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- https://ufdc.ufl.edu/UFE0008378/00001
## Material Information- Title:
- Development of a MEMS-Based Acoustic Energy Harvester
- Creator:
- HOROWITZ, STEPHEN BRIAN (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Acoustic impedance ( jstor )
Acoustic resonance ( jstor ) Diaphragm ( jstor ) Electric potential ( jstor ) Energy ( jstor ) Equivalent circuits ( jstor ) Frequency response ( jstor ) Narrative devices ( jstor ) Resonant frequencies ( jstor ) Structural deflection ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright Stephen Brian Horowitz. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 4/15/2006
- Resource Identifier:
- 495636967 ( OCLC )
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DEVELOPMENT OF A MEMS-BASED 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 Reduced-Order 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 short-circuit mechanical resonant frequency .........50 Case 2: At the short-circuit 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 Quarter-w 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 Response-Laser 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 Response-Laser Scanning Vibrometer ......................114 Frequency response ....................................................................... 114 L in e arity ...............................................................12 4 A acoustical M easurem ents .. ......................................................... ................... 126 Acoustically Actuated Frequency Response Plane-Wave Tube ....................126 Sealed cavity package ............................................... ............... 126 Quarter-w ave resonator package .............. ..... .....................................127 Acoustic Input Impedance Measurements Plane-Wave Tube........................130 Quarter-Wave 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 APPLICATION-PIEZOELECTRIC 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 1-1: Typical material properties of select piezoelectric materials. ......................................7 1-2: Selected thin film piezoelectric materials and corresponding properties as reported in th e lite ratu re ...................................... ............................... ................ 1 0 2-1: Equivalent lumped elements in several common energy domains............................28 2-2: C onjugate pow er variables. ............................................................. .....................29 2-3: General material properties used in modeling ........................................................55 2-4: PZT properties used in m odeling. ........................................ .......................... 55 2-5: Chosen device configurations for fabrication.................................. ............... 81 3-1: Wafer bow measurements and the resulting calculated stress ....................................86 3-2: Measured wafer mass and calculated density of PZT. ...................... .............. 87 5-1: Summary of electrical impedance measurements .............................. ...............1.13 5-2: Comparison of ferroelectric and dielectric properties of thin film PZT.................13 5-3: Summary of electrically actuated frequency response measurements ...................124 6-1: Summary of experimental results of microphone ................................. .............152 LIST OF FIGURES Figure pge 1-1: Roadm ap to Chapter 1. ...................... ..................................... ...... ... . 1-2: Schematic of overall energy harvesting concept ................ ............................... 3 1-3: Conceptual schematic of the acoustic energy harvester.................. .................. 1-4: B asic L C R electrical circuit......... ...................................................... ............... 4 2-1: R oadm ap to Chapter 2. ............. .................................................................... 18 2-2: Thevenin equivalent circuit for purposes of maximizing power transfer ..............21 2-3: Diagram showing side view of a Helmholtz resonator and its mechanical equivalent of a m ass-spring-damper system ........................................................................... 29 2-4: Equivalent circuit representation of a Helmholtz resonator.......................................34 2-5: Theoretical pressure amplification of a conventional Helmholtz resonator. (a) m magnitude (b) phase ...................................... .... .. ...... ........... 35 2-6: Cross-sectional and 3-D schematic of piezoelectric composite circular plate. (not to sc ale) ...................................................... .... ................. 3 6 2-7: Notation of axes used in piezoelectric transduction ................................................37 2-8: Idealized perovskite crystal structure for PZT. a) centrosymmetric structure prior to poling. b)non-centrosymmetric structure after poling..........................................38 2-9: Schematic of the poling process: ........................................ ......... ....39 2-10: Polarization vs. electric field hysteresis loop. .................................. ............... 40 2-11: Electro-acoustic equivalent circuit representation with Cb, Ca, and .................46 2-12: Electro-acoustic equivalent circuit representation with Ce C, and '. ................47 2-13: Dynamic electro-acoustic equivalent circuit. ................................ .................48 2-14: Sensitivity vs. frequency for a piezoelectric composite circular plate ...................55 2-15: Equivalent circuit of piezo-composite plate including the radiation impedance. ....57 2-16: Equivalent circuit with resistive load. ........................................... ............... 60 2-17: Conceptual 3-D and cross-sectional schematic of the circular composite plate .....62 2-18: Effective acoustic short-circuit compliance as a function of RIR,/ and tP/t, .......63 2-19: Effective acoustic mass as a function of R,/R2 and t/t, ......................................64 2-20: Short-circuit resonant frequency as a function of R,/R2 and tp/t, ......................65 2-21: Electro-acoustic transduction coefficient as a function of R,/R] and t /t, ............66 2-22: Electromechanical coupling coefficient, k, as a function ofRK/ R and t/t( .........67 2-23: Equivalent circuit for acoustic energy harvester with resistive load......................68 2-24: Magnitude of the acoustical input impedance for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator .................................70 2-25: Magnitude and phase of the acoustical input power for an input acoustic pressure of P = 94 dB ......... .......................................................... .......... ..... 72 2-26: Electrical output impedance for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. ......................... ............... ......73 2-27: Magnitude and phase of the output voltage for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. .............................74 2-28: Magnitude and phase of the electric output power delivered to the load resistor....75 2-29: Magnitude and phase of the energy harvester efficiency. ...................................76 2-30: Open-circuit voltage as a function of the applied acoustic pressure for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator..........77 2-31: Input and output power as a function of the load resistance placed across the electrodes of the piezoelectric material .............................................................. 78 2-32: Input and output power as a function of the applied acoustic input pressure, while u sing an optim al load resistor ...................................................................... ....... 79 3-1: R oadm ap for C chapter 3. .................................................................... ..................83 3-2: Condensed process sequence in cross-section. ................................................84 3-3: Early DRIE results showing significant sidewall damage. ........................................88 3-4: SEM image showing black silicon at the base of a DRIE-etched trench ...................89 3-5: SEM image of a successful DRIE etch through the thickness of a silicon wafer. .....90 3-6: Illustration of a single diaphragm device.. ...................................... ...............91 3-7: Block layout illustrating numbering scheme for devices. ........................................91 3-8: Wafer layout illustrating numbering scheme for blocks. ........................................92 3-10: Quarter-wave resonator package consisting of acrylic plug, copper leads and vent channel. ....................................................................93 3-11: Sealed cavity package...................................................................... ...................95 3-12: Optical photograph of a packaged energy harvester. .............................................95 4-1: R oadm ap for C chapter 4. .................................................................... ...................99 4-2: Experimental setup for impedance and power measurements. .............................104 4-3: Schematic of incident, reflected and input power. ............. .................................... 106 5-1: R oadm ap for C chapter 5. ........................................... ......................................... 108 5-2: Hysteresis loop for Device 1A-5-4 in a sealed cavity package. The device has a radius of 1200 ium and a central mass. ...................................... ............... 109 5-3: Measured parallel output capacitance, Cp vs. outer radius, R, overlaid with a theoretical capacitance curve based on the average extracted dielectric constant. 111 5-4: Parallel output resistance, Rp, vs. outer radius, R,, overlaid with a theoretical resistance curve based on the average measured conductivity............................112 5-5: Device 1A-6-1, electrically actuated center deflection for the device with a radius of 900 /um and no central mass, packaged in the quarter-wave resonator package.. 115 5-6: Device 1A-6-2, electrically actuated center deflection for the device with a radius of 900 /um and a central mass, packaged in the quarter-wave resonator package....116 5-7: Device 1A-11-3, electrically actuated center deflection for the device with a radius of 1200 /um and no central mass, packaged in the quarter-wave resonator..........117 5-8: Higher order mode image of Device 1A-11-3, taken using scanning laser vibrometer at 12 0 .9 k H z ............... ............ ......................................... .................................. 1 17 5-9: Device 1A-10-4, electrically actuated center deflection for the device with a radius of 1200 um and a central mass, packaged in the quarter-wave resonator ..........118 5-10: Device 1A-6-5, electrically actuated center deflection for the device with a radius of 1500 um and no central mass, packaged in the quarter-wave resonator.......... 119 5-11: Device 1A-10-6, electrically actuated center deflection for a device with a radius of 1500 um and a central mass, packaged in the quarter-wave resonator package. .120 5-12: Device 1A-3-7, electrically actuated center deflection for the device with a radius of 1800 um and no central mass, packaged in the sealed-cavity package............ 121 5-13: Electrically actuated sensitivity at low frequency (well below resonance) (R,/R = 0.95).................................................................. ....... 122 5-14: Summary of electrically actuated resonant frequencies. ....................................... 122 5-15: Drop in resonant frequency due to the addition of the central mass. .....................123 5-16: Magnitude of the resonant center deflection versus driving voltage......................125 5-17: Mechanical sensitivity at resonance versus driving voltage.............................126 5-18: Device 1A-5-4, 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 5-19: Device 1A-10-4 in a quarter-wave 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 5-20: Device 1A-6-5 in a quarter-wave 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 5-21: Device 1A-10-2, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 900 pm and a central mass. ..................... ...............131 5-22: Device 1A-5-4, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1200 pm and a central mass .........................................132 5-23: Device 1A-3-7, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1800 pm and no central mass ............... .................132 5-24: Device 1A-6-1, Normalized acoustic impedance in a quarter-wave resonator package for the device with a radius of 900 pum and no central mass .................133 5-25: Device 1A-11-3, Normalized acoustic impedance in a quarter-wave resonator package for the device with a radius of 1200 pum and no central mass ..............134 5-26: Device 1A-11-3, Initial static deflection resulting from residual stresses for a device with a radius of 1200 pm and no central mass. ............. ..................................... 135 5-27: Device 1A-6-6, Initial static deflection resulting from residual stresses for a device with a radius of 1500 pum and a central mass ...................................................... 136 5-28: Device 1A-3-7, Initial static deflection resulting from residual stresses for a device with a radius of 1800 pm and no central mass. ............. ..................................... 136 5-29: Device 1A-3-8, Initial static deflection resulting from residual stresses for a device with a radius of 1800 pum and a central mass ...................................................... 137 5-30: Measured power delivered to a load as function of the load resistance for Device 1A-4-4 and Device 1A-3-8 as compared against theoretical values ....................140 5-31: Measured output voltage across the load as a function of applied acoustic pressure and com pared to theoretical values.. ........................................... ...............140 5-32: Measured power delivered to load as a function of applied acoustic pressure and com pared to theoretical values .................................................................. ....... 14 1 5-33: Measured overall efficiency of each device overlaid with theoretical values for com prison ...................................................... ................. 142 5-34: Resonant frequency versus applied acoustic pressure, resulting from nonlinear response..................................... ........................... ..... .......... 146 6-1: Linearity of the microphone device at 1 kHz....................................................... 148 6-2: Frequency response spectrum in terms of magnitude and phase. ............................149 6-3: Noise floor spectrum of output voltage when no acoustic signal is applied, as well as noise floor due to measurement setup alone. ................................. ............... 151 6-4: Electrically actuated frequency response of microphone device. ............................152 7-1: R oadm ap to C chapter 7. ............. ..................... ......... .................................... 154 B-l: Step 1: Deposit Ti on SOI Wafer and oxidize to TiO2. .......................................172 B-2: Step 2: Deposit Ti/Platinum(30nm/170nm) -Lift-Off w/ Mask (ElectrodeBot) ......172 B-3: Step 3: Spin PZT 6 times to achieve desired thickness ............... ................. 172 B-4: Step 4: Deposit Platinum(180nm)-Lift-off w/ Mask (ElectrodeTop)......................172 B-5: Step 5: Wet Etch PZT using Pt as etch mask................................... ..................173 B-6: Step 6.1: Spin thick photo resist on bottom (7um). ............................................173 B-7: 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 B-10: Step 6.5: BOE backside to remove oxide. .................................. .................174 B -11: B ackside m etal m ask .................................................. ............................... 174 B-12: Bottom electrode mask ......... .. ..... .. ......... .. ........................ 175 B -13 : T op electrode m ask ................................................................................ .... ... 176 B -14 : C av ity m a sk ..................................................................................................17 7 B-15: Detailed schematic drawing of quarter wave resonator package and mounting.... 178 B-16: 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 MEMS-BASED 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 in-depth 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 1-1. Acoustic Energy Harvester-Concept The overall concept behind the energy harvester is illustrated in Figure 1-2. 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 1-2. 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 Enei-2- H. jj~ll Figure 1-1: 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 1-2, 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 1-2: Schematic of overall energy harvesting concept. A conceptual close-up schematic of the acoustic energy harvester is shown in Figure 1-3. 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 1-4. Both systems have a single degree-of-freedom, 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 1-3: Conceptual schematic of the acoustic energy harvester. R L Figure 1-4: 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 low-loss 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, ink-jet 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 1-1 for comparison; however these properties are highly dependent on the deposition method, as will be described in more detail shortly. Table 1-1: 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[9-11] -4.7 12 12.7 210 5.6 In Table 1-1, d31 is the piezoelectric coefficient relating electrical displacement in the '3' direction (z-axis) to a mechanical stress in the '1' direction (x-axis) and is the relevant coefficient for bending-mode transducers, whereas d33 relates the electrical displacement in the '3' direction (z-axis) to a mechanical stress in that same direction. The coefficient, d33, is thus the relevant coefficient for compression-mode 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 random-access memory (FRAM) [12-14], 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 sol-gel 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 [16-22] followed soon after by CVD [23]. Castellano and Feinstein [16] used an ion-beam sputtering technique to deposit PZT as did Trolier-McKinstry 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 sol-gel 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. [27-30] 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 sol-gel process for deposition of PZT [31-47]. In particular, Bahr et al. [31] used a sol-gel 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 sol-gel process to achieve crack-free 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 sol-gel deposited PZT film used in a triaxial accelerometer. Zurn et al. [48] report similar material properties for sol-gel deposited PZT on a microcantilever, including a d,3 of 120 pC/Nfor a PZT film thickness of 0.5 um deposited on a low-stress silicon nitride layer. A summary table of deposited thin film piezoelectric materials and their properties, as reported in the literature, is given in Table 1-2. 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 single-target RF sputtering and exhibited a piezoelectric coefficient of 2.3 pC/N. Also, Devoe [9] investigated micromechanical beam resonators that use ZnO and a three-mask fabrication process. The resonators are intended for use as electromechanical filters. Table 1-2: 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) Sol-Gel [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/1-X) React. Sputt. Properties -33,r 31 d33 e3 12.7 10.5 900 1100 1300 700-1000 1400 1100 800 125 60-460 200-600 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 ( o-r (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. Reduced-Order 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 physics-based approach for design, a simplified, reduced-order model is necessary that accurately captures the geometric and material dependencies. This reduced-order 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 [54-56]. 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, 61-94] Cho et al. [61, 65] developed a five-port 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 driving-point 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 non-uniqueness 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 multi-modal vibration suppression. In addition, numerous other researchers have investigated piezoelectric means of vibration damping [99-108]. More recently, Caruso [109] discusses electrical shunt circuits for damping of vibrations, as does Wu et al. [110] for damping of panels on an F-15 aircraft. Additionally, a number of papers directly address the issue of obtaining electrical energy from piezoelectric conversion of mechanical energy [3, 4, 111-117]. 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 1-2. 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 [118-120]. Furthermore, a macroscale version of an acoustic energy harvester was previously developed [120] and, through miniaturization, led to the present MEMS-based 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 thin-film PZT with SOI-based 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 2-1. Overview-Energy 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 2-1: 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 non-rechargables, 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 2-2, 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 root-mean-square voltage and current, respectively. I, V TH V oO VL Z Figure 2-2: 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 input-output 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 input-output 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 physics-based 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 2-1. 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 2-1: 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 2-2 for a number of energy domains. Table 2-2: Conjugate power variables. Energy Domain Effort Mechanical translation Force, F Fixed-axis 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 2-3, where V is the cavity or bulb volume, / and S = ra2 are the length and cross-sectional 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 2-3: Diagram showing side view of a Helmholtz resonator and its mechanical equivalent of a mass-spring-damper 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 non-uniform 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, non-linear 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 kl-1 (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 one-third 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 effort-type variable or a flow-type 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 2-4. 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 2-4: 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 2-5, 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 2-5: 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 cross-section in Figure 2-6, 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 2-6: Cross-sectional and 3-D 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 2-7. For example, d31 is the piezoelectric coefficient relating electrical displacement in the '3' direction (z-axis) to a mechanical stress in the '1' direction (x-axis). From Eq. (2.33), it is also seen to relate a mechanical strain in the '3' direction (x-axis) with an electric field in the '1' direction. 