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DEVELOPMENT OF A MEMSBASED ACOUSTIC ENERGY HARVESTER BY STEPHEN BRIAN HOROWITZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Stephen Brian Horowitz I dedicate this dissertation to my wife, Megan. You're my sweetheart, my wife and my best friend. I love you more than words can say. TABLE OF CONTENTS Page L IS T O F T A B L E S .............................................................. .......................................... v ii L IST O F FIG U R E S .............. ............................ ............. ........... ........... viii ABSTRACT ........ .............. ............. ...... ...................... xiv CHAPTER 1 IN TR OD U CTION ............................................... .. ......................... .. Acoustic Energy Harvester Concept....................................................... ................ M otiv action ............................................. 5 B ack g rou n d ................................................................................................. . .6 Piezoelectricity ................................................ ........6 Piezoelectric M materials in M EM S ........................................ ...... ............... 6 M materials and properties ....................................................... .... ........... 7 D position m methods .......................................................................... .... .8 Piezoelectric integration w ith M EM S ........................................ .................9 ReducedOrder Modeling of Piezoelectric Materials.........................................11 Energy Absorption/Vibration Damping and Energy Harvesting ......................13 A p p ro a c h ....................................................................... 14 2 THEORETICAL BACKGROUND ........................................ ....................... 17 Overview Energy H harvesting ............................................................................ 17 Maximum Average Power Transfer .............. ...........................................20 Electrom echanical Transducers................................ ................................... 23 Lum ped Elem ent M odeling ......................................................... ............... 25 H elm holtz R esonator .......................................... ................... ........ 29 Lum ped Elem ent M odel ............................................. ............................. 29 E qu iv alent C ircu it.......... ............................................................... .... .... .... .. 33 Piezoelectric Com posite Plate ............................................................................. 35 P iezoelectricity .....................................................................36 Com posite Plate M odeling ............................................................................ 41 Lum ped elem ent m odel ........................................ .......................... 41 E equivalent circuit ............................... ........ ... .... .. ........ .... 46 Case 1: Well below the shortcircuit mechanical resonant frequency .........50 Case 2: At the shortcircuit mechanical resonant frequency.....................52 General expression vs. simplified cases.................................................54 Effects of including the radiation impedance.............................................55 Equivalent circuit parameters of piezoelectric composite circular plates ....61 Acoustic Energy Harvester Dynamic Behavior .....................................................68 A coustical Input B ehavior.............................................................. ... ............ 68 E electrical O utput B behavior ........................................... ................. ............... 71 Operation at an Optim al Frequency ....................................... ............... 76 D vice C configurations .............................................................................. 79 3 FABRICATION AND PACKAGING .............................................. ...............82 O v e rv ie w ............................................................................................................... 8 2 P process F low .......................................................82 P a c k a g in g .............................................................................................................. 9 2 Packaging Scheme ................................. .......................... ..........92 Quarterw ave resonator package ............................................................... 92 Sealed cavity package ................. ................................94 Chip to Package M counting ............................................................. ............ 94 P o lin g .......................................................9 6 4 EXPERIMENTAL SETUP ........................................ ........ .........98 Parameters and How They are Obtained ........................... ... ...............98 E xperim ental Setup D details ................................................................................ 10 1 Ferroelectric M easurem ents ....................................................................... 101 Electrically Actuated ResponseLaser Scanning Vibrometer ......................102 Acoustic Characterization Plane Wave Tube .................. ..............................103 Electrical Im pedance Characterization.............................. .............................107 Initial Deflection Measurements Wyko Optical Profilometer......................107 5 EXPERIMENTAL RESULTS AND DISCUSSION............... .............108 Ferroelectric M easurem ents ......................................................................................108 Electrical M easurem ents .......................................................... ............... 110 Electrical Output Im pedance ....................................... ............ ..................110 Electrically Actuated ResponseLaser Scanning Vibrometer ......................114 Frequency response ....................................................................... 114 L in e arity ...............................................................12 4 A acoustical M easurem ents .. ......................................................... ................... 126 Acoustically Actuated Frequency Response PlaneWave Tube ....................126 Sealed cavity package ............................................... ............... 126 Quarterw ave resonator package .............. ..... .....................................127 Acoustic Input Impedance Measurements PlaneWave Tube........................130 QuarterWave Resonator Package ............ ......................................... 133 Initial Deflection Measurements Wyko Optical Profilometer................. ...............134 Energy Harvesting Measurements ................................................................. 138 O ptim al R resistance ....... ................................... ............... ............... 138 O ptim al Energy H harvesting ..................................................... .. .. ............ .. 139 v 6 ALTERNATIVE APPLICATIONPIEZOELECTRIC MICROPHONE............ 147 7 CONCLUSION AND FUTURE WORK ............ .................... .... .............153 D evelopm ent Issues ........1.. ... ...... ........ .... .................... .. ............ ... 154 F future W ork .............. ................................................................................ 156 APPENDIX A M A T L A B C O D E ......................................................................... .......................158 le m .m ................................................................................................................... 1 5 8 e h setu p .m ........................................................................................................... 1 6 3 silicon_sw eep.m ............................................................................................. .....163 p latin u m .m .....................................................................................................1 6 3 tio2.m ........... .......... ................................ ............... 163 piezo_sw eep.m ............................................................................................. .... .....164 in itia lise .m ........................................................................................................... 1 6 4 totaldeflection forP.m ............................ ......... 164 totaldeflection forV .m .................................................. 164 initialdeflection.m ................................................................. ... ......... 165 form m atrix C M Fa.m .....................................................165 from m atrix dA .m ...............................................................166 abdpiezo.m ................................................. 166 con stantsl2 34 .m ................................................................167 so lv e r l .m .................................................................................................................. 1 6 8 B DETAILED PROCESS FLOW .................................... .................. 170 Energy Harvester Process Traveler ......... .........................170 Graphical Representation of Process Flow .........................................................172 M ask Layouts .......................................................................... ...... .............. ......... 174 P ack ag e D raw in g s ............................................................................................... 17 8 LIST OF REFERENCES ......................... ......... .........180 BIOGRAPHICAL SKETCH ............... ......... ........ ........195 LIST OF TABLES Table p 11: Typical material properties of select piezoelectric materials. ......................................7 12: Selected thin film piezoelectric materials and corresponding properties as reported in th e lite ratu re ...................................... ............................... ................ 1 0 21: Equivalent lumped elements in several common energy domains............................28 22: C onjugate pow er variables. ............................................................. .....................29 23: General material properties used in modeling ........................................................55 24: PZT properties used in m odeling. ........................................ .......................... 55 25: Chosen device configurations for fabrication.................................. ............... 81 31: Wafer bow measurements and the resulting calculated stress ....................................86 32: Measured wafer mass and calculated density of PZT. ...................... .............. 87 51: Summary of electrical impedance measurements .............................. ...............1.13 52: Comparison of ferroelectric and dielectric properties of thin film PZT.................13 53: Summary of electrically actuated frequency response measurements ...................124 61: Summary of experimental results of microphone ................................. .............152 LIST OF FIGURES Figure pge 11: Roadm ap to Chapter 1. ...................... ..................................... ...... ... . 12: Schematic of overall energy harvesting concept ................ ............................... 3 13: Conceptual schematic of the acoustic energy harvester.................. .................. 14: B asic L C R electrical circuit......... ...................................................... ............... 4 21: R oadm ap to Chapter 2. ............. .................................................................... 18 22: Thevenin equivalent circuit for purposes of maximizing power transfer ..............21 23: Diagram showing side view of a Helmholtz resonator and its mechanical equivalent of a m assspringdamper system ........................................................................... 29 24: Equivalent circuit representation of a Helmholtz resonator.......................................34 25: Theoretical pressure amplification of a conventional Helmholtz resonator. (a) m magnitude (b) phase ...................................... .... .. ...... ........... 35 26: Crosssectional and 3D schematic of piezoelectric composite circular plate. (not to sc ale) ...................................................... .... ................. 3 6 27: Notation of axes used in piezoelectric transduction ................................................37 28: Idealized perovskite crystal structure for PZT. a) centrosymmetric structure prior to poling. b)noncentrosymmetric structure after poling..........................................38 29: Schematic of the poling process: ........................................ ......... ....39 210: Polarization vs. electric field hysteresis loop. .................................. ............... 40 211: Electroacoustic equivalent circuit representation with Cb, Ca, and .................46 212: Electroacoustic equivalent circuit representation with Ce C, and '. ................47 213: Dynamic electroacoustic equivalent circuit. ................................ .................48 214: Sensitivity vs. frequency for a piezoelectric composite circular plate ...................55 215: Equivalent circuit of piezocomposite plate including the radiation impedance. ....57 216: Equivalent circuit with resistive load. ........................................... ............... 60 217: Conceptual 3D and crosssectional schematic of the circular composite plate .....62 218: Effective acoustic shortcircuit compliance as a function of RIR,/ and tP/t, .......63 219: Effective acoustic mass as a function of R,/R2 and t/t, ......................................64 220: Shortcircuit resonant frequency as a function of R,/R2 and tp/t, ......................65 221: Electroacoustic transduction coefficient as a function of R,/R] and t /t, ............66 222: Electromechanical coupling coefficient, k, as a function ofRK/ R and t/t( .........67 223: Equivalent circuit for acoustic energy harvester with resistive load......................68 224: Magnitude of the acoustical input impedance for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator .................................70 225: Magnitude and phase of the acoustical input power for an input acoustic pressure of P = 94 dB ......... .......................................................... .......... ..... 72 226: Electrical output impedance for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. ......................... ............... ......73 227: Magnitude and phase of the output voltage for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. .............................74 228: Magnitude and phase of the electric output power delivered to the load resistor....75 229: Magnitude and phase of the energy harvester efficiency. ...................................76 230: Opencircuit voltage as a function of the applied acoustic pressure for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator..........77 231: Input and output power as a function of the load resistance placed across the electrodes of the piezoelectric material .............................................................. 78 232: Input and output power as a function of the applied acoustic input pressure, while u sing an optim al load resistor ...................................................................... ....... 79 31: R oadm ap for C chapter 3. .................................................................... ..................83 32: Condensed process sequence in crosssection. ................................................84 33: Early DRIE results showing significant sidewall damage. ........................................88 34: SEM image showing black silicon at the base of a DRIEetched trench ...................89 35: SEM image of a successful DRIE etch through the thickness of a silicon wafer. .....90 36: Illustration of a single diaphragm device.. ...................................... ...............91 37: Block layout illustrating numbering scheme for devices. ........................................91 38: Wafer layout illustrating numbering scheme for blocks. ........................................92 310: Quarterwave resonator package consisting of acrylic plug, copper leads and vent channel. ....................................................................93 311: Sealed cavity package...................................................................... ...................95 312: Optical photograph of a packaged energy harvester. .............................................95 41: R oadm ap for C chapter 4. .................................................................... ...................99 42: Experimental setup for impedance and power measurements. .............................104 43: Schematic of incident, reflected and input power. ............. .................................... 