS dynamic shak er (V408) driven using an LDS power amplifier (P A100E-CE) as shown in Figure 5-7. A photograph of the experimental setup is provided in Figure 5-8.

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165 acceleration OPTICAL TABLE Package with PZT Device voltage Power Amplifier SHAKER Spectrum Analyzer Imp. Head Figure 5-7: Experimental setup for open circuit voltage measurements. Figure 5-8: Experimental setup for open circuit voltage measurements. The input acceleration is measured using a Bruel & Kjaer impedance head (model 8001) that simultaneously measures force and accelerati on. The resulting volta ge across the PZT is measured using the spectrum analyzer. The fre quency response is calcul ated between the input acceleration and output open circuit voltage. This extracted response provides an estimate of

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166 how much voltage per unit acceleration can be generated across the range. The data acquisition parameters used for this experi ment are listed in Table 5-4. Table 5-4: Data acquisiton parameters for mechanical characterization. Waveform Periodic chirp Frequency Range 0-1 kHz Number of FFT lines 1600 Window Uniform Resolution 0.625 Hz Number of Averages 50 or 100 Voltage and Power Measurements Finally, a sinusoidal signal wa s generated using the waveform generator at the measured resonance frequency of each device to excite it mechanically through the shaker. The experimental setup, similar to the one described before for the vibration measurements is shown in Figure 5-9. Figure 5-9: Experimental setup for voltage and power measurements. The input acceleration at this frequency was measured at the clamp using the vibrometer, similar to the process described in 0. A low vibration level on the order of 0.1 g is used to

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167 ensure safe operation of the devi ce at resonance. The resulti ng output voltage at the PZT was measured across various resistive loads 1 750 kk to observe the voltage and power characteristics. Plots of rms vo ltage and power normalized to th e input acceleration as a function of the load resistance are generated. The output voltage increases with increasing output resistance and finally approaches the open circ uit voltage. On the ot her hand, the output rms power increases and reaches a maximum for an optimal output load and then decreases with increasing resistance.

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168 CHAPTER 6 EXPERIMENTAL RESULTS AND DISCUSSION A detailed experimental characterization of the MEMS PZT power generators is presented in this chapter. The overall goal of the experi mental procedure is to extract all possible LEM parameters in the electromechanical circuit (F igure 2.2) by performing various experiments and using the extracted values to predict the ove rall voltage and power out put of the device and compare with experimental results. Ferroelectric Characterization Prior to releasing the devices from the process wafer, polari zation tests (Section 5.2) were performed to measure the piezoelectric prope rties. First, polarization hysteresis was characterized by applying a static voltage to the piezoelectric laye r and measuring the current in the film. The current in the PZT is integrated with respect to time to obtain the stored charge. Polarization, which is defined as charge stored per unit area, is usually expressed in units of 2Ccm. An ideal hysteresis loop for a pi ezoelectric material is shown in Figure 6-1 which represents a typical polarization-electric field (P -E) plot. Hysteresis occurs when the material exhibits multi-valued behavior of any property (Megaw 1957). In Figure 6-1, for every value of electric field there exists two distinct values fo r the polarization depending on the direction of the hysteresis cycle. Various important parameters in the hysteresis loop (Cady 1964), denoted by labeled points in the figure are described below.

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169 Pr-PrPm-EcEcPs Polarization (P, C/cm2)Electric Field (E, V/m) Figure 6-1: A typical P-E hystere sis loop for a piezoelectric mate rial (adapted from Cady 1964). Points a and d are maximum polarization points in positive and negative directions represented by mP Another parameter s P, defined as saturation pola rization is obtained as the intersection of the tangent to the hysteresis loop at point a and the polarization axis. Points b and e are called retentivity points represented by rP and rP These points are defined as the residual polarization in the material when the ap plied electric field is zero. Finally, points c and f are defined as coercivi ty points represented by cE and cE These points are defined as the electric coercive fields required to reverse th e direction of polarization in the piezoelectric material. Next, a dielectric characterization is performe d to calculate the capacitance and the relative permittivity of the piezoelectric material. Shown in Figure 6-2 is a typical r-E plot that

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170 represents the behavior of relative permittivity as a function of the electric field. In the figure, cE and cE define the coercive fields when the relative permittivity of the material approaches a minimum. Furthermore, r is the residual relative permittiv ity when the material is not subjected to any field (Cady 1964). Figure 6-2: A typical -E curve for a piezoelectric material. The results for a sample PZT geometry (PZTEH-02-1-1) measured on the process wafer are presented here, and the hysteresis parameters are compared to the values specified by ARL. Figure 6-3 shows the polarizati on and capacitance plot correspond ing to the input triangular voltage waveform. As seen in the plots, when th e polarization crosses zero (corresponding to the coercive voltage) capacitance of PZT reaches its maximum.

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171 Figure 6-3: Polarization, capacitance and i nput voltage waveforms for PZT-EH-02-1-1. Hysteresis tests were perfor med for waveforms with different maximum voltages and the results are presented in the above figure. In Figure 6-4, the polarization in the material is plotted as a function of the applied voltage. As expe cted the PZT layer exhibits highly nonlinear hysteresis behavior similar to the one shown in Figure 6-1.

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172 Figure 6-4: Hysteresis pl ots for PZT-EH-02-1-1. Tests were conducted for different voltages as shown in the above figu re. It can be seen that the polarization saturates with increasing vol tage. The points where the curve intersects the horizontal and vertical axes are extrac ted for each voltage and are plotted in Figure 6-5. Hysteresis parameters such as and mrcPPV defined earlier are shown in the figure below. All parameters are observed to saturate as the voltage increases. The units for r and mPP are 2Ccm while cV is expressed as V in Figure 6-5. These results ar e plotted as a function of increasing input voltage along x axis

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173 Figure 6-5: Pr and Vc for different applied vo ltages for PZT-EH-02-1-1. The measured remnant polari zation for PZT-EH-02-1-1 is 226.34 Ccm. The coercive electric field, cE is obtained from the coercive voltage as cpVt, where pt is the thickness of PZT. Therefore, the measured coercive field is obtained from the saturated 6.6 cVV which corresponds to an electric field of 63.34 kVcm for 1.04 m thick PZT layer. The maximum polarization observed was 253.25 Ccm for a maximum applied voltage of 60 V. From the measured charge, the capacitance of the PZT layer is calculated fo r all the cases. The normalized capacitance expressed as capacitance per unit surf ace area of the PZT la yer is plotted as a function of applied voltage in Figure 6-6. The curve exhibits a t ypical nonlinear behavior similar to Figure 6-2, where the static blocked capacitance is extracted at zero voltage. The shape of the

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174 curve is generally symmetrical about the 0V axis and the maximum capacitance occurs at the coercive field. Figure 6-6: Normalized Ceb for PZT-EH-02-1-1 duri ng the hysteresis test. Finally, a leakage test was conducted on PZT-EH-02-1-1 where 10 Vdc was applied for 1000 ms and the resulting current was measured. Figure 6-7 shows a plot between the measured current and time. The steady state current is 81.1810 A that corresponds to a dielectric resistance of 88.4110 for the applied voltage. It should be noted that the dielectric resistance tends to vary inversely with frequency and the value presented here is measured at dc. The resistivity for the PZT f eature is obtained from p pA R t as 112.6910 cm where pA is the surface area of the PZT and R is the measured dielectric resistance.

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175 Figure 6-7: Leakage current for P ZT-EH-02-1-1 subjected to 10V DC. Next, poling tests were carried out to observe its effect on the piezoelectric properties. Initially, PZT-EH-02-1-1 was poled by subjecting the PZT feature to 5 dcV at room temperature for 5 min, 10 min, and 15 min respectively. After each poling step, hysteresis tests were conducted using the ferroelectric te ster at different vol tages. The remnant polarization, coercive voltages and maximum polarization paramete rs were recorded and are plotted in Figure 6-8. No significant change/improvement in the pola rization was observed wh ile poling at room temperature and the remnant polar ization of the material remain ed unchanged. The results were similar for tests that were carried out at 10 dcV and 15 dcV for the above listed time periods.

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176 Figure 6-8: Poling of PZT-EH-02-1-1 at 5V for different times. Another poling test was carried out to observe the effect of temperature on piezoelectric properties. The wafer was heated to a specifi c temperature, and PZTEH-02-1-1 feature was poled with a constant 5 dcV while the wafer was allowed to cool to room temperature. The experiment was performed at room temperature, o50 C, and o100 C. Previous work in thin film PZT have performed poling at elevated temperatures upto o150 C (Roeder et al. 1998, Kholkin et al. 1998). To achieve maximum polarization, the PZT should be pol ed at a temperature slightly below the curie point. Since the curie temperature for our thin films is not know nor supplied from ARL, polarizatio n tests were onl y conducted at o100 C. Future work can extend poling to higher temperatures to observe any notable improveme nt in remnant polarization. Moreover, ARL recommends room temperature poling for 1015 min with 35 times the

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177 coercive voltage. In addition, there will be some degradation of the piezoelectric coefficients with time after poling. This phenomenon known as aging occurs primar ily since thin film materials are not single domain materials and some degree of self alignment with the ferroelectric domains remains inherent, even after poling. For reliability, it is necessary to keep track of the time allowed between the end of the poling procedure and the beginning of the measurement. If this time is not kept consiste nt, each device may have aged to a different degree prior to the measurement and in between me asurements (Polcawich and Troiler-McKinstry 2000). More information can be found in literatu re on poling and its effect on piezoelectric properties. However, for this dissertation, those different pol ing conditions have not been explored in detail. After each poling step men tioned above, hysteresis te sts were conducted for the feature with the ferroelectric tester. The extracted hysteresis para meters are shown in Figure 6-9. Clearly, the remnant polarization remains within 10 % of the unpoled value, and it is therefore concluded that the PZT is sufficiently poled during the fabrication process as reported by ARL.

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178 Figure 6-9: Poling of PZT-EH-02-1-1 at diffe rent temperatures. Table 6-1 compares the measured values pi ezoelectric hysteresis parameters with those reported by ARL for the PZT process. Since the va lues are similar and not much difference was observed with post-fabrication poling, all subsequent devices were te sted directly after release. Table 6-1: Comparison of ARL's reported hyste resis parameters with measured values. Hysteresis parameters Measured at ARL Measured at UF Remnant polarization rP 222.5 Ccm 226.34 Ccm Coercive field cE 80 kVcm 63.34 kVcm

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179 Blocked Electrical Impedance Measurements Since it was established from the hysteresis that the capacitance is a function of the applied voltage, a similar test was carried out here to observe the variation of ebC and tan (Eq. 5.1) with parameters such as source amplitude and dc bias. First, a constant sinusoidal voltage of 500 mV at 100 Hz was applied to the PZT-EH-1-3-02 featur e, and the dc bias was varied from 0 to 40 dcV. The electrical impedance was simulta neously measured using HP4924a Vector Impedance Analyzer, and the blocked capacitance and dielectric loss were extracted and are plotted in Figure 6-10. A point averaging scheme with 10 samples at each frequency was used to measure the data. Both ebC and tan were observed to monotoni cally decrease with bias. Figure 6-10: variation of Ceb and tan with dc bias and a constant sinusoid, 500 mV at 100 Hz.

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180 A least squares fit was performed to obtain an empirical relation between ebC and tan with dc bias as shown in Figure 6-10. The expression for ebC was observed to fit best with 48.310 92.2110 ebdcCVF and tan was fitted using 0.014 21.2410dcV where dcV is the dc bias applied to the PZT. Another test was performed at zero bias where a sinusoidal voltage at 100 Hz was applied to the PZT with increasing source amplitude. Figure 6-11 shows a plot of ebC and tan along with their curve fits, genera ted using least squares method. Figure 6-11: Variation of Ceb and tan with source amplitude at 100Hz. Here, a quadratic polynomial was used to fit the data and the empirical expressions are 2 922.16100.040.021 ebssCVVF and 2 22tan0.84100.720.291ssVV where

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181 s V is the applied source amplitude. It should be noted that the coefficients of the fitting functions in both cases represent the asymptotic values of ebC and tan However the overall functional form is complicated and needs to be explored further to understand the ge neral dependence. This exercise was carried out only on one device geomet ry just to demonstrate the highly nonlinear dependence of the electrical a nd piezoelectric properties (hyste resis tests) on the dc bias, frequency of operation and voltage amplitudes. More information on the nonlinear dependence of piezoelectric properties can be found in Damjanovic (1997a; 1997b), Takahashi et al. (1998), Masys et al. (2003) etc. For the blocked impedance measurements, an intermediate value for source amplitude, 500 mV, was chosen and applied across the electrodes of the piezo layer, and the resulting current is simultaneously measured using the im pedance analyzer. The electrical impedance was measured across a frequency range of 401000 Hz that was divided into 201 points. A point averaging scheme is employed for the measuremen ts that cycles through 10 averages at every point to measure the final impedance. From the m easured complex electrical impedance, the real and imaginary parts were curve-fitted using the least squares technique to extract an effective blocked capacitance ebC and dielectric loss tangent tan The fitting function used in the technique consists of a parallel combination of a capacitor and a resistor per the lumped element model shown in Figure 2.1. Using th e thickness of the PZT layer 1.020.1 m and the surface areas of the design geometries, the effective diel ectric permittivities are calculated. The results obtained from these experiments are shown in Figure 6-12 and Figure 6-13. A summary of the estimated parameters are listed in Table 6-2. In Table 6-2, the areas are calculated from the theoretical dimensions in the design structures.

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182 Table 6-2: Dielectric parameters of all te sted design geometries on the device wafer. Device Geometry ebCnF 2 pAmmr tan PZT-EH-01 2.070.03 0.26 93115 21.290.1710 PZT-EH-02 27.150.43 3.34 95515 21.690.1510 PZT-EH-03 98.192.05 12.05 95720 21.790.2410 PZT-EH-04 102.441.43 12.39 97214 22.302.1410 PZT-EH-05 35.720.42 4.43 94812 21.550.1910 PZT-EH-06 1.600.02 0.20 93611 21.270.1710 PZT-EH-07 12.080.20 1.47 96216 21.530.2810 PZT-EH-08 43.260.60 5.31 95713 21.590.1510 PZT-EH-09 2.590.04 0.32 95115 21.430.2210 Therefore, from the blocked electrical impe dance measurements performed on the process wafer, the overall dielectric permittivity was estimated as 94318 and the dielectric loss tangent was measured to be 21.410.4610 These values will be used to estimate the blocked capacitance ebC and dielectric resistance e R to complete the LEM representation for individual tested devices. Anot her observation that can be drawn from these results is that the permittivity and dielectric loss tangent both tend to increase with the PZT area. However, this conclusion is just based on thes e specific results and can only be substantiated with more experiments and further investigation.

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183 Figure 6-12: Ceb and r for MEMS PZT devices on wafer befo re release for a) PZT-EH-01 (106 geometries) b) PZT-EH-02 (16 geometries ) c) PZT-EH-03 (15 geometries) d) PZTEH-04 (14 geometries) e) PZT-EH-05 (16 geometries)

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184 Figure 6-13: Ceb and r for MEMS PZT devices on wafer befo re release for a) PZT-EH-06 (150 geometries) b) PZT-EH-07 (12 geometries ) c) PZT-EH-08 (22 geometries) d) PZTEH-09 (108 geometries) Next, the PZT MEMS devices that were rele ased and packaged were characterized to extract all the lumped element parameters described in Section 2.1. Various frequency response measurements were carried out fo r each device and least square e rror curve fits were generated using the functional forms of the response described in 2.3. The LEM parameters were estimated from the functional fits and compared with theory. Finally, output voltage and power measurements were obtained for each device operat ing at resonance, subj ected to fixed source acceleration. These experimental results were comp ared with those predicted using the extracted LEM parameters.

PAGE 185

185 In the next section, various algorithms for lumped element parameter extraction are investigated in detail. The met hods are first presented and discu ssed after which each of these methods was tested on a single device (PZT-EH09-02). The method that produced the best results was then adopted and applied for all other devices. Lumped Element Parameter Extraction In this section, a detailed description of the procedure adopted for the lumped element parameter extraction is provided. First, a list of all the parameters required to complete the electromechanical circuit is presented followed by a list of various ex periments performed to characterize the behavior of the MEMS PZT device. Next, three diffe rent data extraction algorithms are presented along with the corr esponding results and discussion. One of the presented methodologies is selected as the comm on extraction procedure and is applied for all experimental data for the devices. The results thus estimated for the LEM parameters for each of the MEMS devices are presen ted with their uncertainties. The lumped element parameters that need to be estimated for completing the electromechanical circuit (Figure 2.1) are again listed below in Table 6-3. Table 6-3: LEM parameters extr acted using experimental data. m M Effective mechanical mass msC Short circuit mechanical compliance m R Effective mechanical damping md Effective piezoelectric coefficient ebC Blocked electrical capacitance e R Dielectric loss nF Natural frequency The device responses obtained in the charac terization experiments described in Chapter 5 are used to extract these parameters. The equations from the LEM that will be used to represent the behavior of the device are described next These equations (simp lified and applied to

PAGE 186

186 different frequency regions) are us ed to fit the measured experime ntal response with an effective set of LEM parameters. The electromechanical characterization e xperiment is conducte d to measure the frequency response between the input voltage to the PZT and resulting tip displacement. From the LEM described in Section 2.2, the fre quency response can be represented using Eq. (6.1), where tipw and V are the relative tip displacement and input voltage, respectively. 2 0. 12tip m F nnw d V j (6.1) From the short circuit mechanical characte rization where the relative tip displacement tipw was measured as a function of the input base displacement basew, the resulting frequency response is inherently dimensionless and is shown in Eq. (6.2). 2 2 2 0. 12tip n base V nnw w j (6.2) The free electrical impedance of the PZT de vice was experimentally measured and the response obtain from the lumped element model is 2 2 2 2 21 12 tan 21 112 tantantanebnn ef m nnnnmsebj C Z d j CC (6.3)

PAGE 187

187 The final experiment conducted to extract the LEM parameters is the fre quency response between the open circuit voltage across the PZT and the input base acceleration. The model when solved for this acce leration response follows Eq. (6.4) 2 2 2 2tan 21 112 tantantanmm oceb o m nnnnmsebMd VC a d j CC (6.4) These equations and their asymptotic limits will be used in the following section. In particular, they are applied to the measured response in specific frequency regions such as low frequency region , near resonance etc. This approach simplifies the expressions facilitating easy parameter extraction via th e method of least squares. Ne xt, three potential parameter extraction algorithms are presented with a disc ussion on the extraction procedure for each. In addition, flowcharts are provided for each to understand the overall process better. The general extraction algorithm to estimate the LEM parameters is listed below in the order they need to be obtained 1. Extract md (Eq. (6.5)) from the low frequency response of the electromechanical characterization results. 2. Extract efC from the low frequency response of the free electrical impedance measurements. It should be noted th at since the capacitance depends on the source amplitude, these measurements s hould ideally be obtained at similar amplitude levels. 3. Extract the short circuit natural frequency s c f and open circuit natural frequency oc f of the device.

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188 4. Extract 2 from the extracted frequencies using the relation, 21 s c oc f f 5. Calculate msC from md efC and 2 using the relation, 2 2m efmsd CC 6. Calculate m M from msC and n f using Eq. (6.12). 7. Extract and tan by fitting the responses around resonance. In addition, ebC can be extracted from the process wafer before releasing the individual devices. Furthermore, the extraction steps assume that 31d and are constant with frequency which is generally not true. It should be noted here that for damping estimates, experiments can be conducted to separate the individual mechan isms. For example, conducting the short circuit characterization will yield a pur ely mechanical damping factor, while the open ci rcuit condition included electrical losses also. If these experi ments are carried out under vacuum, air flow dissipation can be estimated a nd neglected in the model. However, since the coupling coefficient of thes e devices is very small, the short circuit resonance and open circuit resona nce could not be separated. Hen ce, alternate techniques were devised and implemented. Three different algorith ms were designed to extract the parameters that are described in detail in the following sections. Method 1 This method aims to obtain the LEM paramete rs using the algorithm listed below in the order they are extracted. 1. Extract md (Eq. (6.5)) from the low frequency response of the electromechanical characterization results.

PAGE 189

189 2. Extract ebC (Eq. (6.8)) from the low frequency re sponse of the free electrical impedance measurements. 3. Extract m M (Eq. (6.10)) from the low frequency response of open circuit voltage measurements. 4. Extract an effective n f and by simultaneously fitting both short circuit mechanical (Eq. (6.2)) and electromechanical (Eq. (6.11)) responses near resonance. 5. Calculate msC from m M and n f using Eq. (6.12). 6. Extract an effective tan by fitting the open circuit voltage response (Eq. (6.13)) around resonance. All other prev iously estimated parameters will be included in the model.

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190 Experimental response spectra 1ef ebZ C ocmm oebVMd aC 21 2ms nmC f M oc o mm ebV a F M d C md ebC m M ,n f msC tan 0 f 0 f 0 f r f f r f f ,,,,,,tanmebmnmsdCMfC 0 tip m Fw d V 0 21 12tip F m nnw V d j 2 2 2 012tip n base V nnw w j Figure 6-14: Flowchart for method 1 to extract th e LEM parameters from the experimental data.

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191 A flowchart that shows the extraction procedure pictorially is shown in Figure 6-14. Next, the details of all fitted func tions in different frequency region s in each step are described. From Eq. (6.1) the response at very low frequencies (approaching dc) reduces to 0.m Fz d V (6.5) The data at low frequencies is thus obtained using least squa res via linear regression at low frequencies, which directly provides md the effective piezoelectric coefficient. The uncertainty in the estimate is also obtained from the regression model for the fit. It should be noted that the experimental data is very noi sy at low frequency with low coherence. For example, the coherence measured for PZT-EH-02 was approximately 0.50.6 for frequencies close to 100 Hz. The poor coherence reflects in the un certainty estimates for the extracted parameters. Next, the free electrical impedance, ef Z is simplified for low frequencies to 21 tan 1 11 taneb ef m msebC Z d j CC (6.6) Here, 2 m msebd CC is defined as the coupling factor 2 which is very sma ll and therefore we assume 211m msebd CC This holds true especially for these devices as the coupling between electrical and mechanical domains is unfortuna tely poor, and no significant shift is observed between short circuit and ope n circuit response. This observation was confirmed when

PAGE 192

192 measurements were conducted for short-circuit and open-circuit resonant frequencies and no shift was noted. From Eq. (6.6), the magnitude of the impe dance is further simplified as 21 tan 1 1 taneb efC Z (6.7) The reported tan for PZT in literature is generally 2 %4 % which was verified during the blocked electrical impedance meas urements on MEMS PZT devices (Chapter 6). Hence, it is assumed that 2211 1 tantan simplifying the electri cal impedance at low frequency to 1ef ebZ C (6.8) Consequently, fitting 1ef Z with a straight line on a Bode plot provides an effective ebC The uncertainty in the estimate is directly obtained from the linear regression model. Next, the open circuit volta ge response at low freque ncies is simplified to 2tan 1 11 tanmm oceb o m msebMd VC a d j CC (6.9) Using the same assumptions employed in the response for ef Z the response follows ocmm oebVMd aC (6.10)

PAGE 193

193 Since both md and ebC have already been experimentally extracted, fitting the data at low frequency directly yields the effective mass m M of the device. The short circuit mechanical response is i nherently dimensionless shown earlier in Eq. (6.2). In addition, dividing Eq. (6.1) by md renders it dimensionless as 0 21 12tip F m nnw V d j (6.11) Both functions, Eqs. (6.2) and (6.11), are simultaneously fitted near the short-circuit resonance to extract an effective damping ratio and natural frequency n f Note that if 2n nf is the extracted resonant frequency, it does not, by definitio n match the measured resonance r f They are related as 212rnff r f corresponds to the frequency where the magnitude of the data reaches its maximum. The fits are generated by minimizing the root mean square (rms) error of the data. Here, was estimated to be very small for all the devices. Therefore, it is assumed that th e estimate for the natural freque ncy is exact. Consequently, only the uncertainties in are estimated by adjusting its valu e such that the peaks match at resonance. The upper and lower bounds on damping ratio, namely, u and l are calculated from the functions Eqs. (6.2) and (6.11) such that the peaks at resonance match in both the data and fit. From the previously estimated n f and m M an effective compliance msC is calculated using

PAGE 194

194 21 2ms nmC f M (6.12) Finally, the open circuit voltage response at resonance is fitted using the nondimensionalized form of Eq. (6.4) as shown in Eq. (6.13) to extract an effective tan by minimizing the rms error. The frequency range of interest in this cases depended on the number of points. It was ensured that atleas t 10 points were chosen for the fit. 2 2 2 21 tan 21 112 tantantanoc o mm m eb nnnnmsebV a Md d j C CC (6.13) The fits obtained using the above mentioned steps are shown below along with the actual measured data. The results presented here in Figure 6-15-Figure 6-19 are for just 1 device, PZTEH-02. Figure 6-15: Low frequency electromechanical response data co mpared with curve fit to extract dm.

PAGE 195

195 Figure 6-16: Comparison between experiment and LEM based curve fit around resonance for a) electromechanical response b) short-circuit mechanical response Figure 6-17: Low frequency curve fit comp ared with experiment to extract Ceb.

PAGE 196

196 Figure 6-18: Comparison between experiment a nd curve fit for low frequency open circuit voltage response to extract Mm. Figure 6-19: Experimental data and curve fits for open circuit voltage response compared around resonance. The final extracted parameters for PZT-EH-02 are listed in the following Table 6-4 using the above described method.

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197 Table 6-4: LEM parameters extracted using Method 1. LEM parameter Estimated value Uncertainty md 73.2410mV 3 % n f 502.97Hz 43.2010 41.1510 ebC 97.5110F 1 % m M 62.3410kg 8 % msC 24.2810mN 7 % tan 0.72 0.25 The above method although appears to fit data well, predic ts a much higher tan than expected. This essentially contra dicts the initial assumption that tan is small. Therefore, another formulation was adopt ed to extract the paramete rs and is presented next. Method 2 This method was proposed and implemented primarily to get a better estimate of tan by simultaneously fitting more functions arou nd resonance by minimizing the error. The algorithm is listed below describing the steps involved. 1. Extract md (Eq. (6.5)) from the low frequency response of electromechanical characterization results. 2. Extract ebC (Eq. (6.8)) from the low frequency response of free electrical impedance measurements. 3. Extract m M (Eq. (6.10)) from the low frequency response of open circuit voltage measurements. 4. Extract an effective n f and by simultaneously fitting both short circuit mechanical (Eq. (6.2)) and electromechanical responses (Eq. (6.11)) around resonance. 5. Calculate msC from m M and n f using Eq. (6.12).

PAGE 198

198 6. Extract an effective tan by fitting both open circuit voltage response and free electrical impedance (Eqs. (6.14)-(6.15)) around resonance. All other previously estimated parameters will be included in the model. It was assumed that fitting both functions simultaneously will produce a better estimate for tan

PAGE 199

199 Experimental response spectra 0 tip m Fw d V 1ef ebZ C ocmm oebVMd aC 21 2ms nmC f M 0 21 12tip F m nnw V d j 2 2 2 012tip n base V nnw w j 1 212ebef nnCZ F j 2 oc o mm ebV a F M d C md ebC m M ,n f msC tan 0 f 0 f 0 f r f f r f f ,,,,,,tanmebmnmsdCMfC Figure 6-20: Flowchart for parame ter extraction using Method 2.

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200 Steps 1-5 remain same as the ones described in Method 1 and theref ore are not repeated. The main difference between the two methods is in Step 6, where both el ectrical impedance and open circuit voltage are fitted simultaneously fitted (using Eqs. (6.4) and (6.3)) that results in dimensionless expressions to extract an effective tan minimizing the overall rms error of the fit. The fitted non-dimensional functions are listed in Eqs. (6.14) and (6.15). 1 2 2 2 21 tan 21 112 tantantanoc o mm m eb nnnnmsebV a F Md d j C CC (6.14) 2 22 2 2 21 tan 21 12112 tantantanebef m nnnnnnmsebCZ F d jj CC (6.15) The fits obtained using this al gorithm, remain similar as Me thod 1, except the fit obtained in Step 6 that fits both norma lized electrical impedance and open circuit voltage. The results obtained for PZT-EH-02 using this step are shown in Figure 6-21. As seen in the figure, 2F does not fit as well as 1F. This could be due to the fact that th e absolute values of the magnitudes are lower for 2F. In addition since the frequency resoluti on for the electrical impedance is poor 4.77 Hz, it is possible that the exact resonance is not captured and he nce the low value for the response.

