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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 coadvisor 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 SelfPow 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 PZTEH07 ........................ .......................... 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 EULERBERNOULLI BEAM ANALYSIS: VARIOUS BOUNDARY CONDITIONS..237 E uler B ernoulli B eam .......................................................... ................. 237 Cantilever Beam (ClampedFree Condition) ............................................................237 ClampedClamped Beam (FixedFixed Condition) ...................................... 239 PinPin 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 11 Conjugate power variables for different energy domains.........................................26 12 Vibration based energy harvesters characterterized for power............... ...................40 21 Material properties and dimensions for a homogenous aluminum beam ........................61 22 Material properties and dimensions for a piezoelectric composite aluminum beam.........65 23 Material properties and dimensions for a homogenous aluminum beam ........................70 24 Measured and calculated parameters for the homogenous beam...................................71 25 Measured and calculated parameters for the homogenous beam with a proof mass........74 26 Material properties and dimensions for a piezoelectric composite aluminum beam.........75 27 Measured and calculated values for a PZT composite beam.........................................76 28 Measured and calculated parameters for a PZT composite beam with a proof mass........77 29 Comparison between experimental and theoretical values for power transfer ...............82 31 List of all device variables that are described in the electromechanical model .................88 32 Dimensional representation of all the device variables. ............. ..................................... 89 33 Primary variables used in the dimensional analysis. ................................. ............... 90 34 List of independent H groups. ............. ................. ........................................................... 93 35 Final set of nondimensional groups involving response parameters........................... 109 36 Material dimensions and properties of composite beam for FEM validation................ 117 37 Static lumped element parameters from FEM and LEM to validate the scaling a n a ly sis ................... ......................................................................... 12 1 38 Properties and dimensions used for designing MEMS PZT devices.............................123 39 Material properties of piezoelectric composite beam. ............................................... 128 310 D designed M EM S PZT structures ......................................................... ..................... 129 41 Residual stress measurements for the PZT pattern process (source : ARL)....................133 42 DRIE recipe conditions for top side etch........... ........................... .. ............... 136 43 DRIE recipe conditions for back side etch ............................. ...................... 142 44 Process traveler for the fabrication of micro PZT cantilever arrays.............................147 51 Reported polarization results (ref: ARL) ................. ............... ............ ............... 156 52 Data acquisition parameters for mechanical characterization................. ............... 163 53 Data acquisition parameters for mechanical characterization................. ............... 164 54 Data acquisition parameters for mechanical characterization................. ............... 166 61 Comparison of ARL's reported hysteresis parameters with measured values.................178 62 Dielectric parameters of all tested design geometries on the device wafer. .................... 182 63 LEM parameters extracted using experimental data.......... .... ............... 185 64 LEM parameters extracted using M ethod 1.......................................... ............... 197 65 LEM param eters extracted using M ethod 1.......................................... .....................201 66 LEM parameters extracted using Method 3................................................ 206 67 LEM parameters extracted for PZTEH0901 ..... ........................... 207 68 Extracted LEM parameters for PZTEH0903.................... .........212 69 LEM parameters extracted for PZTEH0702 ...................... .................212 610 Comparison between theory and experiments for PZTEH07. ......................................215 611 Comparison between theory and experiments for PZTEH09 devices ........................216 612 Quality factors for PZT M EM S devices. .............................................. ............... 219 A1 LEM parameters and bending strain for various beams subjected to a point load. .........246 A2 LEM parameters and bending strain for various beams subjected to uniform load. .......247 LIST OF FIGURES Figure page 11 Schematic of a typical vibration to electrical energy converter.................. ...............24 12 An electromagnetic vibrationpowered generator (adapted from GlynneJones and W white 200 1). ..............................................................................30 13 Deformation of a piezoceramic material under the influence of an applied electric fi eld ........................................................... .................................... 3 3 14 A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al, 1997) .......................................................... ...................................38 15 Schematic of the proposed cantilever configuration for energy reclamation ..................44 21 Schematic of a piezoelectric composite beam subject to a base acceleration ..................50 22 Overall equivalent circuit of composite beam. ...................................... ............... 52 23 Schematic of the piezoelectric cantilever composite beam. ............................................55 24 Free body diagram of the overall configuration. ................................................56 25 Free body diagram of the composite beam where the self weights are replaced with equivalent loads. ...........................................................................57 26 Static model verified with the ideal solution for a homogenous beam solved for self w e ig h t ............. ................... .............. ........................................... 6 1 27 Static model verified with the ideal solution for a homogenous beam solved for tip lo a d ..... ........ ................... .................. ......................................... 6 2 28 Deflection modeshape for a composite beam subjected to an input voltage ...................66 29 Experimental setup for verifying the electromechanical lumped element model for m esoscale cantilever beam s .............................................................................. ....... 72 210 Comparison between experiment and theory for tip deflection in a homogenous beam (n o tip m a ss)......................................................................... 7 3 211 Comparison between theory and experiments for the tip deflection in a homogenous beam with tip mass............... .. .. .................................74 212 Frequency response of a piezoelectric composite beam (no tip mass) ...........................77 213 Frequency response for a piezoelectric composite beam (mp=0.476 gm) ........................78 214 Output voltage for an input acceleration at the clamp. ................... ............................. 80 215 Output voltage for varying resistive loads. .............................................. ............... 81 216 Output power across varying resistive loads. ........................................ ............... 