3 (z) 6 4 A O 2 (y) ~-- 5 1(x) Figure 2-7: 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 2-8a. PZT is a typical example of a ferroelectric material, and has a perovskite crystal structure as shown in Figure 2-8. By applying an external electric field to rotate the polar axis, a non-centrosymmetric crystal structure is created and the ferroelectric material becomes piezoelectrically active, as in Figure 2-8b. 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 2-8: Idealized perovskite crystal structure for PZT. a) centrosymmetric structure prior to poling. b)non-centrosymmetric 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 2-9(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 2-9(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 2-9: 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 double-valued 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 2-10. Figure 2-10: 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 one-dimensional (1-D) piezoelectric transduction. This 1-D analysis can then be extended to incorporate effective lumped element values that are calculated from the actual distributed case. In the 1-D 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 1-D 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 1-D 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 1-D 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 time-harmonic signals [64]. The time-harmonic, 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 two-port network theory, a generalized electro-acoustic, reciprocal, two-port 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 electro-acoustic transduction admittance, given both by the ratio of current, I, to pressure, P, when V = 0, and the ratio of volume-velocity, 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 electro-acoustic circuit can be drawn, as shown in Figure 2-11, 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 2-11: Electro-acoustic equivalent circuit representation with Cb, Ca, and . An alternative equivalent circuit can also be drawn, as shown in Figure 2-12. The alternative circuit is simply another representation for the same physical process. 1:+' Co, P V T-_ Figure 2-12: Electro-acoustic equivalent circuit representation with Ce,, C"o and q'. In Figure 2-12, C,, represents the acoustic compliance when an open-circuit is placed across the piezoelectric, and is given by, Cao =C( 1-k2) [ (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 2-13. 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 2-13: Dynamic electro-acoustic 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 open-circuit 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 R-C 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 short-circuit mechanical resonant frequency (compliance dominated) 2. At the short-circuit mechanical resonant frequency (resistance dominated) Case 1: Well below the short-circuit 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 short-circuit 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 = j-q-C,%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 short-circuit 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 short-circuit 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 2-13. 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 short-circuit resonance, but makes no requirement for an actual short-circuit load. As the load across Ceb is increased, the resonance frequency will shift towards the open-circuit resonance. Recalling Eq. (2.64), the open and short-circuit 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 short-circuit. The larger the coupling factor, then the larger the range between open and short-circuit resonant frequencies. General expression vs. simplified cases Figure 2-14 plots the open-circuit and near short-circuit sensitivity versus frequency overlaid with the values of 0' and 1/0 The material properties used in calculating these results are shown in Table 2-3 and Table 2-4. Notice that below resonance, the general open-circuit 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 short-circuit mechanical resonant frequency, whereas the general expression represents the open-circuit frequency response. Thus the peak amplitude occurs at the open-circuit resonant frequency. The frequency at which the general expression for sensitivity reaches the value of 1/\ coincides with the short-circuit resonant frequency, as expected. Also note that this short-circuit sensitivity calculation was performed with an almost-short circuit condition, primarily to illustrate the effect on the resonant frequency. To achieve the almost-short 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 2-3: 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 2-4: 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 2-14: 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 first-order 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 2-15. Q R d aDrad M8a C as I + + P V .e TCe=- Figure 2-15: Equivalent circuit of piezo-composite 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 2-15, 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 second-order 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 short-circuit resonance frequency then reduces to Sres (2.100) J)esse D + MaDrad Upon comparison with Eq. (2.74), it can be seen that the short-circuit resonance frequency has now been shifted downward by the radiation mass, as is also the case with the open-circuit resonance frequency. At the radial frequency of the short-circuit resonance, the short-circuit input impedance reduces to Z,n =RDad, (2.101) because Ceb and R are effectively removed by the short-circuit. 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 2-16. R'd MSld I Cas :l I + + P CT eb load Figure 2-16: 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 short-circuit 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 2-6 into two portions, an inner circular plate, surrounded by an annular composite ring with matching boundary conditions at the interface, as shown in Figure 2-17. 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 ; Ji-ni I I I I Region Figure 2-17: Conceptual 3-D and cross-sectional 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 electro-acoustic transduction coefficient and the blocked electrical capacitance. Using the parameters shown in Table 2-3 and Table 2-4, 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, 12-14, 16-48]. The effective acoustic short-circuit compliance and mass are functions of both R1/IR and t/t and are shown in Figure 2-18 and Figure 2-19, 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 2-18: Effective acoustic short-circuit 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 2-19: 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 2-20 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 2-20: Short-circuit 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 electro-acoustic transduction coefficient, q, was also found as function of these relative dimensions, and is shown in Figure 2-21. 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 2-21, 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 2-21: Electro-acoustic 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 2-22. 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 2-22: 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 2-16) is combined with that of the Helmholtz resonator (Figure 2-4). By using the same lumped element connection rules previously described, the complete circuit can be obtained, as shown in Figure 2-23. S RaN MaN RaDrad MaDrad Ma CaD I ? I Iload- Figure 2-23: 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 2-24 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 2-5. 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 2-24: 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 plane-wave tube near the end-face of the tube. The microphone that measures this pressure is placed as close as possible (- 1/16 ) to the end-face 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 2-25. 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 sC-c .(2.115) lout ( 1 sCb sC )s 1 SCe 0 2 M + MaDrad e) + S + rd SC' 72 10-5 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 2-25: 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 2-26 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 2-26: 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 2-27 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 I-i 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 2-27: 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 2-28 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 L---L-- LL----------------- 1I I 0 1000 Io 3000 4Dfo0 000 Go00 "DO Frequency [Hz] Figure 2-28: 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 2-29 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 plane-wave tube. L 34 C10 ------------ ------------,-------- S10' -- 0 10iD 200 3000 4i000 f6L EOJD 7000 Frequency [Hz] -- with Helmholtz resonalor without Helrnholtz rEsonator -- ---------------- ---- S-100 - SC 0 1000 000 3000 4000 600f OOD 7000 Frequency [Hz] Figure 2-29: 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 2-5 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 2-30. 10 10 Membrane Membrane/HR -1 " 10 > 1-2 10- 115 120 125 130 135 140 145 150 155 160 Input Acoustic Pressure [dB] Figure 2-30: Open-circuit 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 2-5, 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 2-31. 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 10-8 10- -10 2 3 4 10 10 10 Load Resistance [Ohms] Figure 2-31: 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 2-5, 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 2-32. 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- 10-6 10- 10 115 120 125 130 135 140 145 150 155 160 Acoustic Input Pressure [dB] Figure 2-32: Input and output power as a function of the applied acoustic input pressure, while using an optimal load resistor. (Device = 8 from Table 2-5, 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 2-5. 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 first-run 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 2-6. Table 2-5: 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 2-6: 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 -5.80E-07 0.001 1.01E-14 5.58E-15 5663.3 0 17.4 2 -1.81 -5.80E-07 0.001 1.01E-14 5.58E-15 5663.3 19649 17.4 3 -9.29 -7.16E-06 0.008 1.93E-13 2.07E-14 2602.5 0 26.9 4 -9.29 -7.16E-06 0.008 1.93E-13 2.07E-14 2602.5 11052 26.9 5 -12.93 -1.94E-05 0.016 7.43E-13 5.74E-14 1521.8 0 38.2 6 -12.93 -1.94E-05 0.016 7.43E-13 5.74E-14 1521.8 7073.6 38.2 7 -16.92 -4.71E-05 0.028 2.42E-12 1.43E-13 1009.8 0 51.3 8 -16.92 -4.71E-05 0.028 2.42E-12 1.43E-13 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 quarter-wave resonator package and a sealed cavity package. A graphical overview of this chapter is provided by the roadmap of Figure 3-1. Process Flow The devices were batch fabricated on 4" silicon-on-insulator (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 3-1: 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 3-2: Condensed process sequence in cross-section. 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 KOH-etched 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 3-2. 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 [139-142] as shown in Figure 3-2a. 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 3-2b. The Ti layer served as an adhesion layer in this step. The next step was the deposition of a previously mixed 52/48 sol-gel solution of PZT [139, 143, 144] as shown in Figure 3-2c. The solution was spin-cast 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 [145-147]. Table 3-1 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 [145-147] 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 3-1: 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 |

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PAGE 1 DEVELOPMENT OF A MEMS-BASED ACOUSTIC ENERGY HARVESTER BY STEPHEN BRIAN HOROWITZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 PAGE 2 Copyright 2005 by Stephen Brian Horowitz PAGE 3 I dedicate this dissertation to my wife, Megan. Youre my sweetheart, my wife and my best friend. I love you more than words can say. PAGE 4 iv TABLE OF CONTENTS Page LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT.....................................................................................................................xi v CHAPTER 1 INTRODUCTION........................................................................................................1 Acoustic Energy Harvester Concept..........................................................................1 Motivation..................................................................................................................... 5 Background...................................................................................................................6 Piezoelectricity......................................................................................................6 Piezoelectric Materials in MEMS.........................................................................6 Materials and properties.................................................................................7 Deposition methods........................................................................................8 Piezoelectric integration with MEMS............................................................9 Reduced-Order Modeling of Piezoelectric Materials..........................................11 Energy Absorption/Vibration Damping and Energy Harvesting........................13 Approach.....................................................................................................................14 2 THEORETICAL BACKGROUND...........................................................................17 Overview Energy Harvesting..................................................................................17 Maximum Average Power Transfer....................................................................20 Electromechanical Transducers...........................................................................23 Lumped Element Modeling........................................................................................25 Helmholtz Resonator..................................................................................................29 Lumped Element Model......................................................................................29 Equivalent Circuit................................................................................................33 Piezoelectric Composite Plate....................................................................................35 Piezoelectricity....................................................................................................36 Composite Plate Modeling..................................................................................41 Lumped element model................................................................................41 Equivalent circuit.........................................................................................46 Case 1: Well below the short-circu it mechanical resonant frequency.........50 Case 2: At the short-circuit m echanical resonant frequency........................52 General expression vs. simplified cases.......................................................54 PAGE 5 v Effects of including the radiation impedance...............................................55 Equivalent circuit parameters of piez oelectric composite circular plates....61 Acoustic Energy Harvester Dynamic Behavior..........................................................68 Acoustical Input Behavior...................................................................................68 Electrical Output Behavior..................................................................................71 Operation at an Optimal Frequency....................................................................76 Device Configurations................................................................................................79 3 FABRICATION AND PACKAGING.......................................................................82 Overview.....................................................................................................................82 Process Flow...............................................................................................................82 Packaging....................................................................................................................92 Packaging Scheme...............................................................................................92 Quarter-wave resonator package..................................................................92 Sealed cavity package..................................................................................94 Chip to Package Mounting..................................................................................94 Poling...................................................................................................................96 4 EXPERIMENTAL SETUP........................................................................................98 Parameters and How They are Obtained....................................................................98 Experimental Setup Details......................................................................................101 Ferroelectric Measurements..............................................................................101 Electrically Actuated Response Laser Scanning Vibrometer.........................102 Acoustic Characterization Plane Wave Tube.................................................103 Electrical Impedance Characterization..............................................................107 Initial Deflection Measurements Wyko Optical Profilometer........................107 5 EXPERIMENTAL RESULTS AND DISCUSSION...............................................108 Ferroelectric Measurements......................................................................................108 Electrical Measurements...........................................................................................110 Electrical Output Impedance.............................................................................110 Electrically Actuated Response Laser Scanning Vibrometer.........................114 Frequency response....................................................................................114 Linearity.....................................................................................................124 Acoustical Measurements.........................................................................................126 Acoustically Actuated Frequenc y Response Plane-Wave Tube....................126 Sealed cavity package................................................................................126 Quarter-wave resonator package................................................................127 Acoustic Input Impedance Measurements Plane-Wave Tube........................130 Quarter-Wave Resonator Package.....................................................................133 Initial Deflection Measurements Wyko Optical Profilometer...............................134 Energy Harvesting Measurements............................................................................138 Optimal Resistance............................................................................................138 Optimal Energy Harvesting...............................................................................139 PAGE 6 vi 6 ALTERNATIVE APPLICATION PIEZOELECTRIC MICROPHONE.............147 7 CONCLUSION AND FUTURE WORK.................................................................153 Development Issues..................................................................................................154 Future Work..............................................................................................................156 APPENDIX A MATLAB CODE.....................................................................................................158 lem.m........................................................................................................................15 8 eh_setup.m................................................................................................................163 silicon_sweep.m........................................................................................................163 platinum.m................................................................................................................163 tio2.m........................................................................................................................1 63 piezo_sweep.m..........................................................................................................164 initialise.m................................................................................................................164 totaldeflection_forP.m..............................................................................................164 totaldeflection_forV.m..............................................................................................164 initialdeflection.m.....................................................................................................165 formmatrix_CMFa.m................................................................................................165 frommatrix_dA.m.....................................................................................................166 abdpiezo.m................................................................................................................166 constants1234.m.......................................................................................................167 solver1.m..................................................................................................................168 B DETAILED PROCESS FLOW................................................................................170 Energy Harvester Process Traveler........................................................................170 Graphical Representation of Process Flow...............................................................172 Mask Layouts............................................................................................................174 Package Drawings....................................................................................................178 LIST OF REFERENCES.................................................................................................180 BIOGRAPHICAL SKETCH...........................................................................................195 PAGE 7 vii LIST OF TABLES Table page 1-1: Typical material pr operties of select piezoelectric materials.......................................7 1-2: Selected thin film piezoelectric materi als and corresponding prope rties as reported in the literature..............................................................................................................10 2-1: Equivalent lumped elements in several common energy domains............................28 2-2: Conjugate power variables.........................................................................................29 2-3: General materi al properties used in modeling............................................................55 2-4: PZT properties used in modeling...............................................................................55 2-5: Chosen device configur ations for fabrication.............................................................81 3-1: Wafer bow measurements and the resulting calculated stress....................................86 3-2: Measured wafer mass and calculated density of PZT................................................87 5-1: Summary of electrical impedance measurements....................................................113 5-2: Comparison of ferroelectric and diel ectric properties of thin film PZT...................113 5-3: Summary of electrically actuated frequency response measurements.....................124 6-1: Summary of experiment al results of microphone...................................................152 PAGE 8 viii LIST OF FIGURES Figure page 1-1: Roadmap to Chapter 1..................................................................................................2 1-2: Schematic of overall energy harvesting concept..........................................................3 1-3: Conceptual schematic of the acoustic energy harvester...............................................4 1-4: Basic LCR electrical circuit..........................................................................................4 2-1: Roadmap to Chapter 2................................................................................................18 2-2: Thvenin equivalent circuit for purposes of maximizing power transfer...................21 2-3: Diagram showing side view of a Helmho ltz resonator and its mechanical equivalent of a mass-spring-damper system..............................................................................29 2-4: Equivalent circuit representa tion of a Helmholtz resonator.......................................34 2-5: Theoretical pressure amplification of a conventional Helmholtz resonator. (a) magnitude (b) phase.................................................................................................35 2-6: Cross-sectional and 3-D schematic of piezo electric composite circul ar plate. (not to scale)......................................................................................................................... 