106 51: R oadm ap for C chapter 5. ........................................... ......................................... 108 52: Hysteresis loop for Device 1A54 in a sealed cavity package. The device has a radius of 1200 ium and a central mass. ...................................... ............... 109 53: Measured parallel output capacitance, Cp vs. outer radius, R, overlaid with a theoretical capacitance curve based on the average extracted dielectric constant. 111 54: Parallel output resistance, Rp, vs. outer radius, R,, overlaid with a theoretical resistance curve based on the average measured conductivity............................112 55: Device 1A61, electrically actuated center deflection for the device with a radius of 900 /um and no central mass, packaged in the quarterwave resonator package.. 115 56: Device 1A62, electrically actuated center deflection for the device with a radius of 900 /um and a central mass, packaged in the quarterwave resonator package....116 57: Device 1A113, electrically actuated center deflection for the device with a radius of 1200 /um and no central mass, packaged in the quarterwave resonator..........117 58: Higher order mode image of Device 1A113, taken using scanning laser vibrometer at 12 0 .9 k H z ............... ............ ......................................... .................................. 1 17 59: Device 1A104, electrically actuated center deflection for the device with a radius of 1200 um and a central mass, packaged in the quarterwave resonator ..........118 510: Device 1A65, electrically actuated center deflection for the device with a radius of 1500 um and no central mass, packaged in the quarterwave resonator.......... 119 511: Device 1A106, electrically actuated center deflection for a device with a radius of 1500 um and a central mass, packaged in the quarterwave resonator package. .120 512: Device 1A37, electrically actuated center deflection for the device with a radius of 1800 um and no central mass, packaged in the sealedcavity package............ 121 513: Electrically actuated sensitivity at low frequency (well below resonance) (R,/R = 0.95).................................................................. ....... 122 514: Summary of electrically actuated resonant frequencies. ....................................... 122 515: Drop in resonant frequency due to the addition of the central mass. .....................123 516: Magnitude of the resonant center deflection versus driving voltage......................125 517: Mechanical sensitivity at resonance versus driving voltage.............................126 518: Device 1A54, magnitude and phase of the acoustically actuated frequency response in a sealed cavity package for the device with a radius of 1200 /m and a central m ass. ........................................................................128 519: Device 1A104 in a quarterwave resonator package. Magnitude and phase of the acoustically actuated frequency response for the device with a radius of 1200 pm and a central m ass. ........................................ ................... ........ 129 520: Device 1A65 in a quarterwave resonator package. Magnitude and phase of the acoustically actuated frequency response for the device with a radius of 1500 um and no central m ass. ........................... ................ ................... .. .... .. 130 521: Device 1A102, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 900 pm and a central mass. ..................... ...............131 522: Device 1A54, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1200 pm and a central mass .........................................132 523: Device 1A37, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1800 pm and no central mass ............... .................132 524: Device 1A61, Normalized acoustic impedance in a quarterwave resonator package for the device with a radius of 900 pum and no central mass .................133 525: Device 1A113, Normalized acoustic impedance in a quarterwave resonator package for the device with a radius of 1200 pum and no central mass ..............134 526: Device 1A113, Initial static deflection resulting from residual stresses for a device with a radius of 1200 pm and no central mass. ............. ..................................... 135 527: Device 1A66, Initial static deflection resulting from residual stresses for a device with a radius of 1500 pum and a central mass ...................................................... 136 528: Device 1A37, Initial static deflection resulting from residual stresses for a device with a radius of 1800 pm and no central mass. ............. ..................................... 136 529: Device 1A38, Initial static deflection resulting from residual stresses for a device with a radius of 1800 pum and a central mass ...................................................... 137 530: Measured power delivered to a load as function of the load resistance for Device 1A44 and Device 1A38 as compared against theoretical values ....................140 531: Measured output voltage across the load as a function of applied acoustic pressure and com pared to theoretical values.. ........................................... ...............140 532: Measured power delivered to load as a function of applied acoustic pressure and com pared to theoretical values .................................................................. ....... 14 1 533: Measured overall efficiency of each device overlaid with theoretical values for com prison ...................................................... ................. 142 534: Resonant frequency versus applied acoustic pressure, resulting from nonlinear response..................................... ........................... ..... .......... 146 61: Linearity of the microphone device at 1 kHz....................................................... 148 62: Frequency response spectrum in terms of magnitude and phase. ............................149 63: Noise floor spectrum of output voltage when no acoustic signal is applied, as well as noise floor due to measurement setup alone. ................................. ............... 151 64: Electrically actuated frequency response of microphone device. ............................152 71: R oadm ap to C chapter 7. ............. ..................... ......... .................................... 154 Bl: Step 1: Deposit Ti on SOI Wafer and oxidize to TiO2. .......................................172 B2: Step 2: Deposit Ti/Platinum(30nm/170nm) LiftOff w/ Mask (ElectrodeBot) ......172 B3: Step 3: Spin PZT 6 times to achieve desired thickness ............... ................. 172 B4: Step 4: Deposit Platinum(180nm)Liftoff w/ Mask (ElectrodeTop)......................172 B5: Step 5: Wet Etch PZT using Pt as etch mask................................... ..................173 B6: Step 6.1: Spin thick photo resist on bottom (7um). ............................................173 B7: Step 6.2: Pattern using mask (Cavity). ....................................... ............... 173 B 8: Step 6.3: D R IE to B O X ............................ ...................................................... 173 B 9: Step 6.4: A sh R esist. ...................... .................... .......................... 173 B10: Step 6.5: BOE backside to remove oxide. .................................. .................174 B 11: B ackside m etal m ask .................................................. ............................... 174 B12: Bottom electrode mask ......... .. ..... .. ......... .. ........................ 175 B 13 : T op electrode m ask ................................................................................ .... ... 176 B 14 : C av ity m a sk ..................................................................................................17 7 B15: Detailed schematic drawing of quarter wave resonator package and mounting.... 178 B16: Detailed schematic of sealed cavity package and mounting plate........................ 179 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 A MEMSBASED ACOUSTIC ENERGY HARVESTER By Stephen Brian Horowitz December 2005 Chair: Toshikazu Nishida Cochair: Mark Sheplak Major Department: Electrical and Computer Engineering In this dissertation, I discuss the development of an electromechanical acoustic resonator for reclamation of energy using microelectromechanical systems (MEMS) technology. The MEMS device converts acoustical energy to electrical energy that can then either be stored for later use or utilized directly for a particular circuit application. The work presented in this dissertation takes a first step towards that goal, by designing, fabricating and testing several electromechanical acoustic resonators of varying size. The resonators are fabricated using MEMS processing techniques on a silicon wafer. Each resonator consists of a 3 um thick silicon diaphragm with a circular ring of piezoelectric material. The diaphragm transduces an acoustical pressure fluctuation into a mechanical deformation, while the piezoelectric material transduces that mechanical deformation into an electrical signal (charge or voltage). CHAPTER 1 INTRODUCTION The goal of my research was to utilize microelectromechanical systems (MEMS) based fabrication techniques to develop an electromechanical acoustic resonator for harvesting of acoustic energy. The MEMS device converts acoustical energy to electrical energy that can then either be stored or used directly for a particular circuit application. The relatively small geometries possible in MEMS make such a device useful for small, portable devices, where battery requirements are often difficult to meet. Additionally, the device is well suited to applications where wiring is difficult such as remote sensors or unwieldy as in the case of large arrays of devices requiring power. This chapter begins with an overview of the acoustical to electrical energy reclamation process that is employed in this research. Following this, the motivation behind this research is presented. Then, an indepth literature survey is presented to familiarize the reader with the technological and theoretical developments related to energy harvesting and piezoelectricity, followed by a short discussion of the technical approach that was employed. A graphical roadmap for this chapter is shown in Figure 11. Acoustic Energy HarvesterConcept The overall concept behind the energy harvester is illustrated in Figure 12. The main idea is to convert acoustic energy into a usable form of electrical energy. The figure illustrates a plane wave tube as the source of acoustic power. Some of the incident acoustic power is reflected at the end of the tube; however, a portion is transmitted through to the energy harvester that is circled in Figure 12. The energy harvester performs the actual transduction of energy from the acoustical to the electrical energy domains. More details on this transduction technique will be provided later. Piezoelectiicity Piezoelectiic Nhiterials in NIENIS ImpedaC(lnI1ce NIOde(iIng of Piezoelecriics I h r I Eiieiry Absorpfioii/ Eine.tion DHamin2 Enei2 H. jj~ll Figure 11: Roadmap to Chapter 1. The electrical energy that exits the energy harvester has the same temporal form as the input acoustic signal, which we assume for now is sinusoidal. The sinusoidal signal is then routed to the energy harvesting circuitry that serves to rectify and condition the signal before sending it on to the end application. In the example illustrated in Figure 12, the end application is a battery that is charged by the output of the harvesting circuitry. Plane Wave Tube fNo Ac % I. > DC Circuitry Power B i Battery Energy Harvester Figure 12: Schematic of overall energy harvesting concept. A conceptual closeup schematic of the acoustic energy harvester is shown in Figure 13. This device consists of a Helmholtz resonator possessing a piezoelectric composite backplate. A Helmholtz resonator, which is a type of acoustic resonator, consists of a cavity connected to the environment through a small neck. When excited by an acoustic input, a single resonance is seen, whereby the acoustic pressure inside the cavity is amplified to a level much greater than the incident acoustic signal [1, 2]. The Helmholtz resonator is very similar to an LCR resonant circuit in electrical engineering, as shown in Figure 14. Both systems have a single degreeoffreedom, with a single resonant peak, whereby the amplitude of the forcing function is greatly amplified. In the LCR circuit, the voltage is equivalent to the acoustic pressure. Both systems operate through the oscillation of energy between generalized potential and kinetic forms. In electrical systems this storage occurs via capacitors and inductors respectively. These electroacoustic analogies will be explained in more detail in Chapter 2. neck cavity piezoelectric compliant ring plate Figure 13: Conceptual schematic of the acoustic energy harvester. R L Figure 14: Basic LCR electrical circuit. The large acoustic cavity pressure created by the resonance can then be exploited for energy reclamation by converting the acoustic energy to electrical energy. This conversion is performed by the piezoelectric composite plate. First, acoustical to mechanical transduction is accomplished via the compliant diaphragm, followed by mechanical to electrical transduction, due to the piezoelectric response of the composite, whereby a mechanical strain creates an electrical voltage. The voltage that is created through the electromechanical transduction can be harvested by the energy harvester circuit, which is a necessary part of the energy harvesting process. The necessity arises due to the need to match impedances at interfaces. As will be derived in Chapter 2, the power transfer across an interface is maximized when the impedances on each side of the interface are complex conjugates of each other. This is often referred to as impedance matching. One role of the energy harvesting circuit is to provide an input impedance that matches the output impedance of the piezoelectric structure. Additionally, the harvesting circuit converts the form of the electrical energy to a more appropriate form for storage, such as converting from AC to DC. In one implementation, a rectifying switched capacitor circuit is used to efficiently convert the AC signal to DC that is then stored across a lowloss capacitor [3]. Another possibility is based on the Kymissis circuit approach [4]. This approach utilizes passive storage elements for impedance matching coupled with a regulated output, for improved energy harvesting performance. Motivation Acoustic energy is commonly present in a variety of everyday situations. The motivation behind this research is to enable the reclamation of that energy that would otherwise be lost. Although the available acoustic energy may be small in many situations, the energy requirements for certain applications such as microsensing are also correspondingly small. The ability to reclaim acoustic energy and store it in a usable electrical form enables a novel means of supplying power to relatively low power devices. Background Piezoelectricity In 1880, the brothers Pierre and Jacques Curie discovered that some crystals developed surface charges when compressed [5]. They furthermore found that these charges were proportional to the applied pressure. This phenomenon was later named piezoelectricityy" by Wilhelm Gottlieb Hankel, and is historically referred to as the direct piezoelectric effect [5]. Additionally, in these same crystals, a strain is produced under application of an electrical field. This is commonly referred to as the converse piezoelectric effect. The material constant relating strain and charge in a piezoelectric material is known as the piezoelectric charge modulus, d, and is typically quoted in units of pC/N or pm/V. In order for a material to be piezoelectric, it must have a non centrosymmetric crystal structure. It was not until 1921, that a useful application was developed for piezoelectricity. It came in the form of a quartz crystal oscillator that was developed by Walter Cady to provide good frequency stability for radio systems [5, 6]. Then, in 1947, the first commercial phonograph pickups based on barium titanate (BaTiO3), a piezoelectric ceramic, were introduced [6]. Modern applications of piezoelectric materials now include high voltage ignition systems, piezoelectric motors, inkjet printer heads, acoustic speakers, sonar, ultrasonic transducers, frequency filters, acoustic delay lines, electrical transformers, and a wide range of physical sensors, such as acoustic, force, pressure, and acceleration sensors [6]. Piezoelectric Materials in MEMS A number of papers have been published on the use of piezoelectric and ferroelectric materials in MEMS [6, 7]. Piezoelectric materials commonly used include zinc oxide (ZnO), aluminum nitride (A1N), and lead zirconate titanate (PZT). The choice of piezoelectric material depends on several selection factors including deposition methods, process complexity, integrated circuit (IC) compatibility, and material parameters. Materials and properties Of the three commonly used materials, PZT offers the largest piezoelectric modulus [6]. In comparison to PZT, A1N and ZnO have relatively weak piezoelectric coefficients and coupling factors, however they tend to have low dielectric constants and low dielectric losses, making them more attractive for certain applications [8]. A table of typical material properties is shown in Table 11 for comparison; however these properties are highly dependent on the deposition method, as will be described in more detail shortly. Table 11: Typical material properties of select piezoelectric materials. 