PAGE 201

201 Figure 6-21: Experimental data and curve fits for open circuit voltage respons e and free electrical impedance compared around resonance. The final extracted parameters for PZT-EH-02 obtained with method 2 are listed in the following Table 6-5. Table 6-5: LEM parameters extracted using Method 1. LEM parameter Estimated value Uncertainty md 73.2410mV 3 % n f 502.97Hz 43.2010 41.15010 ebC 97.5110 F 1 % m M 62.3410kg 8 % msC 24.2810mN 7 % tan 0.71 0.25 Although Method 2 produces a be tter fit than Method 1 at resonance and is more accurate, it stil l over-predicts tan with no significant improvement from the previous estimate. Therefore, another approach is proposed and presented next to obtain a better estimate for tan while maintaining the goodness of the fits. Method 3 This method was proposed and implem ented to get a better estimate of tan by simultaneously fitting more functions around res onance by minimizing the error. The algorithm

PAGE 202

202 is listed below describing the steps involved. A fl owchart for the steps listed below is provided in Figure 6-22. 1. Extract md (Eq. (6.5)) from the low frequency response of electromechanical characterization results. 2. Extract ebC (Eq. (6.8)) from the low frequency response of free electrical impedance measurements. 3. Extract m M (Eq. (6.10)) from the low frequency response of open circuit voltage measurements. 4. Extract an effective n f and tan by fitting all four experimental responses (Eqs. (6.16)-(6.19)), namely, short circ uit mechanical response, electromechanical response, free elec trical impedance and open circuit voltage response around resonance. Fitting all functions simultaneously should provide the best possible estimate for tan 5. Calculate msC from m M and n f using Eq. (6.12).

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203 Experimental response spectra 0 tip m Fw d V 1ef ebZ C ocmm oebVMd aC 21 2ms nmC f M md ebC m M ,,tannf msC0 f 0 f 0 f r ff r ff ,,,,,,tanmebmnmsdCMfC 3 efeb F ZC 0 2 tip F mw V F d 1 0 tip base Vw F w 4 oc o mm ebV a F M d C Figure 6-22: Flowchart for LEM parame ter extraction implementing Method 3. Since Steps 1-3 essentially remain the same even in this method, they are not repeated. In step 4, all four measured responses are non-dimens ionalized and simultaneously fitted to the four

PAGE 204

204 fitting functions listed below in Eqs. (6.16)-(6.19). The fit are generated such that the overall rms error is minimizing resul ting in the extraction of n f and tan 2 2 1 2 0, 12tip n base V nnw F w j (6.16) 0 2 21 12tip F m nnw V F d j (6.17) 2 22 3 121 12 tannn efebFZC dd (6.18) and 4 121 tan ,oc o mm ebV a F Md dd C (6.19) where 1d and 2d are used to simplify the above e xpressions and are represented as 2 2 1 2 2 2 2 22 1 tan 1 12 tantannn m nnmsebd d d CC (6.20) In 2d, msC is expressed in terms of m M and n f Again, n f is assumed to be an accurate estimate and uncertainties are obtained only for and tan using the same process described

PAGE 205

205 in Method 1. After extracting m M and n f the compliance, msC is obtained using Eq.(6.12). The fitted plots around resonance for all func tions are shown below for PZT-EH-02. Figure 6-23: Comparison between experiment and LEM based curve fit for short circuit mechanical and electromechanical response around resonance. Figure 6-24: Experimental data and curve fits for open circuit voltage response compared around resonance. The final extracted parameters for PZT-EH-02 obtained with method 3 are listed in the following Table 6-6.

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206 Table 6-6: LEM parameters extracted using Method 3. LEM parameter Estimated value Uncertainty md 73.2410mV 3 % n f 503.01 Hz 43.7910 40.9810 ebC 97.5110F 1 % m M 62.3410kg 8 % msC 24.2810mN 8 % tan 0.41 0.12 It was decided to adopt method 3 to extract LEM parameters for all tested devices as it gave the best fit with reasonable parameter estimat es. Using Method 3, fits were generated for all experimental characterization curves and are presented in the next section. Results and Discussion The experimental response curves generated for each device are simultaneously compared with the model predictions in the figures. In a ddition, the output voltage and power measured at resonance across different resistive loads are comp ared with the model. The experimental setups and data acquisition parameters for each characteriz ation were described previously in Chapter 5. PZT-EH-09 Using Method 3, the LEM parameters that describe the device behavior were first extracted and are listed in Table 6-7 for PZT-EH-0901 and PZT-EH-09-02.

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207 Table 6-7: LEM parameters extracted for PZT-EH-09-01. PZT-EH-09-01 PZT-EH-09-02 LEM parameter Estimated value Uncertainty Estimated value Uncertainty md 72.9110mV 4 % 73.2410mV 3 % n f 481.04 Hz 503.01 Hz 42.8110 41.2410 43.7910 40.9810 ebC 97.5610 F 1 % 97.5110 F 1 % m M 72.9010kg 20 % 62.3410kg 8 % msC 0.38 mN 20 % 24.2810mN 8 % tan 0.47 0.24 0.41 0.12 These two devices belong to the same design and were characterized identically with all three PZT layers connected in pa rallel and hence should ideally pe rform similar. The plots that show the comparison between measured and model predictions are shown in Figure 6-25 and Figure 6-26 for the two devices.

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208 Figure 6-25: Comparison between model and experi ments for PZT-EH-09-01. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance

PAGE 209

209 Figure 6-26: Comparison between model and experi ments for PZT-EH-09-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance

PAGE 210

210 Next, the different characterization plots compared with fitted models for PZT-EH-09-03 and PZT-EH-09-04 are shown in Figure 6-27 and Figure 6-28. Figure 6-27: Comparison between model and experi ments for PZT-EH-09-03. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance.

PAGE 211

211 Figure 6-28: Comparison between model and experi ments for PZT-EH-09-04. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance.

PAGE 212

212 The extracted LEM parameters fo r these devices are shown in Table 6-8. Table 6-8: Extracted LEM parameters for PZT-EH-09-03. LEM parameter Estimated value Uncertainty Estimated value Uncertainty md 71.1310mV 5 % 87.6010mV 2 % n f 477.99 Hz 473.76 Hz 43.2910 41.2210 413.3310 41.8710 ebC 92.5210F 1 % 92.3610F 3 % m M 61.6510kg 14 % 61.9210kg 25 % msC 26.7210mN 14 % 25.8910mN 25 % tan 21.4810 20.6210 21.5010 20.5710 PZT-EH-07 Next, similar results were obtained for P ZT-EH-07-02 and PZT-EH-07-03 and are shown in Figure 6-29 and Figure 6-30. The extracted LEM paramete rs for both devices are listed below in Table 6-9. Table 6-9: LEM parameters extracted for PZT-EH-07-02. LEM parameter Estimated value Uncertainty Estimated value Uncertainty md 61.2510mV 8 % 61.6610mV 2 % n f 126.60 Hz 128.85 Hz 411.6310 44.1910 45.4410 40.2210 ebC 912.1910 F 1 % 912.4110 F 1 % m M 64.6810kg 11 % 64.7210kg 17 % msC 233.7610mN 11 % 232.3010mN 17 % tan 21.9410 20.7810 20.9910 20.2510

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213 Figure 6-29: Comparison between model and expe riments for PZT-EH-07-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance.

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214 Figure 6-30: Comparison between model and expe riments for PZT-EH-07-03. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance.

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215 Summary and Discussion of Results All the extracted lumped element parameters using detailed experime ntal characterization are summarized and discussed here. Table 6-10: lists the various lumped elements extracted for PZT-EH-07-2 and PZT-EH-07-3 and compares them with those calculated from the theoretical model described in Chapter 2. The short circuit mechanical parameters such as m M msC and nF match very well with theory, thereby valida ting the beam model for current designs. Furthermore, the electrical parameters, ebC and tan also match well with the model as indicated in Table 6-10. However, the effective piezoelectric coefficient md is over predicted by the model, and the experimentally measured valu es are considerably smaller. The results are discussed in more detail later in this section. Table 6-10: Comparison between theory and experiments for PZT-EH-07. assumed values in the model Table 6-11 summarizes the lumped element pa rameters extracted for individual PZT-EH09 devices. It should be noted that the modeli ng procedure describe in 2.1 involves a single cantilever beam structures with a proof mass at one end. However, PZT-EH-09 consists of 3 PZT cantilever beams attached to a single proof mass. In Table 3-10, which lists the predicted lumped PZT-EH-07Theory 02 03 m M mg 4.71 4.68 4.72 msCmN 0.3371 0.3376 0.3230 0.01* 0.0012 0.0005 n f Hz 126.3 126.6 128.8 mdmV -1.91 -1.25 -1.66 NV 5.66 3.70 5.14 2 9.19 X 10-4 3.79 X 10-4 6.87 X 10-4 ebCnF 11.84 12.19 12.41 tan 0.02* 0.0194 0.0099

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216 element parameters for all designs, the values ar e indicated for single ca ntilever configuration. So, if the design has multiple beams connected to a single proof mass, the proof mass is divided equally between the number of cantilevers al ong its width. This essentially enforces the condition that the beams are mechan ically connected in parallel, an assumption that holds true for small deflections. Therefore, the overall mass of the device is approximately three times the mass of each cantilever configuration. Cons equently, the masses of PZT-EH-09 devices measured match well with the model prediction if the cantilever mass is multiplied by three due to multiple beams. Similar reasoning implies that the compliance of the single cantilever beam is thrice the overall compliance of the device which has thr ee beams. In such a case, the short circuit compliance values measured for the devices matc h with the predicted compliance. However, the natural frequencies remain the same and are close to theory. Table 6-11: Comparison between theory and experiments for PZT-EH-09 devices. assumed values in the model PZT-EH-09-01 and PZT-EH-09-02 had all the PZT beams connected in parallel which would effectively add the indivi dual capacitances. The extracted capacitances for these devices PZT-EH-09Theory 01 02 03 04 m M mg 1.92 0.29 2.34 1.65 1.92 msCmN 0.053 0.380 0.043 0.067 0.059 0.01* 0.0003 0.0004 0.0003 0.0013 n f Hz 501.5 481.0 503.0 477.9 473.7 mdmV -0.65 -0.29 -0.32 -0.11 -0.08 NV 4.15 0.76 7.47 0.72 1.36 2 1.4 X 10-3 2.93 X 10-5 3.18 X 10-4 1.24 X 10-4 0.46 X 10-4 ebCnF 7.71 7.56 7.51 2.52 2.36 tan 0.02* 0.47 0.41 0.0148 0.015

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217 match well with the theoretical model. Alternat ively, PZT-EH-09-03 and PZT-EH-09-04 had just one PZT beam connected for external access and therefore, the capacitances are approximately one-third the total capacitance. From Table 6-10 and Table 6-11, it can be noted th at although the mechanical LEM parameters are reasonably close to the model, th e mechanical damping measured is significantly lower than assumed. In the LEM, an assumed value 0.01 was used based on the measured damping for meso-scale devices. However, the MEMS devices produce a much higher quality factor Q as confirmed by the low s (Candler et al, 2003). The quality factor and the damping ratio are related as 12 Q So, using the various dissipation mechanisms studied in Appendix B, the corresponding damping was obtai ned for the two distinct MEMS devices, namely, PZT-EH-07 and PZT-EH-09. These mechan isms are investigated in some detail to obtain empirical relations. It should be noted here that there are possibly many more uninvestigated mechanisms that contribute to the overall mechanical damping in the device. Only the most common ones such as air damping, thermoelastic damping, surface losses etc have been included here. In addition, many different models for these individual losses exist in literature. However, simple and generalized expr essions were reported in this dissertation to basically understand how these loss mechanisms scale to micro devices and decipher the importance of each of these mechanisms in overall damping. This will help design and fabricate these devices accordingly depending on the Q require ments of the device. In some cases, it may be necessary to opt for a lower Q to expand the bandwidth to be able to operate over a wider frequency range. Higher Q has an inherent disadvantage in resonant energy harvesting because a minor change in either the source vibration fr equency or the device res onance will result in a large drop in power generation.

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218 The expressions used for these loss mechanisms are greatly simplified and in general, are applicable to cantilever beam s with rectangular cross-sec tions. The PZT MEMS devices compose of a beam section in addition to a proof mass at one end that complicates the total effective dimensions. For this analysis, the tota l length of the device was used whereas the width and thickness correspond to the di mensions of the silicon layer in the beam section. Hence, the width and the thickness of the proof mass are neglected, which may result in inaccurate damping estimates. The dissipation due to air flow in the viscous region (up to at mospheric pressure) is derived by solving the flow past a string of s pheres (Appendix B). Usuall y, the cantilever beam is modeled using a string of spheres with an e quivalent radius that is obtained by fitting experimental data. Here, the radius of the sphere is based on the simplest approximation, the half width of the beam. The support loss is smaller comp ared to the other losses primarily due to the high length to thickness ratio. Again, the thickness of the beam is used here and not that of the proof mass. It should also be noted that th e surface and volume losses are considerable. The surface loss primarily occurs due to thin films on a substrate, for e.g., PZT. However since all the thermal properties such as conductivity and lin ear expansion coefficient are unknown for PZT, the thickness of the beam was used which result s in a fairly smaller Q. Therefore, the surface dissipation is not particularly us eful in understanding the physics of the da mping in the device. Furthermore, these relations are purely empirical a nd may not be applicable in that form to these PZT MEMS devices. Hence, another Q was calcula ted using only three mechanisms that could be important for the test structures, namely, di ssipation due to air flow, support and thermoelastic mechanism. These values for the two devices are also presented in the last rows of Table 6-12 using effQ and the corresponding damping ratio eff

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219 The quality factors for different loss mechanisms are calculated using the empirical relations in Appendix B for the two devices and presented in Table 6-12. In addition, the equations used specifically to calculate the indi vidual quality factors ar e listed in the table. Table 6-12: Quality factors for PZT MEMS devices. Quality factor PZT-EH-07 PZT-EH-09 Equation # Air flow dissipation airQ 536 31.0410 B.13 Support dissipation supQ 65.7110 61.6210 B.15 Surface dissipation s urfQ 31.0210 31.0110 B.16 Volume dissipation volQ 36.1210 36.1210 B.17 Thermoelastic dissipation temQ 45.6210 43.4210 B.22 Effective quality factor totQ 330 466 B.5 Effective quality factor effQ* 531 31.0110 B.5 Effective damping ratio eff 49.4110 44.9610 B.1 includes only three damping mechanisms The newly calculated damping ratio is of the same order of magnitude as the extracted value from experiments. Further experiments and a more detailed investig ation of the dissipation mechanisms is required to better understand and accurately predict the resonant behavior of these devices. This exercise also confirms that the assumed value for does not apply for these devices. From the calculated quality factors, it appears that the ai r flow damping is the dominant mechanism followed by thermoelastic damping. Th e air damping can be considerably minimized by performing the characterization under vacuum in which case, the overall material damping will approach the thermoelastic limit as described in Candler et al. (2003). Next, the effective piezoelectric coefficients do not match well with theory. The effective piezoelectric coefficient md is extracted using the low frequency electromechanical response. A very small amplitude voltage signal 0.1 V was used to electrically excite the beam and measure its response from dc to beyond resonan ce. Smaller voltage inputs were applied to

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220 prevent excessive deflections at resonance and, he nce, risk damaging the devices. As a result, the deflections at low frequency are significantly lower and the measured coherence was poor. For better estimates of md, it is advisable to apply single fr equency voltage input s to the PZT and measure deflection. Higher voltages can then be applied to obtain clean measurable deflections with better coherence and theref ore, more accurate values for md. Future work will concentrate on more detailed characterization at low frequencies. In addition, there exists some PZT on the clamp with electrodes that is no t accounted for in the model as ev ident in Figure 4.16 and Figure 4.17. This additional parasitic capacitance was ca lculated to be signifi cant, 32% for PZT-EH-07 and 23% for PZT-EH-09. These values are base d on the added area of the electrode and PZT whose dimensions were obtained fr om the design layouts. It is pos sible that the added PZT area affects the device behavior in general, even for voltage and power generation. The parasitic capacitance is smaller for the larger devices 10 % and will eliminate its affect on the deflection. This may still not explain the difference between th e measured data and model. Characterization of many devi ces of the same type is required to understand the electromechanical behavior and obtain better re peatability of measurements between devices. The extracted coupling factor, 2 is dependent on msC, md and efC. The excessive uncertainties in md and msC and the additional parasitic capacita nces may have resulted in the large differences. Factors such as residual stresse s, non-uniformity in piezoelectric properties on the wafer and further poling of the PZT may play an important role in finally determining the actual piezoelectric parameters. Fu rthermore, the uncertainties a ssociated with the fabrication process for each batch of devices coupled with the dimensional tolerances of the structures need to be considered for the differences in measurements.

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221 The output power results are al so included in the plots compared with the predicted values using the fitted model. Although the m odel fitted the various individual frequency responses well, the output power at resonance does not match we ll with model. Although some explanation can be provided for these differences they may not still compensate for the large discrepancy. First, the extracted damping ratio and the resonant frequency using the model were obtained from the least squares fits. These ex tracted frequencies do not match the resonant frequencies where the data was obtained. The model predictions were based on the extracted LEM parameters. Additionally due to very high 'Qs observed, the frequency resolution may have to be improved considerably to excite th e device at its true resonance to measure better results. All these observations were drawn afte r the experiments were conducted and therefore the option of redoing the measuremen ts was not explored. This will have to be considered in future work to obtain better match between model and experiments. PZT-EH-09-03 and PZT-EH-09-04 appeared to significantly underachieve in its piezoelectric and dielectric properties that led to significantly lower output voltages and power compared to similar candidate devices. These results are evident in all the frequency response plots presented before in Figure 6-25 Figure 6-30. The wirebonding process that employs an ultrasonic impact technique was used to provide electrical contacts for all devices. During this process, one of the beams in PZT-EH-09-04 was damaged near its clamp that may have led to some deterioration in its overall performance. It was observed that method 3 (and other methods too) predict tan as expected (<0.02) for 4 out of 6 devices. Only PZT-EH-09-01 an d PZT-EH-09-02 give un reasonably high values (0.41, 0.47) for tan These devices had the three PZT laye rs connected in parallel, while the other 4 devices that matched theory more closel y were tested only with one PZT layer connected.

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222 This difference in electrical c onnection between the devices needs to be explored further to understand if the parallel excitation adds impedance to change the response. Another variation of method 3 was implemente d to observe if it improved the parameter extraction for tan After the first set of parameters were estimated using method 3, a new ebC was calculated using the low freq uency fit using the extracted tan This step resulted in a lower ebC that was then used to estimate the rest of the parameters usi ng an iterative method. The idea of this procedure was to determine if an iterative proced ure would obtained an improved value for tan However, this new approach resulted in the same value of tan as before. The reason behind this is because m M is extracted from the low frequency voltage response, and a lower ebC results in a lower m M while md remains constant. This maintains the ratio constant (Eq. (6.10)), and so the change in the value for ebC is compensated here and is not propagated further in the curvefits. However, if ebC is extracted simultaneously with other parameters, the problem becomes a complete para metric curve fit. The disadvantage with this approach is that the low fre quency behavior of the device th at can be reduced to simple analytical expressions is not leveraged to extract parameters individually. To summarize, various devices of designs PZT-EH-07 and PZT-EH-09 were completely characterized to obtain the LEM parameters an d subsequently measure the output voltage and power generated by each device oper ating at resonance. The next chapter essentially discusses the conclusions of the presen t work involving the overall modeling, fabrication, and characterization of MEMS PZT micro power generators. A disc ussion on the prediction of the overall behavior with the LEM is provided, whic h explores some of the limitations in the model in terms of predicting the behavior of micro devices, specifically at resonance. In addition, some

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223 key aspects for improvement in design, mode ling and fabrication ar e presented along with directions and scope for future work.

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224 CHAPTER 7 CONCLUSIONS AND FUTURE WORK This chapter presents all th e conclusions drawn from the research conduc ted in this dissertation. Furthermore, suggest ions and directions for future work are provided following the concluding remarks. Conclusions A MEMS-based piezoelectric energy harvesting concept is developed and implemented in this dissertation. Firs t, a detailed literatu re review was conducted to study various energy harvesting mechanisms and the curr ent state of technology in the fi eld. A device consisting of a cantilever piezoelectric composite beam with a proof mass was chosen as the candidate energy harvester design for this work. A detailed anal ytical model was developed using the lumped element modeling technique to represent the structur e in the form of an equivalent circuit. The piezoelectric composite beam was modeled using Eu ler-Bernoulli beam theory that provided an analytical piecewise solution for the deflectio n modeshape. The assumptions governing the analytical solution are that the deflections are sm all compared to the beam length. In addition, shear and rotary inertia affects are neglected. Furthermore, the model does not incorporate any residual stress affects that result in an initial curvature of the device. The resulting modeshape was then utilized by lumping the resulting potential and kinetic energies to obtain the mechanical lumped element parameters. An effective transduc tion factor and piezoelectric coefficient for the device was also obtained that rela tes the conversion of energy from the mechanical to electrical domain and vice versa.

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225 After the lumped element circuit parameters were obtained analytically, the LEM was extended to investigate the dynamic behavior of the device and its overal l response for voltage and power output. The developed model was ve rified using meso-scale composite beams comprised of PSI PZT-5A patches bonded to an Al-6061 beam. The experimental data matched favorably with the model predictions. Furthermore, the model was validated using FEM for static LEM parameters. Next, a dimensional analysis was carried out for the analytical LEM where all the circuit parameters and response functions were expressed in terms of nondimensional groups. This was done because all the response parameters are analytical but unwieldy functions of geometric dimensions and material properties. Therefor e, a scaling analysis was performed using the nondimensional LEM to determine micro-scale e ffects on overall device performance. In the scaling theory, the damping ratio was assumed to be constant as the device is scaled. However, experiments and subsequent literatu re search confirmed that this assumption is invalid especially at MEMS scale involving different materials. Most meso-scale devices use commercially available bulk PZT ceramics attached to shims ma de of Aluminum, Brass, Stainless steel etc. However, in MEMS, most devices are based on Silicon and sol gel PZT which has inferior piezoelectric properties such as lower 31d and The developed LEM forms the basis for effici ently designing such piezoelectric composite beam energy harvesters to maxi mize power output. In the dissertation, a parametric search optimizations strategy was adopted to realize va rious designs intended to harvest energy from specific harmonic vibrations with prescrib ed base frequency and acceleration levels. Following the design stage, these MEMS de vices were fabricated using conventional surface and bulk microfabrication techniques. The fabrication stage involved developing a

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226 process flow and mask sets that were used to realize these device s. The PZT deposition and patterning was performed at ARL using a sol-ge l process. The device wafers were initially characterized for their piezoelect ric and dielectric properties. Tests were also conducted to polarize the PZT layer for different temperatures and voltages. However, not much improvement in remnant polarization was observed from the original values reported by ARL. Two different packages, namely open and vacu um, were implemented for the devices, after fabrication and release from the process wafer. Ho wever, only the open packages were tested in the experiments described in Chapter 6. The vacuum packages were primarily designed to investigate the effect of air damping. Since th e quality factors measured as such in the open packages were very large, the vacuum packages were not tested. However, further experiments are required to obtain quantitative estimates of the effects of air damping and mass loading. A detailed experimental characterization was carried out on several packaged devices to determine their frequency response for vari ous operating conditions. A lumped element parameter extraction was carried out to obtain the full set of parameters that complete the equivalent circuit. Various ex traction algorithms were investig ated on a test device and the method that provide reasonable values for these parameters was adopted. These experimentally estimated parameters were compared with those obtained using the theoretical analytical model using the base dimensions and properties. While the mechanical parameters matched favorably, especially for the devices that were tested with a single PZT beam confi guration, the results did not match well when multiple PZT layers were connected in parallel. To summarize the overall characterization results, the normalized voltage and power generated for specific input accelerations for all the tested devices are shown in the following

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227 table. The output voltage and power corres pond to the optimal load as indicated in Table 7-1. As expected, the normalized output power was found to be higher for devices with higher Q. Table 7-1: Normalized voltage and power outputs for all the tested MEMS PZT energy harvesters. Input accn. (m/s2) Frequency (Hz) Optimal load (k ) Voltage (mV/m/s2) Power (nW/(m/s2)2) PZT-EH-0702 0.28 127 100 65 42 03 0.20 129 100 313 980 PZT-EH-0901 1.51 481 50 41 34 02 0.77 503 50 53 57 03 0.28 478 100 44 19 04 0.86 474 200 5 0.14 As confirmed with the experimental results on these microfabricated devices, the quality factor is higher than anticipated during the design stage. Therefore, the measured deflections for each of these devices were used to calculate the acceleration limits for onset of non-linearity, fracture and endurance limits. The measured norma lized tip deflections of each of the devices with input base acceleration are listed in Table 7-2. First, the moment at the clamp where the strain is a maximum is calculated for the specifi c device using the static analytical model. Assuming that the strain is proportional to the tip deflection, it is scaled linearly with the measured tip deflection at resonance, after wh ich the corresponding bending stress at the clamp is estimated. These stresses are also listed in the table. The fracture stress for silicon used in the design procedure (Section 3.3.1) was 10 % of its yield strength, 7 GPa. This assumption is valid as Baghdan and Sharpe (2002) measured the initial fracture strength of silicon for cyclic tensile testing as 1.1 GPa while the endurance limit for 810 cycles was limited at 0.75 GPa. Using the initial fracture stre ss limit and endurance limit, the acceleration limits are calculated and listed in the following table. In additi on, the limits for the onset of nonlinearity (2 %)

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228 deviation are also included. As expected, the li near limit is reached before the endurance limit of the devices. Table 7-2: Acceleration limits for MEMS PZT ener gy harvesters based on different stress states. Acceleration limit based on stress due to 2ms PZTEH Normalized tip deflection 2mms Normalized bending stress 2 M Pams Endurance limit 0.75 GPaFracture limit 1.1 GPa Linear model limit 0.4tipswL 09-01 41.62710 221 3.4 5.0 1.2 09-02 41.32610 180 4.2 6.1 1.5 09-03 41.54310 209 3.6 5.3 1.3 09-04 53.7110 50 15.0 22 5.4 07-02 42.9510 143 5.2 7.7 1.4 07-03 48.0010 389 1.9 2.8 0.5 Finally, the overall power densi ties of the fabricated and test ed MEMS devices are listed in Table 7-3. The power density can be improved with some optim ization in future generation devices. However, the poor elect romechanical coupling restrict s the overall power generation compared to meso-scale devices. The power densities of some of the reported energy harvesters in literature and commercially available energy harvesters are also listed in the table for comparison. It should be noted here that the po wer density values are mere ratios of power generated per unit volume of the device. Since no universal metric is available that efficiently compares the performance of energy harves ters, the acceleration values and operating frequencies are also listed. As expected, hi gher acceleration produces larger power densities since power scales as the square of base acceleration. In additi on, frequency is also important because, typically, lower frequency acceleration produces higher deflection as opposed to higher frequency acceleration with the same magnitude.