82 31 Thevenin equivalent circuit for the energy reclamation system .....................................85 32 Schematic of the MEMS PZT device. ........... ............. ................. ........ 88 33 Meshed PZT composite cantilever beam for FEM validation..................... ........ 117 34 Short circuit natural frequency for a PZT composite beam.................................. 118 35 Short circuit compliance for a PZT composite beam.................................................119 36 Effective mechanical mass for a PZT composite beam .............................................120 37 Effective piezoelectric coefficient for a PZT composite beam.............. ... ...............121 38 Schematic of a single PZT composite beam. ......................................... ...............123 41 Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer.............................131 42 Sputter deposit Ti/Pt (20 nm/200 nm) as bottom electrode............................................131 43 Spin coat solgel PZT (125/52/48) over the wafer using a spinbakeanneal process.....132 44 Deposit and pattern Pt for top electrode using liftoff. ............................................... 132 45 Pattern opening for access to bottom electrode and wet etch PZT using PZTEtch m ask ......................................................... ....................................133 46 Ion milling of PZT and bottom electrode using Ion Milling mask as pattern..................133 47 Deposit Au (300 nm) and pattern bond pads using Bond Pads mask and wet etching....134 48 Sidewall profiles on topside of a 4" Si test wafer. ................................... ..................... 136 49 Wet etch exposed oxide with BOE and DRIE to BOX from top..................................137 410 Sidewall profiles for backside etching using DRIE.................................... ............... 138 411 Curved edges during backside DRIE............ ...................... ...... ................ 139 412 Onset of silicon grass during a backside etch run............... ...... .................. 140 413 Sidewall profiles for a backside etch on a test wafer................................. .............. 141 414 Pattern proof mass on the backside and DRIE to BOX .............................. 143 415 Schematic of final released device................................... 144 416 SEM pictures of a PZTEH07 released device...................................... ......................145 417 SEM pictures of a PZTEH09 released device.. ............... .............. 145 418 Sidew all profiles of released devices ............................ ........................... .... .......... 146 419 Schematic of the bottom of vacuum package for MEMS PZT devices......................149 420 Schematic of glass top for vacuum package ................................. ..........................149 421 An isometric view of the overall vacuum package............. ....... ...............150 422 Schematic of open package for MEMS PZT devices ............................................... 151 423 Picture of the open package. .................................. ......... ................................... 152 51 Schematic for ferroelectric characterization. ......................................... ...............154 52 Experimental setup for ferroelectric characterization ................................. .............. 155 53 Schematic for blocked electrical impedance measurement. .........................................158 54 Experimental setup for electrical impedance characterization. .......................................158 55 Experimental setup for mechanical and electromechanical characterization ................160 56 Experimental setup for vibration and velocity measurements with LV. .......................161 57 Experimental setup for open circuit voltage measurements. ........................................165 58 Experimental setup for open circuit voltage measurements. ........................................165 59 Experimental setup for voltage and power measurements. ...........................................166 61 A typical PE hysteresis loop for a piezoelectric material (adapted from Cady 1964).... 169 62 A typical EE curve for a piezoelectric material. .................................. ............... 170 63 Polarization, capacitance and input voltage waveforms for PZTEH0211..................171 64 H ysteresis plots for PZ TEH 0211 ...................... .... ................................................ 172 65 Pr and Vc for different applied voltages for PZTEH0211 ............................... 173 66 Normalized Ceb for PZTEH0211 during the hysteresis test. ...................................174 67 Leakage current for PZTEH0211 subjected to 10V DC......................................... 175 68 Poling of PZTEH0211 at 5V for different times............ ............................. 176 69 Poling of PZTEH0211 at different temperatures................................... ............... 178 610 Variation of Ceb and tan6 with dc bias and a constant sinusoid, 500 mV at 100 Hz. ......179 611 Variation of Ceb and tan6 with source amplitude at 100Hz. ....................... .........180 612 Ceb and er for MEMS PZT devices on wafer before release for a) PZTEH01 (106 geometries) b) PZTEH02 (16 geometries) c) PZTEH03 (15 geometries) d) PZT EH04 (14 geometries) e) PZTEH05 (16 geometries) ............................................183 613 Ceb and Er for MEMS PZT devices on wafer before release for a) PZTEH06 (150 geometries) b) PZTEH07 (12 geometries) c) PZTEH08 (22 geometries) d) PZT E H 0 9 (10 8 g eom etries)........................................................................ ..................... 184 614 Flowchart for method 1 to extract the LEM parameters from the experimental data......190 615 Low frequency electromechanical response data compared with curve fit to extract dm ................ ......................................................................19 4 616 Comparison between experiment and LEM based curve fit around resonance for a) electromechanical response b) shortcircuit mechanical response ...............................195 617 Low frequency curve fit compared with experiment to extract Ceb............................... 195 618 Comparison between experiment and curve fit for low frequency open circuit voltage response to extract M m ...................................................... .. ...... .. ............ 196 619 Experimental data and curve fits for open circuit voltage response compared around re so n a n c e ............................. .................................................................... ............... 19 6 620 Flowchart for parameter extraction using Method 2..................... ................................. 199 621 Experimental data and curve fits for open circuit voltage response and free electrical impedance compared around resonance. .............................................. ............... 201 622 Flowchart for LEM parameter extraction implementing Method 3..............................203 623 Comparison between experiment and LEM based curve fit for short circuit mechanical and electromechanical response around resonance. .....................................205 624 Experimental data and curve fits for open circuit voltage response compared around re so n an ce ............................. .................................................................. ............... 2 0 5 625 Comparison between model and experiments for PZTEH0901. 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................................................................................208 626 Comparison between model and experiments for PZTEH0902. 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................................................................................209 627 Comparison between model and experiments for PZTEH0903. 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 ....................................... ............... ............... 210 628 Comparison between model and experiments for PZTEH0904. 