31 33 33,r E P [pC/N] [pC/N] [GPa] [kg/m3] PZT[6] 130 290 1300 96 7.7 A1N[8] n/a 3.4 10.5 330 3.26 ZnO[911] 4.7 12 12.7 210 5.6 In Table 11, d31 is the piezoelectric coefficient relating electrical displacement in the '3' direction (zaxis) to a mechanical stress in the '1' direction (xaxis) and is the relevant coefficient for bendingmode transducers, whereas d33 relates the electrical displacement in the '3' direction (zaxis) to a mechanical stress in that same direction. The coefficient, d33, is thus the relevant coefficient for compressionmode transducers. Additionally, E33,r, E, and p are the relative dielectric constant, Young's modulus and mass density of the materials, respectively. Of the three materials, only A1N is fully compatible with standard IC processing, eliminating the integration difficulties present with PZT and ZnO. PZT poses a particular integration challenge, as lead contamination is always a big concern; however, PZT has been integrated successfully into ferroelectric randomaccess memory (FRAM) [1214], typically using sputtered platinum as a bottom electrode and diffusion barrier layer. Deposition methods Piezoelectric deposition techniques include various sputtering methods, photo ablation [15], hydrothermal and chemical vapor deposition (CVD) techniques, and spin on solgel processing. For deposition of PZT, three popular methods are sputtering, sol gel and hydrothermal techniques. The earliest work used various forms of sputtering to deposit PZT [1622] followed soon after by CVD [23]. Castellano and Feinstein [16] used an ionbeam sputtering technique to deposit PZT as did TrolierMcKinstry et al., [22] while Sreenivas et al. [17] employed DC magnetron sputtering in their PZT deposition. Additionally, sputter deposition of PZT thin films was performed by Kawabata et al.[24] and Li et al. [25]. Dubois and Muralt [8] deposited PZT using two different methods. One method involved a solgel process and produced an average thickness of 900 nm. The other method used reactive sputtering and achieved a thickness of approximately 500 nm. Hydrothermal deposition relies on a chemical reaction between a titanium bottom electrode and ionic solution under high temperature and pressure. Deposition of 10 um thick PZT by hydrothermal techniques was performed by Kanda et al. [26] for use in a touch probe sensor. Additionally, Morita et al. [2730] chose to deposit PZT by the hydrothermal method due to the relatively large thicknesses achievable and the self alignment of the poling direction during deposition. More recently, numerous researchers have utilized the solgel process for deposition of PZT [3147]. In particular, Bahr et al. [31] used a solgel PZT process and investigated the reliability and piezoelectric properties of the resulting material. Using this process for various film thicknesses, the relative dielectric constant, 833,., varied between 700 and 1000 and delamination occurred for an indentation load of 1250 pN for a film sintered for 5 minutes. Higher loads were found to be achievable using longer sinter times. Bernstein et al. [32] and Xu et al. [33] used a solgel process to achieve crackfree PZT films with thicknesses of up to 12 um that yielded piezoelectric properties close to the bulk values for PZT. A dielectric constant of 1400 and a piezoelectric coefficient, d3, of 246 pC/Nwere reported on a 4 ,um thick film. The PZT films were used on an array of membranes for acoustic imaging. Kunz et al. [37] report a piezoelectric coefficient, d,,, of 110 pC/Nfor a solgel deposited PZT film used in a triaxial accelerometer. Zurn et al. [48] report similar material properties for solgel deposited PZT on a microcantilever, including a d,3 of 120 pC/Nfor a PZT film thickness of 0.5 um deposited on a lowstress silicon nitride layer. A summary table of deposited thin film piezoelectric materials and their properties, as reported in the literature, is given in Table 12. In this table, e3, is a piezoelectric coefficient relating mechanical stress and an electric field, tan 8 is called the loss tangent and is a measure of the relative losses in the material, and r,, is the residual stress in the material, while the other properties are as previously defined. Piezoelectric integration with MEMS Some of the earliest integration of piezoelectric materials with MEMS focused primarily on ZnO [49, 50] and A1N [49] as the piezoelectric material. More recently, 10 Devoe and Pisano [10] developed and characterized surface micromachined piezoelectric accelerometers that utilized thin films of ZnO for sensing. Deposition of the ZnO was performed using singletarget RF sputtering and exhibited a piezoelectric coefficient of 2.3 pC/N. Also, Devoe [9] investigated micromechanical beam resonators that use ZnO and a threemask fabrication process. The resonators are intended for use as electromechanical filters. Table 12: Selected thin film piezoelectric materials and corresponding properties as reported in the literature. Ref Material Deposition Method [9, 10] ZnO RF Sputt. [11] ZnO Sputtered [8] A1N DC Sputt. [8] PZT (45/55) React. Sputt. [8] PZT (45/55) SolGel [8] PZT (53/47) React. Sputt. [26] PZT Hydrothermal [31] PZT (52/48) React. Sputt. [32] PZT React. Sputt. [37] PZT (53/47) React. Sputt. [42] PZT React. Sputt. [44, 45] PZT (52/48) React. Sputt. [48] PZT React. Sputt. [16] PZT IB Sputt. [20,21] PZTWCd RF Sputt. [51] PZT(X/1X) React. Sputt. Properties 33,r 31 d33 e3 12.7 10.5 900 1100 1300 7001000 1400 1100 800 125 60460 200600 2.3 34.2 110 3.4 55 50 70 246 200 120 30 C 1  ) 1.02 5.12 8.28 6.83 0.13 E P tan ( or (GPa)k (Pa) 161 5605  1...80 210 5700    0.002 700  0.03 150  0.03 70  0.05 230   0.03 56  0.02 60 7600     0.02 The integration of ZnO with micromachining was also investigated by Indermihle et al. [52], where it was used in an array of silicon micro cantilevers. The end application for the array was parallel atomic force microscopy. Another application of ZnO was performed by Han and Kim [11] in the fabrication of a micromachined piezoelectric ultrasonic transducer. They sputter deposited ZnO on Al, followed by a layer of parylene for insulation. Dubois and Muralt [8] fabricated thin films of RF sputtered A1N and performed measurements of the effective transverse piezoelectric coefficient that yielded a value of 3.4 pm/V. ReducedOrder Modeling of Piezoelectric Materials The development of accurate, practical models of the piezoelectric transduction and associated structural interactions is critical to the design and behavioral prediction of piezoelectric based devices, particularly when micromachining is involved. For a micromachined device, considerable time and expense must be invested in the fabrication, and it is therefore desirable to have some ability to predict the device behavior before proceeding with fabrication. Behavioral prediction can be accurately accomplished using complete analytical methods. However, these methods, which often involving partial differential equations, can often be unwieldy and physically unintuitive. Furthermore, this functional form is not readily conducive to a full systems level design that links the transducer to electronics. Similarly, finite element modeling (FEM) techniques are often used to predict system behavior, numerically. The results produced by this technique can very precisely follow the physical system; however, the physical insight that can be gleaned is limited. Additionally, it is very difficult to determine scaling behavior from FEM results. The scaling behavior, i.e., the change in the system performance as the entire system is scaled up or down, is a critical design issue in the creation of devices using micromachining technology. In order to facilitate a physicsbased approach for design, a simplified, reducedorder model is necessary that accurately captures the geometric and material dependencies. This reducedorder model uses lumped elements to represent the key components that dominate the device behavior. In 1915, Butterworth [53] first showed that any mechanical oscillator, when driven by a periodic voltage across a capacitor, would have an equivalent electrical circuit consisting of a resistance, inductance and capacitance in series, and all in parallel with another capacitance. Working independently, the earliest equivalent circuit model specifically for piezoelectric materials was developed by Van Dyke in 1925 [5456]. Later, Dye [57] proved that Van Dyke's circuit could be derived from Butterworth's theorem. Additionally, Mason [58, 59] and Cady [5] provided thorough reviews of the equivalent circuit model and associated equations for quartz oscillator applications. Finally, Fischer [60] extensively covered equivalent circuit models for electromechanical oscillators. Significant research has been performed on the modeling of structures containing piezoelectric materials. [38, 41, 6194] Cho et al. [61, 65] developed a fiveport generalized equivalent circuit for a piezoelectric bimorph beam. The generalized circuit can be used under a variety of boundary conditions. In these papers, three boundary conditions are specifically analyzed free, simply supported, and cantilevered. For these boundary conditions, the equivalent circuit was found to produce the exact expressions for the beam vibration. Other equivalent circuits were developed by Martin [68] for resonators with low Q values, Sheritt et al. [69] for thickness vibrators, Lin [67] for coupled ceramic disk resonators, and Chen et al. [68] for coupled resonant filters. Tilmans [94] also presented an equivalent circuit approach for modeling distributed parameter systems. Liang et al. [66] developed a generalized electromechanical impedance model that was then used to address issues of energy conversion, power consumption, and dynamic response. The approach described can be used for any system for which the drivingpoint impedance can be found either analytically or experimentally. Additionally, van de Leur [91] provided a critical interpretation of equivalent circuit models obtained from impedance measurements, indicating that care must be taken in identifying individual components contributing to an impedance as resulting from particular structures in a given device. This can be further understood as an example of the nonuniqueness of equivalent circuit representations, i.e., more than one equivalent circuit can represent the same impedance. Additionally, Lesieutre and Davis [87] provided insight into the piezoelectric coupling coefficient, including a technique whereby an outside stress is applied to increase the effective device coupling coefficient beyond the coupling coefficient of the material itself. Research into coupled resonators and their unique properties is a related and also relevant topic since the acoustic energy harvester involves mating the piezoelectric composite diaphragm to a Helmholtz resonator, resulting in a coupled resonant system. Fischer [60] provided an early treatment of coupled resonant systems. Chen et al., [72], Lin [71, 95], and Li et al. [96] also discuss modeling and design of coupled resonant systems. These papers address some of the issues involved in coupled resonators, including the shift in resonant frequencies away from their uncoupled values. Energy Absorption/Vibration Damping and Energy Harvesting The absorption of acoustical and mechanical energy via piezoelectric coupling is closely related to the harvesting of electrical energy from acoustical and mechanical energy sources. From the acoustical point of view, acoustical energy that is converted to the electrical domain and dissipated across a resistive load can be viewed as being absorbed. From the electrical point of view, the incident acoustical energy can be viewed as an energy source that may theoretically be harvested and used immediately for electrical subsystems or stored for later use. Because of the related nature of these two fields, papers that address either of these fields are directly relevant to acoustic energy harvesting. Some of the earliest work in piezoelectric vibration damping was performed by Hagood and von Flotow [97] who used resistive and inductive elements in various shunt networks designed to dissipate mechanical energy. A few years later, Hollkamp [98] discussed the use of resonant electrical shunt circuits for multimodal vibration suppression. In addition, numerous other researchers have investigated piezoelectric means of vibration damping [99108]. More recently, Caruso [109] discusses electrical shunt circuits for damping of vibrations, as does Wu et al. [110] for damping of panels on an F15 aircraft. Additionally, a number of papers directly address the issue of obtaining electrical energy from piezoelectric conversion of mechanical energy [3, 4, 111117]. Early work in this area was performed by Lomenzo et al. [114] and Stein et al. [116]. Later, Kymissis' [4] and Smalser's [3] work focuses on the electrical circuitry necessary for storage of piezoelectrically generated energy, while Meninger [115] discusses energy harvesting from an electrostatic transducer. Goldfarb and Jones, [111] Giurgiutiu and Rogers [112, 113] and Zhou and Rogers [117] utilize impedance modeling of the piezoelectric and associated structure to analyze the ability to harvest energy. Approach The approach taken in this research is to divide the concept of an acoustical energy harvester into three distinct components as shown previously in Figure 12. The first component consists of a piezoelectric composite diaphragm that is responsible for the acoustical to electrical transduction of energy. The second component, which is the packaging, serves as a means to improve the acoustical coupling to the environment. The third and final component is the electrical circuitry that takes the alternating current (AC) electrical signal from the piezoelectric diaphragm and converts it into a direct current (DC) output that can be stored for later use. The first two components of the energy harvester were developed in this research, while the conversion circuitry remains an ongoing research topic in our group [118120]. Furthermore, a macroscale version of an acoustic energy harvester was previously developed [120] and, through miniaturization, led to the present MEMSbased energy harvester. Chapter 2 presents a theoretical background on various aspects of this research including lumped element modeling of the complete system as well as individual components along with predictions of the dynamic behavior. Chapter 3 discusses the fabrication of the piezoelectric diaphragm and the packaging scheme employed. Following this, the experimental setup is provided in Chapter 4, including a discussion of the measurements that were taken. Then, in Chapter 5, the experimental results are presented. Chapter 6 addresses an alternate application that has been explored, while Chapter 7 presents the conclusions and a statement of the contributions to this research. Finally, Appendix A presents the Matlab program code for computing the diaphragm deflection and calculating the lumped element values, while Appendix B provides a detailed process flow, mask layout, and packaging design The main contributions of this dissertation are as follows: Acoustic energy harvesting concept 16 * Comprehensive lumped element model (LEM) of the acoustic energy harvester * First reported integration of thinfilm PZT with SOIbased MEMS * First aeroacoustic capable piezoelectric microphone. CHAPTER 2 THEORETICAL BACKGROUND This chapter focuses on the theory and modeling of the acoustic energy harvester and each of its components. The chapter begins with an overview of energy harvesting, followed by an introduction to lumped element modeling, by discussing both its utility and limitations. This is followed by the development of a lumped element model for a Helmholtz resonator and the corresponding