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229 Table 7-3: Power densities of reported and commercially available energy harvesters. Acceleration (g) Frequency (Hz) Power Density ( W/cm3) Type Source 10.9 13900 37000 a Piezoelectric Jeon et al. 2005 0.25 120 70 Roundy et al 2003 0.25 120 375 Roundy et al 2004 0.24 a 80.1 16.8 Glynne-Jones et al 2001 0.24 120 1.1 Mide Inc 12 120 500 Mide Inc 0.1 57 16 a Microstrain Inc 0.1 21 122 ElectromagneticFerro Solutions Inc 0.1 120 49 Perpetuum Co. 0.1 60 98 Perpetuum Co. 10.4 322 2208 El Hami et al 0.28 127 0.3 Piezoelectric PZT-EH-07-02 0.20 129 3.2 PZT-EH-07-03 1.51 481 17 PZT-EH-09-01 0.77 503 7.6 PZT-EH-09-02 0.28 478 0.3 PZT-EH-09-03 0.86 474 0.02 PZT-EH-09-04 a estimated (Beeby et al, 2006) One of the main contributions of this disserta tion is the development of a complete static analytical model of a piezoelect ric cantilever composite beam th at was validated using FEM and experiments on candidate devices. The equivale nt electromechanical m odel for a piezoelectric energy harvester provides a design tool for specific applications to maximize power transfer and enable complete circuit simulation with power processors. Another major contribution of this research is the realization of a first generation fabrication of a MEMS PZT cantilever array for vibrational energy harvesting. In addition, the complete design, fa brication and testing of a standalone device is demonstrated for energy reclam ation. Furthermore, the novel aspect of this research is the ability to connect multiple PZT la yers either in series or parallel for power enhancement.

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230 Future Work This section provides some suggestions for future work and for potential improvements in piezoelectric cantilever beam energy harvesting, sp ecifically in the areas of device modeling, fabrication process, packaging and characterization. Although the developed model worked well fo r predicting device response at the mesoscale, it failed to match well for the MEMS de vices. There are cons iderable fundamental differences between the devices operating at these different sc ales. The fabrication process involved for MEMS devices is co mpletely different and invalidat es some of the assumptions incorporated in the model. One main factor is th e inclusion of residual st resses in the device that occur during fabrication. The residual stress give s raise to an initial curv ed shape of the device and potentially places the device in the non -linear regime. In such a case, the model cannot be directly applied. Hence, as a fi rst step, residual stress should be considered in the model and a modified analytical solution for the composite beam should be obtained. The resulting modeshape may provide a better estimate for the LEM parameters. Furthermore, a plate theory may have to be carried out as some of thes e devices perform better with higher PZT area in which case, the length and width of th e beam will approach a plate shape. It is also assumed in the model that the widt h of the PZT and substrate are the same, while a significant clearance 510 m is provided along the width in MEMS devices during fabrication to prevent any misalignment and pote ntial shorting of electrodes. Furthermore, the model incorporates a two layer model neglecti ng the effect of the electrodes, whereas many layers are present in MEMS fabrication that may ha ve to be considered in the model. In addition, the electrodes and PZT layer extend on to the clamp of the device for bond pads. This will lead

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231 to a parasitic capacitance that could alter the elec trical performance and he nce, the predictability of the model. Ideally, more detailed experiments need to be conducted for the parallel PZT configurations to investigate the re latively higher observed values for tan compared to the single PZT beams. More experiments need to be conducted for each of the designs to verify the repeatability of their performance and the extracted LEM parameters. It was observed in the experiments that th e electromechanical response consistently measured a lower Q than a pure short-circuit m echanical response. However, the LEM fails to capture this difference and provides only a mechanical It is proposed that there are additional electrical losses that need to be modeled. One of the packaged devices was damaged during operation. The package is generally mounted on a vibrating shaker that, when turned on, provided an initial impulse. This impulse broke the cantilever structure n ear its clamp. For later experime nts, the packaged was mounted after the shaker was setup and operational. However, to mainta in the robustness of the device under such operating conditions, mechanical stops n eed to be provided eith er during fabrication or in packaging. This will prevent from excessive vibration of the beam when subject to sudden impacts etc. Mechanical stops in the form of an outer frame may be provided to prevent vibration due to higher order modes such as twisting moti on. This may require th at the device be in a vacuum package which will benefit in eliminati ng air flow damping. If the gap between the mechanical stop and the device is small, s queeze-film damping may become significant and needs to be considered for overall damping in the system.

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232 Second Generation Design Procedure For the second generation PZT de vice, a suggested design pro cedure that may be adopted is listed below. This procedure is similar to th e one discussed earlier in Section 3.3.1. A more robust design strategy would invol ve a global deterministic or uncertainty based optimization procedure (Gurav et al. 2004) that was not investigated in this dissertation. First, the overall available area of the dye is denoted as A for eg., 1 1 cmcm. This total area is divided into various cells comprised of individual chips. 1. The first step involves deciding the area of each cell. a. Smaller cells for higher frequencies b. Larger cells for lower frequencies 2. Assume the number of cantilevers per proof mass for eg., 1..5 n 3. Assume a size for the clamp, for example, between 0.5 mm and 1 mm, depending on the overall length of the composite beam. 4. After providing enough clearance around the edges, a. choose the width of the proof mass, w b. choose the overall length of the structure, sLlclamp 5. After establishing n and w, the width of the beam can be calculated spw bandb n 6. Use nominal dimension ratios s sLb (5, 10) and s sLt (50, 100, 200) to act as base dimensions for parametric search strategy. 7. To obtain the required natural frequency, very the lengths and thicknesses of the beam and proof mass by trial and error.

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233 a. Prior knowledge from scaling theory will be useful to easily obtain required dimensions 8. For each of the designs, perform parametric an alysis to maximize the output power of the device. An alternate and more robust appr oach is to perform a constrained global optimization with output power as the cost function. a. For each of the designs, it has to be ensure d that the device failure limits such as endurance and fracture limits are not exceeded. b. In addition, the individual damping mechanisms should be estimated for each of the designs. This will help design the device for higher Q (lower bandwidth) or higher bandwidth (lower Q), depending on the application. Electromechanical Conversion Metrics To better understand the output power and elect romechanical efficiency metrics of the PZT energy harvesters, response expressions were deri ved at resonance to observe their behavior and dependence on various device parameters. From th e electromechanical LEM, the expressions for output power and efficiency were obtained analy tically as functions of the circuit parameters. Since the analytical expressions are complicated in their general form, the following assumptions are made to simplify the expressions and to pres ent some physical insight. First, the frequency of operation is assumed to be at its short circuit natural frequency n f It should be noted here that the resonant frequency of the device varies between its short circuit and open circuit limit that is determined by its effective coupling factor. Furt hermore, the resonant frequency depends on the external load that is connected to the device. However, for a poorly coupled device where the difference between the short circuit and open circuit natural frequencies is negligible, for example, in MEMS devices, the natural frequency assumption is reasonable.

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234 Another assumption adopted in this analysis was that the damping due to dielectric loss e R is negligible and is therefore not considered. Since the LEM employs a parallel combination of capacitor and resistor to model the PZT, the an alytical expressions become complicated to derive. In addition, the dielectr ic loss is higher compared to th e mechanical damping in general and is therefore neglected in most piezoelectri c device models. Furthermore, at resonance the mechanical mass and compliance cancel each ot her leaving only the damping in the system. Consequently, the mechanical that is assumed in the model can be used to represent the overall damping in the device incl uding the electrical losses. Using the above assumptions and using the Thev enin equivalent model discussed in Section 3.1, the optimal load for maximum power transfer occurs when the Thevenin impedance is matched. Here, for a purely resistive load, it is eq ual to the magnitude of the impedance resulting in the expression 2 42212 41opt nefR C (7.1) It is evident from Eq. (7.1) that the load is dominated by the capacitive behavior of the piezoelectric material, but is dependent on the coupling and damping in the device. The output power across the load is obtaine d as the ratio between the Thev enin voltage and the overall impedance. These formulas were presented earlier in Section 3.1. 22 2 2422. 441mo L nMa P (7.2) The output power depends on the input acceleration, the overall mass of the device, device coupling and damping in the system. However, th e acceleration is genera lly fixed for a specific

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235 application. The relative motion is dominated by the proof mass, which is amplified at resonance. So, increasing the mass is restricted by limits of device failure due to stress at the clamp during vibration. The output power ideally reaches its maximum when the effective device coupling approaches unity and the damping in the system becomes zero. Since that is practically impossible, the goal in the design should be to minimize the damping and maximize the device coupling independently. Alternatively, the input power to the beam is calculated from the product of the effective force and relative tip velocity, given as 2 422 2 1 2 22 422242281 22 4141mo in nMa P (7.3) The device efficiency that repres ents the power conversion from the beam to the resistive load is obtained from the ratio between Eqs. (7.2) and (7.3) and is expressed as 1 2 2 422 2 22 422242241 2 8141 (7.4) From Eq. (7.4), it can be noted that the device e fficiency is only dependent on the effective coupling factor and effective damping in the devi ce. The device coupling factor is a function of the material coupling coefficient and device geom etry and properties. The functional dependence of the device coupling with its di mensions and properties is not deri ved here due to the analytical complexity of the expressions. In future design strategies, Eqs. (7.2) and (7.4) need to be investigated and maximized for a given set of materials subject to the input conditions and geometrical constraints.

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236 Other potential future work invo lves further investig ation of arrays of devices connected in series/parallel to methodically st udy the power enhancement. An alternate application of this device is as an accelerometer that can be studie d using PZT-EH-06 that was designed to operate beyond 1000 Hz. This will provide a reasonable opera ting range for the accelerometer and, if it provides good performance, the design tool can be used to fabricate one for a larger range and higher sensitivity.

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237 APPENDIX A EULER-BERNOULLI BEAM ANALYSIS WITH VARIOUS BOUNDARY CONDITIONS Euler Bernoulli Beam The governing equation for a beam derived from Euler-Bernoulli theory is represented as 4 40wx EI x (A.1) Let us assume a general so lution to be of the form 23 0123wxaaxaxax (A.2) that describes the transver se deflection of a beam. 012,,aaa and 3a are integration coefficients of the modeshape. In this analytical exercise, beams with various boundary conditions are solved for their exact deflection shapes, natural fr equencies and maximum bending strain in the following sections. Cantilever Beam (Clamped-Free Condition) Figure A-1: Schematic of a cantilever beam. For a cantilever represented in Figure A-1, the following boundary conditions apply

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238 0 2 2 3 300 0 0x xL xLw wx x wx EI x wx EIP x (A.3) They enforce the deflection and slope at the clamp to zero. In addition, the moment at the tip is zero and the vertical shear force represented by the 3rd derivative is the external load, P. Applying the first 2 of the boundary conditions results in 010 aa When the next two boundary conditions are applied, the rest of the coefficients can be calculated as 23; 26 PLP aa E IEI (A.4) Substituting back for these coefficients results in the deflection modeshape as 2323 266 PLPP wxxxxLx EIEIEI (A.5) From the above expression, the tip de flection is obtained by substituting x L as 3. 3tipPL w EI (A.6) The effective mass and compliance using th e deflection are obtained by calculating the stored potential and kinetic energies in the beam and lumping them at the tip (described in detail in Chapter 2). Using this method, the expressi ons for the compliance and mass of the beam are 33msL C EI (A.7) and

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239 33 140mL M L (A.8) It should be noted here that the effective mass is different from what is generally presented in literature mainly because the model here is based on a st atic modeshape, while most predictions of natural frequencie s are based on the eigen modeshape. A static modeshape is used here to be consistent with the actual m odeling technique employed to represent the electromechanical behavior of the piezoelectric composite beam (Chapter 2). From the mass and compliance, the natural fre quency is represented as 21111140 2211n mmsLEI f MCL (A.9) The bending strain can be calculated from the deflection modeshape and is given as 2 2 0 0 x xwx P zzxL xEI (A.10) As indicated in the expression, the strain va ries linearly along the length of the beam. In addition, the bending strain is lin early increasing from zero at the neutral axis to the top and bottom surface where it reaches the maximum (z is the vertical co-ordinate measured from the neutral axis). Hence, the maximum stress and st rain occurs at the clamp on the top or bottom surface, when the bending moment is the highest and its value is max2 PLt EI (A.11) Clamped-Clamped Beam (Fixed-Fixed Condition) Let us now analyze a beam with a differen t set of boundary conditions, clamped at both ends as shown in Figure A-2.

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240 Figure A-2: A schematic of clamped-clamped beam. Drawing the free body diagram for the clamped-clamped beam and performing a force balance results in (see Figure A-3) Figure A-3: Free body iagram of a clamped-clamped beam. In this case, let us assume the following boundary conditions that govern the beam for 0. 2 L x

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241 0 2 3 3 200 0 0 2x L x L xw wx x wx x wx P EI x (A.12) The above conditions mean that the beam is clam ped at one end which enforces zero deflection and slope. At 2 L x the slope is zero and the shear force is 2 P Imposing the first 2 of the boundary conditions yields 010 aa The next conditions when applied result in the coefficients as 23; 1612 PLP aa E IEI (A.13) Substituting back for these coefficients results in the deflection modeshape as 23234 161248 PLPP wxxxxLx EIEIEI (A.14) The above deflection is valid for 0 2 L x and will be symmetrical about the center, 2 L x From the above expression, the maximum defl ection occurs at the center is obtained by substituting 2 L x as 3. 192cenPL w EI (A.15) As described earlier, the eff ective mass and compliance using the deflection are obtained by calculating the stored potential and kinetic energies in the beam and lumping them to the tip. It should be noted here that the de flection shape represents only the half beam and therefore, the

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242 energies are first obtained by inte grating the deflection shape till 2 L x doubled to obtain the total potential and kinetic energy in the beam. The modified equations for the energy are shown in Eq. 2 2 2 2 0 2 2 02 2 2. 2L L Ldwx EI PEdx dx KEwxdx (A.16) Lumping the energies at the tip results in the compliance and mass as 3192msL C EI (A.17) and 13 35mL M L (A.18) From the mass and compliance the natu ral frequency is represented as 211105 8 213n LEI f L (A.19) The bending strain can be calculated from the deflection modeshape and is given as 2 2 0,2 04 8 x L xwx P zzxL xEI (A.20) As indicated in the expression, the strain varies linearly along the length of the beam with its maximum values occurring at the clamp and th e center. In addition, the bending strain approaches zero at 4 L x before it changes from compressi on to tension and vice-versa. The maximum bending strain at the clamp or the center is

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243 max82 PLt EI (A.21) Pin-Pin Beam (Simply Supported) Finally, let us analyze another set of boundary c onditions for a beam, namely, pin-pin as shown in Figure A-4. Figure A-4:: Schematic of a pin-pin beam. Drawing the free body diagram for the pin-pin beam and performing a equilibrium match results in Figure A-5. Figure A-5: Free body diagram for a simply supported beam. In this case, lets assume the following boundary conditions that govern the beam for 0 2 L x as

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244 2 2 0 2 3 3 200 0 0 2x L x L xw wx EI x wx x wx P EI x (A.22) The above conditions mean that th e deflection and moment for the beam at one end is zero. At 2 L x the slope is zero and the shear force is 2 P The main difference between a pin-pin and a clamped-clamped condition is that one imposes a zero moment and the other imposes a zero slope at the end. Imposing the first 2 of the boundary conditions yields 020 aa The next conditions when applied result in the coefficients as 2 13; 1612 PLP aa EIEI (A.23) Substituting back for these coefficients results in the deflection modeshape as 2 32234 161248 PLPP wxxxxLx EIEIEI (A.24) The above deflection is valid for 0 2 L x and will be symmetrical about the center, 2 L x From the above expression, the maximum defl ection occurs at the center is obtained by substituting 2 L x as 3. 48cenPL w EI (A.25)

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245 As described earlier, the eff ective mass and compliance using the deflection are obtained by calculating the stored potential and kinetic energi es in the beam and lumping them to the tip, which results in msC and m M as 348msL C EI (A.26) and 17 35mL M L (A.27) From the mass and compliance the natu ral frequency is represented as 211105 4 217n LEI f L (A.28) The bending strain can be calculated from the deflection modeshape and is given as 2 2 2 22L x L xwx P zzx xEI (A.29) As indicated in the expression, the strain vari es linearly along the length of the beam starting from zero at the clamp which was enforced us ing the boundary conditions. Hence, the maximum stress and strain occurs at the center on the to p or bottom surface, when the bending moment is the highest and its value is max42 PLt E I (A.30) The important parameters for all the various beams for an applied load, P are listed in Table A-1. Similar analysis was carried out for a uniform loading condition on these beams. However, only the results are summarized in Table A-2.

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246 Table A-1: LEM parameters and bending strain for various beam s subjected to a point load. Clamped-Free Pin-Pin Clamped-Clamped wx 23 6 P x Lx EI 2234 48 P x Lx EI 234 48 P x Lx EI maxw 30.333 PL EI 30.021 PL EI 30.005 PL EI msC 30.333 L EI 30.021 L EI 30.005 L EI m M 0.236LL 0.486LL 0.371LL n f 21 0.568 L EI L 21 1.582 L EI L 21 3.619 L EI L x P zxL EI 2 P zx E I 4 8 P zxL EI max 2 PLt EI 0.25 2 PLt EI 0.125 2 PLt EI As evident in the tables, a can tilever beam exhibits much hi gher compliance than beams bounded on both ends. In addition, they possess a signifi cantly lower natural frequency compared to the other beams. Furthermore, a cantilever produ ces a higher bending strain for similar loading condition making it an excellent choice for piezoe lectric energy harvesting applications that require higher strain to convert to voltage and lower frequencie s to match ambient vibrations.

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247 Table A-2: LEM parameters and bending strain for various beam s subjected to uniform load. Clamped-Free Pin-Pin Clamped-Clamped wx 2 2246 24 qx LxxL EI 3232 24 qx LLxx EI 2 224 qx Lx EI maxw 40.125 L q EI 40.013 L q EI 40.003 L q EI msC 30.313 L EI 30.020 L EI 30.005 L EI m M 0.257LL 0.504LL 0.406LL n f 21 0.562 L EI L 21 1.572 L EI L 21 3.573 L EI L x 22 q zxL EI 2 q zxLx EI 2266 12 q zLLxx EI max 20.5 2 Lt q EI 20.125 2 Lt q EI 20.083 2 Lt q EI

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248 APPENDIX B DISSIPATION MECHANISMS FOR A VI BRATING CANTILEVER BEAM Materials can be broadly be cl assified into three categorie s based on how they respond to external load. The first kind, elastic materials will completely recover the stored energy after the load is removed and follow the Hookes law th at relates stress and st rain linearly with its elastic modulus. The second kind called, viscous materials will lose the stored energy fully once the load is removed. Here, the stress is relate d to the strain rate in terms of viscosity. Finally, we have viscoelastic materials that lead to partial recovery of the stored energy and the remaining is lost in the form of heat. To represent this behavior, the elastic modulus of the material is expressed as a comp lex quantity, where the imaginary part corresponds to the lost energy. This loss of energy is generally expressed as mechanical damping. Introduction Mechanical damping relates to the convers ion of mechanical energy into heat. An oscillating structure contains a co mbination of potential and kineti c energy that dissipates a part of it during each cycle of motion. In this pr oject, arrays of micr omachined piezoelectric cantilevers are being fabricated to harness el ectrical energy from source vibrations using a mechano-electrical transduction mechanism by vi rtue of a piezoelectric layer. Since these devices are resonant structures, extracting maximum energy at its resonance is critical for its performance. However, at resonance, the perfor mance of the device is gr eatly influenced by its inherent mechanical damping related to the quality factor, Q The output voltage and consequently power in the device is directly related to the quality factor. Therefore, in the design of MEMS devices, dissipation mechanisms may ha ve detrimental effects on the quality factor.

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249 Some of the main dissipation mechanisms that describe this loss of energy are surface and volume losses, support losses, loss due to air flow (fluid damping), thermoelastic dissipation etc. One of the major dissipation phenomena to consider in such micro-systems is the thermoelastic damping which greatly influences the behavior of the vibrating mi cro-structures. So, a detailed thermoelastic dissipation mechanism in cantilever beams has been studied. In addition, the effects of supports and air dampi ng have been analyzed and empiri cal relations for the individual quality factors are presented. As part of futu re work, the model will enable a more efficient design of resonant micro-structures The overall mechanical damping is generally represented using any of the following parameters, namely, effective quality factor Q, damping ratio loss factor phase angle between cyclic strain and stress tan etc. All these parameters are inter-related using the expression 31112 2tann dBf W Q f W (B.1) Here, n f and 3 dB f are the natural frequency and band-wi dth corresponding to half power points in the frequency spectrum. Overall Mechanical Quality Factor For a vibrating cantilever beam, the loss mechan isms can be broadly classified into two types, namely, external and internal losses. The external losses include loss due to fluid damping (air), radiation of the bending wave at the support (clamp). Losses such as thermoelastic loss and surface loss due to material defect s comprise the internal losses (Yang et al., 2002). The overall quality factor is defined as

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250 111n i iQQ (B.2) where Q is the total quality factor and iQ correspond to the quality factor of each loss mechanism. For the analysis here, le t us define the quality factor as 2 W Q W (B.3) where, W is the stored energy in the beam and W is the dissipated energy. If we include all the dissipation mechanisms listed earlier, Eq. (B.3) can be written as /2TEDsupportairfluidsurfaceotherW Q WWWWW (B.4) where, each W represents an individual damping effect Separating the above expression into individual effects yields / /1 22222 111111supportairfluidsurface other TED TEDsupportairfluidsurfaceotherWWW W W QWWWWW QQQQQQ (B.5) Here, the overall quality factor has the effect of all the individual quality factors connected in a parallel impedance form. Therefore, the smallest quality factor of a ll dictates the overall mechanical damping in the device. Dissipation Mechanisms Although there are various mechanisms by whic h energy is dissipated in a vibrating structure (beam), some of the major phenomena are discussed here. In addition, a detailed derivation is obtained for thermoel astic dissipation in a cantilev er as it contributes a major portion to the overall mechanical damping.

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251 Airflow Damping A simple schematic of the cantilever beam that is analyzed here is shown in Figure B-1. In the figure, the dimensi ons are indicated and ,, EI correspond to the elastic modulus, density and moment of inertia of the homogenous beam. Figure B-1: A simple schematic of the cantilever beam. The drag force on a vibrating cantilever beam su ch as the one shown above due to fluid loading is given as (Blom et. al, 1992) 121212 where are real constants u pjuu (B.6) Consequently, the governing Euler-Bernoulli beam bending equation is modified into 42 1 2 420 wxwxwx EIA L xLtt (B.7) Here, is the angular frequency of vibration, A is the area of cross-section and wx is the resulting transverse deflection at x The stored and dissipated energy in terms of the deflection shape are given as

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252 2 2 01 2LWAwxdx (B.8) and 2 1 00.TLWpudtwxdx L (B.9) respectively. The quality fact or from the ratio of the energies as represented in Eq. (B.3) is given as 1AL Q (B.10) In order to obtain 1 three major regions are analyzed to cover the fluid pressure from vacuum to atmospheric pressure and beyond. Intrinsic region : Very low pressure exists in this region which implies negligible damping due to air flow. Molecular region : Here, damping is caused by collision of air molecules and 1 is expressed as 132 where 9mam M LkbPk R T (B.11) where M is the molecular mass of the gas, T is the temperature and R is the universal gas constant. Now, Q can be obtained as 2 212n mak hE Q kPL (B.12) In Eq.(B.12), aP is the pressure and nk is the constant for thn resonant mode.

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253 Viscous region This region includes pressure upto the atmo spheric pressure and further thereof. The damping here is solved using fluid mechanics pr inciples for an incompressible viscous medium ( constant). An approximate model for this is imp lemented in Blom et al (1992) that consists of a string of spheres vibrating independently with in finite separation. Th is region is fairly complex and can be further divided into su b-regions depending on operation pressure to accurately model the damping. However, for the pur pose of this investigation a more general and simplified empirical re lation is used (Blom et al. 1992) given by 1 2 2212 61n eq eqE kbh Q R RL (B.13) Here, is the dynamic viscosity of the medium and is the width of the boundary layer expressed as 1 22 ,a (B.14) where aaM P R T is the density of the medium. In general, eq R is defined as the equivalent radius of the sphere that is obtained from em pirical measurements. However, in the simplest, case it is assumed as the half width of a beam. Therefore, in the case of a cantilever beam, 2eqb R From Eq. (B.13), it can be inferred that for 1eqR Q is independent of the pressure, while for 1eqR it is proportional to 1aP More information on airflow damping and related work is su mmarized in Lin and Wang (2005).