36 2-7: Notation of axes used in piezoelectric transduction...................................................37 2-8: Idealized perovskite crys tal structure for PZT. a) centrosymmetric structure prior to poling. b)non-centrosymmetric structure after poling.............................................38 2-9: Schematic of the poling process:...............................................................................39 2-10: Polarization vs. electri c field hysteresis loop...........................................................40 2-11: Electro-acoustic equivalent circuit representation with ebC asC and .................46 2-12: Electro-acoustic equivalent circuit representation with efC, asC and ................47 2-13: Dynamic electro-acoustic equivalent circuit............................................................48 PAGE 9 ix 2-14: Sensitivity vs. frequency for a pi ezoelectric composite circular plate.....................55 2-15: Equivalent circuit of piezo-composite plate includi ng the radiation impedance.....57 2-16: Equivalent circuit with resistive load.......................................................................60 2-17: Conceptual 3-D and cr oss-sectional schematic of th e circular composite plate.....62 2-18: Effective acoustic short-circ uit compliance as a function of 12 R R and pstt.......63 2-19: Effective acoustic mass as a function of 12 R Rand pstt.......................................64 2-20: Short-circuit resonant frequency as a function of 12 R R and pstt........................65 2-21: Electro-acoustic transducti on coefficient as a function of 12 R Rand pstt............66 2-22: Electromechanical coupling coefficient, k, as a function of 12 R R and pstt.........67 2-23: Equivalent circuit for acoustic en ergy harvester with resistive load........................68 2-24: Magnitude of the acoustical input im pedance for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator.................................70 2-25: Magnitude and phase of th e acoustical input power for an input acoustic pressure of 94 PdB .................................................................................................................72 2-26: Electrical output impedance for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator..........................................................73 2-27: Magnitude and phase of the output voltage for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator.................................74 2-28: Magnitude and phase of the electric out put power delivered to the load resistor....75 2-29: Magnitude and phase of th e energy harvester efficiency.........................................76 2-30: Open-circuit voltage as a function of the applied acoustic pressure for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator..........77 2-31: Input and output power as a function of the load resistance placed across the electrodes of the piezoelectric material....................................................................78 2-32: Input and output power as a function of the applied acoustic input pressure, while using an optimal load resistor...................................................................................79 3-1: Roadmap for Chapter 3.............................................................................................83 PAGE 10 x 3-2: Condensed process se quence in cross-section. ........................................................84 3-3: Early DRIE results showi ng significant sidewall damage.........................................88 3-4: SEM image showing black silicon at the base of a DRIE-etched trench...................89 3-5: SEM image of a successful DRIE etch through the thickness of a silicon wafer......90 3-6: Illustration of a single diaphragm device...................................................................91 3-7: Block layout illustrating numbering scheme for devices...........................................91 3-8: Wafer layout illustrating numbering scheme for blocks............................................92 3-10: Quarter-wave resonator package consisting of acrylic plug, copper leads and vent channel.....................................................................................................................93 3-11: Sealed cavity package...............................................................................................95 3-12: Optical photograph of a packaged energy harvester................................................95 4-1: Roadmap for Chapter 4..............................................................................................99 4-2: Experimental setup for impe dance and power measurements.................................104 4-3: Schematic of incident, reflected and input power....................................................106 5-1: Roadmap for Chapter 5...........................................................................................108 5-2: Hysteresis loop for Device 1A-5-4 in a sealed cavity package. The device has a radius of 1200 m and a central mass..................................................................109 5-3: Measured paralle l output capacitance, pC vs. outer radius, 2 R overlaid with a theoretical capacitance curve based on the average extracted dielectric constant.111 5-4: Parallel output resistance, p R vs. outer radius, 2 R overlaid with a theoretical resistance curve based on the average measured conductivity...............................112 5-5: Device 1A-6-1, electrically actuated center deflection for the device with a radius of 900 m and no central mass, packaged in the quarter-wave resonator package..115 5-6: Device 1A-6-2, electrically actuated center deflection for the device with a radius of 900 m and a central mass, packaged in th e quarter-wave resonator package....116 5-7: Device 1A-11-3, electrical ly actuated center deflection for the device with a radius of 1200 m and no central mass, packaged in the quarter-wave resonator..........117 PAGE 11 xi 5-8: Higher order mode image of Device 1A11-3, taken using scanning laser vibrometer at 120.9 kHz............................................................................................................117 5-9: Device 1A-10-4, electrical ly actuated center deflection for the device with a radius of 1200 m and a central mass, packaged in the quarter-wave resonator............118 5-10: Device 1A-6-5, electrica lly actuated center deflection fo r the device with a radius of 1500 m and no central mass, packaged in the quarter-wave resonator..........119 5-11: Device 1A-10-6, electricall y actuated center deflection for a device with a radius of 1500 m and a central mass, packaged in the quarter-wave resonator package..120 5-12: Device 1A-3-7, electrica lly actuated center deflection fo r the device with a radius of 1800 m and no central mass, packaged in the sealed-cavity package............121 5 -13: Electrically actuated sensitivity at low frequency (well below resonance) ( 120.95RR )........................................................................................................122 5-14: Summary of electrically actuated resonant frequencies........................................122 5-15: Drop in resonant frequency due to the addition of the central mass......................123 5-16: Magnitude of the resonant center deflection versus driving voltage......................125 5-17: Mechanical sensitivity at re sonance versus driving voltage...................................126 5-18: Device 1A-5-4, magnit ude and phase of the acousti cally actuated frequency response in a sealed cavity package fo r the device with a radius of 1200m and a central mass............................................................................................................128 5-19: Device 1A-10-4 in a quarter-wave resona tor package. Magnitude and phase of the acoustically actuated frequency response for the device with a radius of 1200 m and a central mass..................................................................................................129 5-20: Device 1A-6-5 in a quarter-wave resona tor package. Magnitude and phase of the acoustically actuated frequency response for the device with a radius of 1500 m and no central mass................................................................................................130 5-21: Device 1A-10-2, Normali zed acoustic impedance in a sealed cavity package for the device with a radius of 900 m and a central mass..............................................131 5-22: Device 1A-5-4, Normalized acoustic impe dance in a sealed cavity package for the device with a radius of 1200 m and a central mass............................................132 5-23: Device 1A-3-7, Normalized acoustic impe dance in a sealed cavity package for the device with a radius of 1800 m and no central mass..........................................132 PAGE 12 xii 5-24: Device 1A-6-1, Normalized acousti c impedance in a quarter-wave resonator package for the device with a radius of 900 m and no central mass..................133 5-25: Device 1A-11-3, Norma lized acoustic impedance in a quarter-wave resonator package for the device with a radius of 1200 m and no central mass................134 5-26: Device 1A-11-3, Initial st atic deflection resulting from residual stresses for a device with a radius of 1200 m and no central mass.....................................................135 5-27: Device 1A-6-6, Initial static deflection resulting from residual stresses for a device with a radius of 1500 m and a central mass.......................................................136 5-28: Device 1A-3-7, Initial static deflection resulting from residual stresses for a device with a radius of 1800 m and no central mass.....................................................136 5-29: Device 1A-3-8, Initial static deflection resulting from residual stresses for a device with a radius of 1800 m and a central mass.......................................................137 5-30: Measured power delivered to a load as function of the load resistance for Device 1A-4-4 and Device 1A-3-8 as compar ed against theoretical values......................140 5-31: Measured output voltage across the load as a function of applied acoustic pressure and compared to theoretical values........................................................................140 5-32: Measured power delivered to load as a function of applied acoustic pressure and compared to theoretical values...............................................................................141 5-33: Measured overall efficiency of each de vice overlaid with theoretical values for comparison.............................................................................................................142 5-34: Resonant frequency versus applied acoustic pressure, resulting from nonlinear response..................................................................................................................146 6-1: Linearity of the microphone device at 1 kHz............................................................148 6-2: Frequency response spectrum in terms of magnitude and phase.............................149 6-3: Noise floor spectrum of output voltage wh en no acoustic signal is applied, as well as noise floor due to measurement setup alone..........................................................151 6-4: Electrically actuated fre quency response of microphone device.............................152 7-1: Roadmap to Chapter 7..............................................................................................154 B-1: Step 1: Deposit Ti on SOI Wafer and oxidize to TiO2............................................172 B-2: Step 2: Deposit Ti/Pla tinum(30nm/170nm) -Lift-Off w/ Mask (ElectrodeBot)......172 PAGE 13 xiii B-3: Step 3: Spin PZT 6 times to achieve desired thickness.........................................172 B-4: Step 4: Deposit Plat inum(180nm)-Lift-off w/ Mask (ElectrodeTop)......................172 B-5: Step 5: Wet Etch PZT using Pt as etch mask...........................................................173 B-6: Step 6.1: Spin thick photo resist on bottom (7um)..................................................173 B-7: Step 6.2: Pattern using mask (Cavity).....................................................................173 B-8: Step 6.3: DRIE to BOX...........................................................................................173 B-9: Step 6.4: Ash Resist.................................................................................................173 B-10: Step 6.5: BOE back side to remove oxide..............................................................174 B-11: Backside metal mask.............................................................................................174 B-12: Bottom electrode mask..........................................................................................175 B-13: Top electrode mask................................................................................................176 B-14: Cavity mask...........................................................................................................177 B-15: Detailed schematic drawing of quarter wave resonator package and mounting....178 B-16: Detailed schematic of sealed ca vity package and mounting plate........................ 179 PAGE 14 xiv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEVELOPMENT OF A MEMS-BASED 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 devel opment of an electromechanical acoustic resonator for reclamation of energy using microelectromechanical systems (MEMS) technology. The MEMS device converts acousti cal energy to electri cal energy that can then either be stored for later use or utilized directly for a pa rticular circuit application. The work presented in this dissertation takes a first step to wards that goal, by designing, fabricating and test ing several electromechanical acoustic resonators of varying size. The resonators are fabri cated using MEMS processing techniques on a silicon wafer. Each re sonator consists of a 3 m thick silicon diaphragm with a circular ring of piezoelectric material. The dia phragm transduces an acoustical pressure fluctuation into a mechanical deformation, wh ile the piezoelectric material transduces that mechanical deformation into an el ectrical signal (charge or voltage). PAGE 15 1 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 devi ce converts acoustical en ergy to electrical energy that can then either be stored or used directly for a particular circuit application. The relatively small geometries possible in ME MS make such a device useful for small, portable devices, where battery requirements ar e often difficult to m eet. Additionally, the device is well suited to applications where wiri ng 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 th is research. Following this, the motivation behind this research is presented. Then, an in-depth literature survey is presented to familiarize the reader with the technologica l and theoretical developments related to energy harvesting and piezoelec tricity, followed by a short discussion of the technical approach that was employed. A graphical roadmap for this chapter is shown in Figure 1-1. Acoustic Energy Harvester Concept The overall concept behind the energy harv ester is illustrated in Figure 1-2. The main idea is to convert acoustic energy into a usable form of el ectrical energy. The figure illustrates a plane wave tube as the sour ce of acoustic power. Some of the incident PAGE 16 2 acoustic power is reflected at the end of the tube; however, a portion is transmitted through to the energy harvester that is ci rcled in Figure 1-2. The energy harvester performs the actual transduction of energy fr om the acoustical to the electrical energy domains. More details on this transduc tion technique will be provided later. Overview Motivation Background Piezoelectricity Piezoelectric Materials in MEMS Impedance Modeling of Piezoelectrics Energy Absorption/ Vibration Damping Energy Harvesting Approach Materials and properties Deposition methods Piezoelectric integration with MEMS Figure 1-1: Roadmap to Chapter 1. PAGE 17 3 The electrical energy that exits the ener gy 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 a pplication. In the example illustrated in Figure 1-2, the end application is a battery th at is charged by the out put of the harvesting circuitry. Figure 1-2: Schematic of overa ll energy harvesting concept. A conceptual close-up schematic of the acoustic energy harvester is shown in Figure 1-3. This device cons ists 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 envir onment through a small neck. When excited by an acoustic input, a single resonance is see n, whereby the acoustic pressure inside the cavity is amplified to a level much greater th an the incident acoustic signal [1, 2]. The Helmholtz resonator is very similar to an LCR resonant circuit in electrical engineering, as shown in Figure 1-4. Both systems have a single degree-of-freedom, with a single resonant peak, whereby the amplit ude of the forcing function is greatly PAGE 18 4 amplified. In the LCR circuit, the voltage is equivalent to the acoustic pressure. Both systems operate through the os cillation of energy between generalized potential and kinetic forms. In electrical systems this storage occurs via cap acitors and inductors respectively. These electroacoustic analogies wi ll be explained in more detail in Chapter 2. neck cavity piezoelectric ring compliant plate neck cavity piezoelectric ring compliant plate Figure 1-3: Conceptual schematic of the acoustic en ergy harvester. Figure 1-4: Basic LCR electrical circuit. The large acoustic cavity pressure created by the resonance can then be exploited for energy reclamation by converting the acous tic energy to electr ical energy. This conversion is performed by the piezoelectric composite plate. First, acoustical to mechanical transduction is accomplished vi a the compliant diaphragm, followed by mechanical to electrical transduction, due to the piezoelectric response of the composite, whereby a mechanical strain creat es an electrical voltage. PAGE 19 5 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 impe dance that matches the output impedance of the piezoelectric structure. Additionally, th e harvesting circuit conve rts the form of the electrical energy to a more a ppropriate form for storage, su ch 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 st ored across a low-loss cap acitor [3]. Another possibility is based on the Kymi ssis 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 th at energy that would otherwise be lost. Although the availa ble acoustic energy may be small in many situations, the energy requirements for certain applications such as microsensing are also correspondingly small. The ability to reclai m acoustic energy and store it in a usable electrical form enables a novel means of supplying power to relatively low power devices. PAGE 20 6 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 piezoelectricity by Wilhelm Gottlieb Hankel, and is historically referred to as the direct piezoelectric effect [5]. Add itionally, in these same crysta ls, 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 char ge in a piezoelectric material is known as the piezoelectric charge modulus, d, and is typically quoted in units of p CN or p mV. In order for a material to be piezoelectric, it must have a noncentrosymmetric crystal structure. It was not until 1921, that a useful applica tion was developed for piezoelectricity. It came in the form of a quartz crystal osc illator that was developed by Walter Cady to provide good frequency stabil ity 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 appl ications of piezoelec tric materials now include high voltage ignition systems, piezoel ectric motors, ink-jet printer heads, acoustic speakers, sonar, ultrasonic transducers, fre quency filters, acoustic delay lines, electrical transformers, and a wide range of physical sens ors, such as acoustic, force, pressure, and acceleration sensors [6]. Piezoelectric Materials in MEMS A number of papers have been publ ished on the use of piezoelectric and ferroelectric materials in ME MS [6, 7]. Piezoelectric materials commonly used include PAGE 21 7 zinc oxide (ZnO), aluminum nitride (AlN), a nd lead zirconate titanate (PZT). The choice of piezoelectric material depends on se veral selection fact ors including deposition methods, process complexity, integrated ci rcuit (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, AlN a nd ZnO have relatively weak piezoelectric coefficients and coupling factors, however they tend to have low diel ectric constants and low dielectric losses, making them more attrac tive for certain applications [8]. A table of typical material properties is shown in Table 1-1 for comparison; however these properties are highly dependent on the deposi tion method, as will be described in more detail shortly. Table 1-1: Typical material properties of select piezoelectric materials. 