equivalent circuit. Next, the modeling of piezoelectric composite circular plates is addressed. To achieve this, an equivalent circuit is presented and general equations are discussed. Then specific limiting cases are addressed that simplify the analysis. Finally, the overall dynamic behavior of the system is theoretically investigated, including the acoustical input and electrical output behavior. A graphical overview of this chapter is provided by the roadmap of Figure 21. OverviewEnergy Harvesting The general concept behind energy harvesting is to convert energy from the environment that is in an otherwise unusable form into a more useful form. Often the form of energy that is most useful in modem applications is electrical energy, where it can be stored in a battery or used to power electrical circuitry. The initial form of energy can originate from any number of energy domains, such as optical, thermal, mechanical, acoustical, fluidic, chemical, and biological. Some form of transducer is then required to convert that energy to a usable form of electrical energy. Depending on the situation, that also may require passage of the energy through an intermediary energy domain.. it e Plate Figure 21: Roadmap to Chapter 2. There are pros and cons to each of these different local energy sources [121]. Some energy sources such as solar power utilize the optical energy domain to achieve high power densities on the order of 15,000 uW/cm2 in direct sunlight. The main downside is that direct sunlight is not always available. Vibrational energy, on the other hand, offers power densities up to 250 /W/cm2 While this is considerably lower than solar energy, it is useful in places without sunlight but where high vibrational energy is available. Acoustic energy, in a manner similar to vibrational energy, offers power densities on the order of 1 uW/cm2 for a 100 dB acoustic signal [121], or approximately 964,000 t/W/cm2 at 160 dB. While most signals are typically much lower than 160 dB, there are applications where such high levels are present. Additionally, as with vibrational energy, acoustic energy, does not require the presence of sunlight. Chemical energy sources are commonly employed today in the form of batteries and fuel cells. Batteries typically offer power densities in the range of 45 uW/cm3 for nonrechargables, and 7 uW/cm3 for rechargeable lithium batteries [121]. Fuel cells employing methanol, on the other hand, offer power densities as high as 280 /W/cm3, leading to the current interest in fuel cell development [121]. Additionally, a micro combustion engine that employs hydrocarbons as a fuel source realizes power densities of 333 /jW/cm3 [122]. The downside to these types of chemical energy sources is the limited supply of energy. Used batteries must either be replaced or recharged and fuel cells require refueling for continual operation. By contrast, the scavenged energy sources, such as vibrational, solar, and acoustical do not theoretically have a limited supply of energy, given the right operating environment. For the particular application considered in this dissertation, the initial energy is in the form of acoustical energy. In order to convert acoustical energy into electrical energy, a diaphragm based transducer is utilized. The diaphragm transducer utilizes the mechanical energy domain as an intermediary to transfer energy. Maximum Average Power Transfer Regardless of the route through which the energy passes, certain fundamental issues must be addressed in order to maximize the amount of energy that is harvested. Whenever a change in impedance is encountered by a traveling wave, a reflection of all or part of the energy in that wave occurs, while the remaining energy is transmitted. The procedure for maximizing the energy focuses on minimizing the reflected component and thus maximizing the transmitted component. This is achieved by matching the impedance along the route traveled by the energy. As long as the impedance at a given interface is matched, complete transmission of the energy will occur, regardless of whether or not the two sides of the interface are in the same energy domain. Often it is not possible to match an impedance exactly, due to external and physical constraints on the system, however it is generally still desirable to match the impedances as closely as possible in order to optimize for maximum energy within those constraints [2, 123]. Additionally, one must consider that power is a complex quantity. The total power, which is generally complex, is composed of real power and reactive power. The real power is the physical power that is delivered to the load, while the reactive power represents energy that is temporarily stored in the load, before it returns to the source. When designing an energy harvesting system, it is generally preferable to maximize the real power rather than the total power, as it is only the real power that is ultimately retained by the load. It is useful to look at a Thevenin equivalent circuit, as shown in Figure 22, in order to determine the maximum achievable power transfer to a load [124]. Assuming sinusoidal voltage and current, the average real power delivered to the load is n =VL Cos( L ), (2.1) where VL and IL are the peak magnitudes of the voltage and current at the load, and 0. and 0, are the phase angles of the voltage and current respectively. This can be rewritten as 1 nL = IL (pf ) (2.2) where pf is the power factor, defined by the ratio of average power to apparent power and is given by n pf =r (2.3) V I where V,, and I, are the rootmeansquare voltage and current, respectively. I, V TH V oO VL Z Figure 22: Thevenin equivalent circuit for purposes of maximizing power transfer. Using basic circuit analysis, the load voltage and current can be expressed in terms of the open circuit voltage, Vo, the Thevenin equivalent impedance ZH, and the load impedance, Z The expressions are given as VL ZL (2.4) TH + Z and IL = V1 (2.5) z +z ZTH + ZL where the Thevenin equivalent impedance, ZTH = RH + jXTH, and the load impedance can be written as ZL = RL + jXL Plugging these expressions back into Eq. (2.4) and Eq. (2.5) and separating out the magnitude components yields [(H RL2 (XTH X)2 VL = t. . ^ ,Voo~z +AL2(2.6) and Ic = Vo 1 (2.7) [(H )2+(XTH +XL)2 2 The phase angle between the voltage and current can also be expressed as 6, 0,L = Oz, where Oz, is the phase angle of the impedance. The power factor can be defined as pf =cosZL = RL1 (2.8) (R2 + lx)2 Plugging this expression into Eq. (2.1), together with Eq. (2.6) and Eq. (2.7) yields 1 /V 2 R nL = 2 c2L (2.9) 2 (RH + R)2 +(XTH + XL )2 As, the quantity (XTH + X,) is only in the denominator, any nonzero value reduces the overall power absorbed by the load, thus X = XTH is optimal and reduces Eq. (2.9) to 1 V 2RL HL = oc 2 L (2.10) 2(R +TH)2 This expression can be maximized by differentiating this expression with respect to RL and equating the derivative to zero, yielding an optimal value of RL = RH The average power to the load can be maximized, then, by setting the load impedance to ZL = RL + jXL = RTH jXTH = ZTH*, (2.11) where ZH* is the complex conjugate of the Thevenin impedance. This derivation assumes total freedom in the choice of the load impedance. Under the constraint of a purely resistive load, where XL = 0, the equation given in Eq. (2.9) can be differentiated directly and set equal to zero with XL = 0, in order to find the maximum average power transfer. This procedure yields an optimal value for the load resistance of RL = RTH2 +XTH2 = ZTH (2.12) under the constraint of a purely resistive load. Electromechanical Transducers A transducer can be broadly defined as an interface between any two energy domains, including optical, thermal, mechanical, electrical, acoustical, fluidic, chemical, and biological. Typically, transducers are broadly classified into two main categories: sensors and actuators. The breakdown among these classifications relates to the direction of information flow. Sensors are transducers that convert information from the environment into a form that is more easily analyzed, recorded or processed. Actuators, on the other hand, take information from this form and convert it into a form that more readily interacts with the intended environment. Both types of transducers are mainly concerned with the way in which the information is utilized. The classifications described above focus on the information aspect of a given signal that is encoded in either the effort or flow, but not their product. As such, these types of transducers are optimized for transmission of information in one direction or another. Besides information, a signal carries power. A third type of transducer exists that falls under neither category and can be best described as an energy harvester. The concept behind the energy harvester is to achieve an optimal amount of average power transfer between energy domains. Instead of optimizing sensitivity or dynamic range, an energy harvester is optimized for efficiency and maximum average power transfer. Additionally, actuators and especially sensors are designed as linear devices in order to facilitate a straightforward relationship between input and output. The linear inputoutput relationship ensures that the problem of calculating their behavior is tractable, and is necessitated by the information contained in the signal. Energy harvesters, on the other hand, do not necessarily require a linear inputoutput relationship as only the power is of interest. Some details of the input may get lost in the nonlinear transduction, but only the power at the output is needed. There are a wide range of transduction techniques available that couple the mechanical and electrical energy domains, including electrostatic, electrodynamic, piezoresistive and piezoelectric [1, 125]. Each technique offers various tradeoffs among performance, reliability, and material integration. Transducers can be separated into two types, direct and indirect, based on the way they interact with energy. Direct transducers, as the name implies, directly transduce energy between the two energy domains. Indirect transducers, however, rely upon a second energy source that modulates the primary energy source as it passes through the transducer. The inherent nature of indirect transduction leads to a lower transduction efficiency as compared to direct transduction. Electrodynamic transduction, a direct transduction mechanism, yields high coupling efficiencies. Additionally, this method uses no outside power source in order to operate. The transduction is essentially powered by the input signal itself. Despite these advantages, electrodynamic transduction suffers from difficulty in integrating the particular required materials, such as copper or magnetic materials, into a standard IC process line. Additionally, the necessary fabrication is often more complex than the other techniques discussed here. Piezoelectric transduction requires no outside source of power other than the input signal, and offers a high sensitivity, with a potentially low noise floor. The main disadvantages are the difficulty in integrating the piezoelectric materials into a standard process flow, and the limited coupling efficiency due to indirect transduction of energy. Lumped Element Modeling The most accurate, complete, mathematical description of a physical system is a physicsbased model, supported by an exact analytical expression for the system behavior. Why then are alternative modeling techniques commonly used? These alternative methods, such as lumped element modeling (LEM), and finite element modeling (FEM), excel in predicting system behavior in situations where an exact, analytical approach is unwieldy or impossible. Additionally, it is not uncommon to have an exact mathematical description of a system that allows for precise prediction of device behavior, but is so complex that it offers little in the way of physical insight into the design and scaling system. Often, the exact solution is in a mathematical form that, while convenient for behavioral prediction, conceals the underlying physics. Similarly, FEM techniques can accurately predict system behavior, in this case via a numerical approach. The results produced by this technique can very precisely follow the physical system; however, the physical insight that can be gleaned is limited. Furthermore, the results depend on the numerical mesh and convergence of the iterative calculations. In particular, it is very difficult and/or cumbersome to determine scaling behavior from FEM results. The scaling behavior, i.e. the change in the system performance as the entire system is scaled up or down, is a critical design issue in the creation of devices using micromachining technology. In order to efficiently understand the physics behind a given systems response and also to fully appreciate the scaling laws for that system, it is necessary to employ LEM [1, 60]. The concept behind LEM is to reduce the complexity of an analytical or numerical expression by breaking down a given distributed system into discrete elements based on how the system interacts with energy [60, 126]. More specifically, the total energy going into any given system is divided among three distinct types of interactions: the storage of kinetic energy, the storage of potential energy, and the dissipation of energy. All systems are composed of these three energy processes. In some systems, known as distributed systems, the storage of kinetic and potential energy occurs over a distributed region in space [90, 127]. To accurately represent these systems mathematically requires a partial differential equation, as spatial and temporal components are inherently coupled. Physically, the distribution occurs because the wavelength is on the order of the physical system or smaller. At different points along the period of the wave, differing amounts of energy are split among the energy storage and dissipation mechanisms. As the wavelength of the signal (e. g. acoustical, fluidic, mechanical, optical, etc....) gets larger, to where it is significantly larger than the length scale of interest, very little variation occurs in the distribution of energy as a function of space. Mathematically, under these conditions, the spatial and temporal components can be decoupled, allowing for the use of ordinary differential equations to solve the problem, rather than partial differential equations. Physically, it means each energy storage or dissipation mechanism can be equated to the energy stored or dissipated in an equivalent element that is lumped to a chosen spatial location. In various energy domains, the names for the types of lumped elements vary; however, the concept and mathematics remain the same. In lumped mechanical systems, kinetic energy is stored via mass, potential energy via the compliance of a spring (i.e. inverse of stiffness), and dissipation of energy through the losses of a damper. Similarly, in electrical systems, where lumped element systems are commonplace due to the extremely long wavelengths of electrical signals, kinetic energy is stored in the magnetic field of an inductor, while potential energy is stored in the charge across a capacitor. Additionally, dissipation of energy is modeled via the resistor. Finally, in lumped acoustical systems, the kinetic and potential energy is stored in an acoustical mass and acoustical compliance, respectively, while dissipation of energy is represented by an acoustic resistance. These elements are summarized in Table 21. The commonplace nature of lumped elements in electrical systems has led, over the years, to a large growth in graphical and analytical techniques to solve large networks of interconnected elements. These networks are most commonly represented using electrical circuit notation. A significant benefit of lumped element modeling is that in all of the energy domains, the lumped elements can be represented using an equivalent circuit form. Thus, masses can be represented using inductors, compliances using capacitors, and dissipative components using resistors. Once the complete equivalent circuit is constructed, standard circuit analysis techniques (e.g. Kirchoff s current and voltage laws) can be applied to find the solution of interest. Table 21: Equivalent lumped elements in several common energy domains. Kinetic Energy Potential Energy Energy Storage Storage Dissipation Acoustical Acoustic Mass Acoustic Compliance Acoustic Resistor [kg/m4] [n'/Pa] or [m'/N] [Mns] Mechanical Mass [kg] Compliance [m/N] Frictional Damper (pt mass) (spring) [m s] Electrical Inductance [H] Capacitance [F] Resistance [0] Whenever dealing with more than one lumped element, the concept of power flow between the elements must be considered. If we define the power flow from element A to element B as nAB and the reverse as nBA, then the net power flow from A to B is "net = nAB IBA. (2.13) Now, since