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254 Support Losses In this mechanism, the vibration energy of a resonator is dissi pated by transmission through its support. It therefore exerts both shear and moment on the clamp during flexural vibration. It is assumed that the elastic wave entering the support is propagated away and no energy is returned, which ensures that the reso nant modes are not aff ected. This assumption holds true if the thickness of the beam is much smaller than the wavelength of the elastic wave, which is already enforced in the model that us ed the lumped element approach. The support loss also includes friction between conne cted surfaces such as bolted jo ints, clamps etc. The quality factor due to support loss is give n simply as (Hosaka et al. 1995) 30.23. L Q h (B.15) This loss becomes insignificant for 100. L h Surface Dissipation The dissipation due to surface effects become s significant especially when the cantilever thickness scales down leading to a much highe r surface-to-volume ratio. This dissipation is mostly caused due to induced surface stresses on the cantilever. The loss is enhanced when the device has a thin surface layer (of thickness, s ) with high dissipative properties. The quality factor for this mechanism is defined as 23 s dsbhE Q bhE (B.16) where b is the width and dsE is the dissipation Youngs m odulus of the surface layer. Volume Loss The empirical relation for volum e loss is given in Yasamura et al. (2000) as

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255 ,dsE Q E (B.17) where E is the elastic modulus and dsE is the dissipative modulus of the material. Squeeze Damping Loss The squeeze loss that usually occurs when de vice operates in a thin layer of fluid is modeled using Reynolds equation in Hosaka et al ., (1995). It employs a narrow bearing approximation with no pressure grad ient in the longitudinal directi on. In addition, if we assume that the vibration is much smaller than the gap, the squeeze loss is given as 3 2aongh Q b (B.18) where 0g is size of the gap and is the coefficient of viscosity. Thermoelastic Dissipation During flexural vibration of a structure, sp ecifically a beam, tension and compression on the top and bottom of the neutral axis results in a temperature gradient le ading to thermoelastic dissipation. Zener (1937) approximated this loss with a single relaxation peak and a characteristic time proportional to the thermal diffusion time as shown below 2 21 where and 1E pbk QC (B.19) Here, 2adad E p adEEEE ET EC EE where adE is the unrelaxed adia batic elastic modulus, E is the isothermal elastic modulus and pC is the specific heat at constant pressure. For small vibrations, assuming that adEE and pvCC results in the thermoelastic quality factor as

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256 2 21 1pET QC (B.20) Eq. (B.20) is rewritten as 2 2211 1pET QC to observe two limiting conditions, namely, 1. If 1 the structure remains in equilibrium resulting in no energy dissipation. 2. If 1 the relaxation time is negligible which leads to minimal dissipation. Appreciable dissipation occurs only when and 1 are of the same order in magnitude. For example, a thin beam under flexure (w ith a rectangular cro ss-section) undergoes 98% of thermal relaxation through the first resonant mode whose relaxation time is given as 2 2b (B.21) The quality factor due to thermoelastic dissipation can be represented in the form of an empirical relation as (Lifshitz and Roukes 2000) 2 231 66sinhsin coshcoso pQ ET C (B.22) where n f is the resonant frequency, E is the isothermal Youngs modulus, E is the difference between the adiabatic and the isothermal youngs modulii and is the thermal diffusivity defined as pc Here, is the thermal conductivity and pC is the specific heat at

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257 constant pressure. is defined as 2ob where o is the isothermal value of eigenfrequency. As evident from the above empirical relations, the loss due to the ai rflow can be minimized by using the device in vacuum. In addition, th e support losses can also be reduced by using beams with high length to thickness ratios. Th e internal losses such as the thermoelastic dissipation can be difficult to avoid for a specific beam and therefore need to be studied in detail to model analytically. So, a detailed 2-D coupled beam bending and heat conduction solution is provided for a homogenous beam to accurate ly estimate the thermoelastic damping. Analytical model For the purpose of this exercise, a thermoel astic model is obtained for a homogeneous cantilever beam similar to the one described in Guo and Rogerson (2003) The model is derived based on the following assumptions, 1. Linear Euler-Bernoulli beam theory th at uses a small deflection model. 2. Pure bending with no rotary in ertia and shear deformation. 3. The top and bottom surfaces are thermally insulated, 20.zhdT dz The cantilever beam that is analyzed here is shown in figure 1 with the dimensions indicated. ,, w uzwwxt x (B.23) correspond to the displacement in axial (x) direction and transverse (z) direction. 2 2,,xw EzTxzt x (B.24)

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258 where E is the elastic modulus. is the thermal modulus defined as C where C is the compliance matrix and is the coefficient of linear thermal expansion. In addition, 2 2 h x h M xbzdz (B.25) is defined as the bending moment in the beam. Now substituting the expression for the axial stress in Eq. (B.25) yields the overall moment in the beam as 2 2 2 2,,h hw M xbEzTxztzdz x (B.26) We know that the moment of inertia is given as 2 3 2 212h hbh Ibzdz (B.27) This simplifies Eq. (B.26) to 2 2 2 2,,h hw M xEIbTxztzdz x (B.28) We denote 2 2,,h hKTxztzdz. Hereafter ,, Txzt will be denoted as T for simplicity in the derivation. The equation of motion for a cantilever beam modeled using Euler-Bernoulli beam theory is 242 242 LwwK EIb txx (B.29) where L is the linear mass density. The heat con duction equation includi ng the thermoelastic effect is denoted as (Fung, 1965)

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259 222 2220voTTTw kkCTz xzttx (B.30) where kWmK is the ther mal conductivity, is the material density and vCJkgK is the specific heat at constant volume. Multiplying Eq. (B.30) by and integrating along the thickness of the beam will result in the following conduction equation 2222 222 2 222 22220hhhh vo hhhhw kTzdzkTzdzCTzdzTzdz xzttx (B.31) substituting for K in the above equation simplifies it to 2 2 2 2 2 2 2 20h h o v h hTI KTTKw kkzdzC xzztbtx (B.32) The second term in the expression is expanded using the product rule. Since the top and bottom surface of the beam is assumed to be thermally insulated, it follows that 20zhT z which further simplifies Eq. (B.32) to 22 2 22 20h o v hTI KKw kkTC xtbtx (B.33) Assuming a cubic polynomial dependence of temp erature along the thickness of the beam, it can be represented as 32 1234Tczczczc (B.34) From Eq. (B.34), 1c can be expressed as 22 432 1234 22,hh hhKTzdzczczczczdz (B.35)

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260 which will simplify to 53 13. 8012 hh Kcc (B.36) If we assume that the thickness of the beam is very small, we can rewrite Eq. (B.36) as 533 2 1313 280124h hhhh KccTcch (B.37) This assumption leads to 2 2 212h hh KT for small thickness, when h is very small compared to the length of the beam. Now, the two governing equations can be rewritten by substituting for I as 2342 2420 12 wbhwK bhEb txx (B.38) and 3 22 22212 0 12o vTh KKKw kkC xhttx (B.39) Let us now introduce the following non-dimensi onal parameters to further transform Eqs. (B.38) and (B.39). 2,,,,, TK xwtv E xwvTK LhhE L E (B.40) Here, is the non-dimensional time and v is the non-dimensional velocity. All the parameters represented as ,, and x wTK are the corresponding non-dimensional terms. Equation (B.38) will now be represented in their non-dimensional form as 2342 2 2342 20 12 bhLwbhwK EbE h Lxx v (B.41)

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261 which can be rewritten as 242 1 2420 wwK A xx (B.42) and 22 234 220 KKw AKAA x (B.43) where the coefficients are defined by 2 1 212 h A L 2 2 210 L A h 2 3vL Acv hk and 2 3 4 212oAThv ELk The boundary conditions for the can tilever can be represented as 0,0,0 1,1,0 0,1,0 ww ww KK (B.44) Let us assume a harmonic solution fo r the deflection and temperature as j jwwe KKe (B.45) where is the non-dimensional frequency. Substituting the above harmonic solution into the two governing equations, we obtain 2 10 wAwK (B.46) and 2340 KAKjAKjAw (B.47) Eliminating K from equations (B.46) and (B.47), we obtain 2 1423wAwjAwAjAK (B.48)

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262 Differentiating K twice with respect to and substituting back in Eq. (B.46) wll yield the overall coupled differential equation as 2 14 2 1 230 wAwjAw wAw AjA (B.49) The above equation can be simp ly expressed in the form 12340 awawawaw (B.50) where 1 1 23 4 21 23 2 3 23 2 4A a AjA jA aA AjA a AjA a (B.51) If we assume the solution to Eq. (B.50) is of the form sinhcosh wBC (B.52) the characteristic equation for Eq. (B.50) reduces to 642 12340 aaaa (B.53) Since 1234,, and aaaa are complex constants, the roots of the above equation can be assumed to be 1,2,3 and represented as 3 1sinhcoshiiii iwBC (B.54)

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263 The above solution applies only the distinct roots and when multiple roots occur, the solution is of the form 3 1sinhcoshiiiiii iKBdCd (B.55) where 24 41 23 1,2,3ii iiAA di AiA (B.56) Substituting Eqs. (B.54) and (B.55) 3 1 3 1 3 22 1 3 33 1 3 1 3 10 0 sinhcosh0 coshsinh0 0 sinhcosh0i i ii i iiiiii i iiiiii i ii i iiiiii iC B BC BC Cd BdCd (B.57) Rewriting the above equations in th e form of a matrix will yield 123 112233 222222 111222333111222333 333333 111222333111222333 112233 1110 0 sinhsinhsinhcoshcoshcosh0 coshcoshcoshsinhsinhsinh0 0 sinh CCC BBB BBBCCC BBBCCC CdCdCd BdB 222333111222333sinhsinhcoshcoshcosh0 dBdCdCdCd (B.58) which simplifies to

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264 123 222222 112233112233 123 333333 112233112233 112233112233000111 000 sinhsinhsinhcoshcoshcosh 000 coshcoshcoshsinhsinhsinh sinhsinhsinhcoshcoshcosh ddd dddddd 1 2 3 1 2 30 B B B C C C (B.59) For a non-trivial solution, the de terminant of the 6X6 matrix should be zero. The above set of equations can be solved num erically by assuming a guess for and substituting in the determinant to obtain its value. An iteration pr ocedure will be carried out to minimize the error between the assumed value and calculated value. From the calculated value of which will be a complex number, the real part corresponds to the natural frequency a nd the imaginary part corresponds to the thermo elastic dissipation. A detailed thermoelastic model of a hom ogeneous cantilever beam in bending was obtained analytically using coupl ed set of equations. The solution was non-dimensionalized and it was observed that the non-dimensional freque ncy is scale dependent when thermoelastic effects are considered. The solution was completely formulated from first principles. However, the numerical solution technique was not implemen ted to estimate the thermoelastic dissipation.

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265 APPENDIX C TRANSFORMATION OF COORDINA TES FOR RELATIVE MOTION Let us consider a vibrating cantilever beam th at is excited at its clamp. The resulting configuration lies in an acceleratin g frame of reference with respect to the ground. A simple schematic of a cantilever beam descri bed above is shown in the following Figure C-1. Cantilever Beam Proof Mass x y z X Y Z Global Co-ordinatesLocal Co-ordinates 0sinclampdt Figure C-1: Vibrating cantil ever beam in an accelerating frame of reference. The governing dynamic equation describing th e cantilever beam is expressed as 42 42,, ,, WXtWXt EIApXYZ Xt (C.1) Here, E is the elastic modulus of the material, I is the moment of inertia, is the material density and A is the cross-sectional area. WXt is the transverse dynamic deflection with respect to the global coordinates. ,, p XYZ is the static load on the beam due to its self-weight, where and X YZ are the global coordinates. The cantil ever beam is subjected to an input sinusoidal displacement at the clam p that causes it to vibrate. Let us assign lo cal coordinates and x yz that describe the motion of the beam relative to its clamp position as shown in the figure. Therefore, the relative vertical deflecti on with respect to the clamp can be expressed as

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266 ,,clampWXtwxtdt (C.2) where wxt is the resulting deflection in the beam and clampdt is the applied dynamic displacement to the clamp. Substituting Eq. (C.2) in Eq. (C.1) results in the following expression shown below. 42 42,, ,,clampclampwxtdtwxtdt EIApXYZ Xt (C.3) Eq. (C.3) can be expanded as 2 42 422,, ,,clampdt wxtwxt EIAApXYZ xtt (C.4) For a harmonic excitation at the clamp, 0sinclampclampdtdt where 0clampd is the amplitude and is the angular frequency, Eq. (C.4) can be rewritten as 42 2 0 42,, ,,clampwxtwxt EIAAdpXYZ xt (C.5) 42 2 0 42,, ,,clampwxtwxt EIApXYZAd xt (C.6) As evident in the above equati on, the governing equations rema in the same with the input harmonic displacement acting like an effec tive inertial force on the right side.

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267 APPENDIX D ELECTRICAL IMPEDANCE FOR A PIEZOELECTRIC MATERIAL Let us consider the following circuit representation shown in Figure D-1 that models the piezoelectric part of the device with an effective electrical impedance. CebV I Re Figure D-1: Blocked electri cal impedance in a parallel network representation. The blocked electrical impedance of a piezoelectric material is represented as a parallel network in the above representation. In the shown format, the diel ectric loss represented as e R can be empirically given using the relation 11 tane ebR C (D.1) where ebC is defined as the blocke d electrical capacitance and is the frequency of oscillation given as 2 f tan is defined as the loss tangent for the impedance. In the shown format, the overall impedance of the circuit is given as

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268 1e eb eb e ebR jC Z R j C (D.2) which can simplified as 1e eb ebeR Z jCR (D.3) 21 1eebe eb ebe R jCR Z CR (D.4) Substituting for e R the above equation simplifies to 21111 1 tantan 11 1 taneb ebeb eb eb ebjC CC Z C C (D.5) 21 tan 1taneb eb j C Z (D.6) For small values of loss tangent, 21tan1 which further simplifies the impedance to tantan1eb ebebebebj Z CCCjC (D.7) The above expression looks like a se ries combination of a real quant ity that can be represented as an equivalent resistance and a complex quantity resulting from the capacitance. Hence, the electrical impedance of a piezoel ectric material can be alternat ively represented as a series combination as shown below in Figure D-2.

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269 CebV I Re Figure D-2: Blocked electr ical impedance in a seri es network representation Where, the overall impedance of the circuit is given as '1e ebR j C (D.8) where 'tane ebR C Even though the series representa tion is mathematically valid, it may not accurately represent the physical behavior of a piezoelect ric material for the following reason. When a DC voltage is applied to piezoelectric material, charge accumulate s as the material is completely blocked for any physical motion. Therefore, this charge dissipates due to the dielectric loss as a function of time. Using a series representation breaks the circuit for DC as the impedance due to the capacitor approaches infinity. However, the para llel network facilitates the physical phenomenon of dissipation across the resistor, e R even though the capacitor branch becomes an open circuit in Figure D-1.

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270 APPENDIX E CONJUGATE IMPEDANCE MATCH FO R MAXIMUM POWER TRANSFER Let us consider the following circuit repr esentation shown in electrical impedance thevenin equivalent form (Figure E-1). V I ZL Zth Figure E-1: Thevenin equivalent representation connected to a external complex impedance. The above plot shows a simple circuit represen tation of the electromec hanical lumped element model circuit simplified to its thevenin equivalent circuit. The thevenin voltage, thV and the thevenin impedance, th Z are shown in the figure. L Z represents an arbitrary load at the output across which, the output voltage is measured to recl aim power. The goal of this analysis is to get an expression for L Z to maximize the power output. First, let us assume that the thevenin impedance, th Z is a complex value represented as riththth Z ZjZ (E.1) Similarly, the thevenin voltage can be represented as rithththVVjV (E.2)

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271 If we assume the output load to be complex as well given by L Z RjX (E.3) the voltage across the load is given as ri rithth th LL thLththVjV V VZRjX ZZZjZRjX (E.4) which simplifies to riri rithththth L ththVRVXjVXVR V ZRjZX (E.5) 2 2riri ri rithththth Lthth ththVRVXjVXVR VZRjZX ZRZX (E.6) Collecting the real and imaginary parts results in 2 2rirriirirrii rithththththththththththth L ththVRVXZRVXVRZXjVXVRZRVRVXZX V ZRZX (E.7) The current across the load is given as riri ri rithththth th L thLthth ththVjVVjV V I ZZZjZRjX Z RjZX (E.8) which simplifies to 2 2rriiirri rithththththththth L ththVZRVZXjVZRVZX I ZRZX (E.9) The power measured across the load is the pro duct between the voltage and complex conjugate of the current given as LLLPVI (E.10)

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272 where 2 2rriiirri rithththththththth L ththVZRVZXjVZRVZX I ZRZX (E.11) The real power is obtained by taking the real pa rt of the complex product between the current and voltage. Further simplification results in the following expression for the load power 22 2 2Reri r rithth LL ththVVR PP Z RZX (E.12) For maximum power transfer, 0rLP X and 0rLP R 22 2 22ri r i rithth Lth ththVVR PZX X ZRZX (E.13) Equating the above result to zer o yields the optimal value of X as ith X Z (E.14) 22 3ri rr rthth Lth thVV PZR R ZR (E.15) which results in the value of R as rth R Z (E.16) Therefore, the optimal external load for maximum power transfer is riLthth Z ZjZ (E.17)

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273 However, if we assume that the extern al load is purely resistive (shown in Figure E-2) which is the nature of some st andard circuits such as pulse width moderator circuit (PWM) as discussed in Taylor 2004, then the real power assumes the expression 22 2 2Reri r rithth LL ththVVR PP ZRZ (E.18) Vth Ith RL Zth Figure E-2: Thevenin equivalent repres entation connected to a resistive load. For maximum power transfer, 0rLP R 222 22 2 2ri rri rithth Lthth thth Z ZR PVV R ZRZ (E.19) Equating the above result to zer o yields the optimal value of R as 22rithth R ZZ (E.20) Therefore, the optimal external load for maximum power transfer is 22riLthth Z RZZ (E.21)

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274 APPENDIX F UNDERSTANDING THE PHYSICS OF THE DEVICE The dynamic behavior of the energy harveste r configuration comprised of a composite cantilever beam with a proof mass at one and the clamp subjected a fixed acceleration is discussed here. A schematic of the device is shown in Figure F-1. Cantilever Beam Proof Mass ao PZT Base fn Figure F-1: Schematic of the composite beam energy harvester. In the actual configuration shown above, the i nput acceleration is app lied to the base that translates into a relative motion of the proof ma ss with respect to the base. Consequently, this device becomes a two degree of freedom system and can be represented as shown in Figure F-2. MpCms y x Rm Mb F Figure F-2: Free body representati on of the device as a two degree of freedom system. Here, b M is the mass of the base, and p M is the proof mass. Th e two governing equations describing the system are shown in Eqs. (F.1) and (F.2) 1bm ms M yFRyxyx C (F.1) and 1pm ms M xRyxyx C (F.2)

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275 The device can then be represented using an equiva lent circuit notation (similar to Figure 2-2) as shown in Figure F-3. In the fi gure, the input force F applied to the ba se produces a base velocity, Y, and a velocity of the proof mass, X as shown in the circui t. Bending in the beam occurs due to relative motion between the base and proof mass whose veloci ty is expressed using Z YX As evident from the circuit behavior, when the base mass is large, there is no base velocity, 0Y Similarly, if the impedance of the proof mass is large it does not move, 0X Furthermore, at low frequencies approach ing dc, the device e xhibits rigid body motion as there is no relative motion between the proof mass and base, 0Z F MpCms Rm Mb X Z Y Figure F-3: Electromechanical circuit re presentation of the energy harvester. Replacing zyx the equations of motio n representing the two de gree of freedom (Eqs. (F.1) and (F.2)) become 1bm ms M yFRzz C (F.3) and 1 .pm ms M xRzz C (F.4)

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276 Here, Eq. (F.3) describes the motion of the base and Eq. (F.4) models the proof mass. Substituting for x with y z Eq. (F.4) simplifies to 1 .pm ms M yzRzz C (F.5) Eq. (F.5) can be represented using a single degr ee of freedom system with relative motion between the tip and base, z as the variable. The new governing equation becomes 1 .ppm ms M yMzRzz C (F.6) Here, inpFMy is the effective input inertial force on the composite beam and the proof mass, as shown in Figure 2-2. In practice, the beam also has an effective mass. For the static analytical approach employed in this thesis (S ection 2.1), this mass is lumped with the proof mass (i.e., they are added). Both the two and single degree of freedom re presentations model the behavior of the energy harvesting device. The only limitation with the simpler single degree of freedom system employed in the thesis is that it does not explicitly include the effects of th e base. In essence, it is assumed that the device (including the base mass) does not load the vi brating device to which it is attached. In this sense, it is like an accelerometer.

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277 APPENDIX G FABRICATION LAYOUTS Figure G-1: Top Electrode Et ch Mask Lift-off mask fo r patterning top electrodes

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278 Figure G-2: PZT Etch Mask Wet etching of PZT to reveal the bottom electrode pads

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279 Figure G-3: Ion Milling Mask Ion milling to pattern the PZT, top and bottom electrode features.

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280 Figure G-4: Beam Etch Mask Front side DRIE to pattern the beams.

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281 Figure G-5: Proof Mass Mask Backside DRIE mask for patterning the proof mass.

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290 Stark, I., and Stordeur, M., N ew micro thermoelectric devices based on Bismuth Telluride-type thin solid films, Proc. 18th Intl.Conf. on Th ermoelectrics (IEEE, Piscataway, NJ, 1999), pp. 465-472, 1999. Starner, T., Human-Power ed Wearable Computing. IBM Systems Journal v 35 n 3-4 1996. p 618-629. Starner, T., and Paradiso, J.A., Energy scav enging for mobile and wireless electronics, Pervasive Computing, 2005, 18-27 Takahashi, S., Yamamoto, M., and Sasaki, Y., N onlinear piezoelectric res ponse in ferroelectric ceramics driven at resonant mode, Proceedings of the IEEE, 1998, 381-384 Taylor, G. W., Burns, J. R., Kammann, S. M ., Powers, W. B., and Welsh,T. R., The Energy Harvesting Eel: A Small Subsurface Ocean/River Power Generator, IEEE Journal of Oceanic Engineering v 26 n 4 October 2001 p 539-547. Taylor, R, Liu, F., Horowitz, S., Khai Ngo, Nishida, T., Cattafesta, L., and Sheplak, M., "Technology Development for Electromechanical Acoustic Liners," paper a04-093, Active 04, Williamsburg, VA, September 2004. Terry, S., A miniature silicon accelerometer with built-in damping, Proceedings of the IEEE, 1988, 114-116. Thomson, W. T., Theory of Vibration with Applications, Prentice Hall, 1993, 4th Edition. Tilmans, H. A. C., Equivalent circuit repres entation of electromechanical transducers: I. Lumped-parameter systems, Journal of Microm echanics and Microengineering, v 6, 1996, 157176. Tilmans, H. A. C., Equivalent circuit repres entation of electromechanical transducers: II. Distributed-parameter systems, Journal of Mi cromechanics and Microengineering, v 7, 1997, 285-309. Uchino, K., Nomura, S., Cross, L. E., Jang, S., J., and Newnham, R. E., Electrostrictive effect in lead magnesium niobate single cr ystals, Journal of Applied P hysics, v 51, February 1980, 11421145 Umeda, M., Nakamura, K., and Ueha, S., Analysi s of the transformation of mechanical impact energy to electric energy using piezoelectric vibr ator, Japanese Journal of Applied Physics, v 35, n 5B, May 1996, 3267-3273. Umeda, M., Nakamura, K., and Ueha, S., Energy storage characteristics of a piezo-generator using impact induced vibration, Japanese Jour nal of Applied Physics, v 36, n 5B, May 1997, 3146-3151.

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291 Vizvary, Z., and Kovacs, A., Design optimiza tion for cantilever and bridge type acceleration microsensors using the finite element me thod, European Conference on Computational Mechanics, June 26-29, Cracow, Poland, 2001. Wang, Z-J., Maeda, R., and Kikuchi, K., Prepa ration and characterizati on of sol-gel derived PZT thin films for micro actuators, Sympos ium on Design, Test a nd Microfabrication of MEMS and MOEMS, Proceedings of SPIE, v 3680, n 11, 1999, 948-955. White, N.M., Glynne-Jones, P., and Beeby, S.P ., A novel thick-film piezoelectric microgenerator, Smart Materials a nd Structures, v 10, 2001, 850-852. Williams, C. B., and Yates, R. B., Analysis of a micro-electric generator for microsystems, Sensors and Actuators A, v 52, 1996, 8-11. Xu, S., Ngo, K. D. T., Nishida, T., Chung, GB., Sharma, A., Converter and controller for micro-power energy harvesting, Applied Powe r Electronics Conference and Exposition, Proceedings of the IEEE, 2005, 226-230. Yang J., Ono, T., and Esashi, M., Energy dissipation in submicr on thick single crystal Silicon cantilevers., Journal of micromechanical system s, Vol 11, No 6, December 2002, pp 775-783 Yasumura, K.Y., Stowe, T.D., Chow, E.M., Pfafma n, T., Kenny, T.W., Stipe, B.C., and Rugar, D. Quality factors in micron and submicron thick cantilevers., Journal of Micromechanical systems, v 9, n 1, March 2000, 117-125 Yazdi, N., Ayazi, F., and Najafi, K., Micromachin ed inertial sensors,Proceedings of the IEEE, v 86, n 8, August 1998, 1640-1659 Zener, C., Internal friction in solids: II. General theory of thermoelastic internal friction, Physical Review, v 53, 1937, 90-99. Zurn, S., Hsieh, M., Smith, G., and Markus, D., F abrication and structural characterization of a resonant frequency PZT microcantilever, Smar t Materials and Structures, v 10, 2001, 252.

PAGE 292

292 BIOGRAPHICAL SKETCH Anurag Kasyap V.S. was born on September 17th 1978, in Chittoor, Andhra Pradesh, India. He graduated from Space Central School, Srih arikota, India, in 1995. He obtained his undergraduate degree in B.Tech (naval architecture) from I ndian Institute of Technology, Chennai, Tamil Nadu, India. He entered the Un iversity of Florida in the Fall of 1999 with a graduate research assistantship and received hi s MS in Aerospace Engineering in 2002. He is currently completing his doctoral degree at the Univ ersity of Florida in the area of Piezoelectric based energy harvesting and microe lectromechanical systems (MEMS).


Permanent Link: http://ufdc.ufl.edu/UFE0015646/00001

Material Information

Title: Development of MEMS-Based Piezoelectric Cantilever Arrays for Vibrational Energy Harvesting
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015646:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015646/00001

Material Information

Title: Development of MEMS-Based Piezoelectric Cantilever Arrays for Vibrational Energy Harvesting
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015646:00001


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Copyright 2007

by

Anurag Kasyap V.S.





























To my father









ACKNOWLEDGMENTS

Financial support for the research project was provided by NASA.

First, I thank my advisor Dr. Louis N. Cattafesta for his guidance and support, which was

vital for completing my dissertation. I also thank my co-advisor Dr. Mark Sheplak for advising

and guiding me with various aspects of the project. I would also like to thank Dr. Toshi Nishida

for helping me understand the electrical engineering aspects of the project.

Drs. Khai Ngo and Bhavani Sankar deserve special thanks for finding time to help me out

with the project whenever I approached them. I thank all the members of the Interdisciplinary

Microsystems group, especially fellow students Steve Horowitz and Yawei Li for their help with

my research.

I also thank the University of Florida Department of Aerospace Engineering, Mechanics,

and Engineering Science for their financial support.

Finally, I want to thank my family and friends for their endless support, particularly my

parents whose affection and encouragement has been the driving force for my success as a

student and more importantly as a person.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IS T O F T A B L E S ................................................................................. 8

LIST OF FIGURES .................................. .. .... ..... ................. 10

A B S T R A C T ......... ....................... ............................................................ 16

1 INTRODUCTION ................................................. ....................... .... 18

Energy R eclam ation................ .. .... ........................ ..... .. ................... 18
Energy Resources and Harvesting Technologies .................................... ............... 19
Self-Pow ered Sensors .................. .............................. ....... .. ............ 21
Vibration to Electrical Energy Conversion........................................ ......................... 23
Transduction M mechanism s .................................................................... ............... 24
Electrodynam ic transduction......................................................... ............... 29
Electrostatic transduction ......... ................................ ................................... 31
P iezoelectric tran sdu action ............................................................. .....................33
Microelectromechanical Systems (MEMS)..................................................................... 40
Piezoelectric M EM S .................. ........................................ .............. 45
Objectives of Present W ork ........... ..... ............. ................... ....... ..... ............. 46
O organization of D issertation .......................................................................... ................... 46

2 PIEZOELECTRIC CANTILEVER BEAM MODELING AND VALIDATION ..................48

P iezoelectric C om posite B eam .................................................................... .....................49
Analytical Static Model ................ .............................. .............54
Static Electromechanical Load in the Composite Beam ...........................................55
Experimental Verification of the Lumped Element Model ............... ...............70

3 MEMS PIEZOELECTRIC GENERATOR DESIGN...........................................................84

P ow er T ran sfer A n aly sis............................................................................................. 84
N ondim en sional A naly sis............................................................................. .................... 87
S calling T h eory ...................... .. .. ......... .. .. .......... ....................................110
V alidation of Scaling Theory .............. .......................................................... 116
Extension to MEMS ..................................... ................. ............ 122
D design of Test Structures ............. ............................................................. 122
T e st d ev ic e s ................................................................................................12 7

4 DEVICE FABRICATION AND PACKAGING....................... ....... ................130

P ro c e ss F lo w ................................................................................................................... 1 3 0
P process T traveler ....................................................... 147









P ack ag in g ................... ...................1............................8
V acuum Package ..................................... .. .... ...... .. ............148
O p e n P a c k a g e ...................................................................................................... 1 5 0

5 E X PE R IM E N TA L SE TU P ........................................................................ ..................... 153

Ferroelectric Characterization Setup ............................................................................. 153
Piezoelectric C haracterization ................................................................. ............... 155
E electrical C characterization .................. .. ......... ...........................................................156
Blocked Electrical Capacitance, Cb and Dielectric Loss, R .....................................156
M mechanical C characterization ................................................................. .. ..................... 159
Electrom echanical Characterization ........................................................ .............. 163
Open Circuit V oltage Characterization ........................................ .......................... 164
V oltage and Pow er M easurem ents .............................................. ............................. 166

6 EXPERIMENTAL RESULTS AND DISCUSSION................................168

F erroelectric C haracterization ..................................................................... ...................168
Blocked Electrical Impedance Measurements .......................................... ...............179
Lum ped Elem ent Param eter Extraction.................................................................... ...... 185
M eth o d 1 .............................................................1 8 8
M e th o d 2 .................................................................................................................. 1 9 7
M e th o d 3 .................................................................................................................. 2 0 1
R results and D discussion ...............................................................206
P Z T -E H -0 9 ............. ..... ............ ................. ...................................................2 0 6
PZT-EH-07 ........................ .......................... 212
Summary and Discussion of Results .............. .... ......... ...................215

7 CONCLUSIONS AND FUTURE WORK.................................................. 224

C o n c lu sio n s ........................................................................................................................... 2 2 4
Future W ork ....................................230
Second Generation Design Procedure ......... ............................232
Electrom echanical Conversion M etrics............................................... 233

A EULER-BERNOULLI BEAM ANALYSIS: VARIOUS BOUNDARY CONDITIONS..237

E uler B ernoulli B eam .......................................................... ................. 237
Cantilever Beam (Clamped-Free Condition) ............................................................237
Clamped-Clamped Beam (Fixed-Fixed Condition) ...................................... 239
Pin-Pin B eam (Sim ply Supported) ....................................................... 243

B DISSIPATION MECHANISMS FOR A VIBRATING CANTILEVER BEAM ................248

Introduction .............. ............ .... .........248................
O overall M mechanical Q quality Factor .................................................................... 249
Dissipation Mechanisms ........................... ......... ........................... 250