31d 33d 33, r E p CN p CN GPa 3kgm PZT[6] -130 290 1300 96 7.7 AlN[8] n/a 3.4 10.5 330 3.26 ZnO[9-11] -4.7 12 12.7 210 5.6 In Table 1-1, 31d is the piezoelectric coefficient re lating electrical displacement in the direction (z-axis) to a mechanical st ress in the direction (x-axis) and is the relevant coefficient for bending-mode transducers, whereas 33d relates the electrical displacement in the direction (z-axis) to a mechanical stress in that same direction. The coefficient, 33d, is thus the relevant coefficient for compression-mode transducers. Additionally, 33, r E, and are the relative dielectric constant, Youngs modulus and mass density of the materials, respectively. PAGE 22 8 Of the three materials, only AlN is fully compatible with standard IC processing, eliminating the integration difficulties present with PZT and ZnO. PZT poses a particular integration challenge, as lead contaminati on is always a big concern; however, PZT has been integrated successfully into ferroe lectric random-access memory (FRAM) [12-14], typically using sputtered platinum as a bo ttom electrode and diffusion barrier layer. Deposition methods Piezoelectric deposition t echniques include various sputtering methods, photoablation [15], hydrothermal and chemical va por deposition (CVD) techniques, and spinon sol-gel processing. For deposition of P ZT, three popular methods are sputtering, solgel and hydrothermal techniques. The earliest work used various forms of sputtering to deposit PZT [16-22] followed soon after by CV D [23]. Castellano and Feinstein [16] used an ion-beam sputtering technique to de posit PZT as did Tro lier-McKinstry et al., [22] while Sreenivas et al [17] employed DC magnetr on sputtering in their PZT deposition. Additionally, sputter depositi on 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 sol-gel process and produced an average thickness of 900 nm. The other method used reactive sp uttering 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 m thick PZT by hydrothermal techniques was perf ormed by Kanda et al. [26] for use in a touch probe sensor. Additionally, Morita et al. [27-30] chose to deposit PZT by the hydrothermal method due to the relatively large thicknesses achiev able and the selfalignment of the poling di rection during deposition. PAGE 23 9 More recently, numerous researchers ha ve utilized the so l-gel process for deposition of PZT [31-47]. In particular, Bahr et al. [31] used a sol-gel PZT process and investigated the reliability and piezoelectric properties of the resulting material. Using this process for various film thickness es, the relative dielectric constant, 33, r varied between 700 and 1000 and delamination occu rred for an indentation load of 1250 N for a film sintered for 5 minutes. Higher load s were found to be achievable using longer sinter times. Bernstein et al [32] and Xu et al. [33] used a sol-gel process to achieve crack-free PZT films with thicknesses of up to 12 m that yielded piezoelectric properties close to the bulk values for PZT. A dielectric constant of 1400 and a piezoelectric coefficient, 33d, of 246 pC/N were reported on a 4 m thick film. The PZT films were used on an array of membranes for acousti c imaging. Kunz et al. [37] report a piezoelectric coefficient, 31d, of 110 pC/N for a sol-gel deposited PZT film used in a triaxial accelerometer. Zurn et al. [48] report similar ma terial properties for sol-gel deposited PZT on a micro cantilever, including a 31d of 120 pC/N for a PZT film thickness of 0.5 m deposited on a low-stress silicon ni tride layer. A summary table of deposited thin film piezoelectric materials and their properties, as reported in the literature, is given in Ta ble 1-2. In this table, 31e is a piezoelectric coefficient relating mechanical stress and an electric field, tan is called the loss tangent and is a measure of the relative losses in the material, and res 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 piezo electric materials with MEMS focused primarily on ZnO [49, 50] and AlN [49] as th e piezoelectric material. More recently, PAGE 24 10 Devoe and Pisano [10] developed and charac terized surface micromachined piezoelectric accelerometers that utilized thin films of ZnO for sensing. Deposition of the ZnO was performed using single-target RF sputtering and exhibited a piezoelectric coefficient of 2.3 p CN. Also, Devoe [9] investigated micromech anical beam resonators that use ZnO and a three-mask fabrication process. The resonators are intended for use as electromechanical filters. Table 1-2: Selected thin film piezoelect ric materials and corresponding properties as reported in the literature. Properties Ref. Material Deposition Method 33, r 31d pC N 33d pm V 31e 2C m E GPa 3kg m tan res M Pa [9, 10] ZnO RF Sputt. 12.7 2.3 --161 5605 -1 [11] ZnO Sputtered ----210 5700 --[8] AlN DC Sputt. 10.5 -3.4 -1.02--0.002-700 [8] PZT (45/55) React. Sputt.900 -55 -5.12--0.03150 [8] PZT (45/55) Sol-Gel 1100 -50 -8.28--0.0370 [8] PZT (53/47) React. Sputt.1300 -70 -6.83--0.05230 [26] PZT Hydrothermal--34.2--0.13----[31] PZT (52/48) React. Sputt.700-1000-------[32] PZT React. Sputt.1400 -246 ---0.03-[37] PZT (53/47) React. Sputt.-110------[42] PZT React. Sputt.1100 -------[44, 45] PZT (52/48) React. Sputt.800 --200-56 -0.02-[48] PZT React. Sputt.--120 -60 7600 --[16] PZT IB Sputt. 125 -------[20, 21] PZTWCd RF Sputt. 60-460 -30 -----[51] PZT(X/1-X) React. Sputt.200-600 -----0.02-The integration of ZnO with micromachin ing was also investigated by Indermhle et al. [52], where it was used in an array of silicon micro cantilevers. The end application for the array was parallel atomic force mi croscopy. Another application of ZnO was performed by Han and Kim [11] in the fabr ication of a micromachined piezoelectric ultrasonic transducer. They sputter deposited ZnO on Al, followed by a layer of parylene PAGE 25 11 for insulation. Dubois and Muralt [8] fabri cated thin films of RF sputtered AlN and performed measurements of the effective transv erse piezoelectric coe fficient that yielded a value of 3.4 p mV. Reduced-Order Modeling of Piezoelectric Materials The development of accurate, practical m odels of the piezoelectric transduction and associated structural interact ions 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 therefor e 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 ofte n involving partial differential equations, can often be unwieldy and physically unintuitive. Furthermore, this functional form is not readily conducive to a fu ll systems level design that links the transducer to electronics. Similarly, finite element m odeling (FEM) techniques are often used to predict system behavior, numerically. The re sults 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 difficu lt to determine scaling behavior from FEM results. The scaling behavior, i.e., the cha nge 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 faci litate a physics-based approach for design, a simplified, reduced-order model is necessary that accurately captur es the geometric and PAGE 26 12 material dependencies. This reduced-order m odel 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, woul d have an equivalent electrical circuit consisting of a resistance, inductance and capac itance in series, and all in parallel with another capacitance. Working independen tly, the earliest equivalent circuit model specifically for piezoelectric materials wa s developed by Van Dyke in 1925 [54-56]. Later, Dye [57] proved that Van Dykes ci rcuit could be derive d from Butterworths theorem. Additionally, Mason [58, 59] and Cady [5] provided thorough reviews of the equivalent circuit model and associated equa tions for quartz oscillator applications. Finally, Fischer [60] extensively covered equi valent circuit models for electromechanical oscillators. Significant research has been performed on the modeling of st ructures containing piezoelectric materials. [38, 41, 61-94] C ho et al. [61, 65] developed a five-port generalized equivalent circu it for a piezoelectric bimorph b eam. The generalized circuit can be used under a variety of boundary c onditions. 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. Othe r 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 fo r modeling distributed parameter systems. Liang et al. [66] de veloped a generalized electromechanical PAGE 27 13 impedance model that was then used to address issues of en ergy conversion, power consumption, and dynamic response. The approach described can be used for any system for which the driving-point impedance can be fo und either analytically or experimentally. Additionally, van de Leur [91] provided a critical interpreta tion of equivalent circuit models obtained from impedance measurements indicating that care must be taken in identifying individual compone nts contributing to an impe dance as resulting from particular structures in a gi ven device. This can be further understood as an example of the non-uniqueness 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 c oupling 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 thei r unique properties is a related and also relevant topic since the acoustic energy ha rvester 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 disc uss modeling and design of coupled resonant systems. These papers address some of the issues involved in coupled resonators, including the shift in res onant frequencies away from their uncoupled values. Energy Absorption/Vibration Damping and Energy Harvesting The absorption of acoustical and mechanic al energy via piezoelectric coupling is closely related to the harvesting of electri cal 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 PAGE 28 14 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 relate d nature of these two fields, papers that address eith er of these fields are direc tly relevant to acoustic energy harvesting. Some of the earliest work in piezoelec tric vibration damping was performed by Hagood and von Flotow [97] who us ed 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 multi-modal vibration suppression. In addition, numerous other re searchers have inves tigated piezoelectric means of vibration damping [99-108]. More recently, Caruso [109] discusses electrical shunt circuits for damping of vibrations, as does Wu et al. [110] for damping of panels on an F-15 aircraft. Additionally, a number of pape rs directly address the i ssue of obtaining electrical energy from piezoelectric conve rsion of mechanical energy [3, 4, 111-117]. Early work in this area was performed by Lomenzo et al. [114] and Stein et al. [116]. Later, Kymissis [4] and Smalsers [3] work focuse s on the electrical circuitry necessary for storage of piezoelectrically generated en ergy, 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 1-2. The first PAGE 29 15 component consists of a piezoelectric compos ite diaphragm that is responsible for the acoustical to electrical tr ansduction 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 take s the alternating current (AC) electrical signal from the piezoelectric dia phragm and converts it into a direct current (DC) output that can be stor ed for later use. The first two components of the energy harvester were developed in this researc h, while the conversion circuitry remains an ongoing research topic in our gr oup [118-120]. Furthermore, a macroscale version of an acoustic energy harvester was previously deve loped [120] and, through miniaturization, led to the present MEMS-based energy harvester. Chapter 2 presents a theore tical 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 dyna mic behavior. Chapter 3 discusses the fabrication of the piezoelectric diaphragm and the packaging scheme employed. Following this, the experimental setup is provi ded 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 statem ent of the contributions to this research. Finally, Appendix A presents the Matlab pr ogram code for computing the diaphragm deflection and calculating the lumped elem ent 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 PAGE 30 16 Comprehensive lumped element model (LEM) of the acoustic energy harvester First reported integration of thin -film PZT with SOI-based MEMS First aeroacoustic capable piezoelectric microphone. PAGE 31 17 CHAPTER 2 THEORETICAL BACKGROUND This chapter focuses on the theory and modeling of the acoustic energy harvester and each of its components. The chapter begi ns with an overview of energy harvesting, followed by an introduction to lumped elemen t modeling, by discussi ng both its utility and limitations. This is followed by the development of a lumped element model for a Helmholtz resonator and the corresponding equi valent circuit. Next, the modeling of piezoelectric composite circular plates is addre ssed. To achieve this, an equivalent circuit is presented and general equations are di scussed. Then specific limiting cases are addressed that simplify the anal ysis. Finally, the overall dyna mic behavior of the system is theoretically investigated, including the ac oustical input an d electrical output behavior. A graphical overview of this chapter is provided by th e roadmap of Figure 2-1. Overview Energy Harvesting The general concept behind energy harv esting 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 mode rn applications is electrical energy, where it can be stored in a battery or used to power el ectrical circuitry. The initial form of energy can originate from any number of energy domai ns, such as optical, thermal, mechanical, acoustical, fluidic, chemical, and biological. So me form of transducer is then required to convert that energy to a usable form of elect rical energy. Depending on the situation, that also may require passage of the energy through an intermediary energy domain.. PAGE 32 18 Figure 2-1: Roadmap to Chapter 2. PAGE 33 19 There are pros and cons to each of these di fferent local energy sources [121]. Some energy sources such as solar power utilize the optical energy doma in to achieve high power densities on the order of 15,000 2Wcm in direct sunlight. The main downside is that direct sunlight is not always available. Vibr ational energy, on the other hand, offers power densities up to 250 2Wcm While this is considerably lower than solar energy, it is useful in pla ces without sunlight but wher e high vibrational energy is available. Acoustic energy, in a manner similar to vi brational energy, offers power densities on the order of 1 2Wcm for a 100 dB acoustic signal [121], or approximately 964,000 2Wcm at 160 dB. While most signals are t ypically much lower than 160 dB, there are applications where such high levels are presen t. Additionally, as w ith vibrational energy, acoustic energy, does not require th e presence of sunlight. Chemical energy sources are commonly em ployed today in the form of batteries and fuel cells. Batteries typically o ffer power densities in the range of 45 3Wcm for non-rechargables, and 7 3Wcm for rechargeable lithium ba tteries [121]. Fuel cells employing methanol, on the other hand, offer power densities as high as 280 3Wcm leading to the current interest in fuel ce ll development [121]. Additionally, a microcombustion engine that employs hydrocarbons as a fuel source realizes power densities of 333 3Wcm [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 operat ion. By contrast, the scavenged energy sources, such as vibrational, solar, and acoustical do not theoretically have a limited supply of energy, given the ri ght operating environment. PAGE 34 20 For the particular application considered in this dissertation, the initial energy is in the form of acoustical energy. In order to convert acoustical en ergy into electrical energy, a diaphragm based transducer is utili zed. The diaphragm tr ansducer utilizes the mechanical energy domain as an inte rmediary to transfer energy. Maximum Average Power Transfer Regardless of the route through which th e energy passes, certain fundamental issues must be addressed in order to maximi ze the amount of energy that is harvested. Whenever a change in impedance is encounter ed 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 focuse s on minimizing the reflected component and thus maximizing the transmitted component. This is achieved by matching the impedance along the route traveled by the ener gy. 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, du e to external and physical constraints on the system, however it is genera lly 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 consid er that power is a comple x 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 deliver ed to the load, while the reactive power represents energy that is tempor arily 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. PAGE 35 21 It is useful to look at a Thvenin equivale nt circuit, as shown in Figure 2-2, 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 1 cos 2LLLVi LLVI (2.1) where LV and L I are the peak magnitudes of the vo ltage and current at the load, and L V and L i are the phase angles of the voltage and current respectively. This can be rewritten as 1 2LLLVIpf (2.2) where p f is the power factor, defined by the ra tio of average power to apparent power and is given by L rmsrmspf VI (2.3) where rmsV and rms I are the root-mean-square voltage and current, respectively. Figure 2-2: Thvenin equivalent circuit for purposes of maximi zing power transfer. Using basic circuit analysis, the load vo ltage and current can be expressed in terms of the open circuit voltage, ocV, the Thvenin equivalent impedance TH Z and the load impedance, L Z The expressions are given as PAGE 36 22 ocL L THLVZ V Z Z (2.4) and oc L THLV I Z Z (2.5) where the Thvenin equivalent impedance, THTHTH Z RjX and the load impedance can be written as LLL Z RjX Plugging these expressions b ack into Eq. (2.4) and Eq. (2.5) and separating out the magnitude components yields 1 22 2 1 22 2 ocLL L THLTHLVRX V RRXX (2.6) and 1 22 2 oc L THLTHLV I RRXX (2.7) The phase angle between the voltage a nd current can also be expressed as L LLViZ where L Z is the phase angle of the impedance. The power factor can be defined as 1 22 2cosL Z L LLR pf RX (2.8) Plugging this expression into Eq. (2.1), toge ther with Eq. (2.6) and Eq. (2.7) yields 2 221 2ocL L THLTHLVR RRXX (2.9) As, the quantity THL X X is only in the denominator, any nonzero value reduces the overall power absorbed by the load, thus LTH X X is optimal and reduces Eq. (2.9) to PAGE 37 23 2 21 2ocL L LTHVR RR (2.10) This expression can be maximized by differe ntiating this expression with respect to L R and equating the derivative to ze ro, yielding an optimal value of LTH R R. The average power to the load can be maximized, th en, by setting the load impedance to LLLTHTHTH Z RjXRjXZ (2.11) where TH Z is the complex conjugate of the Thvenin impedance. This derivation assumes total freedom in the choice of the load impedance. Under the constraint of a purely resistive load, where 0LX the equation given in Eq. (2.9) can be differentiated di rectly and set equal to zero with 0LX, in order to find the maximum average power transfer. Th is procedure yields an optimal value for the load resistance of 22 L THTHTH R RXZ (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 br oadly classified into two main categories: sensors and actuators. The breakdown among thes e classifications relates to the direction of information flow. Sensors are trans ducers 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 PAGE 38 24 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 focu s 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 transm ission of information in one direction or another. Besides information, a signal carries power. A third type of transducer exists that falls under neither categor y and can be best described as an energy harvester. The concept behind the energy harvester is to achieve an optimal am ount 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 straight forward relationship between in put and output. The linear input-output 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 necessari ly require a linear i nput-output relationship as only the power is of interest. Some deta ils 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 va rious tradeoffs among performance, reliability, and material integrat ion. 