each of the power flows must be greater than or equal to zero, then they can each be written as the square of a real number, r, and r2 [128]. The net power flow can then be rewritten as net = r2 r =(r, +rz)(r, r2). (2.14) It is therefore seen that the net power flow can be written as the product of two real numbers, which are referred to as conjugate power variables. Moreover, these quantities are more specifically referred to as an effort, e, and a flow f, where the product e f is the power. A table of conjugate power variables, divided into effort and flow, is given in Table 22 for a number of energy domains. Table 22: Conjugate power variables. Energy Domain Effort Mechanical translation Force, F Fixedaxis rotation Torque, r Electric circuits Voltage, V Magnetic circuits MMF, 97N Incompressible fluid flow Thermal (after Senturia, pg 105 [128]) Pressure, P Temperature, T Flow Velocity, v Angular velocity. o) Current. I Flux rate, Volumetric flow, Q Entropy flow rate, S Helmholtz Resonator Lumped Element Model LEM was applied to the Helmholtz resonator in order to better understand the system [1, 125]. A schematic diagram of a Helmholtz resonator is shown below in Figure 23, where V is the cavity or bulb volume, / and S = ra2 are the length and crosssectional area of the neck, respectively, where a is the radius of the neck, P' is the incident acoustic pressure, and P,' is the cavity acoustic pressure. Both PI'and P,' are considered to be functions of the radian frequency, a). neck PH R 7PP cavity Figure 23: Diagram showing side view of a Helmholtz resonator and its mechanical equivalent of a massspringdamper system. A conventional Helmholtz resonator can be lumped into three distinct elements. The neck of the resonator constitutes a pipe through which frictional losses are incurred. Additionally the air that is moving through the neck possesses a finite mass and thus kinetic energy. Therefore the neck has both dissipative and inertial components. The air in the cavity is compressible and stores potential energy, and is therefore modeled as a compliance. The acoustic compliance of the cavity and effective mass of the neck can be derived from first principles. [129] As mass flows into the bulb, the volume, V, remains constant, assuming the walls are rigid, and so the pressure must rise, by conservation of mass. dM dp(t) F kg7 =V Qp mass flow rate kg, (2.15) dt dt T s where Mis the mass in the bulb, po is the mean density of the air, and Q = u(ra2) is the volumetric flow rate or volume velocity, where u is the velocity. If the disturbance is harmonic and isentropic then P2'= c02 = (2.16) jco V where co is the isentropic speed of sound of the medium and j = 1 . The linearized momentum equation for a lossless medium is given by Ou o = VP', (2.17) where P' is the acoustic pressure. Assuming a linear pressure gradient yields P' P' = p 1, (2.18) at where / is the length of the neck. Substituting for P2 yields the following equation. S= 2 (2.19) jcoV S Factoring Q, this can be rewritten as P = O? I + jMO (2.20) where the effective acoustic compliance, C, of the cavity is C, = PoC2 (2.21) and effective acoustic mass, MA,, of the air in the neck is given by M N (2.22) LS m The notation for the lumped elements has been determined through the use of three components. First, the primary variable name is determined by the element type. Next, the first subscript represents, the energy domain in which the element has been defined, where the subscript 'a' represents the acoustic energy domain, 'e' represents the electrical enegy domain, and 'm' represents the mechanical energy domain. The final subscript, which has been capitalized for easier reading, represents the actual structure that is represented at least in part by the lumped element. In this instance, 'C' represents, the cavity, while 'N' stands for the resonator neck. Later, a compliant diaphragm will be introduced that will be represented by a 'D' subscript. The expressions above do not account for any viscous damping effects that occur in the Helmholtz resonator neck. The viscous damping represents a resistance, whose value can be approximated from pressure driven, laminar pipe flow as R 8;L/ kg (2.23) S2S m4S where / is the dynamic viscosity of the air. Furthermore, the viscous damping produces a nonuniform axial velocity profile in the neck that ultimately leads to an additional factor of 4/3 in the expression for the effective acoustic mass [130]. The corrected effective mass is then given as 4pM o kg (2.24) 3S m Additionally, the effective resistance and mass of the neck are, in fact, nonlinear and frequency dependent due to turbulence and entrance/exit effects, [131] however for simplicity in modeling, these are not considered here. The expression for cavity compliance given by Eq. (2.21) can be compared to an approximation based on the exact expression for the impedance in a short closed tube [2]. The exact expression is given by Z,,n = c cot(kl), (2.25) na where k = is the wavenumber. Using a Maclaurin series expansion of the cotangent CO function yields cot(ki) 1 kl1 (kl)3.... (2.26) kl 3 45 For kl << 1, the impedance can be approximated by keeping only the first couple of terms in the expansion, yielding Poco klpoco poc2 poV Z+ kl= p + P + pV (2.27) jkla 2 3ra2 jcoV 3(sra2) From this expression, we once again see that C, C = a. (2.28) We now also have an additional mass term, given by M = poV [kg] (2.29) 3( ;ra2) m 4 which is equal to onethird the acoustic mass of the cavity. This correction term is small for k <<1 but becomes more prominent as kl increases. At kl= 1, the correction term is 33.3% of the primary term, while at kl = 0.1 the correction term is only 0.33% of the primary term. Equivalent Circuit To create an equivalent circuit model for the Helmholtz resonator, knowledge of how to connect these lumped elements is needed. Connection rules between elements are defined based on whether an efforttype variable or a flowtype variable is shared between them [132]. Whenever an effort variable, such as force, voltage or pressure, is shared between two or more elements, those elements are connected in parallel in the equivalent circuit. Conversely, whenever a common flow (i.e., velocity, current, or volume velocity) is shared between elements, those elements are connected in series. These connection rules are used to obtain the equivalent circuit representation for the Helmholtz resonator, as shown in Figure 24. The connection rules, as given, are assuming that what is known as an impedance analogy is employed. If an admittance analogy were used instead, then the connection rules would be reversed from what is described above. RN My PI P2CLL =2 Figure 24: Equivalent circuit representation of a Helmholtz resonator. The frequency response function P2/P represents the pressure amplification of the resonator. It is the ratio of cavity pressure to incident pressure, and is given by 1 P2 SC'C P= sac (2.30) P 1 R RN + sM, + sC, where s = jco. From an analysis of the above circuit, a single resonant peak is expected in this frequency response function, when the sum of the reactances is zero, and is given by 1 fre. = Ic [Hz]. (2.31) At the resonant frequency, the pressure amplification reaches a value of PA. = (2;fref). (2.32) RaN This is shown in Figure 25, for an arbitrary Helmholtz resonator having a neck length and diameter of 3.18 mm and 4.72 mm, respectively, and a cavity volume of 1950 mm3. The single peak in the pressure amplification frequency response represents the single degree of freedom present in the system. 102 (a) 1 10 10 1 " 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] (b) 50  CD C 1 50 200 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] Figure 25: Theoretical pressure amplification of a conventional Helmholtz resonator. (a) magnitude (b) phase. Piezoelectric Composite Plate On the back wall of the Helmholtz resonator, a piezoelectric diaphragm can be placed such that the acoustic pressure in the cavity can be exploited to cause a deflection of the plate and ultimately an electrical signal via piezoelectric transduction. The piezoelectric composite plate, consisting of a circular layer of silicon and an annular ring of PZT, is shown in crosssection in Figure 26, whereE is theYoung's modulus, v is Poisson's ratio and p is the density of the silicon and PZT as indicated by the subscript. Additionally, t is the thickness, and R2 is the outer radius of the silicon, while for the PZT, tp is the thickness, e is the relative permittivity, d,3 is the transduction coefficient for a voltage applied across the piezoelectric causing a displacement in the radial direction, and rp is the residual stress. A region of high stress exists in a circular plate near the clamped boundary during deflection of the plate. By placing the PZT in this region, the electromechanical coupling is increased due to the stress concentration. In addition, the annular structure facilitates the connection of electrodes and bond pads, as the metal lines can be run from the bond pads to the outer radius of the annular structure and therefore do not have to be placed across the surface of the diaphragm. Es ots Figure 26: Crosssectional and 3D schematic of piezoelectric composite circular plate. (not to scale) Piezoelectricity Mathematically, the linear piezoelectric effect is expressed as [133] S, = s Tkd +dkEk (2.33) EP v P, t r F,~ D = dklTkl + lEk, (2.34) where S, is the mechanical strain, s'1 is the elastic compliance (m2/N) at constant electric field, Tk is the mechanical stress (N/m2), and dj is the piezoelectric coefficient (C / N or m / V), Dr is the electric displacement (C / m2), k is the electric permittivity at constant stress (F/m), and Ek is the electric field (V/m) [133]. These equations can also be rewritten using a reduced notation convention, by replacing ij or kl by p or q where i, j, k, and I can only have values of 1,2,or 3 while p and q can have any value between 1 and 6. Using this notation, the resulting equations are S, = fT +dEk (2.35) and D = dqTq + Ek. (2.36) The subscripts in the reduced notation equations refer to the component of each variable in a specified direction as defined by Figure 27. For example, d31 is the piezoelectric coefficient relating electrical displacement in the '3' direction (zaxis) to a mechanical stress in the '1' direction (xaxis). From Eq. (2.33), it is also seen to relate a mechanical strain in the '3' direction (xaxis) with an electric field in the '1' direction. 3 (z) 6 4 A O 2 (y) ~ 5 1(x) Figure 27: Notation of axes used in piezoelectric transduction. One class of materials, ferroelectrics, exhibits the property that the direction of their polar axis can be changed through application of an external electric field [5, 6, 134]. The polar axis is the direction along which a polar molecule exhibits an internal electric field. When the external electric field is then removed, the polar axis remains in an altered direction. This process that causes a long term rotation of the polar axis is commonly referred to as "poling". All ferroelectric materials are also piezoelectric, however they are not naturally piezoelectrically active, as the crystal structure is centrosymmetric, as in Figure 28a. PZT is a typical example of a ferroelectric material, and has a perovskite crystal structure as shown in Figure 28. By applying an external electric field to rotate the polar axis, a noncentrosymmetric crystal structure is created and the ferroelectric material becomes piezoelectrically active, as in Figure 28b. Additionally, as a piezoelectrically active material is brought close to its Curie temperature, it loses its piezoelectric properties as the dipoles relax to their unpoled state. I I I I I I I     T    SIII    4  4 i3 a) Zr/Ti Pb 0 0 b) @ Zr/Ti Pb 0 0 Figure 28: Idealized perovskite crystal structure for PZT. a) centrosymmetric structure prior to poling. b)noncentrosymmetric structure after poling. A piezoelectric ceramic, such as PZT, consists of individual domains. The polarization within each domain is in the same direction; however, the polarization direction varies randomly from domain to domain, leading to a net polarization of zero. This is illustrated in Figure 29(a), for the unpoled material. After poling the material with a sufficiently high electric field, the dipoles are rotated so that the polarization of each domain is in the same general direction, as defined by the poling direction, as is illustrated in Figure 29(b). Raising the temperature during poling enhances the polarization and resulting piezoelectric properties, as the dipoles rotate more readily at higher temperatures [6]. (a) (b) Figure 29: Schematic of the poling process: a) An unpoled piezoelectric material, where the polarization in each domain is randomly oriented. B) The same material, after poling, where the domains are now polarized mostly along the direction of the applied field. (after Setter, pg 6 [6]) An additional property of ferroelectric materials is the doublevalued nature of their response to an electrical excitation, resulting in a hysteretic behavior. Hysteresis is a property of systems that do not react instantly to an applied force and may not return to 40 their original state when the force is removed. In ferroelectrics, when an electric field via a voltage is applied, a polarization is incurred. Upon removal of the voltage, the polarization decreases but does not return to zero. The result is a remanent polarization, PI. If the voltage is swept up and down, the result will be what is known as a hysteresis loop, shown in Figure 210. Figure 210: Polarization vs. electric field hysteresis loop (after Xu, pg 10 [134]). The graph is displayed here as polarization, P, typically given in [I C/cm2], versus applied electric field, E, although other variations do exist. There are four figures of merit shown in the illustration. Pm is the maximum polarization measured, while PI is the remanent polarization (i.e. the polarization which remains when the field is returned to zero). Furthermore, PI is known as the spontaneous polarization and is defined as the straight line extrapolation of the line defined by the upper saturation region. Finally, the coercive field, Ec, represents the magnitude of the field required to cancel out the remanent polarization. Composite Plate Modeling In order to accurately model this structure, the material properties of each of the layers were considered. These include the mechanical properties of the layers such as Young's Modulus, E, and Poisson's ratio, v and the dielectric constant, e, and piezoelectric coefficient, d31, of the PZT. The subscript p or s indicates the layer as PZT or silicon respectively. Furthermore, the geometry of the structure is accounted for in the values for inner PZT radius, R,, outer PZT radius, R2, PZT thickness, t and silicon thickness, t. Lumped element model A pressure applied to the plate creates a deflection of the plate, resulting in a stress in the piezoelectric layer. The stress deforms the piezoelectric layer, creating an electrical charge on the electrodes, thus generating a voltage across the piezoelectric layer. Although the stress and charge are distributed over a finite region of the plate, it is useful to first look at onedimensional (1D) piezoelectric transduction. This 1D analysis can then be extended to incorporate effective lumped element values that are calculated from the actual distributed case. In the 1D piezoelectric transduction, the piezoelectric material displaces longitudinally due to the application of a force, F, and/or a voltage, V, applied in the thickness direction. Additionally, a charge, q, is generated by the application of the same force and/or voltage. The constitutive equations, Eq. (2.33) and Eq. (2.34), can be extended to this situation by modifying their form. When looking at the resulting strain and electric displacement in the '3' direction, for an applied stress and electric field in the same direction, the equations reduce to S33 = s3T+d33E (2.37) and D33 =d33T + 3T3E, (2.38) Then this set of equations can be converted by multiplying both sides by the thickness of the piezoelectric material. t(33 =sT+d33E) (2.39) (2.39) tS = sE tT+d33tE Now, the mechanical compliance of a material under compression in the thickness direction can be defined as s3jt C st (2.40) SA' where t is the thickness, and A is the area over which the force is applied. Additionally, for a constant electric field, E, E = (2.41) t Substituting (2.40) into second part of (2.39) yields, x= CAT+d33V (2.42) Finally, the stress, T, when applied over the area, A, can be equated to a force, F, given by F = TA, (2.43) leading to the final equation given by x= CF+d33V. (2.44) A similar procedure can then be performed on Eq. (2.38), by multiplying both sides by the area, and making similar substitutions as before, giving A(D33 =33T T3E), (2.45) q= d33F+ 3T3AE, (2.46) E3T A q=d33F +33 (2.47) t