6









Airflow D am ping ....................................................... .......... .. ............. 251
Intrinsic region : ......... ................................... .. 252
M olecular region : .............. ............................... .. 252
V iscous region .................................................... 253
S u p p o rt L o sse s ......................................................................................................... 2 5 4
Surface D issipation ............................................ 254
V o lu m e L o ss ............................................................................................................ 2 5 4
Squeeze D am ping L oss .............................................................255
T herm oelastic D issipation ............................................................................................255
A analytical m odel ................................................. .... ................. 257

C TRANSFORMATION OF COORDINATES FOR RELATIVE MOTION ........................265

D ELECTRICAL IMPEDANCE FOR A PIEZOELECTRIC MATERIAL ............................267

E CONJUGATE IMPEDANCE MATCH FOR MAXIMUM POWER TRANSFER .............270

F UNDESTANDING THE PHYSICS OF THE DEVICE .................................................274

G FA B R IC A TIO N LA Y O U T S ................................................................................ ........ 274

L IST O F R E F E R E N C E S ..................................................................................... ..................282

B IO G R A PH IC A L SK E T C H ............................................................................. ....................292









LIST OF TABLES


Table page

1-1 Conjugate power variables for different energy domains.........................................26

1-2 Vibration based energy harvesters characterterized for power............... ...................40

2-1 Material properties and dimensions for a homogenous aluminum beam ........................61

2-2 Material properties and dimensions for a piezoelectric composite aluminum beam.........65

2-3 Material properties and dimensions for a homogenous aluminum beam ........................70

2-4 Measured and calculated parameters for the homogenous beam...................................71

2-5 Measured and calculated parameters for the homogenous beam with a proof mass........74

2-6 Material properties and dimensions for a piezoelectric composite aluminum beam.........75

2-7 Measured and calculated values for a PZT composite beam.........................................76

2-8 Measured and calculated parameters for a PZT composite beam with a proof mass........77

2-9 Comparison between experimental and theoretical values for power transfer ...............82

3-1 List of all device variables that are described in the electromechanical model .................88

3-2 Dimensional representation of all the device variables. ............. ..................................... 89

3-3 Primary variables used in the dimensional analysis. ................................. ............... 90

3-4 List of independent H groups. ............. ................. ........................................................... 93

3-5 Final set of nondimensional groups involving response parameters........................... 109

3-6 Material dimensions and properties of composite beam for FEM validation................ 117

3-7 Static lumped element parameters from FEM and LEM to validate the scaling
a n a ly sis ................... ......................................................................... 12 1

3-8 Properties and dimensions used for designing MEMS PZT devices.............................123

3-9 Material properties of piezoelectric composite beam. ............................................... 128

3-10 D designed M EM S PZT structures ......................................................... ..................... 129

4-1 Residual stress measurements for the PZT pattern process (source : ARL)....................133









4-2 DRIE recipe conditions for top side etch........... ........................... .. ............... 136

4-3 DRIE recipe conditions for back side etch ............................. ...................... 142

4-4 Process traveler for the fabrication of micro PZT cantilever arrays.............................147

5-1 Reported polarization results (ref: ARL) ................. ............... ............ ............... 156

5-2 Data acquisition parameters for mechanical characterization................. ............... 163

5-3 Data acquisition parameters for mechanical characterization................. ............... 164

5-4 Data acquisition parameters for mechanical characterization................. ............... 166

6-1 Comparison of ARL's reported hysteresis parameters with measured values.................178

6-2 Dielectric parameters of all tested design geometries on the device wafer. .................... 182

6-3 LEM parameters extracted using experimental data.......... .... ............... 185

6-4 LEM parameters extracted using M ethod 1.......................................... ............... 197

6-5 LEM param eters extracted using M ethod 1.......................................... .....................201

6-6 LEM parameters extracted using Method 3................................................ 206

6-7 LEM parameters extracted for PZT-EH-09-01 ..... ........................... 207

6-8 Extracted LEM parameters for PZT-EH-09-03.................... .........212

6-9 LEM parameters extracted for PZT-EH-07-02 ...................... .................212

6-10 Comparison between theory and experiments for PZT-EH-07. ......................................215

6-11 Comparison between theory and experiments for PZT-EH-09 devices ........................216

6-12 Quality factors for PZT M EM S devices. .............................................. ............... 219

A-1 LEM parameters and bending strain for various beams subjected to a point load. .........246

A-2 LEM parameters and bending strain for various beams subjected to uniform load. .......247









LIST OF FIGURES


Figure page

1-1 Schematic of a typical vibration to electrical energy converter.................. ...............24

1-2 An electromagnetic vibration-powered generator (adapted from Glynne-Jones and
W white 200 1). ..............................................................................30

1-3 Deformation of a piezoceramic material under the influence of an applied electric
fi eld ........................................................... .................................... 3 3

1-4 A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al,
1997) .......................................................... ...................................38

1-5 Schematic of the proposed cantilever configuration for energy reclamation ..................44

2-1 Schematic of a piezoelectric composite beam subject to a base acceleration ..................50

2-2 Overall equivalent circuit of composite beam. ...................................... ............... 52

2-3 Schematic of the piezoelectric cantilever composite beam. ............................................55

2-4 Free body diagram of the overall configuration. ................................................56

2-5 Free body diagram of the composite beam where the self weights are replaced with
equivalent loads. ...........................................................................57

2-6 Static model verified with the ideal solution for a homogenous beam solved for self
w e ig h t ............. ................... .............. ........................................... 6 1

2-7 Static model verified with the ideal solution for a homogenous beam solved for tip
lo a d ..... ........ ................... .................. ......................................... 6 2

2-8 Deflection modeshape for a composite beam subjected to an input voltage ...................66

2-9 Experimental setup for verifying the electro-mechanical lumped element model for
m eso-scale cantilever beam s .............................................................................. ....... 72

2-10 Comparison between experiment and theory for tip deflection in a homogenous beam
(n o tip m a ss)......................................................................... 7 3

2-11 Comparison between theory and experiments for the tip deflection in a homogenous
beam with tip mass............... .. .. .................................74

2-12 Frequency response of a piezoelectric composite beam (no tip mass) ...........................77

2-13 Frequency response for a piezoelectric composite beam (mp=0.476 gm) ........................78









2-14 Output voltage for an input acceleration at the clamp. ................... ............................. 80

2-15 Output voltage for varying resistive loads. .............................................. ............... 81

2-16 Output power across varying resistive loads. ........................................ ............... 82

3-1 Thevenin equivalent circuit for the energy reclamation system .....................................85

3-2 Schematic of the MEMS PZT device. ........... ............. ................. ........ 88

3-3 Meshed PZT composite cantilever beam for FEM validation..................... ........ 117

3-4 Short circuit natural frequency for a PZT composite beam.................................. 118

3-5 Short circuit compliance for a PZT composite beam.................................................119

3-6 Effective mechanical mass for a PZT composite beam .............................................120

3-7 Effective piezoelectric coefficient for a PZT composite beam.............. ... ...............121

3-8 Schematic of a single PZT composite beam. ......................................... ...............123

4-1 Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer.............................131

4-2 Sputter deposit Ti/Pt (20 nm/200 nm) as bottom electrode............................................131

4-3 Spin coat sol-gel PZT (125/52/48) over the wafer using a spin-bake-anneal process.....132

4-4 Deposit and pattern Pt for top electrode using liftoff. ............................................... 132

4-5 Pattern opening for access to bottom electrode and wet etch PZT using PZTEtch
m ask ......................................................... ....................................133

4-6 Ion milling of PZT and bottom electrode using Ion Milling mask as pattern..................133

4-7 Deposit Au (300 nm) and pattern bond pads using Bond Pads mask and wet etching....134

4-8 Sidewall profiles on topside of a 4" Si test wafer. ................................... ..................... 136

4-9 Wet etch exposed oxide with BOE and DRIE to BOX from top..................................137

4-10 Sidewall profiles for backside etching using DRIE.................................... ............... 138

4-11 Curved edges during backside DRIE............ ...................... ...... ................ 139

4-12 Onset of silicon grass during a backside etch run............... ...... .................. 140

4-13 Sidewall profiles for a backside etch on a test wafer................................. .............. 141

4-14 Pattern proof mass on the backside and DRIE to BOX .............................. 143









4-15 Schematic of final released device................................... 144

4-16 SEM pictures of a PZT-EH-07 released device...................................... ......................145

4-17 SEM pictures of a PZT-EH-09 released device.. ............... .............. 145

4-18 Sidew all profiles of released devices ............................ ........................... .... .......... 146

4-19 Schematic of the bottom of vacuum package for MEMS PZT devices......................149

4-20 Schematic of glass top for vacuum package ................................. ..........................149

4-21 An isometric view of the overall vacuum package............. ....... ...............150

4-22 Schematic of open package for MEMS PZT devices ............................................... 151

4-23 Picture of the open package. .................................. ......... ................................... 152

5-1 Schematic for ferroelectric characterization. ......................................... ...............154

5-2 Experimental setup for ferroelectric characterization ................................. .............. 155

5-3 Schematic for blocked electrical impedance measurement. .........................................158

5-4 Experimental setup for electrical impedance characterization. .......................................158

5-5 Experimental setup for mechanical and electromechanical characterization ................160

5-6 Experimental setup for vibration and velocity measurements with LV. .......................161

5-7 Experimental setup for open circuit voltage measurements. ........................................165

5-8 Experimental setup for open circuit voltage measurements. ........................................165

5-9 Experimental setup for voltage and power measurements. ...........................................166

6-1 A typical P-E hysteresis loop for a piezoelectric material (adapted from Cady 1964).... 169

6-2 A typical E-E curve for a piezoelectric material. .................................. ............... 170

6-3 Polarization, capacitance and input voltage waveforms for PZT-EH-02-1-1..................171

6-4 H ysteresis plots for PZ T-EH -02-1-1 ...................... .... ................................................ 172

6-5 Pr and Vc for different applied voltages for PZT-EH-02-1-1 ............................... 173

6-6 Normalized Ceb for PZT-EH-02-1-1 during the hysteresis test. ...................................174

6-7 Leakage current for PZT-EH-02-1-1 subjected to 10V DC......................................... 175









6-8 Poling of PZT-EH-02-1-1 at 5V for different times............ ............................. 176

6-9 Poling of PZT-EH-02-1-1 at different temperatures................................... ............... 178

6-10 Variation of Ceb and tan6 with dc bias and a constant sinusoid, 500 mV at 100 Hz. ......179

6-11 Variation of Ceb and tan6 with source amplitude at 100Hz. ....................... .........180

6-12 Ceb and er for MEMS PZT devices on wafer before release for a) PZT-EH-01 (106
geometries) b) PZT-EH-02 (16 geometries) c) PZT-EH-03 (15 geometries) d) PZT-
EH-04 (14 geometries) e) PZT-EH-05 (16 geometries) ............................................183

6-13 Ceb and Er for MEMS PZT devices on wafer before release for a) PZT-EH-06 (150
geometries) b) PZT-EH-07 (12 geometries) c) PZT-EH-08 (22 geometries) d) PZT-
E H -0 9 (10 8 g eom etries)........................................................................ ..................... 184

6-14 Flowchart for method 1 to extract the LEM parameters from the experimental data......190

6-15 Low frequency electromechanical response data compared with curve fit to extract
dm ................ ......................................................................19 4

6-16 Comparison between experiment and LEM based curve fit around resonance for a)
electromechanical response b) short-circuit mechanical response ...............................195

6-17 Low frequency curve fit compared with experiment to extract Ceb............................... 195

6-18 Comparison between experiment and curve fit for low frequency open circuit voltage
response to extract M m ...................................................... .. ...... .. ............ 196

6-19 Experimental data and curve fits for open circuit voltage response compared around
re so n a n c e ............................. .................................................................... ............... 19 6

6-20 Flowchart for parameter extraction using Method 2..................... ................................. 199

6-21 Experimental data and curve fits for open circuit voltage response and free electrical
impedance compared around resonance. .............................................. ............... 201

6-22 Flowchart for LEM parameter extraction implementing Method 3..............................203

6-23 Comparison between experiment and LEM based curve fit for short circuit
mechanical and electromechanical response around resonance. .....................................205

6-24 Experimental data and curve fits for open circuit voltage response compared around
re so n an ce ............................. .................................................................. ............... 2 0 5

6-25 Comparison between model and experiments for PZT-EH-09-01. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance









response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance................................................................................208

6-26 Comparison between model and experiments for PZT-EH-09-02. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance
response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance................................................................................209

6-27 Comparison between model and experiments for PZT-EH-09-03. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance
response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance ....................................... ............... ............... 210

6-28 Comparison between model and experiments for PZT-EH-09-04. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance
response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance ........................................................ ............... 211

6-29 Comparison between model and experiments for PZT-EH-07-02. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance
response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance ....................................... ............... ............... 213

6-30 Comparison between model and experiments for PZT-EH-07-03. A) Short circuit
mechanical response B) Electromechanical response C) Free electrical impedance
response D) Open circuit voltage response E) Normalized output voltage and power
across resistive loads at resonance ....................................... ............... ............... 214

A Schem atic of a cantilever beam ........................................................... .....................237

A-2 A schematic of clamped-clamped beam. .............................................. ............... 240

A-3 Free body iagram of a clamped-clamped beam. ................................... ............... 240

A -4 Schem atic of a pin-pin beam ......................................... .............................................243

A-5 Free body diagram for a simply supported beam.................................. ............... 243

B-l A simple schematic of the cantilever beam .................................. ............ ............. 251

C-1 Vibrating cantilever beam in an accelerating frame of reference..............................265

D-1 Blocked electrical impedance in a parallel network representation.............................. 267

D-2 Blocked electrical impedance in a series network representation ................................269

E-l Thevenin equivalent representation connected to a external complex impedance..........270

E-2 Thevenin equivalent representation connected to a resistive load..............................273









F-l Schematic of the composite beam energy harvester................................................275

F-2 Free body representation of the device as a two degree of freedom system....................275

F-3 Electromechanical circuit representation of the energy harvester ................................276









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC CANTILEVER ARRAYS FOR
VIBRATIONAL ENERGY HARVESTING

By

Anurag Kasyap V.S.

May 2007

Chair: Louis Cattafesta
Cochair: Mark Sheplak
Major: Aerospace Engineering

In this dissertation, the development of a first generation MEMS-based piezoelectric

energy harvester is presented that is designed to convert ambient vibrations into storable

electrical energy. The objective of this work was to model, design, fabricate and test MEMS-

based piezoelectric cantilever array structures to harvest power from source vibrations.

The proposed device consists of a piezoelectric composite cantilever beam

(Si/SiO2/Ti/Pt/PZT/Pt) with a proof mass at one end. The proof mass essentially translates the

input base acceleration to an effective deflection at the tip relative to the clamp, thereby

generating a voltage in the piezoelectric layer (using d31 mode) due to the induced strain. An

analytical electromechanical lumped element model (LEM) was formulated to accurately predict

the behavior of the piezoelectric composite beam until the first resonance.

First, macro-scale PZT composite beams were built and tested to validate the LEM. In

addition, a detailed non-dimensional analysis was carried out to observe the overall device

performance with respect to various dimensions and properties. Various first generation test

structures were designed using a parametric search strategy subject to fixed vibration inputs and

constraints.









The proposed test structures thus designed using the electromechanical LEM were

fabricated using standard sol gel PZT and conventional surface and bulk micro processing

techniques. The devices have been characterized with various frequency response measurements

and the lumped element parameters were extracted from experiments. Finally, they were tested

for energy harvesting by measuring the output voltage and power at resonance for varying

resistive loads.














CHAPTER 1
INTRODUCTION

This dissertation discusses the modeling, design, fabrication, and characterization of an

array of micromachined piezoelectric power generators to harness vibration energy. The

reclaimed power is rectified and stored using a power processor (Taylor et al. 2004, Kymissis et

al. 1998) for subsequent use by, for example, sensors. The details of this concept are discussed

in subsequent sections. This chapter begins with an introduction to energy reclamation, various

available resources, and harvesting technologies. Then, a detailed description is presented

concerning energy reclamation from vibration and its uses in various fields such as self-powered

sensors, human-wearable electronics and vibration control. Finally, it concludes with motivation

for microelectromechanical systems (MEMS) and piezoelectricity as the tools for this research.

An in-depth literature survey is presented to familiarize the reader with the previous and current

work in these fields.

Energy Reclamation

Conservation of energy is a fundamental concept in physics along with the conservation of

mass and Newton's laws. The law of conservation of energy states that energy can neither be

created nor destroyed but only converted from one form to another. A useful description of this

law in a thermodynamic system is the first law of thermodynamics. It states that the difference

between the total rate of inflow of energy into a system minus the total rate of outflow of energy

from the system (to the surroundings) equals the time rate of change of energy contained within

the system. Therefore, energy reclamation, by definition, relates to converting any form of

energy that is otherwise lost to the surroundings into some form of useful power.









Energy Resources and Harvesting Technologies

There are two classes of available energy sources, renewable and non-renewable. Non-

renewable sources, as the name suggests, include all that have a limited supply such as oil, coal,

natural gas, etc. These sources take thousands or millions of years to form naturally and cannot

be replaced once consumed. They have constituted the major part of the United States (U.S.)

power supply for a long time. But, with increasing technology and society's ever-growing

consumption of energy, these sources could soon be exhausted (National Energy Policy Report,

2001). Hence, it is an ecological and economical necessity to investigate alternate sources of

energy to meet societal demands. Consequently, research in the past few decades has focused on

using an alternate form, called renewable resources, to meet the demand, such as optical, solar,

tidal, etc.

Jan Krikke, in his editorial article in "Pervasive Computing" (2005) reviews the current

situation in energy harvesting technologies. Many companies in the US, Europe, and Japan are

steadily involved in this area as there exists a general fascination with energy scavenging from

ambient sources. Many energy harvesting concepts are already available such as a self-reliant

house (powered by solar energy that operates all appliances in the house) and a camel fridge,

which uses solar energy to operate a refrigerator used to store (below 8 C) and transport

vaccines in African nations.

Previous studies have successfully shown that energy can be reclaimed from renewable

sources such as solar and tidal energy (Saraiva 1989). Solar cells are an existing technology that

is extensively used in self-powered watches, calculators, and rooftop modules for houses. Solar

energy has also been harvested on a smaller scale from an array of micro-fabricated photovoltaic

cells to produce an overall open circuit voltage of 150 V and a short circuit current of 2.8 /A









(Lee et al. 1995). While solar energy has been widely explored and implemented, it becomes

difficult to generate power in dark areas.

Even though renewable sources can serve as a substitute for the usual power supply

resources, energy is still wasted in the form of heat, sound, light and vibrations that can be

further reclaimed, at least partially, for future use. For example, thermal energy was generated

from a 0.75x0.9 cm2 bismuth-telluride thermoelectric junction to produce 23.5 /W for a

temperature difference of 20 K (Stark and Stordeur, 1999). Qu et al. in 2001 designed and

fabricated a thermoelectric generator, 16 x 20 x 0.05 mm3, consisting of multiple micro Sb-Bi

thermocouples embedded in a 50 um epoxy film capable of producing 0.25 V from a

temperature difference of 30K. Kiely et al. (1991; 1994) designed a low cost miniature

thermoelectric generator consisting of a silicon on sapphire and silicon on quartz substrate.

Another thermoelectric power generator based on silicon technology produced 1.5 pW with a

temperature difference of 10C (Glosch et al. 1999). Of all the renewable sources, optical and

thermal energy have been the most popular and widely implemented, even in micro power

requirements. However, in applications where light and thermal energy are not readily available,

alternate sources need to be considered such as mechanical energy. In addition, an advantage for

mechanical energy conversion over thermal conversion is that, ideally, it does not require any

heat isolation. In addition, scaling thermal systems to microscale possesses fundamental

limitations such as thermal related noise due to thermal fluctuations, temperature based

adsorption, etc. (Devoe 2003).

In recent years, extensive research has been conducted on harvesting undesirable

vibrational energy. Although most efforts have been in the area of mesoscale energy harvesting,

the focus on microscale has gained importance lately. The energy thus claimed from vibrational









sources can be stored and later used to power various devices. In the past, efforts in energy

reclamation from vibrations have largely focused on the available energy in human ambulation

(Starner 1996). Reclaiming energy from human ambulation has generated immense interest

primarily because of its ability to power artificial organs and human wearable electronic devices.

Growing interest in the area of human-wearable electronic devices creates a need for portable

power sources for these devices. Starner and Paradiso (2005) describes various sources from

humans for energy harvesting such as body heat, breath, blood-pressure, walking, etc. In

addition, heel strike, limb movement, and other gait-related activities are useful sources of strain

energy and can be used as alternate methods for powering artificial organs (Antaki et al. 1995).

This could replace conventional portable batteries that are currently restricted by energy

limitations, especially for prolonged usage. In addition, batteries are often bulky and possess a

limited shelf life and could be potentially hazardous due to chemicals. The development of

MEMS technology has led to a wide range of applications for micro actuators and sensors (see,

for example, Senturia 2000). It also has enabled implantation of these devices into various host

structures, such as medical implants and embedded sensors in buildings and bridges (Mehregany

and Bang 1995). In most of these applications, the devices need to be completely isolated from

the outside world. These remote devices, along with their accompanying circuitry, have their

own power supply that is usually powered by batteries. The strides achieved in battery

technology have not sufficiently matched the improvements in integrated circuit technology.

Therefore, developing a micro-scale self-contained power supply offers great potential for

applications in remote systems.

Self-Powered Sensors

The ever-reducing size of CMOS circuitry and correspondingly lower power consumption

have also provided immense opportunities to design and build micro power generators that can









be ideally integrated with CMOS. Simultaneous research is also being carried out to develop

new chip technology to lower the power requirement for electronic equipment (Krikke 2005).

The need for self-contained power generators has led to the development of "self-powered

systems" that is an important application for energy reclamation, and is currently gaining

widespread importance (Shenck and Paradiso 2001). Self-powered systems possess an inherent

mechanism to extract power from the ambient environment for their operation. The main

objective of self-powered systems is to utilize a generator that can convert energy from an

ambient source to electrical energy as long as sufficient energy is available in the ambient

source. Consequently, the primary features of self-powered systems include power generation,

energy extraction, and storage. Ideally, a self-powered device should possess high power density

for given size constraints. Attempts to build perpetual motion machines date back to as early as

the 13th century when the conservation laws had not yet been formulated. Glynne-Jones and

White (2001) provide a review on available energy resources for self-powered sensors such as

vibrations, optical, thermoelectric, etc.

Next, some of the relevant work carried out in the field of self-powered sensors is

examined. As mentioned earlier, heel strike is a resource for strain energy that can be

electromechanically transformed into electrical power. Consequently, shoe-mounted devices

have been developed and tested that convert strain energy induced during heel strike and store it

as electrical energy. Kymissis et al (1998) and Shenck and Paradiso (2001) designed two novel

piezoelectric devices to harness power that were embedded in a shoe. Furthermore, vibrations

when available are excellent potential sources for energy harvesting. Meso-scale energy

reclamation approaches include rotary generators (Lakic 1989), a moving coil electromagnetic

generator (Amirtharajah 1998), and a dielectric elastomer with compliant electrodes (Pelrine









2001). Single meso-scale piezoelectric cantilevers (Ottman et al. 2002) and stacks (Goldfarb and

Jones 1999) have been investigated for energy reclamation but were not operated in a stand-

alone, self-powered mode. Another source for power harvesting is mechanical energy from fluid

flow. Taylor et al. (2001) designed an energy harvesting eel that was approximately 1 m long

using a piezoelectric polymer to convert fluid flow and vortex-induced strain to generate power.

In addition, Allen and Smits (2001) investigated the feasibility of utilizing a piezoelectric

membrane in the wake of a bluff body to induce oscillations in the structure generating a

capacitance build-up that acts as a voltage source to power a battery in a remote location. Power

generation from ocean waves has also been investigated involving very large-scale piezoelectric

generators (Smalser 1997). As a result, there is a clear indication that energy reclamation from

strain energy is a promising field in terms of research and applications. The focus of this

dissertation is to study the possibility of using vibrational mechanical energy as a potential

source for energy reclamation on a micro scale. Ambient vibration sources, such as household

appliances, machinery equipment, and HVAC ducts typically occur at frequencies in the range of

100's of Hz with an acceleration amplitude of 1-10 m/s2 (Roundy et al. 2003).

Vibration to Electrical Energy Conversion

Continuing the discussion on converting vibrational energy to electricity, this can be

achieved using a transduction mechanism that effectively converts energy from the mechanical

domain to the electrical domain. A simple schematic of a power generator based on vibration is

shown in Figure 1-1. The device consists of a spring-mass-damper system acting as a single

degree of freedom system with an input vibration that results in an effective displacement z(t).






























Figure 1-1: Schematic of a typical vibration to electrical energy converter.

The following equation is used to represent the behavior of the above system that basically

converts the kinetic energy of a vibrating structure to electrical energy by virtue of the relative

motion between the base and the inertial mass.

Mz + Rz + Kz = -My (1.1)

where z is the relative deflection, y is the input displacement, M is the inertial mass, K is the

spring constant and R is the effective damping in the system that accounts for mechanical and

electrical losses. The above model does not include nonlinear effects and is thus valid only

under the constraints of linear system theory. It also does not specify the electromechanical

transduction mechanism with which the kinetic energy is converted to electrical power. These

mechanisms are discussed in detail in the following sections.

Transduction Mechanisms

Vibrational energy reclamation can be achieved conceptually using different transduction

mechanisms. Any transduction mechanism relates to energy conversion from one form to









another. For example, it can involve coupling of two or more energy domains such as

electrostrictive coupling (Uchino et al 1980), electromagnetic (Hanagan 1997; Kato 1997) and

electromechanical coupling (Lee 1990). In his Ph.D. dissertation, Roundy (2003) calculates the

theoretical maximum and the practical maximum for the energy densities of various transduction

mechanisms, namely piezoelectric, electrostatic, and electromagnetic. The expressions were

obtained from the basic governing equations of each of the materials and calculated using

maximum yield stress for the piezoelectric, the electric field for capacitive, and the maximum

magnetic field for electromagnetic materials as the respective upper limits. In his summarized

results, he found that piezoelectric materials possess a practical maximum energy density of

17.7 mJ/cm3, which is almost four times that of the other transducers. The following

paragraphs provide some basic discussion on transducer theory and explain electromechanical

transduction mechanism in detail.

A typical transducer is represented using different energy domains associated with power

flow from one domain to another. Modeling the energy transfer between domains enables a

better representation of the transducer behavior. The net power flow between two elements

describing the device is represented as a product of two terms called the conjugate power

variables (Senturia 2000).


P=e.f, (1.2)

where e is effort and f is flow. Next, a generalized momentum can be defined by integrating

the effort over time and is represented as

t
p= fe(t)dt + p(0). (1.3)
0









Similarly, a generalized displacement is defined that is associated with the flow variable, given

by


(1.4)


q= f(t)dt+q(0).
0


Here, p(0) and q(0) are the initial momentum and displacement in the element respectively.

Consequently, the energy in the element is given by the product of flow and momentum or effort

and displacement as

E=q-e=p.f. (1.5)

The ratio between effort and flow results in the generalized complex impedance of the element.


e/f =Z.


(1.6)


Some examples of conjugate power variables for various energy domains (Senturia 2000) are

listed in Table 1-1.

Table 1-1: Conjugate power variables for different energy domains.
Energy domain Effort Flow

Translational Mechanical Force (F)[N] Velocity (u,U)[ms 1]

Rotational Mechanical Torque (r)[Nm] Angular velocity (co)s 1]

Incompressible flow Pressure (P)[Nm 2] Volumetric flow (q,Q)[m's 1]

Thermal Temperature (T)[K] Entropy rate (5)[J(Ks) 1

Electrical Voltage (V)[V] Current (i,J)[A]

Magnetic mmf (7)[A] Flux rate (V)[V]

A transducer is broadly classified into energy conserving and non-energy conserving

transducers (Hunt 1982, Fischer 1955). They can be classified further on factors such as

linearity, reciprocity etc. Electromechanical transducers are classified based on force generation









due to the interaction between electric field and charge or magnetic field and current. For

electromechanical transduction, there are five major linear energy conserving transducers,

namely, electrodynamic, electrostatic, piezoelectric, magnetic, and magnetostrictive.