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 ........................................................ ............... 211 629 Comparison between model and experiments for PZTEH0702. 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 ....................................... ............... ............... 213 630 Comparison between model and experiments for PZTEH0703. 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 A2 A schematic of clampedclamped beam. .............................................. ............... 240 A3 Free body iagram of a clampedclamped beam. ................................... ............... 240 A 4 Schem atic of a pinpin beam ......................................... .............................................243 A5 Free body diagram for a simply supported beam.................................. ............... 243 Bl A simple schematic of the cantilever beam .................................. ............ ............. 251 C1 Vibrating cantilever beam in an accelerating frame of reference..............................265 D1 Blocked electrical impedance in a parallel network representation.............................. 267 D2 Blocked electrical impedance in a series network representation ................................269 El Thevenin equivalent representation connected to a external complex impedance..........270 E2 Thevenin equivalent representation connected to a resistive load..............................273 Fl Schematic of the composite beam energy harvester................................................275 F2 Free body representation of the device as a two degree of freedom system....................275 F3 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 MEMSBASED 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 MEMSbased 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, macroscale PZT composite beams were built and tested to validate the LEM. In addition, a detailed nondimensional 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 selfpowered sensors, humanwearable electronics and vibration control. Finally, it concludes with motivation for microelectromechanical systems (MEMS) and piezoelectricity as the tools for this research. An indepth 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 nonrenewable. 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 evergrowing 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 selfreliant 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 selfpowered watches, calculators, and rooftop modules for houses. Solar energy has also been harvested on a smaller scale from an array of microfabricated 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 bismuthtelluride 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 SbBi 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 humanwearable 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, bloodpressure, walking, etc. In addition, heel strike, limb movement, and other gaitrelated 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 microscale selfcontained power supply offers great potential for applications in remote systems. SelfPowered Sensors The everreducing 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 selfcontained power generators has led to the development of "selfpowered systems" that is an important application for energy reclamation, and is currently gaining widespread importance (Shenck and Paradiso 2001). Selfpowered systems possess an inherent mechanism to extract power from the ambient environment for their operation. The main objective of selfpowered 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 selfpowered systems include power generation, energy extraction, and storage. Ideally, a selfpowered 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. GlynneJones and White (2001) provide a review on available energy resources for selfpowered sensors such as vibrations, optical, thermoelectric, etc. Next, some of the relevant work carried out in the field of selfpowered sensors is examined. As mentioned earlier, heel strike is a resource for strain energy that can be electromechanically transformed into electrical power. Consequently, shoemounted 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. Mesoscale 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 mesoscale 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, selfpowered 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 vortexinduced 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 buildup 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 largescale 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 110 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 11. The device consists of a springmassdamper system acting as a single degree of freedom system with an input vibration that results in an effective displacement z(t). Figure 11: 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=qe=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 11. Table 11: 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 nonenergy 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 twoport 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 freeelectrical 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 mechanicalelectro 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 opencircuit 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 Twoport 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 twoport 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 12 (GlynneJones and White 2001, GlynneJones 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 12: An electromagnetic vibrationpowered generator (adapted from GlynneJones and White 2001). ElHami 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 'smW '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 twoport 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 microscale, 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 13, 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 13: 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 3direction, while the stress acts in the 1direction 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 onedimensional 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 twoport 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 14 for reference. Ramsay and Clark (2001) performed a detailed design study on piezoelectric energy harvesting for bioMEMS applications. Their design employed a simple geometry for harnessing energy from blood flow in the body. The proposed structure consisted of a square PZT5A 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 31mode than in the 33mode. pivot / piezoelectric membrane mass of rod nput vibration Figure 14: A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al, 1997). GlynneJones 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 macroscale 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 selfpowered sensors, namely a liquid crystal display and an infrared 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 microwatts 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 interdigitated 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 fullbridge 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 selfpowered 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 selfpowered vibration damper. Table 12 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 12: 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 @ 60110Hz 1cm3 200830 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 15. 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 15: 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, Tilx)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 mesoscale 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 nondimensional 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 selfpowered sensors. This chapter describes the static electromechanical modeling of the composite beam using conventional EulerBernoulli 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, fabricationinduced 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 21, 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 21. Energy reclamation S circuit (electric load) Clamp / V S / Cantilever Beam Proof Mass aoext / Vibrating Surface Figure 21: Schematic of a piezoelectric composite beam subject to a base acceleration. In Figure 21, 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 22. 