so q= d33F + CV. (2.48) Thus a pair of equations can be written describing the 1D piezoelectric transduction, and is given by x Cs, d3I F [)]= [C, IF (2.49) q d33 C_ V In this pair of equations, C,,, the mechanical compliance when a short is placed across the electrodes of the piezoelectric, is defined as Cm (2.50) F while Ce the electrical capacitance when the piezoelectric is free to move, is defined by Eq o 33 AP C = F _33Ap (2.51) Vtp where q is the resulting charge from the applied voltage V, E33 is the electrical permittivity in the thickness direction, Ap is the area of the piezoelectric, and tp is the thickness of the piezoelectric. Furthermore, d33 is the piezoelectric coefficient relating the displacement in the thickness direction for an applied voltage in the same direction, when no mechanical force is applied across the piezoelectric, defined by d33 F= (2.52) For the case of the piezoelectric composite plate that is presented here, though, there is a distributed, nonuniform deflection across the plate. In order to apply the 1D model developed above to this situation, it is necessary to lump the actual distributed deflection to a single point and compute effective values by equating the energy in the distributed system to the energy in a corresponding lumped component. The 1D electromechanical transduction described above can then be extended to relate the acoustical and electrical energy domains by integrating over the areas and switching to acoustic conjugate power values, as well as incorporating timeharmonic signals [64]. The timeharmonic, electroacoustical equivalent of Eq. (2.49) is given by SL] d=oC jodA][V] [ AS] (2.53) Q jod, jcoCa P M1 S whereP is the acoustic pressure, Q is the volume velocity of the plate, I is the electrical current. Furthermore, dA is the piezoelectric coefficient relating the volumetric displacement to the applied voltage, when pressure equals zero, and defined by R2 fw(r)\ 2,rrdr AVol PJw ( 3 d, = (2.54) SP>O V v where w (r) is the deflection as a function of the radial position, R2 is the radius of the plate and AVol is the volumetric displacement defined by R2 AVol= 27rrw(r)dr [m3. (2.55) 0 Additionally, Ca, the acoustical compliance when a short circuit is placed across the piezoelectric, is defined by R2 w (r) 2Irrdr AVol o m CI o V " o .Pm ,a(2.56) From twoport network theory, a generalized electroacoustic, reciprocal, twoport network can be written as I Y G V [ f G (2.57) Q G Y, P_ where YeJ is the electrical admittance when the acoustic terminal is free to move (i.e no pressure exists, P = 0), Y, is the acoustical admittance when the electrical terminal is shorted (i.e. no voltage exists, V = 0), and G is the electroacoustic transduction admittance, given both by the ratio of current, I, to pressure, P, when V = 0, and the ratio of volumevelocity, Q, to voltage, V, when P = 0. Comparing Eq. (2.53) and Eq. (2.57), it can be seen that G= jcodA, (2.58) Yef = j 0CeC, (2.59) and Y = jo)C". (2.60) Equivalent circuit Using Eq. (2.57), an equivalent electroacoustic circuit can be drawn, as shown in Figure 211, where Ceb, the electrical capacitance when the plate is blocked from moving is given by Ceb C (1 k) [F], (2.61) the transduction factor, q, is given as G jcdA d, PaJ ~= (2.62) Y, jmC C V and the coupling factor, k, is given by k = = (2.63) Physically, the coupling factor, k, represents the fraction of energy that is coupled between the acoustical and electrical energy domains. It is seen here to be a function of the piezoelectric coefficient, dA, which represents coupled energy and the two elements which store potential energy associated with the transduction, Cef and Ca,. The coupling factor is therefore related to the ratio of the coupled energy to the stored energy. Cr v Figure 211: Electroacoustic equivalent circuit representation with Cb, Ca, and . An alternative equivalent circuit can also be drawn, as shown in Figure 212. The alternative circuit is simply another representation for the same physical process. 1:+' Co, P V T_ Figure 212: Electroacoustic equivalent circuit representation with Ce,, C"o and q'. In Figure 212, C,, represents the acoustic compliance when an opencircuit is placed across the piezoelectric, and is given by, Cao =C( 1k2) [ (2.64) Additionally, the transduction factor, 0', is given by G jod dA V (2.65) Yf j oCef Cef Pa The two circuits above represent the transduction under static conditions, thus they do not take into account the dynamics of the system (i.e. mass). At frequencies greater than zero, a mass must be added to the equivalent circuit. As the mass stores kinetic energy, it is represented by its electrical equivalent, which is an inductor. Furthermore, this mass must be in series with the compliance, as both experience the same motion (i.e displacement or velocity). The acoustic mass, M,, is determined by equating the lumped kinetic energy of a point mass moving with the center velocity to the total kinetic energy of the vibrating diaphragm and is therefore given by , R2r, ( rAV 2 (2. M = 27r pA \rdr (2.66) 0 AV=0 where p, is the areal density of the piezoelectric composite plate defined by, PA =J pdz [kg/m2], (2.67) where p is the density of the corresponding layer. In addition to the above mentioned elements, it is also necessary to include an additional element that represents the dielectric losses that are incurred during transduction, yielding the circuit shown in Figure 213. Generally this is expressed through a term known as the loss tangent, also known as the dielectric loss factor. Represented by the ratio of the parallel reactance to the parallel resistance in the dielectric, the loss tangent is formally given by X 1 tan = P (2.68) R, 2zrfC ,R where R is a resistor in parallel with a capacitance, Cp, and f is the frequency at which Rp and Cp were measured. The loss tangent is also equal to the inverse of the quality factor, Q. "a Cas + [ + P = ZID Figure 213: Dynamic electroacoustic equivalent circuit. In this circuit, MaD represents the acoustic mass of the diaphragm. The input impedance from the acoustical side is then given by 1 R Z= = sMD + + 2 P (2.69) sC.s 1+ SCebR By voltage division we get 02 Rp 02 Rp R R 1+ sCRp 1+ sCR Ve = P b= P (2.70) Z 1 R SsM^ + + p sCD 1 + SCebR, which, after rearranging yields, V 1 S= sCSR 2 (2.71) P s MaDC sCebR, +S MDCs + sRp (C + ) +1 This is the general expression for the opencircuit sensitivity of the circular composite piezoelectric plate. From a physical standpoint it is also useful to look at certain conditions that allow for additional insight. The presence of the dielectric loss resistance, Rp, complicates some of the underlying behaviors of the device, by adding an additional RC time constant. Looking at a situation in which this parallel loss resistance is infinite (i.e. R > oo), the above expression can be reduced to V C 1 == (2.72) S^XsMC "+1 P Cb S2M oDC", 2 CIS + Ceb Several important things can be gleamed from this expression. First of all, the equation describes the behavior of a second order system, with a primary resonance occurring when the denominator goes to zero at Sre^ Cb (2.73) r MaD Cos From this expression, the resonance is seen to depend on the ratio 2C ,/Ce in addition to the standard dependence on the product MADC, If q = 0, this reduces to a simple plate. For 0 # 0, electrical energy is stored across the piezoelectric, resulting in a stiffer device possessing a larger o),  The full expression that includes the dielectric loss, given by Eq. (2.71), is the general expression for the sensitivity of the circular composite piezoelectric plate. It can be further simplified for two important cases: 1. Well below the shortcircuit mechanical resonant frequency (compliance dominated) 2. At the shortcircuit mechanical resonant frequency (resistance dominated) Case 1: Well below the shortcircuit mechanical resonant frequency When a short is placed across Ceb, it is equivalent to assigning R > 0 Physically, this implies that no electrical energy storage is allowed, as the voltage across the capacitor, Cb, must be zero. The expression for the shortcircuit mechanical resonant frequency is given by l 1 (2.74) which is the same resonant frequency as described above when 0 = 0. Note that both cases describe a situation where no electrical energy is stored. The general expression for sensitivity can then be rewritten in terms of o)re, as V 1 P = jqC,%RP 2 2 (2.75) oK j CbR Cw o + + j (Ceb 2C) W res, b ressc Well below resonance, we have jo 1. (2.76) This term can then be dropped from the expression, yielding V C 1 =jco (2.77) P Ceb 2 Ceb RpCeb To simplify this further, the transduction factor, q, can be written in terms of the effective piezoelectric modulus, dA, by recalling Eq. (2.62), yielding V jcodA C, 1 P C, Ceb dA )2C' jeO) 1+ a + C, Ceb RCeb (278) (2.78) dA 1 Ceb 1+ dA21+ 1 C C,Ceb jo RpCeb Furthermore, the blocked electrical capacitance, Ceb, can be written in terms of the free electrical capacitance, Ce and the coupling coefficient, k, using Eq. (2.61) and Eq. (2.63) to yield the an expression for sensitivity, given by V d A I 1 P Ceb 1+ dA2 + I A 1d(2.79) I C 1CJ j^ Ceb d 1 SCasC 1 k2 j R Cef 1 k2 d 1 (l1 k2 jR Ce1 k2) which can be simplified further to yield d C 1 1 eb 2 + eb (1 k2) jR dA Cef + j coRp (2.80) 1+ j ) CfR Now under the condition of a high value for the loss resistor, Rp, Eq. (2.80) can be reduced approximately to V dA P Cef This corresponds to the alternative transduction coefficient, 0', given by d, ci Cef (2.81) (2.82) and thus under the conditions of a high loss resistor and operation well below a S I, the sensitivity can be given by V P Case 2: At the shortcircuit mechanical resonant frequency The general expression for sensitivity is once again given by V 1 S= joC, RP 2 2C, j j CebR + + j p ( Ceb +Ca)+ )resc )ress (2.83) (2.84) Now, since we are only considering the system at the shortcircuit resonance, where (0 = ),,se we have o = 1, (2.85) yielding an exact expression for the sensitivity as V 1 1 V= j1o CRp1 (2.86) P jc CebRP 1+ jR(Ceb +2CaL)+1 We therefore find that the sensitivity at resonance reduces to V 1 #' V I (2.87) P q k2 ' as k2 = '. (2.88) This can be understood from the equivalent circuit of Figure 213. At a = o)reS', the impedance of the mass is canceled by the impedance of the compliance, permitting a direct transduction of energy from the acoustic to electrical energy domain. Note that 0),o exists whether or not R, = 0, therefore Eq. (2.86) is always valid whenever o = )se In other words, Eq. (2.86) holds whenever the operating frequency coincides with the frequency that is defined by the shortcircuit resonance, but makes no requirement for an actual shortcircuit load. As the load across Ceb is increased, the resonance frequency will shift towards the opencircuit resonance. Recalling Eq. (2.64), the open and shortcircuit acoustic compliances are related by the coupling factor and define the limits of the resonant frequency as all resistive loads fall somewhere between open and shortcircuit. The larger the coupling factor, then the larger the range between open and shortcircuit resonant frequencies. General expression vs. simplified cases Figure 214 plots the opencircuit and near shortcircuit sensitivity versus frequency overlaid with the values of 0' and 1/0 The material properties used in calculating these results are shown in Table 23 and Table 24. Notice that below resonance, the general opencircuit sensitivity expression asymptotes to the value of q'. Additionally, notice that the general expression reaches the value of 1/0 at a frequency below where the peak amplitude occurs. This is because 1/0 is the simplified expression for the sensitivity at the shortcircuit mechanical resonant frequency, whereas the general expression represents the opencircuit frequency response. Thus the peak amplitude occurs at the opencircuit resonant frequency. The frequency at which the general expression for sensitivity reaches the value of 1/\ coincides with the shortcircuit resonant frequency, as expected. Also note that this shortcircuit sensitivity calculation was performed with an almostshort circuit condition, primarily to illustrate the effect on the resonant frequency. To achieve the almostshort circuit condition a resistive load of 1 Q was used as the value is much less than the impedance due to Ceb. One final feature of the frequency response is the low frequency rolloff that is visible in the open circuit case. This rolloff is due to the dielectric loss resistor, Rp, and the resulting time constant due to the combination of that resistor and the electrical capacitance, as evidenced in Eq. (2.80) for the low frequency regime. Table 23: General material properties used in modeling. Material E[GPa] v p[ Silicon 150 0.27 Platinum 170 0.38 TiO2 283 0.28 [135, 136] Table 24: PZT properties used in modeling. EA [GPa] v, p [kg/m3] d 30 0.3 7600 1000 [9, 10, 16, 31] 0 10 1 1 10 + Near short circuit frequent 2 Open circuit frequency re 10 S10 1 >o a^ ^^_B^nnn^ kg/nm3 2300 21440 2150 S[pm/V] 50 cy response sponse 10  10 10 0 1 2 3 4 10 10 10 10 10 Frequency [Hz] Figure 214: Sensitivity vs. frequency for a piezoelectric composite circular plate. Effects of including the radiation impedance Since the clamped circular plate is vibrating in a medium, the radiation impedance of the plate must be taken into account, and consists of a radiation mass and radiation resistance. The radiation mass accounts for the inertial mass of the fluid that is vibrating in unison with the plate, while the radiation resistance accounts for the acoustic radiation t[nm] 3 0.170 0.100 tan 8 0.02 uo [MPa] 30 e am,  5; of energy away from the plate. In addition to the radiation resistance, energy is lost via structural radiation to the supports. The radiation resistance and radiation mass is determined to firstorder by approximating the backplate as a piston in an infinite baffle, as given by Blackstock [2] in terms of specific acoustic impedance, Z as 2J, (2ka) 2K, (2ka) , ZP =Pc 2J 2ka+ = 2K2ka pc[R(2ka) +jX (2ka)], (2.89) 2ka 2ka where R1 (2ka) and X1 (2ka) indicate that they are functions of 2ka, a = R' is the radius of the piston, J, is a Bessel function of the first kind of order one, and K, is a first order Struve function. The Maclaurin expansions of Eq. (2.89) are also given by [2]as (ka)2 (ka)4 (ka)6 R, +  (2.90) 1.2 1.22 3 1.22.32 4 and 4 2ka (2ka)+ 2 .... (2.91) X, (2.91)2 2 I r 3 3 5 3 5 .7 For small values of ka, where ka << 1, the resistance and reactance can be approximated by keeping only the first terms of Eq. (2.90) and Eq. (2.91). The radiation impedance as given in Eq. (2.89) is in terms of specific acoustic impedance. This is converted to an acoustic impedance via the effective area, Aff, of the piezoelectric plate. As the circular composite diaphragm does not deflect uniformly over the entire surface (i.e. the deflection is a function of radial distance), the volume displaced by the deflection of the diaphragm is less than that of a circular piston of the same area. The effective area, Af, is therefore defined to represent the equivalent area that a circular piston with uniform deflection would need to have to create the same volumetric displacement as the composite diaphragm. This is necessitated by the need to maintain continuity of volume velocity across the interface between the mechanical and acoustical domains. It can be calculated by integrating the distributed deflection of the diaphragm over the entire surface and then dividing by the center deflection of the diaphragm. By using the center deflection as the reference value, the distributed deflection is then considered to be lumped to the center. For an axisymmetric deflection, such as would occur for the fundamental mode, the effective area is given by w (r)27rrdr Aw() [m2] (2.92) The radiation resistance can then be approximated, for low values of ka as S(ka) 2 P [ kg] RaDrad 2Af ,ka << 1 (2.93) while the radiation mass is approximated as M 8ra k c ka < 1 (2.94) These two elements add in series to create a radiation impedance, Zra, that is defined by ZaDad = RaD +sM d (2.95) The equivalent circuit of the composite plate including the radiation impedance is shown in Figure 215. Q R d aDrad M8a C as I + + P V .e TCe= Figure 215: Equivalent circuit of piezocomposite plate including the radiation impedance. 