All linear conservative transducers are generally represented using simple two-port

network theory (Rossi 1988) expressed in impedance or admittance notation. Here, the

impedance form is explained to discuss the various transduction mechanisms. The governing

equations for an electromechanical transducer are


V TZ LU I


The blocked electrical impedance is defined as,


Ze= (1.8)


where U = 0 indicates that the device is mechanically restricted or "blocked" from any motion.

Alternatively, the free-electrical impedance,


JZ, = (1.9)
Zef = V F(1.9)


is defined as the electrical impedance when the device is "free" or not subjected to any

mechanical load. The coupling terms are defined as open circuit electromechanical transduction

impedance and the blocked mechanical-electro transduction impedance, represented as


=em (1.10)


and


T = (1.11)
I0









respectively. The electromechanical transducer is defined to be reciprocal when the cross

diagonal coefficients in Eq. (1.7) are equal, T,, =T. Z,, is defined as the open-circuit

mechanical impedance expressed as the ratio between mechanical force and resulting velocity for

zero current


Z (1.12)
SU =0

Alternatively, the ratio between the force and velocity while preventing any voltage from

building up defines the short circuit mechanical impedance


Z (1.13)
SU v=

Both forms of mechanical and electrical impedances expressed in Eqs. (1.8)-(1.9) and Eqs. (1.12)

-(1.13) are related to each other as


Z, Zmo(=1- K2) (1.14)

and

Z = Z(1 -2), (1.15)

where K2 is defined as the electromechanical coupling coefficient that relates the amount of

energy converted from electric domain to mechanical domain. The coupling coefficient

represents the ideal effectiveness of an electromechanical transducer is defined as


K= TTme (1.16)
ZebZmo

Two-port network theory can also be represented with a corresponding set of coefficients

in the admittance form. For reciprocal transducers, Tm =T ,, which implies that the









electromechanical conversion from an applied voltage to velocity and applied force to resulting

current are equal.

Electromechanical transducers are commonly represented using equivalent circuits with

lumped elements and will be explained in detail in Section 2.1. Some of the widely used

electromechanical transduction mechanisms for energy harvesting involve electromagnetic

(specifically electrodynamic), electrostatic and piezoelectric phenomenon that are explained

next.

Electrodynamic transduction

Electrodynamic transduction occurs when energy conversion is produced by motion of a

current carrying electric conductor subject to a constant magnetic field. This phenomenon is

characterized by Laplace's law (Beranek 1986, Tilmans 1996), which defines the force on the

electric conductor in terms of the current and the magnetic field through the relation


mag L(xB) (1.17)

Here, Fg is termed as 'Lorentz force', I is the current, B is the magnetic field and L is the

length of the conductor. Conversely, the motion of the conductor in the presence of a magnetic

field leads to a voltage generation across its terminals, given by Lenz's law


S=L(UxB). (1.18)


In Eq. (1.18), U is the velocity of the conductor and V is the generated voltage. Combining

these two laws in a two-port representation yields,


~Vl 0L BL (I~
F BL (1.19)









Since Zeb and Z,, for this system are identically zero a direct coupling between electrical and

mechanical domains exists. So, an electrodynamic transducer is linear, reciprocal, and direct.

Another mechanism called the electromagnetic transduction is proposed in Figure 1-2

(Glynne-Jones and White 2001, Glynne-Jones et al, 2004). This transduction in nonlinear, but

can be linearized about its mean state to be represented as a linear, reciprocal transducer. The

linearization is valid for small variations in current and magnetic field that are possible by

biasing the electrical conductor with an initial current (Tilmans 1997).



mass
spring displaced



mass


magnet I
S
coil




input vibration
out

Figure 1-2: An electromagnetic vibration-powered generator (adapted from Glynne-Jones and
White 2001).

El-Hami et al. (2001) designed an electromagnetic generator comprised of a magnetic core

mounted on the tip of a steel beam. When an input vibration is supplied to the structure, the

beam vibrates, thereby inducing current in the coil. They report an output power of 0.53 mW

for an input displacement magnitude of 25 um at 322 Hz The overall volume of the device

was 0.24 cm3. In 2000, Li et al. presented a micromachined generator that had a permanent









magnet mounted on a spring structure and generated 10 uW at 2 V DC for an input vibration

amplitude of 100 ,um at 64 Hz from a volume of 1 cm3. Williams and Yates in 1996 designed

an electromagnetic generator (5 mm x 5 mm x 1 mm) that had a predicted power output of 1 uW

at 70 Hz and 0.1 mW at 330 Hz for an input vibration amplitude of 50 ium. Shearwood and

Yates in 1997 designed an electromagnetic generator based on a polyimide membrane 2 mm in

diameter that could generate 3 pW of RMS power at a resonant frequency of 4.4 kHz.

Rodriguez et al. (2005) presented their work on the design optimization of an

electromagnetic vibrational generator to scavenge pW 's-mW 's of power in the frequency range

between 10 Hz to 5 kHz. The design proposed in their work consists of a movable magnet

mounted on a resonant membrane that induces a current in a fixed planar coil.

Electrostatic transduction

Electrostatic transduction is the conversion of energy that is produced by varying the

mechanical stress to generate a potential difference between two electrodes. An example for this

transduction is a simple parallel plate capacitor.

If we assume that one plate is moving relative to the other (generally stationary), due to an

external load, the variation in gap generates a capacitance given by

EA
Ce (t) (1.20)
x(t)

where E is the permittivity of the medium separating the plates, A is the area and x(t) is the

distance between the plates that changes about an initial mean distance. The voltage generated

between the terminals due to this is


e(t)
E (t) = (1.21)
C, Wf









where Q(t) is the accumulated charge in the capacitor. From Eq. (1.21), we know that the field

has a nonlinear relation with charge and displacement, which implies that it is nonlinear with

current and velocity. In addition, the force generated also follows a nonlinear relation with the

flow variables. However, the coupled equations can be linearized for small variations about a

mean initial condition, generally achieved by applying a bias voltage to the plates (Rossi 1988,

Tilmans 1997) or by storing a permanent charge using an electret (Boland et al 2003). The final

linearized set of equations are expressed in the two-port form as


L1
jC c (1.22)
F V, 1 U


Here, Eo and x, are electric field and distance between the plates. Cm is the mechanical

compliance that relates the force and velocity and Ceo is the mean capacitance. Since the effort

variables are originally calculated using charge and distance, jco is the integration factor in the

frequency domain to convert them to current and velocity. Although the cross terms in the

matrix are same, diagonal terms do exist, which implies indirect coupling between the electrical

and mechanical domains for an electrostatic transducer. Hence, this system of equations

represents a linear, reciprocal and indirect transduction mechanism.

In electrostatic transduction, a relative deflection induces charge between the electrodes

that can be converted to power. For example, at the micro-scale, a MEMS variable capacitor has

been designed and fabricated to harvest vibrational energy with a chip area of 1.5 x 1.5 cm2 and

a reported net power output of approximately 8 pW (Meninger et al. 2001).









Piezoelectric transduction

Piezoelectricity, by definition, is a property of certain materials to physically deform in the

presence of an electric field or, conversely, to produce an electric charge when mechanically

deformed. Piezoelectricity occurs due to the spontaneous separation of charge within the crystal

lattice (Cady 1964). This phenomenon, referred to as spontaneous polarization, is caused by a

displacement of the electron clouds relative to their individual atoms, as well as a displacement

of the positive ions relative to the negative ions within the crystal structure, resulting in an

electric dipole. There are a wide variety of materials that exhibit this phenomenon, including

natural quartz crystals and even human bone. During electrical polarization, the material

becomes permanently elongated in the direction of the poling field (polar axis) and

correspondingly reduced in the transverse direction. Applying a voltage in the direction of the

poling voltage produces further elongation along the axis and a corresponding contraction in the

transverse direction subject to its Poisson's ratio. This effect is depicted in Figure 1-3, which

shows a piezoelectric material under the influence of an electric field; P is the poling direction

and V is the externally applied voltage.





v=of V p 4
V=0 P P P


Contraction
Expansion

Figure 1-3: Deformation of a piezoceramic material under the influence of an applied electric
field.

Piezopolymers and piezoceramic materials are typically used as transducers for

piezoelectric energy harvesting applications. Piezoelectric materials possess a unique property









that makes them a viable option for electromechanical transducers. Applying an external electric

field across the piezoelectric material induces a mechanical strain in the material, thereby

enabling them to function as actuators. Conversely, when the piezoelectric material is

mechanically deformed, the resulting strain produces a voltage that allows them to operate as a

sensor. This strain/electric field characteristic of a piezoelectric material is termed as the

piezoelectricc effect." Materials with good piezoelectric properties possess high coupling

between the mechanical and electrical domains. This effect can be generated using

piezopolymers, such as polyvinyledene fluoride (PVDF), or piezoceramics, such as lead

zirconium titanate (PZT), Zinc Oxide (ZnO), Aluminum Nitride (A1N) and Barium Titanate.

For any linear piezoceramic material (IEEE Standard on Piezoelectricity, 1987), the

constitutive governing equations can be expressed as


sk = Skj +d kTE, (1.23)

and
D, = dqq+ yEj. (1.24)

In the above equations, Ek is the mechanical strain, a, is the stress, D, is the electric

displacement, E, is the electric field applied to the ceramic, Sk is the proportionality constant

between the stress and strain (and is the reciprocal of the elastic modulus of any material), y, is

defined as the dielectric permittivity at constant stress, and dck is the piezoelectric coefficient.

The material constants S, d, and y are defined as shown below for a piezoceramic due to its

crystal structure (IEEE Standard on Piezoelectricity, 1987)









s- si s13 o o o
S S1,, s1 o o o
S S1,, S 0 0 0
S 13 S13 S33 0 (1.25)0
S = ,(1.25)
0 0 0 S44 0 0
0 0 0 0 S44 0

0 0 0 0 0 S66

O0 0 0 0 15 0
d 0 0 0 d15 0 (1.26)
dA dzi d 0 0 0o

and

/ 0 0
7= 0 Y, 0 (1.27)
0 0 733

For a typical piezoceramic patch, the electric field is often applied vertically across the

ends of the piezoceramic in the 3-direction, while the stress acts in the 1-direction for the

composite beam. Therefore, we extract index k = 1 from Eq. (1.23) and i = 3 from Eq. (1.24),

since -1 0, a0 0, E3 0, and E, E 0. Substituting the matrices for the constants

and expanding the constitutive equations for the one-dimensional case results in

_- = SS1- +d31E3 (1.28)

and
D3 = d311 + 33,E,. (1.29)

Rewriting the above equations to express strain in terms of deflection x, stress in terms of

the force applied F, electric field in terms of an applied voltage V, and the electric

displacement in terms of charge q induced in the piezoceramic simplifies them to


x = C, F +dm .V (1.30)

and









q=dm -F +Cf V,


x a
where C = is the short circuit compliance, C = is the free electrical capacitance,
SF_ V F
iV=0 F=0


and d = is an effective piezoelectric constant. Equations (1.30) and (1.31) will be used to


model the composite cantilever beam in this dissertation. Equations (1.30) and (1.31) when

expressed in frequency domain provide the two-port network equations in admittance matrix

form as


IU I jCdm jcodC ] v IF. (1.32)
I jmd] j Co Vf

Piezoelectric materials, especially PZT, exhibit good strain sensitivity and possess an

elastic modulus (e.g., 60 GPa) that is comparable to many structural materials. This property is

essential for effective strain transfer between the layers, which occurs when there is a good

impedance match between the piezoceramic and the shim material. However, PZT is a brittle

material and cannot withstand large strains without fracturing unlike PVDF, which is very

flexible and easy to handle and shape (Starner 1996). PVDF can sustain higher strains and

exhibits higher stability over long periods of time. However, the disadvantage of using PVDF

instead of PZT is the fact that it has a very low electrical permittivity and, therefore, a much

lower coupling factor. Due to this, the electrical response of the device, such as output voltage,

power, and overall efficiency are significantly lower. Also, the working frequency range, which

can be defined as the difference between the open and short circuit resonance for the device is

greatly decreased due to poor electromechanical coupling.

A very common application of piezoceramics is that of a bending motor composed of a

layer of piezoceramic bonded to a host material. The piezoelectric material is assumed to be


(1.31)









firmly attached to the cantilever beam to ensure continuity in strain across the interface (Crawley

and deLuis 1987). Thus, when a voltage is applied to the piezoceramic, an induced moment is

concentrated at the ends of the piezoceramic patch. The maximum induced strain is given by the

expression


Sp = d3 Eefld (1.33)

where d31 is the piezoelectric constant, Efeld is the externally applied electric field, and p, is the

strain induced in the piezoceramic. The curvature of a bending motor is due to the expansion of

one layer and the contraction of the other. This phenomenon occurs due to an induced moment

(Crawley and De Luis 1987) when voltage is applied to the piezoceramic.

Umeda et al. (1996, 1997) performed theoretical and experimental characterizations of a

piezoelectric generator based on impact energy reclamation. In their studies, an oscillating

output voltage resulting from an input mechanical impact was rectified and stored in a capacitor.

With an initial voltage of over 5 V a maximum efficiency of 35 % was achieved with a

prototype generator. The working principle employed in their design is based on a steel ball that

freely falls toward the center of a circular membrane consisting of bronze and piezoceramic that

vibrates on impact resulting in an alternating current in the ceramic. A schematic representing

their structure is redrawn in Figure 1-4 for reference.

Ramsay and Clark (2001) performed a detailed design study on piezoelectric energy

harvesting for bio-MEMS applications. Their design employed a simple geometry for

harnessing energy from blood flow in the body. The proposed structure consisted of a square

PZT-5A plate that is connected to the blood pressure on one side and a chamber with constant

pressure on the other. Preliminary results reported an output power of 2.3 /W from a

(1 cm x 1 cm x 9 pum) plate. It was also reported in their work that the device has a mechanical









advantage in converting applied pressure to working stress for piezoelectric conversion, when it

functions in the 31-mode than in the 33-mode.

pivot






/ piezoelectric
membrane






mass of rod nput vibration


Figure 1-4: A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al,
1997).

Glynne-Jones et al. (2001) and White et al. (2001) designed a thick film piezoelectric

composite beam structure that generated 3 pW of power at 90 Hz from ambient vibrations. An

another paper by the same authors measured 2 pW at 80 Hz for a maximum amplitude of 0.9 mm

across an optimal resistive load of 333 k.(2 Their device consisted of a macro-scale piezoelectric

composite beam that was tapered along its length to ensure constant stress distribution at any

point on its length. In 2004, James et al. investigated two applications for two self-powered

sensors, namely a liquid crystal display and an infra-red link to transmit the data output. The

required energy for the prototypes was derived from a 0.17 g 0.23 g vibrating source at 102 Hz.

In another application of piezoelectric energy harvesting, Hausler and Stein (1984)

proposed a device that basically consisted of a roll of PVDF material that can be attached

between body ribs. They were designed in such a way that regular breathing induced a strain in









the material thereby producing power. It was tested on a dog by surgically implanting the

device, thereby generating micro-watts of power from the breathing.

Roundy and Wright in 2004 designed a piezoelectric vibration generator consisting of a

cantilever bimorph bender with a proof mass at its end. Their design was aimed at generating

enough energy from a 1 cm3 to power a 1.9 GHz radio transmitter from the same vibration

source. Their design was predicted to produce 375 /W from a vibration source of 2.5 m/s at 120

Hz. The lumped element model (LEM) introduced in their work was unconventional and used

stress as the effort variable unlike force which is the standard effort function for LEM

representation. Correspondingly, strain rate was used as the flow variable in the representation.

Sood et al. (2004) developed a piezoelectric micro power generator (PMPG) that is based

on a piezoelectric layer deposited and patterned on a membrane consisting of SiO2 and SiNx,

followed by a ZrO2 diffusion barrier. The two electrodes for the PZT layer are formed using an

inter-digitated top electrode (IDT) with Pt/Ti that makes use of the d33 mode (described later in

this chapter) to extract power. The premise governing their device was that the d33 coefficient is

much higher than d,3 of a piezoelectric material. This potentially results in a higher voltage, but

the power density and input acceleration levels are not available directly for comparison with

other available d31 configurations. The maximum measured power using a direct charging circuit

consisting of a full-bridge rectifier and a capacitor occurred at 5 MQ of load resistance. The

corresponding output voltage and power were 2.4 V, and 1.01 pW respectively (Jeon et al.

2005).

Another application for a self-powered piezoelectric device is a Strain Amplitude

Minimisation Patch (STAMP) damper that uses piezoelectric elements as sensor, actuator and

power source. Konak and Powlesland (2001) presented their analysis on this device that









combined the vibration control aspect of a piezoelectric element along with its energy generation

characteristic producing a self-powered vibration damper.

Table 1-2 compiles all the reported energy harvesters discussed in this chapter that

generated power from vibration sources using different transduction mechanisms. The columns

list the authors, the vibration source (which was mostly resonant in nature), the size of the

device, and the overall power harvested.

Table 1-2: Vibration based energy harvesters characterterized for power.
Ambient source Size or Mass Power
Sood et al. 10 g @ 13.9 kHz 170 um x 260 im 1.01 uW
Shearwood et al. 500 nm @ 4.4 kHz 2.5 mm x 2.5 mm x 700 /um 0.3 /uW
Chandrakasan et al. 500 nm @ 2.5 kHz 500 mg 8 jW
Li et al. 100 im @ 64 Hz 1 cm3 10 W
Roundy et al. 0.25 g @ 120 Hz 28 mm x 3.6 mm x 8.1 mm 375 uW
White et al. 0.9 mm @ 80 Hz 2.2 W
Marzencki et al. 0.5 g @ 204 Hz 2 mm x 2 mm x 0.5 mm 38 nW
El Hami et al. 25um @ 322 Hz 0.24 cm3 0.53 mW
Ching et al. -200 um @ 60-110Hz 1cm3 200-830 pW
Stark et al. AT = 20K 67 mm2 20 pW


Next, a brief introduction to the application of piezoelectric materials in microsystems is

presented followed by the proposed PZT based micro energy harvester.

Microelectromechanical Systems (MEMS)

Some of the earliest ideas about MEMS were initiated by Richard Feynman in his popular

speech "There is plenty of room at the bottom" delivered in 1960 (Feynman 1992) followed by

"Infinitesimal machinery" (Feynman 1993). In the early 1960's, silicon gained a lot of attention

as a material for microsystems due to its excellent properties that suit both electrical and

mechanical applications (Peterson 1982). Micromachining is based on fabrication techniques

that are used in silicon integrated chips but adds numerous other fabrication techniques as well.









This ability to batch fabricate numerous such devices in each step is a potentially significant

advantage of microfabrication in MEMS.

Another major advantage of MEMS is that their small size enables suitability for micro

applications that were not possible prior to the advent of MEMS. However, there are some

significant considerations, such as packaging for structural robustness, operation in harsh

environment, and power requirements that may limit their feasibility in certain applications

(Angell et al. 1983). Recently, smart structures that incorporate MEMS devices were

investigated for their importance and use in aerodynamic structures, spacecraft, and vehicles for

structural health monitoring (Schoess 1995).

The structural configuration adopted for the device described in this dissertation is that of a

piezoelectric composite cantilever beam with an integrated proof mass that functions along the

lines of conventional accelerometers. Significant research has been invested in understanding a

cantilever beam arrangement for energy harvesting (Kim et al. 2004).

The performance of a piezoelectric cantilever bimorph in the flexural mode has also been

analyzed for scavenging ambient vibration energy (Jiang et al. 2005). Their analysis calculates

the output voltage, power, and the device efficiency of the composite beam with a concentrated

tip mass subjected to a harmonic clamp motion. The analytical dynamic model implemented in

their work can be used to design the device appropriately to tune the frequency and increase the

power. However, their work is purely theoretical and does not provide any experimental data for

validation. In addition, model assumes the end mass as a concentrated point load and does not

account for its finite stiffness. This dissertation also aims to first develop an analytical model that

can be used as a design tool for specific energy harvesting applications. Furthermore, the validity

of the model is investigated for various canonical structures both at mesoscale and MEMS. It









uses a different modeling technique called lumped element modeling that, subject to the

assumption that it is valid until the first bending mode, is analytically simpler. This technique is

applicable when the device is small compared to the characteristic length scale of the distributed

physical system.

A cantilever configuration is chosen for our energy harvester because it provides the

maximum average strain when subjected to a specific load (Appendix A). In addition, a

cantilever beam has a lower natural frequency compared to beams with other boundary

conditions (Roundy et al 2003). An explanation of these reasons along with a proof is provided

in Appendix A, where beams subject to different boundary conditions and loads are analyzed to

estimate their average strain and natural frequencies. Therefore, it provides an opportunity to

model a slightly different configuration with variable piezoelectric dimensions from the shim

layer. In addition, the proof mass, which is generally large (especially for MEMS structures), is

modeled to account for its mass and its stiffness providing a complete accurate model. In

addition, the analytical model developed can be utilized as a tool to design cantilever based PZT

energy harvesters for specific applications. The lumped element modeling technique is

investigated in more detail in 2.1.

A simple schematic of the proposed configuration is shown in Figure 1-5. The structure

basically consists of a cantilever beam with a proof mass and a thin film of piezoelectric material

deposited on the beam. When the device is subjected to base vibrations, the inertial mass

vibrates relative to the base causing bending in the beam. The strain thus resulting from this

relative motion is converted to an effective output voltage by virtue of the piezoceramic

transducing element. The piezoceramic layer converts the mechanical strain induced due to the

vibrations into voltage due to the piezoelectric effect. However, even though the design of the









device is similar to an accelerometer, it is implemented and operated as a resonant sensor. In

other words, the device needs to be "tuned" to the input vibration frequency so that it operates

near its resonance frequency to generate maximum power, unlike a conventional accelerometer

that operates across a wide bandwidth far removed from its resonance. Therefore, the goal of our

design is to maximize the performance of the accelerometer device at its resonance. To provide

a brief insight in this area, several investigations have demonstrated the feasibility of fabricating

silicon accelerometers. The basic structure usually consisted of a silicon cantilever with a proof

mass made of silicon or is gold plated to increase the sensitivity (Seidel and Csepregi 1984).

Different transduction mechanisms for accelerometers such as piezoresistors, piezoelectric films,

and electrostatic coupling have been studied in detail, and the advantages and disadvantages of

these transducers have been already been published in the literature (Polla 1995).

Piezoelectric accelerometers are of interest to us due to their low power dissipation and

high electromechanical coupling (Polla 1995). However, the major drawback of this design is

the difficulty in processing and integration with electrical circuitry. Piezoresistive sensors have

much higher dissipation and noise floor even though the processing is relatively straightforward

and CMOS compatible. Capacitive accelerometers are favorable in many aspects such as noise,

power of dissipation and ease of processing, but are sensitive to dimensional tolerance (Polla

1995; Polla et al 1996). In MEMS, it is difficult to achieve small and accurate dimensions and a

considerable uncertainty exists in the material properties and final dimensions of the device.














ta 6 v Cantilever Beam
ao
Proof Mass




Cms

Mm

Rm
y x



Figure 1-5: Schematic of the proposed cantilever configuration for energy reclamation.

Initially accelerometers were fabricated using conventional bulk micromachining

techniques. This approach has a clear advantage in the fact that large proof masses can be etched

out of a silicon substrate. However, disadvantages with this approach arise during front to back

alignment and passivation for integrated circuitry (DeVoe and Pisano 2001). Additionally, this

process consumes a larger die area for bulk etching which is undesirable for batch fabrication.

On the other hand, surface micromachining uses standard VLSI techniques and therefore does

not pose the above problems. DeVoe and Pisano presented their work on the design, fabrication,

and characterization of surface micromachined piezoelectric accelerometers (PiXLs) that

consisted of thin film Zinc oxide (ZnO) as the piezoelectric material. In addition, they describe

some guidelines for robust design based on device sensitivity and resonant frequency. Using a

cantilever without a proof mass whose resonant frequency was 3.3 kHz, their results reported a

sensitivity of 0.95 fC/g. Addition of a proof mass significantly improved the sensitivity to

13.3 fC/g and 44.7 fC/g, but decreased the corresponding resonant frequencies to 2.23 kHz









and 1.02 kHz respectively. The cantilever accelerometer was modeled using classical Euler-

Bernoulli beam theory similar to the method adopted in our design. However, the model

described in their work assumed that the thickness of the piezoelectric layer is negligible

compared to the thickness of the beam. Additionally, it assumes that the ZnO is deposited across

the length of the structure and that the elastic moduli of the two materials are comparable in

magnitude. This assumption holds true when the active layer is ZnO and the beam is made of

silicon. However, in our design where PZT is the piezoelectric layer, the elastic moduli of the

two materials are significantly different, and therefore a detailed static electromechanical model

is derived for our cantilever composite beam.

Piezoelectric MEMS

Silicon is an excellent material for MEMS due to its good mechanical properties such as

elastic modulus and density (Peterson 1982). As a result, most of the devices that are fabricated

in MEMS consist of a silicon substrate. Initially, thick film piezoelectric layers were imprinted

on micromachined silicon substrates to form the desired structure (Allen et al 1989).

Thicknesses in the range of 100 /m can be achieved with this process leading to much higher

actuation forces compared to conventional thin film piezoelectric micro actuators (Barth el al

1988, Terry 1988). Zinc oxide was often used as the piezoelectric material for most applications

until Lead Zirconate Titanate (Pb(Zrx, Til-x)O3) gained acceptance. x is the percentage

composition of Zr in PZT. It was observed that when x lies between 0.52 and 0.55, the material

exhibited high dielectric constants and electromechanical coupling (Wang et al, 1999). PZT has

been extensively studied and used lately due to its excellent electromechanical coupling and

piezoelectric properties. Piezoelectric thin films in micro systems are used in a wide variety of

applications such as micro actuators (Lee et al. 1998; Zurn et al. 2001), micro mirrors (Cheng et









al. 2001), micropumps (Nguyen et al 2002), microphones (Lee et al. 1996), micro accelerometers

(DeVoe and Pisano 2001), fiber bulk wave acoustic resonators (Nguyen et al. 1998) etc.

Objectives of Present Work

The following chapters describe in detail the lumped element modeling technique used to

represent the composite beam and discuss its use in designing an optimal energy harvesting

device to harness maximum power from a vibrating device. The electromechanical lumped

element model thus developed is validated using meso-scale experiments. Furthermore, a

scaling theory is developed to observe the device behavior as it is reduced in size to a MEMS

scale, which is verified using finite element analysis. In addition, the fabrication process adopted

to build the devices and their characterization will be presented.

The main contributions for this dissertation are as follows:

A complete static analytical model of a cantilever composite beam validated using FEM
and experiments on candidate devices.
Electromechanical lumped element model of a piezoelectric energy harvester, intended to
provide a design optimization tool for complete circuit simulation with power processors.
A first generation fabrication of a MEMS PZT cantilever array is realized for vibrational
energy harvesting.
Design, fabrication and testing of a stand alone MEMS device to demonstrate energy
reclamation.

Organization of Dissertation

The dissertation is organized as follows. Chapter 2 describes in detail the static

electromechanical model of the composite beam structure. In particular, lumped element

modeling is used to obtain the various electromechanical parameters that represent the system.

Chapter 3 discusses the detailed non-dimensional analysis and the design formulation for the

device. Chapter 4 discusses the fabrication process adopted to build arrays of the MEMS

piezoelectric generators. Chapter 5 describes the experimental setup and characterization









procedure for testing energy reclamation devices. Chapter 6 describes the experimental results.

Chapter 7 concludes the dissertation with a summary and discussion of future work.