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 22 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 22, 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. Shortcircuit 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 22: Overall equivalent circuit of composite beam. The beam is represented as a springmassdamper 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 velocitydamped resonant generators (VDRGs) or Coulombdamped 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 closedform solution is available (Den Hartog 1931; Levitan 1960). An alternate class, namely, Coulombforce parameteric generator (CFPG) is also suggested in their work that operates in a nonresonant 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 nonideality, 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 22), 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 22 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 EulerBemoulli 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 22. 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 EulerBemoulli 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 23. 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 23: 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 24, 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 24: 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 24 is assumed to be a linear system, it can be simplified and solved analytically for the deflection using EulerBernoulli beam theory. Figure 25 represents the simplified free body diagram for the composite beam. In Figure 25, the clamp is replaced with a reaction bending moment (MA) and reaction force (R). qIL q2(Lp) M r\ I I F Figure 25: Free body diagram of the composite beam where the self weights are replaced with equivalent loads. As indicated in Figure 25, 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 24 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 25, 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 25 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 EulerBernoulli 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 EulerBernoulli 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 (xL L<_ 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 EulerBernoulli 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(32a))+qL(8L 6L,)(1a)], = [ql(Ia) (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) (17), 6 2 C, =C47 [qL] +qlL(1y)(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 shortcircuit 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 selfweight. 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 21. As indicated in Figure 26, the static solutions are identical. Table 21: 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 26: 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 21, and the deflections are plotted in Figure 27. Again, the solutions match. 0 0.2 0.4 0.6 0.8 1 x/L +1 S Figure 27: 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 22. 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 22, leading to shortcircuit 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 selfweight 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 22. Table 22: 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 1012 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 28 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 28: 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 22, 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) Cefms 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 indepth 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 612 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 22, 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 22. 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 mesoscale 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 21 Table 23: 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 22) 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 MicroEpsilon 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 24. 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 24: 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 29 that was used to excite the composite beam over a frequency range. Figure 29: Experimental setup for verifying the electromechanical lumped element model for mesoscale 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 210 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 ...l1 0.6 o 0  0 20 40 60 80 100 frequency (Hz) Figure 210: 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 211 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 211: Comparison between theory and experiments for the tip deflection in a homogenous beam with tip mass. Table 25: 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 25 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 26. The same tip mass was used for these experiments. Table 26: 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 1012 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 27. 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 27: 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 1.28e6 m/V 1.18e6 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 212 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 104 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 212: 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 28. Table 28: 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 1.30e6 m/V 1.18e6 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 213 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 102 ........_. E 104 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 213: 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 mesoscale 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 26 and therefore not reproduced again. . . ?i'"" "* ""* y ________________ ^_____________________________'!* All the lumped element parameters obtained previously in Table 27 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 212). 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 27 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 214: 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 216 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 215. 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 29. Table 29: 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 216: 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 microscale 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 31. 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 31 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 22. ith Zl Figure 31: 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 22. 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 31 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 32: Schematic of the MEMS PZT device. Table 31: 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 32. Table 32: Dimensional representation of all the device variables. Variable Dimensional units Es ML 'T2 P, FML3 E, [ML 'T2 P, IML d31 'M1L 'T2Q E M 1L3T2 2 LI [L] b [L] Lp [L] L [L] / [L] ao LT2 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 33: 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. 2.92.11. 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=[ML 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 34. 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 34: 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 EulerBernoulli 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 +C3)" + 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 (= 2rf) 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 AE The nondimensional form of the quality factor can be expressed as k E/ 1 Qt = 80 (3.80) nIA C p E tA 