58 When the radiation impedance of the diaphragm is added to the equivalent circuit, the analysis becomes only slightly more complex. The acoustical input impedance, Zn , defined as P/Q, can be represented in terms of the equivalent circuit parameters, and is now given by 1 R Z = sMaD +sM ,,d +R d + +2 P (2.96) C a Drad a 1+ sC R Then, the general expression for sensitivity is now given via the voltage divider method as S R RP 1+ CebRp 1+sCR VP = P = P (2.97) Z sMaD + SMD ad + RDrad + 1 + 2 Rp sC l+sCebRp or 1r _ SCa e R (2.98) P 1 R sMaD + SMaDad + RaDad + p + 02 R sC 1+ sCebR, From Figure 215, it can be seen that the radiation mass, Marad, adds directly to the acoustical mass of the plate, Ma>, as is evidenced in Eq. (2.97), while Rrad provides damping to this secondorder system. Note that here, the acoustical resistance, Rra , damps the resonance, while the electrical resistance, R leads to a low frequency rolloff The different effects arise because Rrad is in series with the reactive elements, C and MaD, while Rp is in parallel with the capacitance, Cb. The general expression for the undamped resonance frequency is now given by S 2 ( +1 Sreb (2.99) (aD + MaDrad )Cas When a short is placed across Ce, it is effectively removed. This is mathematically equivalent to assigning Ceb> o as the impedance of a capacitor is inversely proportional to the capacitance. The shortcircuit resonance frequency then reduces to Sres (2.100) J)esse D + MaDrad Upon comparison with Eq. (2.74), it can be seen that the shortcircuit resonance frequency has now been shifted downward by the radiation mass, as is also the case with the opencircuit resonance frequency. At the radial frequency of the shortcircuit resonance, the shortcircuit input impedance reduces to Z,n =RDad, (2.101) because Ceb and R are effectively removed by the shortcircuit. The volume velocity, Q, is then given by P Q = (2.102) RaDrad Via the piezoelectric transduction, the current, I, in the piezoelectric material, is then given by OP I = Q = (2.103) RaDrad The output voltage in this case, however, remains at zero due to the short circuit placed across the output capacitance. To achieve real power at the output, a finite resistive load is necessary. Adding a resistive load, Rload, across the output yields the equivalent circuit shown in Figure 216. R'd MSld I Cas :l I + + P CT eb load Figure 216: Equivalent circuit with resistive load. The input impedance, Z4n, is then given by 1 + loadp Z1 = S (M +M + +d) + RI + + 2 RloadR (2.104) SC Rload Rp +RloadRpCeb Now, if we choose Rload such that Road <<1/SCeb and Road << R, then the parallel combination of these three elements can be approximated with just Rloa. In addition, the resonant frequency will be very close to the shortcircuit resonant frequency. At this frequency, the input impedance reduces to Z,, = RaDrad + 2Rload, (2.105) and the volume velocity, Q, is given by P P Q = (2.106) Z, RaDrad + 2Rload Through piezoelectric transduction, a current, I, is created in the piezoelectric given by I = Q = PO (2.107) RaDrad + 2Rload As the resistance, Rlod, is much less than 1/sCeb most of the current goes through it, leading to a voltage drop given by V = IRoad = P ad (2.108) RaDrad + 2Rload Thus the power absorbed by the load resistance is purely real and is given by P2 2ROd Re{)}= Re{IV} = 2 Rload (2.109) (RaDrad +2 Rload An optimal solution to this equation is found by setting d = 0 = _p22 (R d (2.110) dRIoad (Drad + 2Rload) Solving for the optimal load resistance yields R Road a d, (2.111) which is just the impedance matching condition at the interface. Note that is for the special case of Roa << 1/sCb Rloa < R, and operation at resonance. Equivalent circuit parameters of piezoelectric composite circular plates In order to obtain the equivalent circuit parameters, an analytical model was developed for the piezoelectric composite circular plate by Wang et al. [62, 63]. Using this approach, analytical modeling was accomplished by dividing the problem of Figure 26 into two portions, an inner circular plate, surrounded by an annular composite ring with matching boundary conditions at the interface, as shown in Figure 217. The boundary conditions consist of equal moments and forces at the interface as well as equal slope and transverse displacement. After solving for the deflection in each region, the deflection equation for each region can then be combined [62, 63]. Outer Region ; Jini I I I I Region Figure 217: Conceptual 3D and crosssectional schematic of the circular composite plate. (Not to scale.) The deflection equation can then be utilized to determine the potential and kinetic energy stored in the plate, leading to expressions for the acoustic compliance and mass of the composite plate. Similarly, expressions can be found for the electroacoustic transduction coefficient and the blocked electrical capacitance. Using the parameters shown in Table 23 and Table 24, plots were obtained for the lumped elements as a function of both R1,/R and tp /t As many of the material parameters for PZT are highly dependent on actual processing conditions and techniques, typical values were chosen as a 'best guess' estimate [8, 1214, 1648]. The effective acoustic shortcircuit compliance and mass are functions of both R1/IR and t/t and are shown in Figure 218 and Figure 219, respectively. 12 x 10 3.5 3 2.5 S2 E O 1.5 t =0.6 pm P 1.5 t =3.0 am 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R /R2 R1R2 Figure 218: Effective acoustic shortcircuit compliance as a function of R1 /R and tp /t For these calculations, t = 3j/m and R, = 2mm, while the piezoelectric layer has thicknesses of tP = 0.6, 1.2, 1.8, 2.4, 3.0 [pjm]. The acoustic compliance is found to increase with increasing R, /P and decrease with increasing tp /t however, the acoustic mass is found to generally decrease with R/R2t and increase with t /tI It is also useful to look at the physical limits and effects of R /R As R1R / 0, the piezoelectric ring covers the entire surface of the diaphragm and the compliance is at a relative minimum for a given thickness, while the mass is at relative maximum. Furthermore, note that the acoustic mass is most sensitive to changes in the piezoelectric layer thickness under this condition. Meanwhile, as R, IR2 1, the piezoelectric ring would have infinitesimal width and so would be essentially nonexistent. Under this condition, the piezoelectric layer thickness has no effect on the compliance, as would be expected. 3500 3000 2500 tp=3.0 p m 2000  1500 / 1000 t =0.6 tm 500 0 0.2 0.4 0.6 0.8 1 R1/R2 Figure 219: Effective acoustic mass as a function of R1/R2 and tt, For these calculations, t = 3/um and R, = 2mm, while the piezoelectric layer has thicknesses of tp = 0.6, 1.2, 1.8, 2.4, 3.0 [pum]. Shown in Figure 220 is a graph of the resonant frequency as a function of both R, /R and t /t From this figure, it can be seen that the resonant frequency increases as the thickness of the piezoelectric layer increases or generally as the inner radius of the piezoelectric layer decreases. The resonant frequency trend with respect to thickness arises because the effective acoustic mass is found to increase with the thickness while the effective acoustic compliance decreased, but by a much larger amount. Again, it should be noted that at the limit of R/R2 > 1, the resonant frequency is no longer affected by the thickness of the piezoelectric layer. Also, for a given tp, there is a maximum resonant frequency at R /R2 0.4. 14000 12000 /12000 t p=3.0 \tm N 10000 8000 0 t =0.6 am On P ( 6000 4000 2000 0 0.2 0.4 0.6 0.8 1 R /R2 R1R2 Figure 220: Shortcircuit resonant frequency as a function of R,/R2 and tp/t For these calculations, t = 3/um and R, = 2mm, while the piezoelectric layer has thicknesses of tp = 0.6, 1.2, 1.8, 2.4, 3.0 [pum]. The electroacoustic transduction coefficient, q, was also found as function of these relative dimensions, and is shown in Figure 221. The transduction coefficient represents the conversion of the voltage across the piezoelectric to the acoustic pressure produced by the resulting motion of the composite plate. It should be noted that the transduction coefficient is negative, implying a 1800 phase shift between pressure and voltage, as computed relative to an assumed poling direction. As can be seen in Figure 221, a maximal magnitude occurs when R1/R2 is around 0.45. 1 2 / t =0.6 tm (3 3 t =3.0 \tm 4 5\ 6 0 0.2 0.4 0.6 0.8 1 R1/R2 Figure 221: Electroacoustic transduction coefficient as a function of R, /R and t /t . For these calculations, t = 3/um and R, = 2mm, while the piezoelectric layer has thicknesses of tp = 0.6, 1.2, 1.8, 2.4, 3.0 [jum]. Additionally, the magnitude of the transduction coefficient increases with increasing piezoelectric thickness. Looking at the limit as R, /R > 1, the transduction is seen to decrease to zero, as would be expected of a structure with no piezoelectric material. Furthermore, as RR, > 0 and the piezoelectric material covers the entire diaphragm, the transduction factor is seen to go to zero as well. Physically, this results from cancellations between different regions of the diaphragm that are undergoing opposite polarities of stress, such that the net electric displacement (charge) is zero. Another parameter of interest is the coupling coefficient, k. It is defined as the ratio of energy converted by the transducer to the energy supplied to the transducer, thus providing a measure of the coupling. For the purposes of harvesting energy, this parameter is more important than the transduction coefficient as the primary interest is in maximizing the coupled energy, rather than maximizing only the output voltage. A plot of k as a function of R1/ R and tpt, is shown in Figure 222. 0.014 0.012 0.01 t =3.0 pm 0.008 0.006 0.004 t =0.6 atm 0.002 0 0 0.2 0.4 0.6 0.8 1 R1/R2 Figure 222: Electromechanical coupling coefficient, k, as a function of R/R2 and tp/t, For these calculations, t = 3pm and RP = 2mm, while the piezoelectric layer has thicknesses of tp = 0.6, 1.2, 1.8, 2.4, 3.0 [jim]. From the figure, the maximum k of 0.013 is obtained for an R,/R2 ratio of 0.95 and a tp/t ratio of unity. Physically, the increasing trend in k as R1/RP increases, arises from the stress distribution within the diaphragm. There is a stress concentration near the clamped boundary of the diaphragm that provides for a high level of coupling. By concentrating the piezoelectric material in this high stress region, the averaged coupling factor over the ring will be higher than a piezoelectric ring that is spread over a larger area. Acoustic Energy Harvester Dynamic Behavior When the piezoelectric diaphragm is mounted in the wall of a Helmholtz resonator, the equivalent circuit for the diaphragm (Figure 216) is combined with that of the Helmholtz resonator (Figure 24). By using the same lumped element connection rules previously described, the complete circuit can be obtained, as shown in Figure 223. S RaN MaN RaDrad MaDrad Ma CaD I ? I Iload Figure 223: Equivalent circuit for acoustic energy harvester with resistive load. Notice that the equivalent circuit for the diaphragm is in parallel with the cavity compliance, C, This occurs because the pressure in the cavity is responsible both for compressing the air in the cavity as well as deflecting the diaphragm, i.e. the cavity and the diaphragm both see the same pressure. Additionally, it is assumed that Rloa << R< and therefore the parallel combination can be approximated as Road HR = Rlo . Acoustical Input Behavior An expression for the acoustical input impedance can be obtained from the equivalent circuit and is given by S(MD +MaDrad ) + RaDrad + R Z = R sM + C SCaCaD o + RloadSCb .(2.112) 1 1 R + 2R/,,d + (MaD +M Drad ) + +a Drad load sC c sCD 1 + Rload SCeb From this equation, it can be seen that the total input impedance is simply the impedance of the Helmholtz resonator neck in series with a parallel combination of the Helmholtz resonator cavity impedance and the piezoelectric composite diaphragm impedance with a resistive load attached. Many energy harvesting devices with varying geometries were designed for this dissertation, but for illustrative purposes, I will only explore the behavior of one, which serves as a typical example. More explicitly, eight different devices were designed and created to give a range in performance, but to illustrate the qualitative and typical behavior of the devices, only a single "representative" device was plotted for the remainder of this chapter. A plot of the acoustical input impedance versus frequency is shown in Figure 224 for the piezoelectric composite diaphragm, both with and without the Helmholtz resonator, to elucidate the behavior of the individual components. For this plot and those that follow, the Helmholtz resonator has a neck length of L = 3.18 mm, and a radius of R = 2.36 mm along with a cavity volume of V = 1950 mm3, in addition to a diaphragm with a thickness of t, = 3 /m and a outer and inner radius of R, =1.95 mm and R, = 1.85 mm, respectively. These dimensions were chosen as typical values in the range of what was expected of the final device and package design. From this plot, it can be seen that, by itself, the piezoelectric composite diaphragm has a single resonance near 3.6 kHz, where the impedance reaches a local minimum. When combined with the Helmholtz resonator, two minima are seen. The lower resonance that occurs near 1.8 kHz is dominated by the Helmholtz resonator that has an uncoupled resonance of 2 kHz, as evidenced by the peak in Figure 25. The upper resonance at 3.9 kHz is 70 dominated by the piezoelectric composite diaphragm. Additionally, an antiresonance, where the impedance reaches a local maxima, occurs between the two resonances. S0  with Helmholtz resonator 10  without Helmholtz resonator 7 10 1000 2000 3000 4000 5000 6000 7000 Frequency [Hz] Figure 224: Magnitude of the acoustical input impedance for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. (L = 3.18 mm, R 2.36 mm, V = 1950 mm3, t = 3 im, R, =1.95 mm, R, =1.85 mm) The acoustical input power can be obtained from the input acoustic pressure, P, and is given by P2 Re{ rI, Re{Z} Re1 ; e{Z (2.113) The input acoustic pressure, P, is measured inside the planewave tube near the endface of the tube. The microphone that measures this pressure is placed as close as possible ( 1/16 ) to the endface so that it serves as a measure of the pressure that is incident on the energy harvesting device. In the case of the device that includes a Helmholtz resonator, the incident pressure is the pressure incident upon the resonator neck, while for the diaphragm only device, it is the pressure incident upon the face of the diaphragm. A plot of the magnitude and phase of the input power is shown in Figure 225. This plot assumes an ideal acoustic source, i.e. that the applied acoustic pressure is constant and independent of the input impedance. In reality, this is not the case, and the applied acoustic pressure will vary as a function of frequency due to a frequency dependent impedance as seen by the speaker. The important point to take away from this plot, however, is that there are frequencies at which the input power will be maximized. These frequencies occur when the input impedance is minimized thus allowing for maximum power flow from an ideal source. These do not necessarily correspond to frequencies where the output power is maximized, as will be shown in the next section. Electrical Output Behavior The electrical output impedance can be found in a similar manner and is given by Zo, = Rload// // (MD +M rad + +RDd + I//(R, +MI ) .(2.114) sCeb SCOD SC, Expanding this equation out yields the full expression for the output impedance, given by F 1 S(R, +Ma) 1 1 1 sC ac (RiCad (MaD + MaD rad) + +RaDrad 1 +RC + Seb 02 Sca I + R, +M a 7 sCc .(2.115) lout ( 1 sCb sC )s 1 SCe 0 2 M + MaDrad e) + S + rd SC' 72 105 S with Helmholtz resonator  without Helmholtz resonator 10  10 10 I I I I I I 0 1000 2000 3000 4000 5000 6000 7000 Frequency [Hz] 50  C3 S0  a 50 0 1000 2000 3000 4000 5000 6000 7000 Frequency [Hz] Figure 225: Magnitude and phase of the acoustical input power for an input acoustic pressure of P = 94 dB. (L = 3.18 mm, R = 2.36 mm, V = 1950 mm3, t, = 3 jm, R2 =1.95 mm, R, =1.85 mm) A graphical plot of the electrical output impedance is shown in Figure 226 in terms of real and imaginary components. The solid curve in the figure corresponds to the expression given by Eq. (2.115), while the dotted curve represents the impedance of the piezoelectric composite diaphragm by itself. Notice that only the real component shows any significant difference between the two cases. Physically, this is due to the "poor" coupling that is typical of indirect transducers. The result is that acoustical components have minimal effect on the electrical impedance. 73 with Helnhollz resonalor S1011 ..... i withool Helmholtz esPnator 10' tn 1000 2000 3000 4000 5000 000 7000 Frequency 1Hz] 0 DO o / 10000 / / 15000 1000 201 3000 4000 5000 GOfl 7000 Frequency 1Hz] Figure 226: Electrical output impedance for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. (L = 3.18 mm, R = 2.36 mm, v = 1950 mm3, t = 3 um, R, =1.95 mm, R, =1.85 mm) The output voltage can easily be found from the equivalent circuit to be Rload (P0) 1 + Rload V = Cb .