CHAPTER 2
PIEZOELECTRIC CANTILEVER BEAM MODELING AND VALIDATION

The objective of this dissertation is to model and design MEMS piezoelectric cantilever

composite beams with an integrated proof mass to reclaim energy from base vibrations.

Consequently, these structures will be optimally designed to extract maximum power from the

vibrations, subject to some design constraints. The ultimate goal is to eventually use an array of

such structures to obtain sufficient power to operate self-powered sensors. This chapter describes

the static electromechanical modeling of the composite beam using conventional Euler-Bernoulli

beam theory. The shortcomings of this approach are that it does not model nonlinear effects due

to large deflections and neglects rotary inertia effects. In addition, fabrication-induced inplane

residual stresses are neglected in the model. These stresses exist in MEMS structures due to

thermal stresses and other sources that mainly occur during layer depositions and other high

temperature treatment. For the purpose of this first generation effort, we assume that Euler-

Bernoulli theory is adequate to model, design and characterize the device. However, future

models may include the above effects to implement a more complete model.

Some of the earlier work in this area was concentrated on modeling and testing a canonical

cantilever mesoscale composite beam without any proof mass that was excited at its tip with a

load (Kasyap 2002). For a known force input, the amount of power generated at the ends of the

PZT was used in a flyback converter circuit to reclaim power (Kasyap et al. 2002). However, the

previous configuration cannot be directly used in real applications because of the nature of its

loading condition. In all practical applications, the energy reclamation device should be directly









attached to the vibrating surface. This alters the complete setup as the device is loaded at the

clamp due to the vibrations, instead of the tip.

In the new configuration shown in Figure 2-1, the test structure modeled consists of a

piezoelectric composite cantilever beam with a proof mass attached to its tip. The composite

beam, when directly attached to a vibrating surface, places the whole structure in an accelerating

frame of reference. The proof mass essentially converts the input base acceleration into an

effective inertial force at the tip that deflects the beam, thereby inducing mechanical strain in the

piezoceramic (Yazdi et al. 1998). This strain produces a voltage in the piezoceramic that is

converted into usable power with the help of an energy reclamation circuit. The motion of the

beam depends on the size of the proof mass. If the proof mass is relatively small compared to

the effective mass of the beam, it reduces to a cantilever beam subject to an acceleration at the

clamp instead of its tip. Alternatively, if the proof mass is very large compared to the effective

mass of the beam, it results in large deflections in the beam and consequently, large strains at the

clamp. This configuration will be favorable for energy reclamation because a piezoelectric

patch, when attached to the beam, converts the induced strain into electrical charge. However, if

the proof mass is comparable to the actual effective mass of the beam, the motion of the beam

resembles that of a rigid body and, therefore, might not induce any strain in the beam. These

issues are clarified via the model described below.

Piezoelectric Composite Beam

In this analysis, the test structure consists of a piezoelectric (PZT) composite cantilever

beam with a proof mass attached to it as shown in Figure 2-1.









Energy reclamation
S circuit (electric load)

Clamp / V
S / Cantilever Beam

Proof Mass


aoext /
Vibrating Surface



Figure 2-1: Schematic of a piezoelectric composite beam subject to a base acceleration.

In Figure 2-1, ao is the input acceleration, fet is the excitation frequency of the base

vibration and V is the resulting voltage from the piezoceramic. The composite beam is modeled

using the lumped element modeling technique described in Hunt (1982) and Rossi (1988). This

approach is valid in general when the characteristic wavelength of the bending waves is very

large compared to the geometric length scale and, in the case of a cantilever composite beam, is

valid up to at least the fundamental bending resonance frequency (Merhaut, 1979). This

approach simplifies the partial differential equations governing the system to coupled ordinary

differential equations.

In addition, the lumped element modeling technique is useful in analyzing and designing

coupled energy domain transducer systems. In this approach, we use equivalent circuit elements

to effectively represent the coupled electromechanical behavior of the device. These circuit

analogies enable efficient modeling of the interaction between different energy domains in a

system. Furthermore, the tools developed for circuit analysis can be utilized for representing and

solving a coupled system with different energy domains.









A piezoelectric composite beam represents an electromechanical system that can be

separated primarily into two energy domains consisting of electrical and mechanical parts.

These two energy domains interact in the equivalent circuit via a transformer as shown in Figure

2-2. The circuit is obtained by lumping the distributed energy stored and dissipated in the

system into simple circuit elements. In this electromechanical circuit, force and voltage are the

generalized effort variables, while velocity and current are the generalized flow variables

(Senturia 2000). An impedance analogy is used to represent the circuit, in which case all

elements that share a common effort are connected in parallel, and the elements that share a

common flow are connected in series. When the composite beam is subject to a mechanical

load, the strain induced in the piezoelectric material generates a voltage, which represents the

conversion from the mechanical to the electrical domain. Conversely, the composite beam can

be driven with an ac voltage that causes it to vibrate due to the piezoelectric effect. This

represents a conversion from the electrical to the mechanical domain.

Figure 2-2 represents the entire equivalent circuit consisting of mechanical and electrical

lumped elements representing the composite beam. All elements are labeled and defined in the

figure. In the notation shown in Figure 2-2, the first subscript denotes the domain (m for

mechanical and e for electric), while the second subscript denotes the condition (s for short

circuit and b for blocked). Using the described notation, for example, Cm, is defined as short-

circuit mechanical compliance, and Ceb is the blocked electrical capacitance of the piezoceramic.

F is the effective force applied to the structure that is obtained by the product of input

acceleration and effective mass lumped at the tip, U is the relative tip velocity with respect to

the base, V is the voltage, and I is the current generated at the ends of the piezoceramic. All










the parameters are obtained by lumping the energy at the tip using the relative motion of the tip

with respect to the clamp/base.


Short-circuit Blocked
Mechanical Vchanical Electrical
Mass of Compliance of Electromechaic Capacitance of
Beam Beam Trasduction Factor piezocenamic


Velocity M C R I

SIVechanical + cure
damping of eb
beam
F V
I putFoice e voltage
across the
pienxerarnuc

Dielectric loss in
the piezoceramic

Figure 2-2: Overall equivalent circuit of composite beam.

The beam is represented as a spring-mass-damper system by lumping the energy (kinetic

and potential) in the beam to an equivalent mass and compliance. The mechanical mass and

compliance of the structure can be equated to an equivalent electrical inductance and

capacitance. Similarly, mechanical damping is analogous to electrical resistance. However,

mechanical damping cannot be easily estimated from first principles although it is a critical

parameter for resonant behavior in structures. The same holds true for electrical losses in the

device, modeled using Re.

In principle, the fundamental operation of any power generator is effectively dependent on

the nature of the mechanism by which the energy is extracted. Most microgenerators reported to

date can be classified into velocity-damped resonant generators (VDRGs) or Coulomb-damped

resonant generators (CDRGs) as described in Mitcheson et al. (2004). VDRG represents the

damping effect as a function of the velocity characterized by a viscous force, while CDRG









represents the same effect using a coulomb frictional force. Analytical expressions for the

dissipated power for these two cases are derived in Mitcheson et al. (2004) that provide an

estimate for the available power. It should be noted that the aforementioned damping

mechanisms are resonant in nature and that while VDRG is widely used and linear, CDRG is a

nonlinear representation, although a closed-form solution is available (Den Hartog 1931; Levitan

1960). An alternate class, namely, Coulomb-force parameteric generator (CFPG) is also

suggested in their work that operates in a non-resonant manner.

For the purpose of our analysis, a VDRG implementation is adopted that represents the

damping phenomenon using a viscous effect with an effective damping coefficient. Damping

coefficients are typically estimated from experimental modal analysis and include effects such as

viscous dissipation, boundary condition non-ideality, thermoelastic dissipation, etc. (Srikar and

Senturia 2002). A detailed analysis of various damping mechanisms is discussed and

corresponding empirical relations are presented in Appendix B. The mechanical damping in the

system is obtained from the damping ratio using the expression


Rm = 2- (2.1)


where M is the effective mechanical mass of the composite beam (discussed in Section 2-2),

and ; is the mechanical damping ratio. However, the dielectric loss of the piezoelectric material

can be estimated using an empirical expression provided in Jonscher (1999)

1
R= -- (2.2)
tan (2'rfCeb)'

where f, is the natural frequency of the system. tan 3 is the loss tangent defined as the ratio of

resistive and reactive parts of the impedance. The theory behind dielectric loss in piezoelectric









materials is described in Mayergoyz and Bertotti (2005) and Jonscher (1999). In this case, the

electrical damping is assumed to be in parallel with the capacitor. A discussion about this and an

alternate representation for electrical impedance is provided in Appendix D.

All the other parameters in the circuit in Figure 2-2 are obtained analytically. The main

purpose for modeling the device as a beam is to obtain the lumped parameters such as C,,,, M,,,

and q5 that characterize the circuit. The following sections describe in detail the process of

lumped parameter extraction.

Analytical Static Model

The composite beam clamped at one end is analyzed from first principles using linear

Euler-Bemoulli beam theory described in, for example, Beer and Johnston (1992). Therefore,

shear and rotary inertia effects are neglected. Another assumption is that plane sections remain

planar and no geometric nonlinearity exists in the structure. The following section presents the

static analytical model for the composite beam to calculate the lumped element parameters

represented in Figure 2-2. In this analysis, the composite beam is solved for its static equilibrium

to obtain its transverse deflection for all the static loads acting on it, which permits calculation of

the lumped element parameters.

When a base acceleration is applied to the structure, the Euler-Bemoulli governing

equations used in our analysis remain valid. However, a Galilean transformation of coordinates

is carried out, described in detail in Appendix C, to transform to a local coordinate system that

treats relative motion of the beam with respect to the clamp. In addition, the effect of the base

mass is also analyzed in Appendix F.









Static Electromechanical Load in the Composite Beam

The static electromechanical model is used to calculate the effective short circuit

compliance, mass and piezoelectric coefficient of the composite beam. A simple schematic of

the composite beam configuration for this case is shown in Figure 2-3. The cantilever composite

beam is analyzed for all the static mechanical loads acting on it from first principles using the

bending beam equation.


C,,Mm,dm4
V/pp2____
a Shim
11111 11 IM 11 1 I I I 11 11 ----I


Proof Mass





Figure 2-3: Schematic of the piezoelectric cantilever composite beam.

In the figure, Lp, L, and I are the lengths of piezoceramic, beam (i.e., shim) and the proof mass,

respectively. The applied voltage will induce a mechanical strain and, hence, a bending moment

at the ends of the piezoceramic as described in Cattafesta et al. (2000). The induced moment

Mo, in the composite beam due to an applied voltage to the piezoceramic is given by the

expression


Mo =-E31 (2c2-). (2.3)
2 ~ 1)









Here, Ep is the elastic modulus of piezoceramic, d31 is the piezoelectric coefficient, V is the

applied voltage to the piezoceramic, c2 is the location of the neutral axis from the bottom of

piezoceramic given by the expression


Et t t+ 2+E tp
C2 = (2.4)
E t + Eptp

where E5 and t, represent the elastic modulus and the thickness of the shim. Similarly, bp and

tp are the width and thickness of piezoceramic, respectively. The free body diagram for the

above configuration, shown in Figure 2-4, essentially replaces the mass of the composite beam as

an equivalent uniform load due to its weight.


q





ShIm
Mr O L
"--.E-- -
ProofMn a ss




Figure 2-4: Free body diagram of the overall configuration.

where q,, q2 and q3 are the equivalent linear load densities (N/m) in the composite, shim, and

proof mass sections, respectively.

Since the configuration represented in Figure 2-4 is assumed to be a linear system, it can

be simplified and solved analytically for the deflection using Euler-Bernoulli beam theory.









Figure 2-5 represents the simplified free body diagram for the composite beam. In Figure 2-5,

the clamp is replaced with a reaction bending moment (MA) and reaction force (R).




qIL q2(-Lp)


M r\ I I F




Figure 2-5: Free body diagram of the composite beam where the self weights are replaced with
equivalent loads.

As indicated in Figure 2-5, the composite beam is uniformly loaded in a piecewise fashion

over its total length, (L, +1). Consequently, each of the uniform loads shown in Figure 2-4 can

be replaced with an effective static load density defined as


q = pwhg, (2.5)

q = (pt b + ptpbp) g, (2.6)

and


q2 = ptbg. (2.7)

Here, b and t are the width and thickness of the shim, w and h are the width and thickness of

the proof mass, and p,, p, and pm are the densities of piezoceramic, shim, and proof mass,

respectively. Assuming static equilibrium for the beam in the Figure 2-5, we can obtain

expressions for the reaction force and bending moment at the clamp as


R = qL +q(q2 L )+ ql (2.8)









and


M, L2 + qL2 p +ql L+ l (2.9)
2=2 2

Let us now divide the composite beam in Figure 2-5 into 3 sections. Section 1 consists of

the composite (0 < x < L ), the second section consists of the shim (Lp < x < L ), and the third

section is the proof mass (L, < x < L, +1). The Euler-Bernoulli equation for the beam is then

solved using free body diagrams to obtain the bending moment and shear force in each of the

sections (Beer and Johnston 1992). Furthermore, the bending moment can be integrated using

the Euler-Bernoulli equation to obtain the mode shape. The governing equations for the sections

are



2 2
E 2 MM, O<_x<_Lp, (2.10)



2 x2 2


2 (L, ( L L +LP -2
E 2w'3(x) M -+R qL L 2 (x-L L<_ (EI. 8x2 r 1 2 2 q 2

Here, (El)c, (EI) and (EI)m are the flexural rigidity in each of the three sections. Further,

w, (x), w2 (x) and w3 (x) are the transverse deflections in each of the sections at a distance x

from the clamp. The two clamped boundary conditions and four matching conditions, shown in

Eq. (2.13), are obtained from the clamped boundary condition and by matching the deflection

and slope at each of the interfaces between the sections










w1(w, (x 0,
x=0
w, (LP)= W2 (L),
w, (x) 2() (213)
--'- x =Lp 2 (2.13)
Dx x
x=L x=L "


w2 (xL) = w3 (L x)
x=Ls x=L

The Euler-Bernoulli equations shown in Eqs. (2.10)-(2.12) can now be solved to obtain a

piecewise continuous deflection mode shape for the beam, that is represented as



S(X) = q + q2 ,- +q 2 2 2 (2.14)
24(EI), 6(EI)- 2(EI),



42X4 f L, +L +l 2 1L+ x
S q2 4 {2L +ql}x3 2 X2 (C +C3)x C2 C4 (2.15)
24(EI) 6(EI) 2((EI) EI, (EI) '

and


(qx4 q(L, +) x3 q(L, +1)2 X2 (C5 C7x C6 C8
w3 (x)=- + + + (2.16)
24(EJ) m 6(EI)m 4(EI)m (E)m (E )m

where the integration constants C,, C2 ,C3 ,C4 C5 ,C6 ,C7 and C8 are given by the following

expressions









C,= L[ (q, qla)+ 3qL, (L,- L)( a)],

C2= E[Lp (q -q(3-2a))+qL(8L -6L,)(1a)],

= [ql(I-a) (2L,- L+1)-2aM],

Moa LP L
C4= LK2 +ql(1 -a){ l




C6' C2+ qL3 12
C6 =y C2 -


C, = C32 + q +Iq/L, (L, +1) (1-7),
6 2
C, =C47 [qL] +qlL(1-y)(4L,+61)]. (2.17)

In the above equation, a is defined as the ratio between the flexural rigidity moduli in sections 1

and 2, [(EI) /(EI)c, and y is the ratio between the flexural rigidity modulus in sections 2 and

3, [(EI)m/(EI) .

To check the validity of this general result, the deflection modeshape thus obtained is

verified by comparing with a few simple special cases. The first ideal case used to verify the

static electromechanical solution is that of a homogenous cantilever beam subjected to its self-

weight. The resulting deflection for this ideal case is given as (Beer and Johnston 1992)


w' (x) q + .L (2.18)
(HEI) 4 24 6

where, q,, is the uniform load acting on the homogenous beam as a result of its own weight.

The static short-circuit solution is verified by setting the input voltage and the proof mass size to

zero. The piezoelectric patch is thus absent in this solution and represents a homogenous beam









that deflects due to its self-weight. The two solutions are plotted for a test case consisting of an

Aluminum beam (Al 6061). The properties and dimensions used in the simulations are listed in

Table 2-1. As indicated in Figure 2-6, the static solutions are identical.

Table 2-1: Material properties and dimensions for a homogenous aluminum beam.
Elastic modulus (E,) 73 GPa
Density (p,) 2718 kg/m3
Length of the beam (L,) 127 mm
Width of the beam (b,) 6.35 mm
Thickness of the beam (t,) 1.02 mm


x/L +1
S


Figure 2-6: Static model verified with the ideal solution for a homogenous beam solved for self-
weight.

Furthermore, the piecewise solution can be verified for a homogenous beam subjected to a

tip load. From conventional theory, for this ideal case, the deflection modeshape in a beam due

to point static tip load is given as (Thomson, 1993)










q X2 x3 )
Wtpload (x) (2.19)


where qpl is the equivalent tip load. In the piecewise static solution, the piezoelectric patch is

absent and the input voltage is set to zero to generate a similar configuration as before but with a

proof mass that contributes an effective tip load. Similar to the previous case, Eq. (2.19) was

calculated for the test beam listed in Table 2-1, and the deflections are plotted in Figure 2-7.

Again, the solutions match.


0 0.2 0.4 0.6 0.8 1
x/L +1
S


Figure 2-7: Static model verified with the ideal solution for a homogenous beam solved for tip
load.

Now, the complete static mechanical model representing the PZT composite beam has

been verified for various test cases. As described in Section 0, the purpose of obtaining a

complete electromechanical model is to calculate the lumped element parameters in the circuit








shown in Figure 2-2. The advantage of this solution model is that all the parameters are

analytical and their scaling dependence on the dimensions can also be obtained which will be

further useful in optimizing the structure for maximum power output.

The static deflection mode shape can now be used to estimate an equivalent effective mass

and compliance that can replace the composite beam as a simple single degree of freedom

system. For this configuration, we emphasize that only the mechanical loads are considered, and

the piezoceramic is electrically shorted. This configuration effectively eliminates the electrical

side from the lumped element circuit represented in Figure 2-2, leading to short-circuit electrical

condition for the piezoceramic. The potential energy associated with distributed strain energy in

the composite beam is given by the expression (Thomson 1993)


PE dx, (2.20)


where E(x) and I(x) are the local elastic modulus and moment of inertia of the section. The

above integral equation is determined in each of the three sections and summed to obtain the

total potential energy in the beam,


(El) L d2x) (EI) (x) (E2l) 2 w) (x)2d
PE =df L (x) dx + dx d+. (2.21)
2 dx 12 dx2 2 dx2

Similarly, the total kinetic energy in the composite beam is given by the integral

expression


KE = L P (w(x)) dx, (2.22)
K 0a









where p, is mass/length of the section and w (x) is the velocity in the section. We calculate

the kinetic energies in the individual sections and add them to obtain the total kinetic energy in

the composite beam

LP Ls L!+1
KE = P (x)2dx+ 1Pw2 ()2 d+ Lm 3(x)dx. (2.23)
2 L 2 L
LP L,

Lumping the overall potential strain energy at the tip yields an effective short circuit

mechanical compliance for the composite beam


(Wtp )2
C = (2.24)
2PE

Using the same analogy, an effective mass for the composite beam from its deflection shape is

obtained by lumping the kinetic energy of the beam at its tip

2KE
M = 2K- (2.25)
(flp, )2

where, Wp =w3 (L +1) is the resulting tip deflection of the beam due to its self-weight

calculated from Eq. (2.16). The natural frequency of the composite beam is calculated from the

effective mass and compliance using the expression


f =1(2.26)


Next, the electromechanical behavior of the general solution, Eq. (2.14) (2.17), is

validated for the case when the piezoceramic composite beam is subjected to an applied voltage.

In oder to validate the solution, the proof mass is neglected for this special case. This

configuration corresponds to the cantilever piezoelectric actuator described in Kasyap (2002).

The actuator deflection is determined for a test specimen comprised of a piezoceramic patch








attached to an aluminum shim. The dimensions and properties for this beam are listed in Table

2-2.

Table 2-2: Material properties and dimensions for a piezoelectric composite aluminum beam.
Length of the beam (L,) 101.60 mm
Width of the beam (b,) 6.35 mm
Thickness of the beam (t,) 1.02 mm
Elastic modulus ofPZT (E) 62 GPa
Density ofPZT (pp) 2500 kg/m3
Length of the PZT patch (L,) 25.40 mm
Width of PZT (bp) 6.35 mm
Thickness of PZT (tp) 0.51 mm
Piezoelectric coefficient (d31) -274 X 10-12 m/V
Relative permittivity (-,) 3400


Rewriting the static solution from Eq. (2.14)

loads are neglected in the composite beam yields


(2.16) for the case when all mechanical


(2.27)


w2(x)= w3(x) P x (2.28)
(EI),I 2 )

Figure 2-8 compares the deflection using the two methods mentioned above. As indicated

in the figure, the modeshapes match exactly, which indicates that the electromechanical static

model accurately represents the structure in the absence of a proof mass.


MI \ x2}
w (x)I=- -
(E), 2
































x/L +1
S


Figure 2-8: Deflection modeshape for a composite beam subjected to an input voltage.

Next, we can use the modeshape for the piezoelectric actuator with a proof mass to

calculate the effective piezoelectric coefficient, which is defined as the tip deflection resulting

from an applied unit voltage. Since we need to obtain the electromechanical coupling between

the input voltage and the resulting deflection, the static deflection of the composite beam due to

all the mechanical loads is subtracted from the overall deflection. However, we assume the

system to be linear, and the solutions can be superimposed. Consequently, the effect of the

voltage on the deflection can be decoupled from the overall equations. Therefore, the resulting

tip deflection due to an input voltage is given as


d -W -- A L L (2.29)
vapp I) 2









where, wV is the tip deflection due to the applied voltage. After obtaining Mm, Cm,, and dm,

the rest of the parameters in the circuit, such as Rm, 0, Ceb and R, can be easily obtained as

shown below, since they are simple analytical expressions related to these elements. In the

electromechanical circuit shown in Figure 2-2, q is defined as the turns ratio for the

transformation between the electrical and mechanical domains and is given by the expression


Sm- (2.30)
Cms

Next, as described earlier Ceb is given by



Ceb =f I- C ,,, (2.31)
Cef-ms

where Ce is the free capacitance of the piezoceramic


A
Cef = E (2.32)
P

Here, E is the dielectric permittivity in the piezoceramic and Ap is the surface area of the

piezoceramic. The resistive elements in the circuit are calculated using Eqs. (2.1) and (2.2).

Therefore, all but two of the lumped element parameters in the circuit have been analytically

obtained from the static electromechanical model. Only the mechanical damping and electrical

loss are estimated using the empirical relations provided in Eqs. (2.1) and (2.2). It should be

noted here that a viscous damping model assumed in this model, represented with an effective

damping ratio does not capture all loss mechanisms. A more detailed in-depth study of various

damping mechanisms is provided in Appendix B. The empirical relations and the estimated









values for the tested MEMS devices are also presented in Table 6-12 along with their

experimentally extracted damping.

Now, we can represent the composite beam in the lumped element circuit and simulate it

for various loading conditions. In the overall configuration, the input to the system is an

effective acceleration applied at the clamp, which is replaced with an equivalent inertial force in

the circuit (Yazdi et al. 1998). This effective force in the single degree of freedom system is

defined as the product of the effective mass and the acceleration of the center of mass of the

system. In this analysis, it has been assumed that the input acceleration is equal to the

acceleration of the center of mass. As will be demonstrated, this assumption has proven to be

fairly accurate in predicting the dynamic response until the first resonance with experimental

results. Therefore, the equivalent force is given as

F =Ma, +Mg. (2.33)

In the above equation, the first term corresponds to the dynamic input force due to applied

acceleration at the clamp. The second term is the static load on the beam acting due to gravity

which indicates the static deflection of the beam. This term is, however, not used for dynamic

simulations to predict the output voltage and current in the equivalent circuit. Therefore, the

input dynamic mechanical power is given as

,, = FU, (2.34)

where U is the relative instantaneous velocity of the tip with respect to the base. In Figure 2-2,

the device is connected to an external load circuit to reclaim power in a real application. Solving

for the input velocity in the circuit from U = F/overall impedance, we obtain


1 Mao (Zeb +RL)
U ZbRL ( (2.35)
/ ZebRL +Z" (Ze +RL)









In the above expression, R, is the external load which is assumed to be purely resistive in

our analysis (Taylor 2004), Z, is the blocked electrical impedance, and Z' is the short circuit

mechanical impedance represented in the electrical domain that are given by the following

expressions


Re

Z'= j-)cb (2.36)
-+R
joeb

and

ZR = j)M +R, +- (2.37)


where ) is the frequency of excitation in [rad/s]. Therefore, the input power supplied to the

composite beam is obtained from Eq. (2.34) as



SZ R (2.38)
\, = Z ebL + Z"
eb +RL

The input power to these structures when calculated using Eq. (2.38) based on the input

base acceleration gives an understanding about the amount of mechanical energy available for

conversion to the electrical domain. The conversion of mechanical power to electrical energy is

related to the coupling factor defined as

d2
K2 = m (2.39)
Cefms

The coupling factor determines the amount of electrical energy available in the

piezoceramic that can be reclaimed (Ikeda 1990).









All the analytical expressions for the electromechanical circuit elements have been derived

and presented for a composite beam to complete the circuit in Figure 2-2. Since all the lumped

parameters excluding the damping ratio in the circuit are obtained analytically and are dependent

on the material dimensions and properties, a detailed scaling analysis is carried out in the next

chapter to provide a motivation for designing MEMS devices. Furthermore, a simple design

strategy is presented to model and design these devices for characterizing energy reclamation

from vibrations.

Experimental Verification of the Lumped Element Model

This section summarizes meso-scale experiments to validate the lumped element model

and the corresponding electromechanical circuit for power generation. First, results are

presented for experiments carried out to verify the electromechanical lumped element model for

its mechanical and electrical behavior. Finally, overall power transfer estimates are obtained

experimentally and compared with the theoretical predictions to validate the model.

Initially, experiments were conducted with a clamped aluminum beam that was mounted

on a vibrating surface (LDS dynamic shaker model V408) to verify the dynamic and static

lumped element model. The material dimensions of the test specimen used are listed in Table 2-

3. The properties are listed in Table 2-1

Table 2-3: Material properties and dimensions for a homogenous aluminum beam.
Length of the beam 127 mm
Width of the beam 6.35 mm

Thickness of the beam 1.02 mm

Length of the proof mass 3.17 mm

Width of proof mass 6.35 mm

Thickness of proof mass 8.64 mm









All the lumped element parameters (Figure 2-2) were obtained experimentally to validate

the model as follows. Static tests were carried out by loading the tip with known masses (that

were measured using an OHAUS mass balance with resolution of +0.1 mg) and the tip

deflection was measured using a Micro-Epsilon laser displacement sensor (OPTONCDT series

2000). An average compliance of the composite beam was obtained by calculating the ratio

between the resulting tip deflection and the static load at the tip for all the masses. The difference

between the estimated and calculated theoretical value using the properties and dimensions is

listed in Table 2-4. A simple impact test was carried out to obtain a damped impulse response to

estimate the natural frequency of the specimen. The natural frequency thus obtained using the

logarithmic decrement method (Craig 1981) was 50.5 Hz. From the measured natural frequency

and the effective compliance, the effective mass was calculated to be 0.523 gm.

Table 2-4: Measured and calculated parameters for the homogenous beam.
CALCULATED MEASURED UNCERTAINTY

Effective mass of the 0.540 gm 0.523 gm 3.1 %
beam, MM
Effective compliance of 0.018 m/N 0.019 m/N 5.5 %
the beam, CMS
Natural Frequency, Fn 50.9 Hz 50.5 Hz 0.8 %

The structure was then mounted on a vibration shaker as shown in Figure 2-9 that was used

to excite the composite beam over a frequency range.

































Figure 2-9: Experimental setup for verifying the electro-mechanical lumped element model for
meso-scale cantilever beams.