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 ener gy between the two energy domains. Indirect PAGE 39 25 transducers, however, rely upon a second en ergy source that modulates the primary energy source as it passes through the transd ucer. The inherent nature of indirect transduction leads to a lower transduction efficien cy as compared to direct transduction. Electrodynamic transduction, a direct transduction mechanism, yields high coupling efficiencies. Additionally, this met hod 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. Additiona lly, the necessary fabrication is often more complex than the other techniques discussed here. Piezoelectric transduction requires no outside source of power ot her than the input signal, and offers a high sensitivity, with a potentially low noise floor. The main disadvantages are the difficulty in integrati ng the piezoelectric materials into a standard process flow, and the limited coupling efficien cy due to indirect transduction of energy. Lumped Element Modeling The most accurate, complete, mathematical description of a physical system is a physics-based model, supported by an exact analytical expressi on for the system behavior. Why then are alternative m odeling techniques commonly used? These alternative methods, such as lumped elem ent 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 lit tle in the way of physical insight into the PAGE 40 26 design and scaling system. Ofte n, 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 behavi or, in this case via a numerical approach. The results produced by this technique can very precisely follow the physical system; however, the physical in sight that can be gleaned is limited. Furthermore, the results depend on the numer ical 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 be havior, 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 ener gy [60, 126]. More sp ecifically, the total energy going into any given system is divide d among three distinct t ypes of interactions: the storage of kinetic energy, the storage of potential energy, a nd the dissipation of energy. All systems are composed of these three energy processes. In some systems, known as distributed syst ems, the storage of kinetic and potential energy occurs over a distributed region in sp ace [90, 127]. To accura tely represent these systems mathematically requires a partial differential equation, as spatial and temporal components are inherently coupled. Physi cally, the distribution occurs because the wavelength is on the order of the physical syst em or smaller. At different points along PAGE 41 27 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 significantl y larger than the length scale of interest, very little variation occurs in the distributi on 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 di fferential equations to solve the problem, rather than partial differential equations. Physically, it means each energy storage or dissipation mechanism can be equa ted 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 th e 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 ener gy through the losses of a damper. Similarly, in electrical systems, where lumped element systems are commonplace due to the extremely long wavelengths of el ectrical signals, kinetic ener gy is stored in the magnetic field of an inductor, while pot ential energy is stored in th e charge across a capacitor. Additionally, dissipation of energy is modeled via the re sistor. Finally, in lumped acoustical systems, the kinetic and potential energy is stored in an acoustical mass and acoustical compliance, respectively, while di ssipation of energy is represented by an acoustic resistance. These elements are summarized in Table 2-1. The commonplace nature of lumped elements in electrical systems has led, over the years, to a large growth in graphical and anal ytical techniques to so lve large networks of PAGE 42 28 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 repres ented using inductors, compliances using capacitors, and dissipativ e components using resistors. Once the complete equivalent circuit is constructed, standard circuit an alysis techniques (e.g. Kirchoffs current and voltage laws) can be applied to find the solution of interest. Table 2-1: Equivalent lumped elements in several common energy domains. Kinetic Energy Storage Potential Energy Storage Energy Dissipation Acoustical Acoustic Mass 4kgm Acoustic Compliance 3mPa or 5mN Acoustic Resistor 4ms Mechanical Mass kg (pt mass) Compliance mN (spring) Frictional Damper ms Electrical Inductance H Capacitance F Resistance Whenever dealing with more than one lump ed 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 AB and the reverse as BA then the net power flow from A to B is netABBA (2.13) Now, since each of the power flows must be gr eater than or equal to zero, then they can each be written as the square of a real number, 1r and 2r [128]. The net power flow can then be rewritten as 22 21212 netlrrrrrr (2.14) It is therefore seen that the net power flow can be written as th e product of two real numbers, which are referred to as conjugate power variables. Moreover, these quantities PAGE 43 29 are more specifically referred to as an effort, e, and a flow f where the product ef is the power. A table of conjugate power variables, divi ded into effort and flow, is given in Table 2-2 for a number of energy domains. Table 2-2: Conjugate power variables Energy Domain Effort Flow Mechanical translation Force, F Velocity, v Fixed-axis rotation Torque, Angular velocity. Electric circuits Voltage, V Current. I Magnetic circuits MMF, M Flux rate, Incompressible fluid flow Pressure, P Volumetric flow, Q Thermal Temperature, T Entropy flow rate, S (after Senturia, pg 105 [128]) Helmholtz Resonator Lumped Element Model LEM was applied to the Helmholtz resona tor in order to better understand the system [1, 125]. A schematic diagram of a Helmholtz resonator is shown below in Figure 2-3, where V is the cavity or bulb volume, l and 2Sa are the length and cross-sectional area of the neck, respectively, where a is the radius of the neck, 1P is the incident acoustic pressure, and 2P is the cavity acoustic pressure. Both 1P and 2P are considered to be functions of the radian frequency, Figure 2-3: Diagram showing si de view of a Helmholtz resonator and its mechanical equivalent of a mass-spring-damper system. PAGE 44 30 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 di ssipative and inertial components. The air in the cavity is compressible and stores pot ential energy, and is th erefore 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. kg sodt dM VQmassflowrate dtdt (2.15) where M is the mass in the bulb, o is the mean density of the air, and 2Qua is the volumetric flow rate or volume velocity, where u is the velocity. If the disturbance is harmonic and isentropic then 2 2 00 200cQ Pc jV (2.16) where oc is the isentropic speed of sound of the medium and 1j The linearized momentum equation fo r a lossless medium is given by 0u P t (2.17) where P is the acoustic pressure. Assuming a linear pressure gradient yields 120u PPl t (2.18) PAGE 45 31 where l is the length of the neck. Substituting for 2P yields the following equation. 2 000 1QcjQl P jVS (2.19) Factoring Q, this can be rewritten as 11aN aCPQjM jC (2.20) where the effective acoustic compliance, aCC, of the cavity is 3 2 00 aCVm C cPa (2.21) and effective acoustic mass, aN M of the air in the neck is given by 4 o aNl kg M Sm (2.22) 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 acousti c 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 e xpressions above do not account for any viscous damping effects that occur in the He lmholtz resonator neck. The PAGE 46 32 viscous damping represents a resistance, whos e value can be approximated from pressure driven, laminar pipe flow as 248 aNlkg R Sms (2.23) where is the dynamic viscosity of the air. Furthermore, the viscous damping produces a non-uniform axial velocity profile in the n eck that ultimately leads to an additional factor of 4/3 in the expression for the e ffective acoustic mass [ 130]. The corrected effective mass is then given as 44 3o aNl kg M Sm (2.24) Additionally, the effec tive resistance and ma ss of the neck are, in fact, non-linear and frequency dependent due to turbulence a nd entrance/exit effect s, [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 ex pression for the impedance in a short closed tube [2]. The exact expression is given by 2cotoo injc Z kl a (2.25) where 0k c is the wavenumber. Using a Maclauri n series expansion of the cotangent function yields 3111 cot... 345 klklkl kl (2.26) For kl << 1, the impedance can be approxima ted by keeping only the first couple of terms in the expansion, yielding PAGE 47 33 2 2 22 23 3ooooooo incklccV Zjj jklaajV a (2.27) From this expression, we once again see that 3 2 00 aCVm C cPa (2.28) We now also have an additional mass term, given by 0 2 4 2kg m 3aCV M a (2.29) which is equal to one-third the acoustic mass of the cavity. This correction term is small for 1kl but becomes more prominent as kl increases. At 1kl the correction term is 33.3% of the primary term, while at 0.1 kl the correction term is only 0.33% of the primary term. Equivalent Circuit To create an equivalent circuit model fo r the Helmholtz resonator, knowledge of how to connect these lumped elements is ne eded. Connection rules between elements are defined based on whether an effort-type vari able or a flow-type variable is shared between them [132]. Whenever an effort variab le, 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 be tween elements, those elements are connected in series. These connection rules are used to obtain th e equivalent circuit representation for the Helmholtz resonator, as shown in Figure 2-4. The connection rules, as given, are assuming that what is known as an impeda nce analogy is employed. If an admittance PAGE 48 34 analogy were used instead, then the connecti on rules would be reve rsed from what is described above. Figure 2-4: Equivale nt circuit representation of a Helmholtz resonator. The frequency response function 21PP represents the pressure amplification of the resonator. It is the ratio of cavity pre ssure to incident pressure, and is given by 2 11 1aC aNaN aCsC P P RsM sC (2.30) where sj 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 2res aNaCf M C Hz. (2.31) At the resonant frequency, the pressu re amplification reaches a value of 2aN resres aNM PAf R. (2.32) This is shown in Figure 2-5, for an arbi trary 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 pr essure amplification freque ncy response represents the single degree of freedom present in the system. PAGE 49 35 Figure 2-5: Theoretical pressure amplification of a conve ntional Helmholtz resonator. (a) magnitude (b) phase. Piezoelectric Composite Plate On the back wall of the Helmholtz res onator, 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 ci rcular layer of silic on and an annular ring of PZT, is shown in crosssection in Figure 2-6, where E is theYoungs modulus, is Poissons ratio and is the density of the silicon and P ZT as indicated by the subscript. Additionally, s t is the thickness, and 2 R is the outer radius of the silicon, while for the PZT, pt is the thickness, is the relative permittivity, 31d is the transduction coefficient for a voltage applied across the piezoelect ric causing a displacement in the radial (a) (b) PAGE 50 36 direction, and p 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 c oupling is increased due to th e stress concentration. In addition, the annular structure facilitates the connection of elec trodes 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. Figure 2-6: Cross-sectional a nd 3-D schematic of piezoelectric composite circular plate. (not to scale) Piezoelectricity Mathematically, the linear piezoelect ric effect is expressed as [133] E ijijklklkijkSsTdE (2.33) and PAGE 51 37 T iiklklikkDdTE, (2.34) where ijS is the mechanical strain, E ijklsis the elastic compliance 2mN at constant electric field, klT is the mechanical stress 2/Nm, and kijd is the piezoelectric coefficient / or /CNmV, iD is the electric displacement 2/Cm, T ik is the electric permittivity at constant stress Fm, and kE is the electric field /Vm [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 l 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 E ppqqkpkSsTdE (2.35) and T iiqqikkDdTE. (2.36) The subscripts in the reduced notation equatio ns refer to the compone nt of each variable in a specified direction as define d by Figure 2-7. For example, 31d is the piezoelectric coefficient relating electrical displacement in the direction (z-a xis) to a mechanical stress in the direction (x-axi s). From Eq. (2.33), it is also seen to relate a mechanical strain in the direction (x-axis) with an electric field in the direction. 2 (y) 3 (z) 1 (x) 4 5 6 Figure 2-7: Notation of axes us ed in piezoelectric transduction. PAGE 52 38 One class of materials, ferroelectrics, e xhibits the property th at the direction of their polar axis can be change d through application of an exte rnal electric field [5, 6, 134]. The polar axis is the direction along wh ich a polar molecule exhibits an internal electric field. When the external electric fi eld is then removed, the polar axis remains in an altered direction. This process that cau ses a long term rotation of the polar axis is commonly referred to as poling. All ferro electric materials are also piezoelectric, however they are not naturally piezoelectri cally active, as the crystal structure is centrosymmetric, as in Figure 2-8a. PZT is a typical example of a ferroelectric material, and has a perovskite crystal structure as show n in Figure 2-8. By applying an external electric field to rotate the polar axis, a non-centrosymmetric crystal structure is created and the ferroelectric material becomes piezoel ectrically active, as in Figure 2-8b. 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. a) b) Figure 2-8: Idealized perovskite crystal stru cture for PZT. a) centrosymmetric structure prior to poling. b)non-centrosymmetric structure after poling. PAGE 53 39 A piezoelectric ceramic, such as PZT, consists of individual domains. The polarization within each domain is in th e same direction; however, the polarization direction varies randomly from domain to do main, leading to a net polarization of zero. This is illustrated in Figure 2-9(a), for th e unpoled material. After poling the material with a sufficiently high electric field, the di poles 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 2-9(b). Raising the temperature during poling enhances the polarization and resulting piezoel ectric properties, as the dipoles rotate more readily at higher temperatures [6]. Figure 2-9: Schematic of the poling process: a) An unpoled piezoel ectric 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 materi als is the double-valued nature of their response to an electrical excitation, resulti ng in a hysteretic behavi or. Hysteresis is a property of systems that do not react instantly to an applied force and may not return to PAGE 54 40 their original state when the force is removed. In ferroelectrics, when an electric field via a voltage is applied, a polarization is in curred. Upon removal of the voltage, the polarization decreases but does no t return to zero. The result is a remanent polarization, rP. If the voltage is swept up and down, the result will be what is known as a hysteresis loop, shown in Figure 2-10. Figure 2-10: Polarization vs. electric field hysteresis l oop (after Xu, pg 10 [134]). The graph is displayed here as polarization, P, typically given in 2Ccm versus applied electric field, E although other variations do exist. There are four figures of merit shown in the illustration. mP is the maximum polarization measured, while rP is PAGE 55 41 the remanent polarization (i.e. the polarizatio n which remains when the field is returned to zero). Furthermore, s P is known as the spontaneous pol arization and is defined as the straight line extrapolation of the line defined by the upper saturation region. Finally, the coercive field, cE, represents the magnitude of th e field required to cancel out the remanent polarization. Composite Plate Modeling In order to accurately model this structure, the material properti es of each of the layers were considered. These include the m echanical properties of the layers such as Youngs Modulus, E and Poissons ratio, and the dielectric constant, and piezoelectric coefficient, 31d, of the PZT. The subscript p or s indicates the layer as PZT or silicon respectively. Furthermore, th e geometry of the structure is accounted for in the values for inner PZT radius, 1 R outer PZT radius, 2 R PZT thickness, pt, and silicon thickness, s t. Lumped element model A pressure applied to the plat e creates a deflection of the plate, resulting in a stress in the piezoelectric layer. The stress de forms the piezoelectric layer, creating an electrical charge on the elec trodes, thus generating a vo ltage across the piezoelectric layer. Although the stress and charge are dist ributed over a finite regi on of the plate, it is useful to first look at one-dimensional (1-D) piezoelectric transduction. This 1-D analysis can then be extended to incorporat e effective lumped element values that are calculated from the actual distributed case. In the 1-D piezoelectric transduction, the piezoelectric material displaces longitudinally due to the application of a force, F, and/or a voltage, V, applied in the PAGE 56 42 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 el ectric displacement in the direction, for an applied stress and electric field in the same direction, the equations reduce to 333333 ESsTdE (2.37) and 333333 TDdTE (2.38) Then this set of equations can be converted by multiplying both sides by the thickness of the piezoelectric material. 333333 3333 E EtSsTdE tSstTdtE (2.39) Now, the mechanical compliance of a ma terial under compression in the thickness direction can be defined as 33 E mst C A, (2.40) where t is the thickness, and A is the area over which the force is applied. Additionally, for a constant electric field, E, V E t (2.41) Substituting (2.40) into second part of (2.39) yields, 33 m x CATdV (2.42) Finally, the stress, T, when applied over the area, A can be equated to a force, F, given by PAGE 57 43 FTA (2.43) leading to the final equation given by 33 m x CFdV (2.44) A similar procedure can then be performed on Eq. (2.38), by multiplying both sides by the area, and making similar su bstitutions as before, giving 333333TADdTE, (2.45) 3333TqdFAE, (2.46) 33 33TA qdFV t, (2.47) so 33efqdFCV (2.48) Thus a pair of equations can be written desc ribing the 1-D piezoelec tric transduction, and is given by 33 33ms efCd x F dC qV (2.49) In this pair of equations, msC, the mechanical compliance when a short is placed across the electrodes of the piezoelectric, is defined as 0|V msx C F, (2.50) while efC, the electrical capacitance when the piezoelectric is free to move, is defined by 33 0p F ef pq A C Vt, (2.51) PAGE 58 44 where q is the resulting charge from the applied voltage V, 33 is the electrical permittivity in the thickness direction, pA is the area of the piezoelectric, and ptis the thickness of the piezoelectric. Furthermore, 33d is the piezoelectric coefficient relating the displacement in the thickness direction for an applied voltage in the same direction, when no mechanical force is applie d across the piezoelectric, defined by 0 33|Fx d V (2.52) For the case of the piezoelec tric composite plate that is presented here, though, there is a distributed, nonuniform deflection acr oss the plate. In or der to apply the 1-D model developed above to this situation, it is necessary to lump th e actual distributed deflection to a single point and compute eff ective values by equating the energy in the distributed system to the energy in a corresponding lumped component. The 1-D 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 time-harmonic signals [64]. The time-harmonic, electroacoustical equi valent of Eq. (2.49) is given by efA AasjCjd I V jdjC QP 3A ms (2.53) whereP is the acoustic pressure, Q is the volume velocity of the plate, I is the electrical current. Furthermore, Ad is the piezoelectric coefficient relating the volumetric displacement to the applied voltage, when pressure equals zero, and defined by 20 0 02R P A Pwrrdr Vol d VV 3m V (2.54) PAGE 59 45 where wr is the deflection as a func tion of the radial position, 2 R is the radius of the plate and Vol is the volumetric displacement defined by 202RVolrwrdr3m (2.55) Additionally, asC the acoustical compliance when a s hort circuit is placed across the piezoelectric, is defined by 20 0 02R V as Vwrrdr Vol C PP 3m Pa (2.56) From two-port network theory, a generali zed electro-acoustic, reciprocal, two-port network can be written as ef asYG I V GY QP (2.57) where efY is the electrical admittance when the aco ustic terminal is free to move (i.e no pressure exists, 0P ), asY is the acoustical admittance when the electrical terminal is shorted (i.e. no voltage exists, 0V ), and G is the electro-acoustic transduction admittance, given both by the ratio of current, I to pressure, P when 0V, and the ratio of volume-velocity, Q, to voltage, V, when 0P Comparing Eq. (2.53) and Eq. (2.57), it can be seen that AGjd (2.58) efefYjC (2.59) PAGE 60 46 and asasYjC (2.60) Equivalent circuit Using Eq. (2.57), an equivalent electro-ac oustic circuit can be drawn, as shown in Figure 2-11, where ebC the electrical capacitance when the plate is blocked from moving is given by 21ebefCCk F, (2.61) the transduction factor, is given as AA asasas j dd G YjCC Pa V (2.62) and the coupling factor, k, is given by 2 2 A efasefasd G k YYCC (2.63) Physically, the coupling factor, k, represents the fraction of energy that is coupled between the acoustical and electr ical energy domains. It is se en here to be a function of the piezoelectric coefficient, Ad, which represents coupled energy and the two elements which store potential energy asso ciated with the transduction, efC and asC. The coupling factor is therefore related to the ratio of the coupled energy to the stored energy. Figure 2-11: Electro-acous tic equivalent circuit representation with ebC, asC and PAGE 61 47 An alternative equivalent circuit can also be drawn, as shown in Figure 2-12. The alternative circuit is simply another repr esentation for the same physical process. Figure 2-12: Electro-acous tic equivalent circuit representation with efC, aoC and In Figure 2-12, aoCrepresents the acoustic complia nce when an open-circuit is placed across the piezoelectric, and is given by, 21aoasCCk 3m Pa (2.64) Additionally, the transduction factor, is given by AA efefefjdd G YjCC V Pa (2.65) The two circuits above repres ent the transduction under sta tic conditions, thus they do not take into account the dynamics of the sy stem (i.e. mass). At frequencies greater than zero, a mass must be added to the equiva lent circuit. As the mass stores kinetic energy, it is represented by its electrical equi valent, which is an i nductor. Furthermore, this mass must be in series with the complian ce, as both experience the same motion (i.e displacement or velocity). The acoustic mass,aD 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 2 2 0 0 02R V aDA Vwr M rdr V (2.66) PAGE 62 48 where A is the areal density of the piezoe lectric composite plate defined by, 2 12 z A zdzkgm (2.67) where is the density of th e corresponding layer. In addition to the above mentioned elemen ts, it is also necessary to include an additional element that represents the di electric losses that are incurred during transduction, yielding the circuit shown in Figure 2-13. Generall y this is expressed through a term known as the loss tangent, also known as th e dielectric loss factor. Represented by the ratio of the parallel re actance to the parallel resistance in the dielectric, the loss tangent is formally given by 1 tan 2p pppX R fCR (2.68) where p R is a resistor in para llel with a capacitance, pC and f is the frequency at which p R and pC were measured. The loss tangent is also equal to the inverse of the quality factor, Q Figure 2-13: Dynamic electro-ac oustic equivalent circuit. In this circuit, aD M represents the acous tic mass of the diaphragm. The input impedance from the acoustical side is then given by 21 1p inaD asebpR ZsM sCsCR (2.69) PAGE 63 49 By voltage division we get 22 211 1 1pp ebpebp p in aD asebpRR sCRsCR VPP R Z sM sCsCR (2.70) which, after rearranging yields, 3221 1asp aDasebpaDaspebasV sCR P sMCCRsMCsRCC (2.71) This is the general expressi on for the open-circuit sensitivit y of the circular composite piezoelectric plate. From a physical standpoi nt it is also useful to look at certain conditions that allow for additi onal insight. The presence of the dielectric loss resistance, p R complicates some of the underlying behavi ors of the device, by adding an additional R-C time constant. Looking at a situation in which this parallel loss resistance is infinite (i.e.pR ), the above expression can be reduced to 221 1as as eb aDas ebC V C PC sMC C (2.72) Several important things can be gleamed from this expression. First of all, the equation describes the behavior of a sec ond order system, with a primary resonance occurring when the denominator goes to zero at 21as eb res aDasC C MC (2.73) From this expression, the resonanc e is seen to depend on the ratio 2 asebCC in addition to the standard dependence on the product aDas M C If 0 this reduces to a PAGE 64 50 simple plate. For 0 electrical energy is stored acro ss the piezoelectric, resulting in a stiffer device possessing a larger res The full expression that includes the dielec tric loss, given by Eq. (2.71), is the general expression for the sensitiv ity of the circular composite piezoelectric plate. It can be further simplified for two important cases: 1. Well below the short-circuit mechani cal resonant frequency (compliance dominated) 2. At the short-circuit mechanical resona nt frequency (resistance dominated) Case 1: Well below the short-circu it mechanical resonant frequency When a short is placed acrossebC, it is equivalent to assigning 0pR Physically, this implies that no electrical energy stor age is allowed, as the voltage across the capacitor, ebC, must be zero. The expression for th e short-circuit mechanical resonant frequency is given by 1scres aDas M C, (2.74) which is the same resonant fre quency as described above when 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 s cres as 22 21 1scscasp ebppebas resresV jCR P jj jCRjRCC (2.75) Well below resonance, we have PAGE 65 51 21scresj (2.76) This term can then be dropped from the expression, yielding 21 1 1as eb as ebpebC V j PC C j CRC (2.77) To simplify this further, the transduction factor, can be written in terms of the effective piezoelectric modulus, Ad, by