(2.116) RaN + sMN 1 R RaN + sa +1 (MaD + MaDrad ) + I + RaDrad + oa + + sMaA SSCaD) 1 + Rload eb S\aC The output voltage is displayed graphically in Figure 227 for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. For both cases, the magnitude has a similar shape to the output impedance, although a few differences remain. In the output impedance curve, the second resonant peak is higher than the first, whereas for the output voltage curve shown below, the first resonant peak 74 is higher than the second. The voltage curve shown was computed for a load resistance, Road,, equal to the electrical output resistance at the diaphragm resonance frequency, and an input acoustic pressure of 1 Pa. 10"2   . A  with HeliiholIr resonalor  without Helmholtz resonator 10 i~ i Ii 1 ' 1 0 1000 2000 30 400D 5000 600 71OD Frequency [Hz] " .  a 2100 I 0 1000 2000 A000 6000 GDOOi 700o Frequency [Hz] Figure 227: Magnitude and phase of the output voltage for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. (L = 3.18 mm, R = 2.36 mm, V = 1950 mm3, t = 3 im, R, =1.95 mm, R, =1.85 mm, P = 94 dB1) The electrical power delivered to the resistive load, RiQod, can be found from V2 Re{H}= (2.117) Rload A plot of the electrical output power delivered to the resistive load is shown in Figure 228 for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator, for the case of an incident acoustic pressure of 1 Pa. 1 dB re 20uPa throughout this document. I I . without Helmholiz resonator 0 1000 200D 300 400A 00D 6000 7000 Frequency [Hz] 200 100 _ .. ...  0  ,100 LL LL 1I I 0 1000 Io 3000 4Dfo0 000 Go00 "DO Frequency [Hz] Figure 228: Magnitude and phase of the electric output power delivered to the load resistor. (L = 3.18 mm, R = 2.36 mm, V = 1950 mm3, ts = 3 pm, R, =1.95 mm, R, = 1.85 mm, P= 94 dB) The overall power conversion efficiency, F, can then be found as the ratio of output electrical power to input acoustical power, given by Re{f} =Re{Ho} (2.118) Re{n The magnitude and phase of the efficiency is shown in Figure 229 for the piezoelectric diaphragm by itself and in combination with the Helmholtz resonator. As seen in the plot, the efficiency of the composite diaphragm reaches a peak at the diaphragm resonance, while the efficiency of the coupled system reaches a peak at the second resonance, which is dominated by the diaphragm resonance. It is important to note that the peak magnitude is similar both with and without the Helmholtz resonator, suggesting that little benefit is gained through the use of the Helmholtz resonator, as this 76 Helmholtz resonator was not optimally designed for impedance matching. This need not always be the case, as the relative benefits of impedance matching are somewhat offset in varying amounts by the increased resistive losses due to the Helmholtz resonator. The benefits can thus be enhanced by maintaining the improved impedance matching while minimizing the additional losses. A perfect impedance match between the plane wave tube and the Helmholtz resonator would improve the efficiency by approximately 40% over the values shown in the figure. This value is estimated based on comparison to a Helmholtz resonator that is perfectly impedance matched to the planewave tube. L 34 C10  , S10'  0 10iD 200 3000 4i000 f6L EOJD 7000 Frequency [Hz]  with Helmholtz resonalor without Helrnholtz rEsonator    S100  SC 0 1000 000 3000 4000 600f OOD 7000 Frequency [Hz] Figure 229: Magnitude and phase of the energy harvester efficiency. (L = 3.18 mm, R =2.36 mm, V =1950 mm3, t =3 /um, R, =1.95 mm, R, =1.85 mm, P = 94 dB) Operation at an Optimal Frequency To get a better feel for these devices in their intended application as harvesters of energy, it helps to look at the theoretical behavior for a single frequency. For the purposes of illustration, a device with the geometry of Device 8, as given by Table 25 on page 81 was used and the frequency at which the efficiency is maximized was chosen. This optimal frequency is different between the diaphragm mounted on the Helmholtz resonator and the diaphragm strictly by itself. For the diaphragm mounted on the Helmholtz resonator, a sinusoidal signal at 1966 Hz was applied, while for the diaphragm mounted by itself, a sinusoidal signal at 13.11 kHz was used. For both cases, the signal was swept over a range of acoustic pressures while the open circuit output voltage was determined. A plot of the open circuit voltage versus acoustic pressure for both cases is shown in Figure 230. 10 10 Membrane Membrane/HR 1 " 10 > 12 10 115 120 125 130 135 140 145 150 155 160 Input Acoustic Pressure [dB] Figure 230: Opencircuit voltage as a function of the applied acoustic pressure for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator. (Device = 8 from Table 25, f =1966 Hz, f, =13.11 kHz) Note the steadily increasing output voltage as the acoustic pressure increases, as would be expected for a linear system such as this. This model, however, does not take into account any nonlinearities that may occur at high acoustic levels. Such nonlinearities may include, but are not limited to, piezoelectric saturation effects on the output voltage and large deflection effects on the diaphragm compliance, in addition to acoustic nonlinearities introduced by the Helmholtz resonator orifice at high amplitudes. These nonlinearities will act to limit the maximum voltage in a real world application. Operating at these same optimal frequencies, a load resistor was placed across the output terminals and the power was determined through that resistor. Repeating this over a range of load resistances yields the plot shown in Figure 231. Note that while the input power remains constant, the output power has a peak at a particular load resistance. This corresponds to the optimal resistance as given by Eq. (2.12). The input and output power using an optimal load are also shown on this graph and are indicated by an 'x' for both the diaphragm with the Helmholtz resonator and without. 4 10 in Membrane 6 H. Membrane/HR 10 in out Membrane S out Membrane/HR 108 10 10 2 3 4 10 10 10 Load Resistance [Ohms] Figure 231: Input and output power as a function of the load resistance. Shown for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator. (Device = 8 from Table 25, f =1966 Hz, fm, =13.11 kHz, and P= 114 dB) Using the optimal values for the load resistance in each case, the input power and output power was determined as a function of the applied acoustic input pressure, and is shown in Figure 232. Once again, it is important to note that nonlinear effects are not included in the model used to calculate the output power. 2 Hn. Membrane 10 in Membrane/HR 0 H out Membrane 0in 10 nout Membrane/HR out 2 ^ ^ ^ 10 ( 104 0  10 8 106 10 10 115 120 125 130 135 140 145 150 155 160 Acoustic Input Pressure [dB] Figure 232: Input and output power as a function of the applied acoustic input pressure, while using an optimal load resistor. (Device = 8 from Table 25, f =1966 Hz, f,, =13.11 kHz, Road,Mem = 486 Q, Rioad,MemHR = 3242 Q) Device Configurations The devices were designed based on the theoretical plots, along with estimates of the output power based on the lumped element model. Eight configurations were chosen based on their resonant frequency and maximum power before the onset of nonlinear behavior. The criteria for choosing the designs were to keep the resonant frequencies inside of the testable range, below 6.4 kHz, while maximizing the power. Four outer radii, R2, were chosen, consisting of 0.9, 1.2, 1.5, and 1.8 mm, while the inner radii were all chosen to be 0.95 times their outer counterpart. The second four configurations had the same dimensions as the first four, with the addition of a central pillar of silicon designed to act as a rigid mass. It should be noted that the addition of the central mass invalidates the structural model that was used, however, the model is still expected to provide some guidance over the behavior of these particular devices as well. The central mass was added in order to reduce the resonant frequency of the diaphragms. It was fixed at 0.2 times the outer radius of each diaphragm. This value was chosen such that the central mass was located primarily over a region of the diaphragm that experiences little stress. By restricting the mass to this region, only a minimal increase in the diaphragm stiffness was expected, while the mass of the diaphragm could be greatly increased, thereby lowering the resonant frequency. The addition of this central pillar was predicted to lower the resonant frequency of each device by an average of 57.76 %. Additionally, the diaphragm thickness was chosen to be 3 um. Only one thickness for the diaphragm was chosen as all the devices were going to be made on the same wafer. The chosen device configurations are summarized in Table 25. Overall, the geometries were chosen as to create a proof of concept set of devices. Manual parameter adjustment was performed in order to improve performance, however, strong constraints were placed on the geometry in order to ensure fabrication ability and improve the chances of firstrun success. Also listed in the table are the predicted resonant frequencies and maximum power before the onset of nonlinearities. Note that this does not define an upper limit for the output power but merely provides a figure of merit upon which to compare devices. Also, for the purposes of predicting the behavior of these specific devices and comparing them to experimental data, the lumped element parameters were computed using the equations from Chapter 2 and the Matlab code given in Appendix A. The calculated parameters are given in Table 26. Table 25: Chosen device configurations for fabrication. Device t, [pm] t [pUm] R2 [um] Rj [Pm] R,,m [Um] fr [kHz] Powo,,[nW] 1 3 0.5 900 830 0 65.68 15.56 2 3 0.5 900 830 180 28.08 36.12 3 3 0.5 1200 1115 0 48.14 1.73 4 3 0.5 1200 1115 240 20.37 3.99 5 3 0.5 1500 1400 0 38.01 0.302 6 3 0.5 1500 1400 300 15.98 0.667 7 3 0.5 1800 1685 0 31.41 0.070 8 3 0.5 1800 1685 360 13.15 0.162 Table 26: Lumped element parameters used for theoretical models. k d, CaD aD MaDma. Cb Pa V kg kg Device V] Pal [] [ Pa] nm] m4 [nF] 1 1.81 5.80E07 0.001 1.01E14 5.58E15 5663.3 0 17.4 2 1.81 5.80E07 0.001 1.01E14 5.58E15 5663.3 19649 17.4 3 9.29 7.16E06 0.008 1.93E13 2.07E14 2602.5 0 26.9 4 9.29 7.16E06 0.008 1.93E13 2.07E14 2602.5 11052 26.9 5 12.93 1.94E05 0.016 7.43E13 5.74E14 1521.8 0 38.2 6 12.93 1.94E05 0.016 7.43E13 5.74E14 1521.8 7073.6 38.2 7 16.92 4.71E05 0.028 2.42E12 1.43E13 1009.8 0 51.3 8 16.92 4.71E05 0.028 2.42E12 1.43E13 1009.8 4912.2 51.3 CHAPTER 3 FABRICATION AND PACKAGING Overview This chapter focuses on the device fabrication and process flow, as well as the device packaging design and implementation. First, a detailed discussion of the process flow is presented, including fabrication methods, equipment, and recipes. This is followed by an overview of the packaging schemes that were employed. The packages were designed to flush mount the device in order to expose the diaphragm directly to the acoustic input, thereby enabling direct measurement of the diaphragm parameters. Two variations in package design were employed, namely a quarterwave resonator package and a sealed cavity package. A graphical overview of this chapter is provided by the roadmap of Figure 31. Process Flow The devices were batch fabricated on 4" silicononinsulator (SOI) wafers. The process sequence is given in detail in Appendix B. All of the processing steps up through the deposition of the top electrode were performed at Sandia National Laboratories. The remainder of the steps were formed at the University of Florida, with the exception of the PZT etch step which was performed at the Army Research Laboratory. Packaging Scheme I Sealed Cavity jj1i ^ n^J Package Chip to Package Package h Mounting S Polin I Figure 31: Roadmap for Chapter 3. A diaphragm thickness of 3 um was desired in order to achieve a testable resonant frequency and therefore required a top silicon thickness in that range, as the top silicon layer of the SOI wafer ultimately forms the bulk of the diaphragm. Unfortunately, the only available SOI wafers contained a top silicon layer of 12 um thickness. In order to reduce this thickness, a timed KOH (Potassium Hydroxide) etch [137] was performed, however a residue was left behind on most of the wafers, that was difficult to remove. The composition of the residue was not identifiable, however a 1 minute dip in 10% Nitric acid, followed by a 2:1 Piranha etch for 5 minutes had no noticeable effect. ........................................................ ................................................... . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ^^ ^^^ m ^ ^  a I^^ ^ d ) ..... .... ...... ... .. .. .. h ) TiO2 I1 PZT ] Si 77 Bottom Electrode Pt =I Buried Oxide (BOX) SiO2 E Photoresist f7M Top Electrode Pt Figure 32: Condensed process sequence in crosssection. a) Deposit 100 nm of Ti on SOI Wafer and oxidize to form TiO2. b) Deposit Ti/Pt (40/180 nm) and liftoff with "ElectrodeBot" mask. c) Spin coat PZT 52/48 solution & pyrolize (6 layers for 400 nm total). d) Deposit Pt (180 nm) and liftoff with "ElectrodeTop" mask. e) Wet Etch PZT in 3:1:1 of (NH4)HF2/HC1/DI water. f) Spin & pattern thick photoresist on bottom with "Cavity" mask. g) DRIE (deep reactive ion etch) to buried oxide layer. h) Ash resist and BOE (buffered oxide etch) backside to remove to buried oxide layer. Two of the original KOHetched wafers were replaced with two new wafers that were etched in an HNA bath to avoid the residue issue [138]. The advantage of the HNA etch was a smooth, mirror finished surface to the wafer with no visible residue. The disadvantage was a less consistent etch rate than the KOH procedure. For comparison the HNA etch had an average etch rate of 2.6 /ummin with a standard deviation of 0.33 /ummin, while the KOH etch had an etch rate of 1.69 /ummin with a standard deviation of 0.03 m//min. .. . .. . . . . . . . Following the wafer thinning, the process flow proceeded as given by Figure 32. The first step was to deposit a 100 nm thick layer of Ti, and oxidize in a tube furnace to create a TiO2 layer that serves as a diffusion barrier for the PZT [139142] as shown in Figure 32a. A liftoff process was then used to pattern the bottom electrodes with 40 nm of Ti followed by 180 nm of Pt as shown in Figure 32b. The Ti layer served as an adhesion layer in this step. The next step was the deposition of a previously mixed 52/48 solgel solution of PZT [139, 143, 144] as shown in Figure 32c. The solution was spincast at 2500 RPM for 30 sec, then pyrolized at 350 C in air for 2 min on a hot plate. Repeating the spin and pyrolize step 6 times yielded a total PZT thickness of approximately 400 nm. The wafers were then furnace annealed at 650 C for 30 min to achieve a perovskite phase of PZT that has the most attractive piezoelectric properties. Wafer bow measurements were performed after deposition and oxidation of the TiO2 as well as after annealing of the PZT. By recording the wafer curvature in terms of radius or bow, the stress in the layers can be determined [145147]. Table 31 shows the results and calculations from these measurements for all of the fabricated wafers. Stress levels were first calculated for the combination of TiO2 and backside SiO2 that was formed during the titanium oxidation step. Then the stress contribution from the TiO2 was calculated. Finally, using the wafer curvature measurements after deposition and annealing of the PZT, the stress in the PZT layer was calculated. In order the calculate the stress from the wafer curvature, a relationship known as Stoney's equation [145147] was used and is given by E h3 s ,(3.1) 6(1 v)Rh (1 hh where 1/R is the curvature, h,,h, is the substrate and film thickness respectively, E. is Young's modulus of the substrate and v, is Poisson's ratio of the substrate. Table 31: Wafer bow measurements and the resulting calculated stress. Initial After TiO2 Dep. + Ox. TiO2 + TiO2 After PZT PZT SiO2 W r Radius Bow Radius Bow Stress Stress Stress Radius Bow Stress Wafer # [m] [pu] [m] [pu] [MPa] [MPa] [MPa] [m] [pu] [MPa] 1 43.8 18.51 137.8 6.15 1367.1 844.7 1217.0 149.9 5.65 12.8 1A 40.1 20.5 103.2 8.4 1335.3 1239.7 1809.6 1144 1.52 176.3 2 38.9 20.45 99.0 8.03 1371.1 848.8 1223.1 1338 0.72 187.3 3 60.8 13.05 291.4 3.12 1140.6 678.6 967.9 n/a n/a n/a 4 44.2 18.24 121.2 6.93 1259.8 772.9 1109.4 279.3 2.82 249.9 5A 63.7 12.84 717.3 1.48 1254.5 776.3 1114.5 156.3 4.74 159.4 6 43.0 18.86 109.3 7.6 1235.4 769.5 1104.2 809.9 0.77 214.5 7 44.3 18.3 106.1 7.84 1154.2 714.8 1022.2 662.9 1.27 233.5 Before the above equation can be used to find the stress in the TiO2 layer, the thickness of the SiO2 layer that was formed during the oxidation step must be determined. For silicon oxidation, the relationship between oxide thickness and oxidation time is approximated by [148] t + At = B(t+r), (3.2) where A and B are coefficients which depend on material properties and operating conditions, to is the total oxide thickness, t is the oxidation time, and r is a time shift due to an initial oxide thickness. For short oxidation times, where t < A2/4B, this equation reduces to t (t+r). (3.3) A 