The input acceleration to the structure was measured using an impedance head (Bruel &

Kjaer type 8001). The resulting tip deflection was measured using the displacement sensor. To

check mass loading effects in the impedance head, the input acceleration measured with the

impedance head was initially compared with the results obtained from a displacement sensor

measurement at the same point. It was observed that the results matched very well over the

frequency range. The measured resonant frequency and the compliance were then used to adjust

the mass of the model to match the predicted natural frequency. Figure 2-10 shows a plot of the

frequency response function between the tip deflection (measured with the displacement sensor)

and input acceleration. The magnitude, phase and coherence are indicated in the plot along with

a comparison with the lumped element model predictions. The results were found to match well

until beyond the first resonance. The parameter plotted in the figure is the transfer function









between the input acceleration and resulting tip deflection. The observed resonant frequency was

50.5 Hz. The frequency response using the LEM is calculated using the expression


w (-) 1 1 (2.40)
1 + j(C OwM1 + Rm j


The damping ratio was estimated to be 0.005 by matching the response peaks at the resonant

frequency.


- 100

20
l10.

E 10
0
-ntnt


UU


........20 40 60 expt
. 40 -- ELEM




20 40 60 80 l


r~:v


0 20 40 60 80 100
1 ...----------l----1-------


0.6
o
0 -

0 20 40 60 80 100
frequency (Hz)

Figure 2-10: Comparison between experiment and theory for tip deflection in a homogenous
beam (no tip mass).

As indicated in the above plots, the response was accurately predicted using the lumped

element model. Similar experiments were carried out for a homogenous beam with a known

proof mass attached at its tip. The addition of the tip mass to the system leads to an inertial tip

load during the vibration. This load will act as shear force along with the static load of the tip


-Inn










mass due to gravity. The measured tip mass was 0.476 g + 0.1 mg. A similar vibration

experiment was carried out for this device, and the tip deflection was measured as a function of

frequency for an input acceleration to the clamp. Figure 2-11 shows a comparison of the

frequency response function between theory and experiment. As seen in the figure, the plots

match well and the resonant frequency was measured to be approximately 37 Hz. The reduction

in the resonant frequency is due to the addition of the tip mass.


100
......... expt
E -- LEM

E .........................
1 0 10....


E 10.
0 20 40 60 80 11
200


-2001
0
1.

0
o
S0.5
0
A ..


20 40 60 80


0 20 40 60 80 100
frequency (Hz)

Figure 2-11: Comparison between theory and experiments for the tip deflection in a homogenous
beam with tip mass.

Table 2-5: Measured and calculated parameters for the homogenous beam with a proof mass.
CALCULATED MEASURED UNCERTAINTY

Effective mass of the 0.948 gm 0.974 gm 2.7 %
beam, MM
Effective compliance 0.020 m/N 0.019 m/N 5 %
of the beam, CMS
Natural Frequency, Fn 36.4 Hz 37.0 Hz 1.6 %









The estimated and calculated lumped element parameters for the homogenous beam with a

proof mass are listed in Table 2-5 along with the estimated uncertainty in the values.

After verifying the lumped element model for the two cases mentioned, they were

extended to a piezoelectric composite beam. The dimensions of the piezoelectric composite

beam are listed in Table 2-6. The same tip mass was used for these experiments.

Table 2-6: Material properties and dimensions for a )iezoelectric composite aluminum beam.
Length of the beam 103.38 mm
Width of the beam 6.35 mm

Thickness of the beam 0.51 mm

Elastic modulus of PZT 66 GPa

Density of PZT 7800 kg/m3

Length of the PZT patch 25.40 mm
Width of PZT 6.35 mm

Thickness of PZT 0.51 mm

Piezoelectric coefficient -190 X 10-12 m/V

Relative permittivity 1800

Length of the proof mass 3.17 mm

Width of proof mass 6.35 mm

Thickness of proof mass 8.64 mm


Before performing the vibration experiments, the static loading test and the impact test

were conducted as before to measure the mechanical compliance and the natural frequency.

These values are compared with the theoretically calculated parameters using the dimensions and

properties in Table 2-7. The relative uncertainties are observed to be higher than the

homogenous beam and can be attributed to the bond layer and the uncertainties in the PZT

dimensions (Mathew 2001). A detailed uncertainty analysis along the lines of what was









described in Kasyap (2002) can be carried out to obtain better estimates. However, for the

purpose of this validation, we use the values measured.

Table 2-7: Measured and calculated values for a PZT composite beam.
CALCULATED MEASURED UNCERTAINTY

Effective mass of the 0.183 gm 0.176 gm 3.9 %
beam, MM
Effective compliance 0.038 m/N 0.041 m/N 7.8 %
of the beam, CMS
Natural Frequency, Fn 60.55 Hz 59.25 Hz 2.1 %

Effective piezoelectric -1.28e-6 m/V -1.18e-6 m/V 7.8 %
coefficient, deff
Blocked electrical 5.01 nF 4.86 nF 3.1 %
capacitance (Cef = 5.06 nF) (Cef = 4.88 nF)


A vibration experiment was then carried out using the composite beam by mounting it on

the vibrating shaker. The clamped base was harmonically excited, and the tip deflection was

measured using a displacement sensor. Figure 2-12 plots the frequency response and a

comparison with the LEM. The response was observed to match well through the first

resonance. The mass of the composite beam was calculated using the compliance and measured

natural frequency. Some of the reasons for higher discrepancy in the composite beam are

attributed to the fact that the LEM does not include the epoxy bond layer that was used to attach

the PZT with shim. In addition, it does not incorporate the small gap that is provided between the

clamp and PZT to prevent any potential shorting during vibration. However, this will not occur

in the MEMS device as the clamp will form a part of the substrate itself. This will be understood

better in Figures 4.16 and 4.17.











10 ....... expt
C -- LEM
10
10



E 10-4
0 20 40 60 80 10
200


0


-200

1


0 20 40 60 80 10
I ,*"


0 20 40 60 80 100
frequency (Hz)

Figure 2-12: Frequency response of a piezoelectric composite beam (no tip mass)

A similar experiment was conducted with the composite beam that has a proof mass

attached to its tip. The measured and calculated parameters for the specimen are listed along

with the uncertainties in Table 2-8.

Table 2-8: Measured and calculated parameters for a PZT composite beam with a proof mass.
CALCULATED MEASURED UNCERTAINTY

Effective mass of the 0.598 gm 0.623 gm 4.1 %
beam, MM
Effective compliance 0.043 m/N 0.041 m/N 4.6 %
of the beam, CMS
Natural Frequency, Fn 31.37 Hz 31.50 Hz 0.5 %

Effective piezoelectric -1.30e-6 m/V -1.18e-6 m/V 7.8 %
coefficient, deff
Blocked electrical 5.01 nF 4.86 nF 3.1 %
capacitance (Co = 5.06 nF) (Co = 4.88 nF)











Figure 2-13 shows a plot of the frequency response function between the tip deflection and

input acceleration and, compares with the theoretical predictions using static LEM. As is evident

from the plot, the response matches well with the predictions and the resonant frequency was

found to be at 30 Hz.



100
S........ expt
SLEM
E 10-2 ........_.


E 10-4
0 20 40 60 80 100
200


-1)

C-
r 0

a.
-9nn


0 20 40 60 80 100
1 o... ......, .. ...... ......




0 *:
~ 0.6
0
0 4

0 20 40 60 80 100
frequency (Hz)

Figure 2-13: Frequency response for a piezoelectric composite beam (mp=0.476 gm).

Based on the above observations, it can be concluded that the lumped mechanical model

that was developed is sufficiently accurate in predicting the dynamic behavior of the meso-scale

composite beam.

To further verify and validate the electromechanical LEM, the same PZT aluminum

composite beam (without the proof mass) was characterized for both its mechanical and

electrical response. The dimensions and properties of the composite beam are listed in Table 2-6

and therefore not reproduced again.


. .

?i'"" "* ""* y

________________ ^_____________________________'!*









All the lumped element parameters obtained previously in Table 2-7 are used for

subsequent validation. The damping ratio for the system was adjusted to match the peaks at

resonance in the response obtained both experimentally and the lumped element model (Figure

2-12). The resulting damping ratio was estimated to be 0.015 and this value was subsequently

used in the analysis. The increase in damping ratio from a homogeneous beam to the composite

beam is attributed to the added losses in the piezoelectric material and the epoxy layer.

To measure the effective piezoelectric coefficient, an ac voltage was applied to the PZT

and the resulting response at the tip was measured using the laser displacement sensor. Ideally,

the deflection needs to be measured at dc, but since it is difficult to perform this experimentally,

the response was measured at very low frequencies (- 20 Hz) where the response is flat. This

value was used as the effective piezoelectric coefficient (di) for subsequent calculations.

The free electrical impedance of the composite beam was measured using a vector

impedance meter, and an effective free capacitance was obtained as a result. However, the value

for dielectric loss was not measured experimentally and an empirical relation was used (Eq. 2.5)

to estimate its value. Therefore, all the lumped elements that can be estimated experimentally

were thus obtained and these values were used in the lumped element model to generate the

overall response and predict its output characteristics. The resulting values are shown in Table

2-7 and were compared with the theoretical values. Reasonable agreement (better than 8%) was

obtained between the measured and calculated values.

To validate the electrical behavior of the composite beam, another experiment was

conducted wherein the resulting voltage across the PZT was measured as a function of the input

acceleration at the clamp. The frequency response function thus obtained is shown in Figure 2-









14. As indicated in the figure, the model matched well with the measured response indicating the

validity of the complete lumped element model.





0
10



E I I I I
0 20 40 60 80 100
200

0
.... ---LEM
r-
-200
0 20 40 60 80 100
1
C
P 0.5
0
0 20 40 60 80 100
frequency (Hz)

Figure 2-14: Output voltage for an input acceleration at the clamp.

Finally, after verifying the lumped element model for the frequency response, a sinusoidal

acceleration signal was input at resonance, and the resulting output voltage was measured across

a range of resistive loads to measure the output power. The results for the measured RMS

voltage are shown in the following figure. The plot indicates the voltage generated for unit

acceleration input (1 m/s2 RMS) as a function of different resistive loads varying from 10 K 2to

1 MD. As indicated in the figure, the output voltage increases and saturates to a constant value

called the open circuit voltage as the load increases. In the shown plot, the output voltage is

normalized with the input acceleration to compare with the experimental values.













1.2 -----


I+

N 0.8



S- LEM

0.4-


0.2 -









Figure 2-16 shows the output power (V2/RL) generated at the PZT calculated from the

measured voltage across the resistive loads for the same input conditions as in Figure 2-15. As

indicated in the plot, the power reaches a maximum value at an optimal resistance which occurs

when it is equal to the input impedance of the composite beam. It was observed that the optimal

load thus estimated is approximately 404 kf2 which is close to the theoretical value, 450 k(2.

These measured and calculated values for the voltage and power are listed in the following Table

2-9.









Table 2-9: Comparison between experimental and theoretical values for power transfer.
Estimated Measured

Optimal load 450 k(2 404 k(2

Voltage (per unit acceleration) 0.91 V/m/s2 0.85 V/m/s2

Corresponding output power 1.83 uW/m2/s4 1.78 u W/m2/s4


1.5





- 1


0.5


0 200 400 600 800 1000
load (kQ)


Figure 2-16: Output power across varying resistive loads.

As seen in the above plots, the lumped element model results agree within 10%. This

discrepancy is either within experimental uncertainty or acceptably small for design purposes. It

should be noted here that the mechanical and electrical damping are not accurately characterized

or known. Using these results and conclusions, the LEM is extended next to the micro-scale by

scaling down all the dimensions of the structure proportionally. The input acceleration is also

scaled proportionally to operate the composite beam in the linear region so that the model can be

used to predict its behavior. The next chapter describes in detail the scaling analysis for the









composite beam indicating the dependence of all the lumped element parameters with the

dimensions as they scale down. The next chapter also describes the motivation for fabricating

these structures using MEMS and the inherent advantages in their performance.














CHAPTER 3
MEMS PIEZOELECTRIC GENERATOR DESIGN

In this chapter, a detailed dimensional analysis is presented for the piezoelectric composite

beam. Then, a scaling theory is developed based on the dimensional analysis to determine the

response of the structure when it is scaled down in size. The objective behind developing a

dimensional analysis and scaling theory is to provide a tool that enables better understanding of

the device behavior as a function of dimensions and properties. In addition, it can be used as a

tool to optimally design a first generation device aimed at specific applications. Next, a design

strategy is formulated for the composite beam based on a given set of input parameters. In

addition, each of the proposed designs is optimized using a parametric search procedure

described in this chapter, subject to design and fabrication constraints, but without any

conventional optimization techniques.

Power Transfer Analysis

Recall that all of the lumped parameters calculated in the previous chapter, with the

exception of the empirical damping coefficient, are analytical functions of the material properties

and device dimensions. The equivalent circuit model for the composite beam can now be

attached to an external circuit to harness power. The external circuit has an electrical impedance

associated with it, which determines the amount of power that can be reclaimed from the

composite beam. For the sake of our simulations, we assume that the external circuit is purely

resistive (Taylor et al. 2004; Horowitz et al. 2002) and is represented as shown in Figure 3-1.

Since most energy harvesters seek to reclaim and store energy (e.g., via a battery) that is later

dissipated, a resistive load works best for analyses. In addition, most energy reclamation circuits









present a purely resistive load to the generator. Figure 3-1 is the overall equivalent

electromechanical circuit drawn as its Thevenin equivalent. From elementary circuit analysis, it

can be proven that maximum power transfer occurs when the complex load impedance Z, is the

complex conjugate of the Thevenin impedance ZH (derived in Appendix D). In the present

case, in which the external circuit presents a purely resistive impedance, the optimal load

resistance R, equals the magnitude of the Thevenin impedance in order to maximize power

transfer (Appendix D). The Thevenin voltage is defined as the open circuit voltage, and the

Thevenin impedance as the short circuit impedance (Irwin 1996) across the output and is

calculated from the original representation shown in Figure 2-2.


ith Zl









Figure 3-1: Thevenin equivalent circuit for the energy reclamation system

In the circuit, Vh is the equivalent Thevenin voltage, which is



j oCeb F
R+ 1 C
VTh e ceb, (3.1)

jCo)Cb jMo) m 1 Rm
+ 1 02 jo)2C.m 02
Re C+
jo Ce b









and the Thevenin impedance, Zr, is


+ 1 I 2 02C. 02
jM j 1 R1


Z eJ= Cb (3.2)

jCo) C jo), 1 R
S+ + +
R + 1 J2 j+ot2Cm f2


All the parameters in the above equations are defined in the equivalent circuit in Figure 2-2.

Consistent with the above discussion, we assume that the output load is optimal and is therefore

equal to the Thevenin impedance as given by the expression (Appendix E)

RL = Zr (3.3)

The current across the load can be obtained from Ohm's law as


IL = VTh (3.4)
Z,, + RL

The rms power across the load is defined as the product of the load and the square of the rms

current, given by the expression

12
PL rs 2 L RL (3.5)

where ILRL is the voltage across the load. In addition, the input rms mechanical power to the

device was calculated using the expression


P I = eU = F U. (3.6)


From Eq. (3.5) and Eq. (3.6), we can calculate the overall electromechanical efficiency of the

power transfer across the resistive load










P I, 2
r (3.7)
P9n FU

Substituting the lumped element expressions for these parameters in terms of material

properties and device dimensions provides the desired scaling dependence of power and

efficiency but results in expressions that, because of their algebraic complexity, do not provide

any significant physical insight. Instead, dimensional analysis is used below for the scaling

analysis to optimally design an energy reclamation device and corresponding external circuit that

can harness maximum energy from the piezoelectric composite beam.

Nondimensional Analysis

A list of the all the variables in the electromechanical model are listed below in Table 3-1

that describe the dynamic behavior of a piezoelectric composite beam. First, a set of primary

variables are selected that incorporate the basic dimensions such as length, time, etc. Next, all the

other variables used to describe the composite beam are expressed as nondimensional groups.

These groups are later used to nondimensionalize the response functions such as modeshape,

LEM parameters, etc. providing the dependent H groups.

A schematic of the device with all dimensions and properties is shown below in Figure 3-

2. The dimensions and properties have already been discussed in Section 2.2.1.












bs=bp


Figure 3-2: Schematic of the MEMS PZT device.


Table 3-1: List of all de e


variables that are des l


Variable Description
E, p, Material properties of shim
Ep ,pd31, r Material properties of piezoelectric layer
Em, p Material properties of proof mass
Ls, b, ts Geometric dimensions of shim
L b tP Geometric dimensions of piezoelectric layer
/, w, h Geometric dimensions of proof mass
tan 3, 4 Dielectric loss tangent and mechanical damping coefficient
ao, f Vibration parameters, acceleration and frequency


Due to the fabrication process that was designed for the devices, the following conditions

hold true, namely,


b= bp =b


(3.8)


The width of the piezoelectric layer and shim are assumed to be same to simplify the analysis. In

addition, the shim and proof mass are assumed to be made from the same material. Therefore,


ES =Em and p,= pm


Es,-S


Ep,--,d31?-^


(3.9)









For the scaling analysis carried out here, we make the following assumption that simplifies

the derivation: namely, tp and h are fixed in the analysis due to fabrication constraints that

restrict the thickness of the piezoelectric layer and the proof mass. The thickness of the proof

mass is formed from the substrate and therefore is equal to the wafer thickness. The thickness of

the PZT layer was restricted by ARL process capability, which was 1 um at the time. Listing the

remaining variables, we obtain the following tabulated parameters with their dimensions as

indicated in Table 3-2.

Table 3-2: Dimensional representation of all the device variables.
Variable Dimensional units
Es ML 'T-2
P, FML-3
E, [ML 'T2
P, IML-
d31 'M-1L 'T2Q
E M 1-L-3T2 2
LI [L]
b [L]

Lp [L]
L [L]

/ [L]
ao LT-2


For the dimensional analysis, the following independent primary variables were defined.

These parameters were chosen to include the primary dimensions of length, time, mass, and

charge. All the other parameters will be expressed using these primary variables.









Table 3-3: Primary variables used in the dimensional analysis.
Independent Dimension
variable

L [L] corresponds to "length" dimension

P, L 3M 1 includes the "mass" dimension
E ML T T21 includes the "time" dimension

d31 M1L 1T2Q] corresponds to the "charge" dimension


The remaining variables are now nondimensionalized using the 4 repeating variables to

obtain independent dimensionless "H" groups as listed below. These H groups will be used to

nondimensionalize the piecewise deflection solution obtained in Eq. 2.9-2.11. Furthermore, the

analysis will be extended to nondimensionalize the LEM parameters in the equivalent circuit

model to finally investigate the device behavior for various topologies.


E
= n,
E


PP= 2
A,


(3.10)



(3.11)


A F
We know that E = Ce which implies that E is dimensionally represented as .
t

Therefore,


S=[M-L 3 2Q2]

and the corresponding nondimensional H group is


E 3
Ed21


(3.12)





(3.13)









Equation (3.13) is a measure of the coupling between the electrical and mechanical domain

and in the ideal case, can be reduced to Eq. 2.33. Furthermore, the device dimensions are scaled

as


(3.14)



(3.15)


(3.16)


(3.17)


From Eq. (3.15) and Eq. (3.17), we obtain


L


-A7
115


(3.18)



(3.19)



(3.20)


From Eq. (3.14) and Eq. (3.20),


w w Ls
b L b

h
L

From Eq. (3.15) and Eq. (3.22),


A9
174


(3.21)



(3.22)









h hL A
h h L 1= 10 (3.23)
t, L t n, 1

Similarly, the external vibration parameters, such as the acceleration and excitation

frequency, are nondimensionalized as

a
a = Hn (3.24)



fe =I12 (3.25)

L, / I P,1 n

All the above derived independent primary H groups are listed below in Table 3-4. Next,

the location of the neutral axis in the composite section (0 < x < L) measured from the bottom

can be expressed as (Chapter 2)

Et 2 +2E t t +Et 2
c2 =PE t (3.26)
2(Et, +EPtP)

Dividing the above expression by E L2 and nondimensionalizing with respect to shim

thickness yields


t 2 +2 t tP t2

S2 + E t
L, E L,

Similarly, c, is denoted as the position of the neutral axis from the top


c t, +t C2 (3.28)

and in nondimensional form becomes









t C
= 1+ =2 f(A, ,A 7)= f(n1,n 5,7)
L L, L L


Table 3-4: List of independent H groups.
H group


Dependent variables


The bending moments of inertia in each section, 1 1,s21 p and Im are nondimensionalized


(3.30)


I = 3 +(Cp


(3.29)


1n E

H2

n1 3
3 E 321





Hn L P


t

Hi
H9 w

H10 h

11 9



12 e 1
1 E j
171 12










b c1 +

3L
3


- = f (r,,,47 5, 7)


' 3 (c2
I =3 c2
3-


3
b C2 C2


3


- = f(H,,H4n,, n,,7)


bt 3
2 = 12
12


: f (H4, n5)


wh3
I 12
" 12


w h
L1
12


f/(r4,7, 5, 9, n10)


The linear mass densities of each section are now nondimensionalized as


Plen = pt b


pLen
PsL


t, b
L L
S S


f (ni4,75)


Pien, =(Ps, +ptp)b


Plens
PA


+ PP
Ls p,


P- f ( 2,14,15,17


Plen = Pmwh


(3.38)


(3.39)


(3.40)


(3.41)


(3.42)


(3.31)


(3.32)


(3.33)



(3.34)


(3.35)



(3.36)




(3.37)









Plen p wh
= -- = f( 4,,1151g,910 (3.z
pL p, L L

Next, the rigidity moduli in each of the beam sections are nondimensionalized as


(EI) = EJ~,2 (3.z


(EI) 2
EL4 L4


(EI)
EL4


f (n4,1 5)


(EI) = Ejl, +EI


I1 EP IP
L- E L f(,


3)


\4)


(3.45)


(3.46)


(3.47)


(3.48)


(EI), = E


\tl)m Em I
EI, E_ Lm= f(H4,Hs,9 H1 ) (3.z
EL4 E L4

Let us now define another nondimensional parameter for acceleration due to gravity as


g
E
L' Pj


(3.50)


Nondimensionalizing the uniform line load on the beam due to its weight, we obtain


q = qen g (3.51)


p EP
) yLp


f (n4,1 n) 5


q, = Een, g



Plenc I =f ( 2,, 74, 1,, 1 ,) Y


19)


EL,


(3.52)


q,


(3.53)



(3.54)


/T-,T\









qm = gq,(. g


L- p

In the static solution for the deflection of the composite beam described in Chapter 2, we

defined two constants

(El)
C = ) = E L f (1, 4,1 5, 7) (3.57)
(EI) (El
/Es2

and

(EISm
D = (E- = f(H4', 5, 910) (3.58)
(EH) (E)
E L2

The Euler-Bernoulli equations that were solved earlier to obtain a piecewise continuous

deflection modeshape for the beam (in Chapter 2), are rewritten here as

x4 2 T T\ 3 L2( L2 L+1 x +M 2
-q -+ L +qjL, -L +q c\- q, c +q,-- flL+- \L, +Mo
24 6 2 2 2 2 2
w, () =El)

(3.59)


2q sx 4 { q sL s q m l I X 3 q q m / L }ml( )
+\+ 2 2 + (l +C-3)" + 2 C 44
24 (El) 6(El) 2 (El) (El), (El)
(3.60)

and


qm x4 qm,(L + l)x3 q (L + 1) 2 2 (C+C7)x C6+C
W3 24 (X)(= + + + (3.61)
24(E1)m 6(E)m 4(El)m (El)m (El)m


(3.55)









where the integration constants C1,C2,C3,C4,C5,C6, C7 and C, are given by the following


expressions


q,a) +3q,L, (L,


Li) (1 a)],


C2 = [L (q -q, (3 -a))+ qL (8Lp

C3 mq (1- a)(2L,- L +l) 2aM0

C4( L 1 -a)LP L
2 3L42


6L,)(1- a)],


C5 = L C q

C6=r L2+


C7 = C3+ qlL (L +)(1- ),
6 2
C8 = C4 I [qmI'L + qmlL (1- y)(4L, +6l/)].


Removing the effect of moment due to applied voltage to the piezoelectric layer and

nondimensionalizing the deflections with length yields,


f (1, 12, n 4,1 5 ,16, n 7, 8, n 9, 7n10,f)E


Substituting for w, = w3 (L, +1) and nondimensionalizing, we obtain


'PF f(1,z1,2, 4,15,116117,1181 9 10)E9


Integrating the deflection across the length of the beam results in the total potential and

kinetic energies, represented as


C, =L (q,


(3.62)


(3.63)


(3.64)


,


w, (W) w, (V) W3 (3)









(EI) L) dw2 1x) (2 E) 2 (d2w2 (x)2 (E) L d ()d2
PE=- f +dx + f dx + m dx (3.65)
Po 2 dx2 2 dx 2 2 dx 2

and

K = pn Lp P Ls Pn Ls+l
KEf 2 d l Pen d eni 2
2KE 2(x)2dx 3(x)dx (3.66)
o LP L,

Nondimensionalizing the energies yields

PE KE
L= f ( (H1,H2,4,15,H6,H7,H8,10,H, -)E2 (3.67)
EsL3 pLs

From the energies, the short circuit compliance and mass are extracted and

nondimensionalized as


CnELs= m = f(r,1r2, 4 5 H6 7 8 ,nnnH,0) (3.68)
pL3,
PsL

The natural frequency shown in Eq. 2.24 is nondimensionalized as


f _1 1 = f(I, 1 2, 4 ,h 6,,8,,7 n g,n10) (3.69)
1 E 2r pL Ms
SE s s L

Furthermore, the angular frequency c (= 2-rf) is a function of the same H groups and

follows Eq. (3.69). The Rayleigh mechanical damping in the system using its empirical relation

listed in Eq. 2.1 is nondimensionalized as


Rm 2 M2 m f(H,,H2 4 5 6 7 10 ) (3.70)
L2S) Vs /


It should be noted here that the mechanical damping model in the system does not

accurately represent all loss mechanisms. Some of the general damping losses were investigated









and presented in Appendix B. Although the LEM in this dissertation still assumes a Raleigh

viscous damping effect with an equivalent damping ratio, other damping losses are also studied

here for their scaling behavior. The loss due to air flow in the viscous region due to device

vibration is given by an empirical relation derived in Appendix B as



Qa = (3.71)
6ZuRL + 1


All the variables are defined in Appendix B. For any structure operating at fixed conditions,

kn and u/ remain constant. So, the nondimensional form simplifies to


k Ls~ E (n1)2 4
Q = + (3.72)
a 120, p J Revi

LL


The losses at the support that arise due to the transmitted energy through the clamp during

flexural vibration is empirically given as

3
Q = 0.23 (3.73)
St,

and in nondimensional form is


Qc =0.23K1 (3.74)


The surface losses are given as


Q E(3.75)
S23(3b+t,) AE,









where AE is the difference between the adiabatic and isothermal Young's modulus of the

material. It is also known as the dissipation modulus of the surface layer whose thickness is

given by 3. The nondimensional form of Eq. (3.75) is


Q1 (41 E (3.76)
2 L (3r4 +r5) AE

Similarly, the volume losses are


Q, = E (3.77)
AE,

For scaling purposes, we assume that the dissipation modulus scales proportional to the elastic

modulus of the material. In this analysis, the squeeze film damping is neglected as the vibrations

occur in free space without influence of walls around the device. Finally the empirical form for

thermoelastic loss in a vibrating structure is given as

k 1 E
Qt = 80 k (3.78)
p,CP t, AE

where k is thermal conductivity and C,, the specific heat capacity at constant pressure for the

material. This expression in Eq. (3.78) is a simpler form of the actual expression derived in Eq.

B.22. Expanding for the natural frequency and simplifying leads to


k L2 JEA
Q, = 80 (3.79)
PsCP t A-E

The nondimensional form of the quality factor can be expressed as

k E/ 1
Qt = 80 (3.80)
nIA C p E tA