recalling Eq. (2.62), yielding 2 21 1 1 1 1 1as A aseb as A asebpeb A eb A asebpebC jd V PCC C d j CCRC d C d CCjRC (2.78) Furthermore, the blocked electrical capacitance, ebC, can be written in terms of the free electrical capacitance,efC and the coupling coefficient, k, using Eq. (2.61) and Eq. (2.63) to yield the an expre ssion for sensitivity, given by 2 2 22 2 221 1 1 1 1 1 11 1 1 1 11A eb A asebpeb A eb A asefpef A eb pefd V PC d CCjRC d C d CCkjRCk d C k kjRCk (2.79) which can be simplified further to yield PAGE 66 52 22 21 11 11 11 1 1 1 1A eb pef A eb p A ef p efpd V PC kjRCk d C jR k d C jR jCR (2.80) Now under the condition of a high value for the loss resistor, p R Eq. (2.80) can be reduced approximately to A efd V PC (2.81) This corresponds to the alternative trans duction coefficient, given by A efd C (2.82) and thus under the conditions of a high loss resistor and operation well below s cres the sensitivity can be given by V P (2.83) Case 2: At the short-circuit mechanical resonant frequency The general expression for sens itivity is once again given by 22 21 1scscasp ebppebas resresV jCR P jj jCRjRCC (2.84) PAGE 67 53 Now, since we are only considering the syst em at the short-circuit resonance, where s cres we have 21scresj (2.85) yielding an exact expression for the sensitivity as 211 11asp ebppebasV jCR P jCRjRCC (2.86) We therefore find that the sensi tivity at resonance reduces to 21V Pk (2.87) as 2k (2.88) This can be understood from the equi valent circuit of Figure 2-13. At s cres the impedance of the mass is canceled by the impedance of the compliance, permitting a direct transduction of energy fr om the acoustic to electrical energy domain. Note that s cres exists whether or not 0pR therefore Eq. (2.86) is always valid whenever s cres In other words, Eq. (2.86) holds whenever the operating frequency coincides with the frequency that is defined by th e short-circuit resonance, but makes no requirement for an actual short-circuit load. As the load across ebC is increased, the resonance frequency will shift towards the open-circuit resonance. R ecalling Eq. (2.64), the open and short-circuit acoustic compliances are related by the coupling factor and define the limits of the resonant frequency as all resistive loads fall somewh ere between open and short-circuit. The PAGE 68 54 larger the coupling factor, th en the larger the range betw een open and short-circuit resonant frequencies. General expression vs. simplified cases Figure 2-14 plots the opencircuit and near short-circuit sensitivity versus frequency overlaid with the values of and 1 The material properties used in calculating these results are shown in Tabl e 2-3 and Table 2-4. Notice that below resonance, the general open-circuit sensitiv ity expression asymptot es to the value of Additionally, notice that the genera l expression reaches the value of 1 at a frequency below where the peak amplitude occurs. This is because 1 is the simplified expression for the sensitivity at the short-circuit mechan ical resonant frequency, whereas the general expression represents the open-circuit frequency response. Thus the peak amplitude occurs at the open-circuit resonant frequenc y. The frequency at which the general expression for sensitivity reaches the value of 1 coincides with the short-circuit resonant frequency, as expected. Also note th at this short-circuit sensitivity calculation was performed with an almost-short circuit cond ition, primarily to illustrate the effect on the resonant frequency. To achieve the almost-short circuit condition a resistive load of 1 was used as the value is much less than the impedance due to ebC. One final feature of the frequency response is the low frequenc y rolloff that is visibl e in the open circuit case. This rolloff is due to the dielectric loss resistor,p R and the resulting time constant due to the combination of that resistor and the electrical cap acitance, as evidenced in Eq. (2.80) for the low frequency regime. PAGE 69 55 Table 2-3: General material properties used in modeling. Material E GPa 3kgm tm Silicon 150 0.27 2300 3 Platinum 170 0.38 21440 0.170 TiO2 283 0.28 2150 0.100 [135, 136] Table 2-4: PZT properties used in modeling. pEGPa p 3 pkgm 31dpmVtan p M Pa 30 0.3 7600 1000 -50 0.02 30 [9, 10, 16, 31] 100 101 102 103 104 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Frequency [Hz]|Sensitivity| [V/Pa] 1/ Near short circuit frequency response Open circuit frequency response Figure 2-14: Sensitivity vs. frequency for a piezoelectric composite circular plate. Effects of including th e radiation impedance Since the clamped circular plate is vibrat ing in a medium, the radiation impedance of the plate must be taken into account, a nd consists of a radiation mass and radiation resistance. The radiation mass accounts for the in ertial mass of the fluid that is vibrating in unison with the plate, while the radiation resistance accoun ts for the acoustic radiation PAGE 70 56 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 first-order by approximating the ba ckplate as a piston in an infinite baffle, as given by Blackstock [2] in term s of specific acoustic impedance, p Z as 11 112222 122 22pooooJkaKka Z cjcRkajXka kaka (2.89) where 12 R ka and 12 X ka indicate that they are functions of 2ka, 2aR is the radius of the piston,1J is a Bessel function of the first kind of order one, and 1K is a firstorder Struve function. The Maclaurin expansio ns of Eq. (2.89) are also given by [2]as 246 1 222121231234kakaka R (2.90) and 35 1 22222 42 335357kaka ka X (2.91) For small values of ka, where 1ka, 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, effA, of the piezoelectric plate. As the ci rcular composite diaphragm doe s 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, effA, is therefore defined to represent the equivalent area that a circular piston with uniform deflecti on would need to have to create the same PAGE 71 57 volumetric displacement as the composite diaphr agm. This is necessitated by the need to maintain continuity of volume velocity ac ross the interface between the mechanical and acoustical domains. It can be calculated by integrating the distri buted 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 22 0effwrrdr Am w (2.92) The radiation resistance can then be approximated, for low values of kaas 2 4,1 2o aDrad effkac kg Rka Ams (2.93) while the radiation mass is approximated as 48 ,1 3o aDrad effkac kg Mka Am (2.94) These two elements add in series to create a radiation impedance, rad Z that is defined by aDradaDradaDradZRsM (2.95) The equivalent circuit of the composite plat e including the radiati on impedance is shown in Figure 2-15. Figure 2-15: Equivalent circ uit of piezo-composite plat e including the radiation impedance. PAGE 72 58 When the radiation impedance of the diaphr agm is added to the equivalent circuit, the analysis becomes only slightly more co mplex. The acoustical input impedance,in Z defined as PQ, can be represented in terms of the equivalent circuit parameters, and is now given by 21 1p inaDaDradaDrad asebpR ZsMsMR sCsCR (2.96) Then, the general expression for sensitivity is now given via the voltage divider method as 22 211 1 1pp ebpebp p in aDaDradaDrad asebpRR sCRsCR VPP R Z sMsMR sCsCR (2.97) or 21 1 1p ebp p aDaDradaDrad asebpR sCR V R P sMsMR sCsCR (2.98) From Figure 2-15, it can be s een that the radiation mass,aDradM adds directly to the acoustical mass of the plate,aD M as is evidenced in Eq. (2.97), while aDradR provides damping to this second-order system. Note that here, the acoustical resistance, aDradR damps the resonance, while the electrical resistance, p R leads to a low frequency rolloff. The different effects arise because aDradR is in series with the reactive elements, asC and PAGE 73 59 aD M while p R is in parallel with the capacitance, ebC The general expression for the undamped resonance frequency is now given by 21as eb res aDaDradasC C M MC (2.99) When a short is placed acrossebC it is effectively removed. This is mathematically equivalent to assigningebC as the impedance of a capacitor is inversely proportional to the capacitance. Th e short-circuit resonance frequency then reduces to 1res sc aDaDradas M MC (2.100) Upon comparison with Eq. (2.74), it can be seen that the shor t-circuit resonance frequency has now been shifted downward by the radiation mass, as is also the case with the open-circuit resonance frequency. At the radial frequency of the short-circuit resonance, the short-circuit input impedance reduces to inaDradZR (2.101) because ebC and p R are effectively removed by the short-circuit. The volume velocity, Q is then given by aDradP Q R (2.102) Via the piezoelectric tr ansduction, the current, I in the piezoelectric material, is then given by aDradP IQ R (2.103) PAGE 74 60 The output voltage in this case, however, rema ins 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, load R across the output yields the equivalent circuit shown in Figure 2-16. Figure 2-16: Equivalent circ uit with resistive load. The input impedance, in Z is then given by 21loadp inaDaDradaDrad asloadploadpebRR ZsMMR sCRRRRsC (2.104) Now, if we choose load R such that 1loadeb R sC and loadp R R, then the parallel combination of these three elements can be approximated with just load R In addition, the resonant frequency will be very close to the short-circuit resonant frequency. At this frequency, the input impedance reduces to 2 inaDradload Z RR (2.105) and the volume velocity, Q is given by 2inaDradloadPP Q ZRR (2.106) Through piezoelectric tran sduction, a current, I is created in the piezoelectric given by 2aDradloadP IQ R R (2.107) PAGE 75 61 As the resistance,load R is much less than 1ebsC most of the current goes through it, leading to a voltage drop given by 2load load aDradloadPR VIR RR (2.108) Thus the power absorbed by the load resi stance is purely r eal and is given by 22 2 2ReReload aDradloadPR IV RR (2.109) An optimal solution to this equation is found by setting 2 22 3 20aDradload load aDradloadRR d P dR RR (2.110) Solving for the optimal load resistance yields 2aDrad loadR R (2.111) which is just the impedance matching condition at the interface. Note that is for the special case of 1loadeb R sC, loadp R R and operation at resonance. Equivalent circuit parameters of piez oelectric composite circular plates In order to obtain the equivalent circ uit parameters, an analytical model was developed for the piezoelectric composite circ ular plate by Wang et al. [62, 63]. Using this approach, analytical modeling was acco mplished by dividing the problem of Figure 2-6 into two portions, an inne r circular plate, surrounded by an annular composite ring with matching boundary conditions at the interface, as shown in Figure 2-17. The boundary conditions consist of equal moments a nd forces at the interface as well as equal slope and transverse displacement. After so lving for the deflection in each region, the deflection equation for each region can then be combined [62, 63]. PAGE 76 62 Figure 2-17: Conceptual 3-D and cross-sec tional schematic of th e circular composite plate. (Not to scale.) The deflection equation can then be utili zed 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, expre ssions can be found for the electro-acoustic transduction coefficient and the blocked elec trical capacitance. Using the parameters shown in Table 2-3 and Table 2-4, plots were obtained for the lumped elements as a function of both 12 R R and pstt As many of the material parameters for PZT are highly dependent on actual processing conditi ons and techniques, typical values were chosen as a best guess estimate [8, 12-14, 1648]. The effective acoustic short-circuit PAGE 77 63 compliance and mass are functions of both 12 R R and pstt and are shown in Figure 2-18 and Figure 2-19, respectively. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 x 10-12 R1/R2Cas [m5/N] tp=0.6 m tp=3.0 m Figure 2-18: Effective acoustic shortcircuit compliance as a function of 12 R R and pstt For these calculations, 3stm and 22 R mm while the piezoelectric layer has thicknesses of 0.6, 1.2, 1.8, 2.4, 3.0 ptm The acoustic compliance is found to increase with increasing 12 R R and decrease with increasing pstt however, the acoustic mass is f ound to generally decrease with 12 R R and increase with pstt It is also useful to look at the physical limits and effects of 12 R R As 120 RR 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 PAGE 78 64 to changes in the piezoelectric layer thic kness under this condition. Meanwhile, as 121 RR the piezoelectric ring would have in finitesimal width and so would be essentially nonexistent. U nder this condition, the piezoel ectric layer thickness has no effect on the compliance, as would be expected. 0 0.2 0.4 0.6 0.8 1 500 1000 1500 2000 2500 3000 3500 R1/R2MaD [kg/m4] tp=0.6 m tp=3.0 m Figure 2-19: Effective acoustic mass as a function of 12 R R and pstt For these calculations, 3stm and 22 R mm while the piezoelectric layer has thicknesses of 0.6, 1.2, 1.8, 2.4, 3.0 ptm Shown in Figure 2-20 is a graph of the resonant frequency as a function of both 12 R R and pstt From this figure, it can be seen that the resonant frequency increases as the thickness of the piezoelectric layer incr eases 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 PAGE 79 65 should be noted that at the limit of 121 RR the resonant frequency is no longer affected by the thickness of the piezoelectric layer. Also, for a given pt there is a maximum resonant frequency at 120.4 RR. 0 0.2 0.4 0.6 0.8 1 2000 4000 6000 8000 10000 12000 14000 R1/R2Resonant Freq [Hz] tp=0.6 m tp=3.0 m Figure 2-20: Short-circuit resona nt frequency as a function of 12 R R and pstt For these calculations, 3stm and 22 R mm while the piezoelectric layer has thicknesses of 0.6, 1.2, 1.8, 2.4, 3.0 ptm The electro-acoustic transduction coefficient, 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 th e acoustic pressure produced by the resulting motion of the composite plate. It should be noted that the transduction coefficient is negative, implying a 180 phase shift between pressure and voltage, as PAGE 80 66 computed relative to an assumed poling direct ion. As can be seen in Figure 2-21, a maximal magnitude occurs when 12 R R is around 0.45. 0 0.2 0.4 0.6 0.8 1 -6 -5 -4 -3 -2 -1 0 R1/R2a [Pa/V] tp=0.6 m tp=3.0 m Figure 2-21: Electro-acoustic transduc tion coefficient as a function of 12 R R and pstt For these calculations, 3stm and 22 R mm while the piezoelectric layer has thicknesses of 0.6, 1.2, 1.8, 2.4, 3.0 ptm Additionally, the magnitude of the transduction coefficient increases with increasing piezoelectric thickne ss. Looking at the limit as 121 RR the transduction is seen to decrease to zero, as would be expected of a st ructure with no piezoelectric material. Furthermore, as 120 RR 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 regi ons of the diaphragm that are undergoing opposite polarities of stress, such that the net electr ic displacement (charge) is zero. PAGE 81 67 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 12 R R and pstt is shown in Figure 2-22. 0 0.2 0.4 0.6 0.8 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 R1/R2k tp=0.6 m tp=3.0 m Figure 2-22: Electromechanical coupling coefficient, k, as a function of 12 R R and pstt For these calculations, 3stm and 22 R mm while the piezoelectric layer has thicknesses of 0.6, 1.2, 1.8, 2.4, 3.0 ptm From the figure, the maximum k of 0.013 is obtained for an 12 R R ratio of 0.95 and a pstt ratio of unity. Physically, the increasing trend in k as 12 R R increases, arises from the stress distribution within the diaphragm. There is a stress concentration near the clamped boundary of the diaphragm that provi des for a high level of coupling. By PAGE 82 68 concentrating the piezoelectric material in th is high stress region, the averaged coupling factor over the ring will be hi gher 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 (Fi gure 2-16) is combined with that of the Helmholtz resonator (Figure 2-4). By usi ng the same lumped element connection rules previously described, the complete circuit can be obtained, as shown in Figure 2-23. MaDCaD :1 Ceb MaDrad RaDrad + P + V Q I Rload RaNMaN CaC Figure 2-23: Equivalent circ uit for acoustic energy harvester with resistive load. Notice that the equivalent circuit for the diaphragm is in parallel with the cavity compliance, aCC This occurs because the pressure in the cavity is responsible both for compressing the air in the cavity as well as de flecting the diaphragm, i.e. the cavity and the diaphragm both see the same pressure Additionally, it is assumed that loadp R R and therefore the parallel combination can be approximated as //loadpload R RR Acoustical Input Behavior An expression for the acoustical input impedance can be obtained from the equivalent circu it and is given by 2 211 1 11 1load aDaDradaDrad aCaDloadeb inaNaN load aDaDradaDrad aCaDloadebR sMMR sCsCRsC ZRsM R sMMR sCsCRsC (2.112) PAGE 83 69 From this equation, it can be seen that the total input impedance is simply the impedance of the Helmholtz resonator neck in series wi th a parallel combination of the Helmholtz resonator cavity impedance and the piezoelect ric composite diaphragm impedance with a resistive load attached. Many energy harvesting devices with vary ing geometries were designed for this dissertation, but for illustrati ve purposes, I will only expl ore 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 representativ e device was plotted for the remainder of this chapter. A plot of th e acoustical input impedance versus frequency is shown in Figure 2-24 for the piezoelectric co mposite diaphragm, both with and without the Helmholtz resonator, to elucidate the beha vior of the individual components. For this plot and those that follow, the Helm holtz resonator has a neck length of 3.18 L mm, and a radius of 2.36 n R mm along with a cavity volume of 31950 cavVmm in addition to a diaphragm with a thickness of 3 sitm and a outer and inner radius of 21.95 R mm and 21.85 R 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, th e piezoelectric composite diaphragm has a single resonance near 3.6 kHz where the impedance reaches a lo cal 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 resonato r that has an uncoupled resonance of 2 kHz as evidenced by the peak in Figure 2-5 The upper resonance at 3.9 kHz is PAGE 84 70 dominated by the piezoelectric composite di aphragm. Additionally, an antiresonance, where the impedance reaches a local maxima, occurs be tween the two resonances. Figure 2-24: Magnitude of the acoustical input impedance for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. (3.18 Lmm 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 11.85 R mm ) The acoustical input power can be obtain ed from the input acoustic pressure, P, and is given by 2Re Re{}in inP Z (2.113) The input acoustic pressure, P is measured inside the planewave tube near the end-face of the tube. The microphone that measures th is pressure is placed as close as possible ( ~116 ) to the end-face so that it se rves as a measure of the pressure that is incident on the energy harvesting device. In the case of the device that includes a Helmholtz PAGE 85 71 resonator, the incident pressure is the pre ssure 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 th e input power is shown in Figure 2-25. This plot assumes an ideal ac oustic 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. Th e 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 sour ce. 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 impeda nce can be found in a similar manner and is given by 21111 //////outloadaDaDradaDradaNaN ebaDaCZRsMMRRM sCsCsC .(2.114) Expanding this equation out yields the full e xpression for the output impedance, given by 2 21 111 1 1 1111 1aNaN aC loadaDaDradaDrad ebaD aNaN aC out aNaN aC loadloadaDaDradaDrad ebebaD aNaN aCRM sC RsMMR sCsC RM sC Z RM sC RRsMMR sCsCsC RM sC .(2.115) PAGE 86 72 Figure 2-25: Magnitude and phase of the ac oustical input power for an input acoustic pressure of 94 PdB. (3.18 L mm 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 11.85 R mm ) A graphical plot of the electrical output impedance is shown in Figure 2-26 in terms of real and imaginary components. Th e solid curve in the figure corresponds to the expression given by Eq. (2.115), while the dotte d 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 transducer s. The result is that acoustical components have minimal effect on the electrical impedance. PAGE 87 73 Figure 2-26: Electrical output impedance fo r the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. (3.18 L mm, 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 11.85 R mm ) The output voltage can easily be found from the equivalent circuit to be 21 1 1 1 1 1load eb load eb aNaN load aDaDradaDradaNaN aDloadeb aCR sC P R sC V RsMR sMMRRsM sCRC sC .(2.116) The output voltage is displayed graphical ly in Figure 2-27 for the piezoelectric composite diaphragm by itself and in combination with the Helmholtz resonator. For both cases, the magnitude has a similar shap e 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 PAGE 88 74 is higher than the second. The voltage curve shown was computed for a load resistance, load R equal to the electrical output resistance at the diaphragm resonance frequency, and an input acoustic pressure of 1 Pa. Figure 2-27: Magnitude and phase of the outpu t voltage for the piezoelectric composite diaphragm and in combination with the Helmholtz resonator. (3.18 Lmm 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 11.85 R mm 94 PdB1) The electrical power delive red to the resistive load, load R can be found from 2Re{}out loadV R (2.117) A plot of the electrical out put power delivered to the resistive load is shown in Figure 2-28 for the piezoelectric composite diaphrag m by itself and in combination with the Helmholtz resonator, for the case of an incident acoustic pressure of 1 Pa 1 re 20dBPa throughout this document. PAGE 89 75 Figure 2-28: Magnitude and phase of the el ectric output power delivered to the load resistor. (3.18 Lmm 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 21.85 R mm 94 PdB ) The overall power conversion efficiency, can then be found as the ratio of output electrical power to i nput acoustical power, given by Re{} Re{} Re{}out in (2.118) The magnitude and phase of the efficiency is shown in Figure 2-29 for the piezoelectric diaphragm by itself and in combin ation 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 di aphragm 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 PAGE 90 76 Helmholtz resonator was not optimally designe d 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 loss es due to the Helmholtz resonator. The benefits can thus be enhanced by mainta ining the improved impedance matching while minimizing the additional losses. A perfect impedance match between the plane wave tube and the Helmholtz resonator would im prove 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 plane-wave tube. Figure 2-29: Magnitude and phase of the energy harvester efficiency. (3.18 L mm, 2.36 n R mm 31950 cavVmm 3 sitm 21.95 R mm 21.85 R mm 94 PdB ) Operation at an Optimal Frequency To get a better feel for these devices in th eir 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 geom etry of Device 8, as given by Table 2-5 on PAGE 91 77 page 81 was used and the frequency at which the efficiency is maximized was chosen. This optimal frequency is different betw een the diaphragm mounted on the Helmholtz resonator and the diaphragm strictly by its elf. 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 pressu re for both cases is shown in Figure 2-30. 115 120 125 130 135 140 145 150 155 160 10-2 10-1 100 Input Acoustic Pressure [dB]Vout [V] Membrane Membrane/HR Figure 2-30: Open-circuit volta ge as a function of the appl ied acoustic pressure for the piezoelectric diaphragm by itself and packaged with a Helmholtz resonator. (Device = 8 from Table 2-5, H R f =1966 Hz mem 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 PAGE 92 78 nonlinearities may include, but are not limited to, piezoelectric satura tion effects on the output voltage and large deflec tion 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 maxi mum 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 s hown in Figure 2-31. Note that while the input power remains constant, the output power has a p eak at a particular load resistance. This corresponds to the optimal resi stance 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. 102 103 104 10-10 10-8 10-6 10-4 Load Resistance [Ohms] [W] in Membrane in Membrane/HR out Membrane out Membrane/HR Figure 2-31: 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 2-5, H R f =1966 Hz mem f =13.11 kHz, and 114 PdB ) PAGE 93 79 Using the optimal values for the load resi stance in each case, the input power and output power was determined as a function of the applied acoustic i nput pressure, and is shown in Figure 2-32. Once again, it is impor tant to note that nonlinear effects are not included in the model used to calculate the output power. 115 120 125 130 135 140 145 150 155 160 10-10 10-8 10-6 10-4 10-2 100 102 Acoustic Input Pressure [dB]Power [W] in Membrane in Membrane/HR out Membrane out Membrane/HR Figure 2-32: Input and output power as a function of the a pplied acoustic input pressure, while using an optimal load resist or. (Device = 8 from Table 2-5, H R f =1966 Hz mem f =13.11 kHz ,486 loadMemR ,/3242 loadMemHRR ) Device Configurations The devices were designed based on the theo retical plots, along with estimates of the output power based on the lumped element m odel. Eight configurations were chosen based on their resonant frequency and maxi mum power before the onset of nonlinear behavior. The criteria for choosing the desi gns were to keep the resonant frequencies PAGE 94 80 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 counter part. 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 th e addition of the central mass invalidates the structural model that was us ed, however, the model is still expected to provide some guidance over the behavior of th ese particular devices as well. The central mass was added in order to reduce the resona nt 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 fre quency. 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 summ arized in Table 2-5. Overall, the geometries were chosen as to create a proof of concept set of devices. Manual parameter adjustment was performed in order to impr ove performance, however, strong constraints were placed on the geometry in order to ensure fabrication ability and improve the chances of first-run success. Also listed in the table are the predicted resonant frequencies and maximum power before the ons et of nonlinearities. Note that this does not define an upper limit for the output power but merely provides a figure of merit upon PAGE 95 81 which to compare devices. Also, for the pur poses 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 2-6. Table 2-5: Chosen device conf igurations for fabrication. Device sitm ptm 2Rm 1Rm massRm res f kHz outPownW 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 2-6: Lumped element paramete rs used for theoretical models. k Ad aDC aD M aDmassM ebC Device Pa V V Pa 3m V 3m Pa 4kg m 4kg m nF1 -1.81 -5.80E-07 0.001 1. 01E-14 5.58E-15 5663.3 0 17.4 2 -1.81 -5.80E-07 0.001 1.01E -14 5.58E-15 5663.3 19649 17.4 3 -9.29 -7.16E-06 0.008 1. 93E-13 2.07E-14 2602.5 0 26.9 4 -9.29 -7.16E-06 0.008 1.93E -13 2.07E-14 2602.5 11052 26.9 5 -12.93 -1.94E-05 0.016 7. 43E-13 5.74E-14 1521.8 0 38.2 6 -12.93 -1.94E-05 0.016 7.43E -13 5.74E-14 1521.8 7073.6 38.2 7 -16.92 -4.71E-05 0.028 2. 42E-12 1.43E-13 1009.8 0 51.3 8 -16.92 -4.71E-05 0.028 2.42E -12 1.43E-13 1009.8 4912.2 51.3 PAGE 96 82 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 discus sion of the process flow is presented, including fabrication me thods, 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 or der to expose the diaphr agm directly to the acoustic input, thereby enabling direct measurement of the diaphragm parameters. Two variations in package design were employed, namely a quarter-wave resonator package and a sealed cavity package. A graphical overview of this chapter is provided by the roadmap of Figure 3-1. Process Flow The devices were batch fabricated on 4 silicon-on-insulator (SOI) wafers. The process sequence is given in detail in Appendix B. A ll of the processing steps up through the deposition of the top electr ode 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. PAGE 97 83 Overview Process Flow Packaging Packaging Scheme Chip to Package Mounting Sealed Cavity Package Quarter-Wave Resonator Package Poling Figure 3-1: Roadma p 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 contai ned a top silicon layer of 12 um thickness. In order to reduce this thickness, a timed KOH (Potassi um 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 iden tifiable, however a 1 minute dip in 10% Nitric acid, followed by a 2:1 Piranha etch fo r 5 minutes had no noticeable effect. PAGE 98 84 Figure 3-2: Condensed process sequence in cr oss-section. a) Depos it 100 nm of Ti on SOI Wafer and oxidize to form TiO 2 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) De posit Pt (180 nm) and liftoff with ElectrodeTop mask. e) Wet Etch PZT in 3:1:1 of (NH 4 )HF 2 /HCl/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 KOH-etched wafers we re replaced with two new wafers that were etched in an HNA bath to avoid the re sidue issue [138]. The advantage of the HNA etch was a smooth, mirror finished surf ace to the wafer with no visible residue. The disadvantage was a less consistent et ch rate than the KOH procedure. For comparison the HNA etch had an average etch rate of 2.6 minm with a standard deviation of 0.33 minm while the KOH etch had an etch rate of 1.69 minm with a standard deviation of 0.03 minm a) b) c) d) e) f) g) h) PAGE 99 85 Following the wafer thinning, the process fl ow proceeded as given by Figure 3-2. The first step was to deposit a 100 nm thick layer of Ti, and oxi dize in a tube furnace to create a TiO2 layer that serves as a diffusion barri er for the PZT [139-142] as shown in Figure 3-2a. 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 3-2b. The Ti layer served as an adhesion layer in this step. The next step was the deposition of a previous ly mixed 52/48 sol-gel solution of PZT [139, 143, 144] as shown in Figure 3-2c. The solution was spin-cast at 2500 RPM for 30 sec, then pyrolized at 350C 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 650C 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 [145-147]. Table 3-1 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 th e wafer curvature, a relationship known as Stoneys equation [145-147] was used and is given by PAGE 100 86 3 2611ss f s sf fEh h Rh h (3.1) where 1 R is the curvature, s fhh is the substrate and film thickness respectively, s E is Youngs modulus of the substrate and s is Poissons ratio of the substrate. Table 3-1: Wafer bow m easurements and the resulting calculated stress. Initial After TiO2 Dep. + Ox. TiO2 + SiO2 TiO2 After PZT PZT Wafer # Radius m Bow m Radius m Bow m Stress M Pa Stress M Pa Stress M Pa Radius m Bow m Stress M Pa 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 us ed 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 be tween oxide thickness and oxidation time is approximated by [148] 2 oxoxtAtBt (3.2) where A and B are coefficients which depe nd on material properties and operating conditions, oxt is the total oxide thickness, t is the oxidation time, and is a time shift due to an initial oxide thickness For short oxidation times, where 24 tAB, this equation reduces to oxB tt A (3.3) PAGE 101 87 For <100> silicon oxidized in dry O2 at 650 C, 0.0001 B Amhr, and 20.0001 B mhr Under these conditions, the a pproximation of short oxidation times holds, as the total oxidation time wa s 30 min, yielding an oxide thickness of approximately 50 nm. This thickness was then utilized in Stoneys equation to help determine the contribution due to each of the layers. In addition to wafer bow measurements, the density of the PZT was calculated by measuring the mass of the wafers both before and after deposition. The results of these measurements are shown in Table 3-2. Table 3-2: Measured wafer mass and calculated density of PZT. Mass [g] PZT Density Wafer Pre-PZTPost-PZT 3kgm 1A 9.752 9.7734 10217 2 9.48169.5129 9963 4 9.50499.5369 10185 5A 9.74459.7616 8164 6 9.556 9.5771 10074 7 9.55659.5764 9501 Avg. 9684 Following the PZT deposition, the top electr odes were deposited and patterned in a liftoff process similar to the bottom electr odes as shown in Figure 3-2d. The top electrodes also served as a hard etch mask for etching the PZT. The PZT was etched using a 3:1:1 solution of ammonium biflouride ((NH 4 )HF 2 ), hydrochloric acid (HCl), and deionized water as shown in Figure 3-2e. The etchant leaves a residue that then must be removed with a dilute nitric acid and hydrogen peroxide solution. The next step was to deposit a thick phot oresist on the backside, and pattern the backside release etch as shown in Figure 3-2f. Then the sili con wafer is backside etched using a deep reactive ion etch (DRIE) proces s, also known as a Bosch etch [148-152] as PAGE 102 88 shown in Figure 3-2g. This is a plasma-bas ed etch process that alternates between a plasma etch, and a polymer deposition. The polymer is easily removed from the bottom of the etch holes during the ensuing etch cycl e, but serves to significantly slow down any lateral etching, leading to highaspect ratio etching. The ratio of etch time to passivation time during each cycle, along with a number of other process parame ters, greatly affects the resulting etch quality. Shown in Figure 3-3 is a scanning electron micrograph (SEM) image of the sidewall of one of the test devi ces after an early stage, through-wafer, DRIE test, prior to working out the optimal parameters In this figure, signi ficant erosion of the sidewall can be seen, in the form of both pitting and undercutting. For comparison, the upper right region of the image shows what a smooth silicon sidewall looks like. The damage shown here is a result of an etch -to-passivation ratio that is too high. The passivation layer is then insufficient to protec t the sidewalls during the entire etch cycle. Figure 3-3: Early DRIE results s howing significant sidewall damage. PAGE 103 89 A different problem occurs when the etch-topassivation ratio is too low. In this instance, some portions of the passivation la yer are not removed from the bottom of the etch holes during the etch cycle, leading to a micromasking effect and the ultimate formation of what is known as silicon grass or black silicon [153]. This was also encountered during the early tria l stages of our DRIE attempts. Shown in Figure 3-4 is an SEM image showing the formation of black silicon at the base of a trench. Figure 3-4: SEM image showi ng black silicon at the base of a DRIE-etched trench. Finally, after adjustment of the various process parameters a successful DRIE was performed through the thickness of the silicon wafer. This is shown in Figure 3-5. Notice, a nearly vertical sidewall was achieved with little cratering and erosion, leaving a smooth surface with the exception of the large debris particle. PAGE 104 90 Figure 3-5: SEM image of a successful DRIE etch through the thickness of a wafer. Once the DRIE recipe was worked out, the DRIE process was used here to etch through the bulk of the silicon wafer, stopping on the burie d oxide layer, leaving a thin diaphragm as well as creating channels that served to later separate the wafer. Additionally, the central pillar that serves as the point mass was created in this process by masking the central portion of the diaphr agm backside. Following the DRIE, the photoresist was stripped from the backside. A composite top view schematic and crosssection of the device are shown in Figure 3-6. The devices were arranged on the wafer in blocks as shown in Figure 3-7. Each block c onsists of the eight device configurations. The blocks were then arranged on the wafer as shown in Figure 3-8. The blocks were arranged across the wafer such that the wafer would maintain structural stability after the DRIE release of the die with in the blocks. A photograph of a completed energy harvester chip taken under a microscope with a 1.25x lens is shown in Figure 3-6c. PAGE 105 91 c) Figure 3-6: Illustration of a single diaphragm device. a)Composite top view schematic of single diaphragm device showing ring shaped electrodes, with bottom electrode extending to lower left and top electrode extending to lower right. Also shown is the die separation cha nnel around the perimeter. b) Completed cross-section showing the Pt/PZT/Pt sa ndwich. c) Optical photograph of a completed device, taken under a micros cope with a 1.25x magnification. Figure 3-7: Block layout illu strating numbering scheme fo r devices, per Table 2-5. a) b) Piezoelectric Ring Diaphragm Bottom Electrode Top Electrode PAGE 106 92 Figure 3-8: Wafer layout illustrati ng numbering scheme for blocks. Packaging Following the processing steps listed above each die was separated from the wafer by scribing across the small tab holding each die and cleaving at the scribe line. Then each die was mounted in a Lucite package, described next, and bonded to a pair of electrical leads. The devices to be tested inside of an acoustic plane wave tube were packaged accordingly so that a tight seal was created with the plane wave tube while allowing external electri cal access to the devices. Packaging Scheme Quarter-wave resonator package In order to provide a convenient pack age for measurement of the diaphragm parameters, an acrylic plug for the acoustic waveguide was designed and constructed. PAGE 107 93 The plug allows for flush mounting of the de vice on the topside, and acoustic venting on the backside to prevent a cavity stiffening e ffect. The cavity stiffening effect describes an overall lowering of sensitivity due to the low compliance (i.e. high stiffness) of a small cavity (Eq. (2.21)). This however introduced a quarter-wave resonance effect that will be demonstrated in the experimental results. Additionally, two, 18 gauge, copper wires are embedded in the plug to allow for electri cal connection to th e electrodes of the piezoelectric ring. A schematic of the package is shown in Figure 3-9. Figure 3-9: Quarter-wave resonator package consisting of acrylic plug, copper leads and vent channel. The quarter-wave resonance effect arises because of the impedance mismatch that occurs at the interface between the backside vent channel and the environment on the exterior of the package. An acoustic wave th at is traveling down the channel is reflected upon encountering the open end of the channel. This sets up a standing wave within the PAGE 108 94 channel. Both the pressure, P, and velocity, U, exhibit this spatial pattern, and the ratio of PU, which is the impedance, Z shows a spatial dependency as well. Mathematically, the impedance at a distance x along the length of the channel can be written as ()tano Z xjZkx (3.4) where o Z is the specific acoustic impedance of the medium. When the distance, x, is equal to of the wavelength, this equation reduces to 2 ()tantan0 444oofc ZjZkjZ cf (3.5) Thus, the impedance goes to zero at fr equency-position combinations where 4 x This zero impedance allows for large velocity responses for small pressure inputs, i. e. a resonance. Similarly, ther e are frequency-position combinations where the impedance becomes infinite. In practice, various losses serve to limit the impedance from becoming exactly zero or infinity. Sealed cavity package This package was designed so that the parameters of the piezoelectric composite diaphragm could be measured independently of effects due to the quarter-wave resonator. The design is identical to the quarter-wave res onator except the vent channel is sealed. A cross-sectional schematic of the sealed cav ity package is given in Figure 3-10 and a corresponding optical photograph of the devi ce and package is shown in Figure 3-11. Chip to Package Mounting The chip was flush mounted within the provided recess in the package, and epoxy bonded to the package to provide an airtight seal. The el ectrical connections to the PAGE 109 95 package leads were accomplished via short le ngths of flattened bare wire that were bonded using two-part, silver epoxy (Epotek H20E) on the chip bond pads as well as to the copper pads on the package. The silver epoxy was then cured at 90 C for one and a half hours. A relatively low temperature, long curing time recipe was chosen so as to remain below the glass transition temperat ure of the Lucite, which occurs at 100 C. Figure 3-10: Sealed cavity package. Figure 3-11: Optical photograph of a packaged energy harvester. PAGE 110 96 The silver epoxy method, as outlined above, was used because of problems encountered when trying to wire bond from the package to the die. The problems were believed to have arisen from overhanging top electrode metal on the individual die. The liftoff step that produced the top electrode was found to have left stringers around the edges of the electrode. These stringers we re the result of performing a liftoff using positive resist, as the positive sidewall slope of the resist does not allow for a clean break in the metal deposition, thereby leading to tear ing during the liftoff portion of this step. The overhanging metal was found to cause s hort-circuits on all of the devices of several wafers. On two of the wafers, no short-circuits were evident initially; however, after wire bonding, all devices would then exhibit a short-circ uit. It was concluded that the vibrations and impact of the wire bonding tool were causing the overhanging top electrode to fall into contact w ith the bottom electrode directly beneath it. This problem was avoided by utilizing a more gentle appr oach to bonding, namely silver epoxy. The entire issue, however, could be avoided in th e future by using a negative resist and an image reversal mask to create a negative slopi ng photoresist sidewall fo r this liftoff step. Poling Following the completion of the packaging steps, the devices were poled so as to align the domains along the poling axis. It should be noted that the devices were found to exhibit a small amount of pi ezoelectric behavior prior to poli ng. This is believed to be due to a partially pre-aligned structure resulting from the deposition process. Poling was accomplished by applying an electric field across the device for a specified amount of time. The devices were poled at room temperature at 20 Vm that corresponds to an appl ied voltage of 5.34 V for the given device thickness. This poling PAGE 111 97 field was held for 10 min A typical maximum polarization of 16.9 2Ccm was achieved with a remanent polarization of 6.26 2Ccm. More details of this measurement are provided in Chapter 5. Typically, poling is accomplished at higher temperatures for improved piezoelectric properties [8, 145], although Bernst ein et al [28] reported acceptable poling results at room temperature. While it wa s desirable to try a hi gher temperature poling procedure, temperature constraints caused by the glass transition temperature of the acrylic package prohibited the use of temperatures above 100 C. Prior the successful poling of the devices, several othe r variations of the poling method were tested. For instance, initially, 10 Vm was utilized as a poling field, however piezoelectric properties were found to be relatively unchanged from the unpoled state. Additionally, higher elect ric fields were tested, above 20 Vm however this consistently led to the development of a s hort-circuit in the device. The short was believed to be a result of the many stringer s that overhung from the top electrode layer down to the bottom electrode. At sufficientl y high voltages, the force of attraction between these stringers and the bottom elect rode is hypothesized to pull them into contact. PAGE 112 98 CHAPTER 4 EXPERIMENTAL SETUP This chapter discusses the various setups for the experimental work. A series of measurements were performed to experimenta lly determine the various parameters of the piezoelectric composite diaphragm. This allo ws for further analysis and verification of the models. The measurements were performed using four different experimental setups. First, the electrical output impedance of each device was measured using an impedance meter. Then an electrically actuated re sponse was measured using a scanning laser vibrometer (LV), to determine a voltage driven frequency res ponse and corresponding mode shapes. Following this, an acoustically actuated response was measured using an acoustic plane wave tube. This was used to determine a pressure driven frequency response, input and output power, output voltage and efficiency. Finally, the static mode shapes were measured using an optical prof ilometer to determine the initial deflection due to residual stresses. A graphical overvie w of this chapter is provided by the roadmap of Figure 4-1. Parameters and How They are Obtained A structured methodology was develope d for obtaining all of the desired measurement parameters for the composite plat e. Using a predefined set of experiments and a few relations between various paramete rs, a method was derived for obtaining each of the parameters. PAGE 113 99 Figure 4-1: Roadma p for Chapter 4. The series electrical output capacitance, s C and resistance, s R can be experimentally obtained through use of an impe dance meter. This measurement needs to be performed after poling the de vice, as the poling procedure can affect the capacitance. The measured value can then be compared to theory, given by sCAd if totally free, (Eq. (2.51)) and 21sCAdk if totally blocked, (Eq. (2.61)). Now in the case of the energy harvesters, 1 k so that the free and blocked cases are approximately equal. For the series resistance, the theoretical value can be obtained from a predicted loss tangent for the material. The resonant frequency of the devices can be obtained through use of an acoustic plane-wave tube (PWT) that is used to excite the diaphragm acoustically, while the output voltage from the device is measured The measured frequency can then be compared to a theoretical value that is based on the lumped element model for aDC and aD M PAGE 114 100 There are four readily definable sensitivitie s of interest for the energy harvester. Two of these sensitivities are defined at reso nance, while the other two represent static sensitivities. EA resSens is the electrical-to-acoustical se nsitivity at resonance and can be experimentally determined by applying a volta ge signal to the device at the resonant frequency while the diaphragm deflection is measured via a laser Doppler velocimeter (LDV). AE resSens is the acoustical-to-electrical sens itivity at resonance and can be found experimentally by acoustically exciting the diaphragm at resonance via the PWT while measuring the voltage produced by the device. Both of these resonant sensitivities can then be compared to theory which is comput ed directly from the lumped element models forrad R aDC ,aD M s R and The static electrical to acoustical sensitivity, EA lowSens is approximated using the LDV at low frequencies. The LDV requires a finite velocity, and so a low frequency approximation is necessary for estimation of th e static value. Similarly, the static acoustical to electrical sensitivity, AE lowSens is measured in the PWT using an acoustic input signal at low frequencie s. The same limitation to th e static case occurs in this measurement as well. Both of these values can then be compared to theoretical values computed directly from the lumped element model using the equations for aDC ,ef R ,efC and The initial deflection of the diaphragm th at results from residual stress in the composite can be measured using a Wyko opti cal profilometer without any excitation to the device. Theoretical values are obtained by numerical calculations accounting for the residual stress. PAGE 115 101 The coupling factor, k, is measured in two different ways. First it may be obtained from the difference between the open and short ci rcuit resonant frequencies, or it can be found via k Theoretical values are obtained from and or from from Ad aDC and efC Finally, the electroacoustic, and acoustoelectric, transduction factors are extracted from a combination of EA lowSens and aDC and a combination of AE lowSens and efC respectively. The theoretical values can be found directly from the numerical model. Experimental Setup Details Ferroelectric Measurements The ferroelectric properties of the devi ces were measured by attaching the two electrodes to a Precision LC, Precision Ma terials Analyzer manufactured by Radiant Technologies. This ferroelectric tester appl ies a bipolar triangular waveform, while it measures the resulting charge, from which it can calculate the polarization of the material as a function of applied voltage. The bipol ar waveform excitation to the ferroelectric material leads to the familiar hysteresis loop. For these measurements, the voltage was swept first between -0.5 V and 0.5 V Subsequent loops swept to higher voltages in steps of 0.25 V up to a maximum of 5 V. For each loop, 101 points were measured and the total loop time was set to 10 ms In addition, for each loop, a pre-loop was performed, where the same sweep was performed without measurements taken. Then the loop was repeated with the actual measurements taken. This process is de signed to preset the conditions for the measurement. A delay of 1 ms was used between the preloop and the measurement loop. PAGE 116 102 Electrically Actuated Response Laser Scanning Vibrometer In order to determine the voltage actua ted mode shapes and frequency response, laser scanning vibrometry was performed. The packaged energy harvesters were mounted under an Olympus BX60 Microscope, [154] with a 5x objective lens. The composite diaphragms were excited using either an HP 33120A Arbitrary Waveform Generator [155] or a Stanford Research Systems SRS-785 Dynamic Signal Analyzer. Displacement data were obtained over the su rface of the composite diaphragms using a Polytec OFV 3001S laser scanning vibromet er, with a Polytec OFV-074 Microscope Adapter. [156] The laser vibrometer was e quipped with a velocity decoder, from which the displacement was calculated. To determine the frequency response, a wh ite noise input was applied. The center deflection was then measured as a function of frequency, while the reference voltage was simultaneously recorded. From this meas urement, the resonant frequencies were determined. A Hanning window was applied to all incoming data and 6400 FFT lines, with 100 averages were taken. Depending on the particular device, the measurement bandwidth was adjusted as was the laser sens itivity and range in order to optimize the signal level and resulting data quality. A slow tracking filter was applied to all incoming laser measurements. Additionally, due to an inherent time delay in the laser vibrometry channel, there is a fixed linear phase trend in any measured frequency response data. All resulting plots were detrended by taking this time delay into account. The amount of the delay varies, depending on the gain setting, how ever for a given setting it is fixed and defined by the manufacturer. The input was then changed to a sine wave at the primary resonant frequency and the laser was scanned over the surface to m easure the primary mode shape. Higher order PAGE 117 103 modes were also measured, although primarily to identify the nature of the higher resonances, and will not be presented in much detail. Acoustic Characterization Plane Wave Tube The packaged energy harvesters were mounted one at a time, to the end of the 1 x 1 cross section plane wave tube (PWT). Figure 4-2 shows this experimental setup for obtaining acoustical input impedance and power to the energy harvester. A Bruel and Kjr (B&K) microphone type 4138 [157] was mounted on the wall of the PWT near the end of the tube to record the incident sound pressure level (SPL). Additionally, two B&K microphones were mounted in a rotating plug at 3.19 cm and 5.26 cm from the end of the tube. These two microphones were used to record the acoustic field in the tube, extract the standing wave component, and de termine the correspondi ng impedance of the energy harvester. The data was recorded by a B&K PULSE Acoustic Characterization System that also supplied the periodic random source signal through a Techron 7540 Power Supply Amplifier [158] to the BMS 50.8 mm coaxial compression driver (Model 4590P) mounted at the far end of the PWT. The tests were then repeated with a 1 k load shunted across the electrode s of the piezoelectric compos ite diaphragm. For all of the tests, the equipment was setup to measure 6400 FFT lines from 0 kHz to 12.8 kHz with uniform windowing, no overlap, and 500 averages. To determine the normal incidence acoustic impedance, frequency response measurements (later defined as 12H ) were taken using the two microphones in the rotating plug. [159, 160] The plug was then rotated and the measurement repeated. The two measurements were then averaged to re move any differences due to the individual PAGE 118 104 microphone calibrations. This averaged freque ncy response was then used with the twomicrophone method (TMM) to determine the acoustic impedance [159, 161-163]. Figure 4-2: Experimental setup for impedance and power measurements. The TMM is a method for estimating the re flection coefficient of an object or material mounted at the end of an acoustic wave guide. The TMM requires less than a half-wavelength spacing between the two microphone locations [159-161, 163, 164]. Although the waveguide used is physically capable of suppor ting plane waves up to 6.7 kHz testing was limited to 6.4 kHz because of this microphone spacing constraint. An alternative, single-microphone method [157] exists but was not used due to greater errors associated with the technique. Additionally the uncertainties were estimated for the acoustic impedance measurements using a t echnique presented by Schultz et al [165]. Under the assumption of planar waves in the waveguide, the sound field inside the waveguide can be described by ReRjtkdjtkd ippee (3.6) PAGE 119 105 where the j tkde term represents the right-propagating waves, the j tkde term represents the left-propagating waves, and d is the axial distance from the specimen surface. From Eq. (3.6), an expression for the reflection co efficient is found by taking the ratio of two pressure measurements at different axial locations, 12 12 21 12jkdjkd jkdjkdHee R eHe (3.7) where 1d and 2d are the distances from the specimen test surface to the measurement locations, is the radian frequency, k is the wavenumber, R is the complex reflection coefficient, and 12H is an estimate of the frequenc y response function between the two microphones. The frequency response function is defined as [166] 12 12 1212 11jG HHe G (3.8) where 12G is the cross-spectral density function and 11G is the autospectral density function. After calculation of the reflection coefficient from the measured data, the normalized acoustic impedance is then found to be a function of the reflection coefficient and is given by 1 01 1 ZR j ZR (3.9) where 0 Z and 1 Z are the specific acoustic impedance of the acoustic medium and the test specimen, respectively and and are the normalized resistance and reactance, respectively. Using the same test setup as desc ribed above, the input acoustic power, input can also be found. A schematic showing the incide nt and input power is given in Figure 4-3. PAGE 120 106 Figure 4-3: Schematic of incide nt, reflected and input power. There are two main methods for obtaining the input acoustic power. First it can be calculated from the power reflection coefficient, 2rR and the incident acoustic power, 2tube p Z as 2 211input tubep rR Z (3.10) where tube Z is the acoustic impedance of the plane-wave tube, and p is the acoustic pressure of the right-propagating (incident) waves. The second method for obtaining the input acoustic power is to u tilize the input impedance, in Z that was also calculated from the two-microphone measurements via R Using this method yields 2tr input inP Z (3.11) where trPPP is the transmitted acoustic pressu re. Under ideal conditions, both methods yield equivalent answers; however, when R approaches 1, the value for R is known with significantly greater certainty than the value for in Z In addition to measuring the input impe dance and power, the frequency response function of the devices can be measured. Th e resonant frequency of the diaphragm can PAGE 121 107 be found from the frequency response functi on, as can the maximum output voltage and power. As the input power is also known, th e efficiency can be calculated. Additionally, a load can be placed across the electrical term inals of the energy harvester and the effects monitored. The resistance of the load can be varied to experimentally determine the optimal loads for maximum power and maximum efficiency. Electrical Impedance Characterization The electrical impedance was measured using a HP 4294A Impedance Analyzer [155] after first poling the de vices. Measurements of the parallel capacitance and resistance were obtained at 1 kHz for each of the packaged devices. Additionally, the impedance was measured over a sufficient sp ectral bandwidth to capture the first resonance that varied from device to device. Initial Deflection Measurements Wyko Optica l Profilometer Due to residual stresses in the composite diaphragms and compliant boundary conditions, the devices exhibited an initial static deflection. This deflection was measured by mounting the devices under a Wyko Optical Profilometer [131] with a 5x optical lens and a 0.5x reducer, yieldi ng a total magnification of 2.5x. PAGE 122 108 CHAPTER 5 EXPERIMENTAL RESULTS AND DISCUSSION This chapter discusses the experimental resu lts that were obtained from the various measurements. First, the electrical impedan ce results are shown and explained. This is followed by the electrically actuated freque ncy response measurements. Following this, the acoustically actuated frequency response and impedance results are discussed for both package configurations. Finally, the initial deflection measurements are presented for several devices. A graphical overview of th is chapter is provided by the roadmap of Figure 5-1. Figure 5-1: Roadma p for Chapter 5. Ferroelectric Measurements The ferroelectric properties of several de vices were measured using the Radiant Technologies, Precision LC Analyzer. Show n in Figure 5-2, is a graph of the polarization versus applied voltage over severa l loops of steadily in creasing voltage. For PAGE 123 109 this measurement, the voltage maxima were swept from 1 V to 5 V in 0.25 V increments, corresponding to a maximum applied electric field of 187.95 kVcm.. -200 -150 -100 -50 0 50 100 150 200 -20 -15 -10 -5 0 5 10 15 20 Electric Field [kV/cm]Polarization [ C/cm2] Pr=6.26 Pm=16.9 Figure 5-2: Hysteresis loop for Device 1A-5-4 in a sealed cavity pack age. The device has a radius of 1200 m and a central mass. The resulting polarization reached a maximum of 16.9 2Ccm with a remanent polarization of 6.26 2Ccm for the 5 V sweep. A typical hysteresis curve shows flattened regions near the upper and lower ends of the sweep, where the response becomes saturated. It was not possible to ex tend the measured hysteresis curve into those regions as the maximum voltage that c ould place across the piezoelectric was approximately 5 V Beyond this voltage, unexpected eff ects occurred in the material that produced a short circuit between the two electrodes. The nature of this short circuit was PAGE 124 110 not resolutely identified, but it is believed to be related to the ove rhanging top electrode resulting from problems in the fabrication. Electrical Measurements Electrical Output Impedance Electrical measurements were taken using a HP 4294A Impedance Analyzer. Data was generally recorded from 1 kHz to 50 kHz however, the results shown below were extracted from the 1 kHz values. The parallel capacitan ce and parallel resistance were measured in this manner, and the correspondi ng relative dielectric constants and loss tangents were extracted, by taking into account the respective geometries of each device. Shown in Figure 5-3 is the measured parall el capacitance as a function of the outer radius, 2 R All eight device configurations, as given in Table 2-5, were measured, resulting in two data points for each outer radius. The capacitance was found to vary between 8.65 nF and 27.4 nF and increases with increasing radius as expected. The individual corresponding relative diel ectric constants were calculated using, 22 21pp r otC RR (5.1) where pt is the piezoelectric thickness, 2 R and 1 R are the outer and inner radii respectively, and o is the permittivity in a vacuum. The relative dielectric constant was found to have an average value of 483.47 with a standard deviation of 23.48. Others have reported dielectric constants for PZT from 125 [16] to 1400 [32], although the average value in the literature is approximately 839. PAGE 125 111 Figure 5-3: Measured parallel output capacitance, pC vs. outer radius, 2 R overlaid with a theoretical capacitance curve based on the average extracted dielectric constant. The error bars indicate th e 95% confidence intervals based on the variance across multiple devices. In a manner similar to the capacitance measurements, the parallel output resistance was measured at 1 kHz for each of the devices and is shown in Figure 5-4. As can be seen from the figure, it was found to decrease with increasing radius. This effect was investigated further by looking at the conductance per unit ar ea of each of the devices, given by 2222 21211pG R RRRR (5.2) where G is the conductance. It was found th at the conductance per unit area was relatively constant with respect to radius, with the exception of one outlier point measured for the device with an 1800 m outer radius. The average value was found to PAGE 126 112 be 2.40 2Siemensm, with a standard deviation of 0.46 2Siemensm, when excluding the outlier point. Figure 5-4: Parallel output resistance, p R vs. outer radius, 2 R overlaid with a theoretical resistance curve based on the average measured conductivity. The error bars indicate the 95% confidence intervals of the variance across multiple devices. The loss tangent was then calculated base d on the measured electrical reactance, p X and resistance, p R and is given by, tanp p X R (5.3) The average loss tangent, as measured at 1 kHz and once again ignoring the outlier, was found to be approximately 0.024, which matche s well with previously published values ranging from 0.02 to 0.05 for thin film PZT [12, 26, 34, 35, 37]. A summary of all the electrical impedance measurements is provide d by Table 5-1. The average values are given at the bottom of the table and were co mputed without the outlier points that are marked in the dotted box. PAGE 127 113 Table 5-1: Summary of electrical impedance measurements. Device # 1Rm 2Rm massRm pCnF pRk r tan 2Gm 1A-5-1 830 900 0 8.72 665.89 507.91 0.027 2.867 1A-6-1 830 900 0 8.65 837.09 503.77 0.022 2.281 1A-6-2 830 900 180 8.45 1120.20 492.42 0.017 1.704 1A-10-2 830 900 180 7.76 900.73 451.84 0.023 2.120 1A-11-3 1115 1200 0 12.68 652.73 478.21 0.019 1.895 1A-5-4 1115 1200 240 12.14 730.94 458.04 0.018 1.692 1A-10-4 1115 1200 240 12.59 414.63 474.99 0.030 2.983 1A-6-5 1400 1500 0 18.33 348.04 486.75 0.025 2.501 1A-6-6 1400 1500 300 17.64 306.31 468.60 0.029 2.842 1A-10-6 1400 1500 300 18.32 367.91 486.67 0.024 2.366 1A-3-7 1685 1800 0 26.70 52.60 527.59 0.113 12.314 1A-6-7 1685 1800 0 22.82 237.78 450.94 0.029 2.724 1A-3-8 1685 1800 360 27.40 233.55 541.54 0.025 2.773 Average n/a n/a n/a n/a n/a 483.47 0.024 2.40 The devices are numbered according to the device type and location on the wafer, in the form of WW-BB-DD, where WW is th e wafer number, BB is the block number as defined in Figure 3-8 and DD is the device number as defined in Figure 3-7. Note that all the devices that were tested came from the sa me wafer, referred to here as wafer 1A. Shown in Table 5-2 is a comparison of the properties of the PZT presented in this dissertation along with the proper ties of thin-film PZT as repor ted in the literature. Upon comparison, the relative permittivity and loss tangent are within range of previously reported values, however the remanent polarizati on is lower. This is most likely a direct result of the room temperature poling proces s, and future work will seek to understand the precise nature of this result. Table 5-2: Comparison of ferroelectric and dielectric properties of thin film PZT. Author rP 2Ccm r tan Tuttle and Schwartz [167] 27 1000 -Morita et al. [169] 9.2 --Nunes et al. [160] 20 436 0.07 Xia et al. [135] 25 860 0.03 Kueppers et al. [161] 22 480 -Horowitz 6.3 500 0.03 PAGE 128 114 Electrically Actuated Response Laser Scanning Vibrometer Frequency response The results of the electri cally actuated frequency res ponse in terms of magnitude and phase are shown in Figure 5-5 to Figure 512 for a number of devices. Along with each frequency response plot is the corres ponding uncertainty for each measurement, based on the measured coherence. A minimu m of one resonance was detected for each device. Some of the larger devices also show a second resonance within the measured frequency range. Note that the phase shifts by 180 at the resonance, close to the observed peak in the magnitude response. At low frequencies, well below resonance, the devices exhibit a relativel y flat response, although more significant scatter is evident in the data resulting from poor coherence due to signal levels close to the noise floor. Also note that the frequency range varies from graph to graph to focus in on the features of interest. All of the devices tested below have the same materials and layer thicknesses, and only vary with respect to the radius as well as the presence or absence of a central mass. The electrically actuated frequency res ponse that is defined here to be the frequency dependent center deflection, 0 w, that results from an applied voltage, can be found from the circuit of Figure 2-15 to be 0effaDaDradw VsAZZ (5.4) where 1aDaDaD Z sMsC aDradZ is defined in Eq. (2.95), and effA is the effective area of transduction. PAGE 129 115 Shown in Figure 5-5 is the electrically act uated frequency response for the smallest device, with a radius of 900 m and without a central mass. Note the resonant peak occurring around 50 kHz with a peak response of 0.71 mV 20 40 60 80 10-4 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 20 40 60 80 10-4 10-2 100 Uncertainty 20 40 60 80 -150 -100 -50 0 Freq [kHz]Phase(EAFR) [Deg] 20 40 60 80 5 10 15 Freq [kHz] Data Theory Figure 5-5: Device 1A-6-1, elec trically actuated center deflection for the device with a radius of 900 m and no central mass, pack aged in the quarter-wave resonator package. Frequency response is shown in the left column with the corresponding uncertainty gi ven in the right column. For an identical size device, except with the addition of a central mass, the electrically actuated frequency response is shown in Figure 5-6. With the addition of the central mass, the resonant frequency drops from 50 kHz down to roughly 23 kHz, while the peak response increases to 6.66 mV This demonstrates that the central mass works as intended. Namely, it increases the overall effective mass without a PAGE 130 116 correspondingly large d ecrease in effective compliance, thereby leading to an overall decrease in the resonant freque ncy. There had been some a pprehension prior to testing as to the net effect of the centra l mass, mainly because significant modeling of the effect of the mass had not been performed prior to fabrication. 10 20 30 40 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 10 20 30 40 10-2 100 Uncertainty 10 20 30 40 -300 -200 -100 0 Freq [kHz]Phase(EAFR) [Deg] 10 20 30 40 10 20 30 40 Freq [kHz] Data Theory Figure 5-6: Device 1A-6-2, elec trically actuated center deflection for the device with a radius of 900 m and a central mass, packaged in the quarter-wave resonator package. Frequency response is sh own in the left column with the corresponding uncertainty gi ven in the right column. Shown in Figure 5-7 is the frequency response for the second smallest device, with an outer radius of 1200 m Similar to Figure 5-5, it does not have a central mass. Note two resonant frequencies are visible. The first resonance occurs near 34 kHz with peak value of 0.44 mV and represents the fundamental m ode of the device, as were the resonances in the previous graphs. The second resonance occurs near 120 kHz and PAGE 131 117 corresponds to a higher order mode with a peak value of 0.98 mV This was verified through use of the scanning feature on the lase r vibrometer, and is shown in Figure 5-8. 50 100 150 10-3 10-2 10-1 100 Frequency ResponseMag(EAFR) [ m/V] 50 100 150 10-3 10-2 10-1 100 Uncertainty 50 100 150 -300 -200 -100 0 Freq [kHz]Phase(EAFR) [Deg] 50 100 150 20 40 60 80 Freq [kHz] Data Theory Figure 5-7: Device 1A-11-3, el ectrically actuated center defl ection for the device with a radius of 1200 m and no central mass, packaged in the quarter-wave resonator package. Frequency response is shown in the left column with the corresponding uncertainty gi ven in the right column. Figure 5-8: Higher order mode image of Device 1A-11-3, taken using scanning laser vibrometer at 120.9 kHz PAGE 132 118 Once again, the addition of a central ma ss lowers the resonant frequency, as illustrated in Figure 5-9. In this instan ce, the resonant frequency drops from 34 kHz as measured above to 14 kHz as shown below. It should be noted here that these are physically different devices and the use of th e phrase addition of a central mass is not meant to convey that the same physical stru cture was modified, but simply that the central mass is the main distinguis hing feature between the two devices. 10 20 30 40 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 10 20 30 40 10-2 100 Uncertainty 10 20 30 40 -300 -200 -100 0 Freq [kHz]Phase(EAFR) [Deg] 10 20 30 40 100 200 300 400 Freq [kHz] Data Theory Figure 5-9: Device 1A-10-4, el ectrically actuated center defl ection for the device with a radius of 1200 m and a central mass, pack aged in the quarter-wave resonator package. Frequency response is shown in the left column with the corresponding uncertainty gi ven in the right column. Shown in Figure 5-10 is the frequency res ponse for the second largest radius device without a central mass. This devi ce has an outer radius of 1500 m It exhibits a resonant frequency of around 25 kHz, compared to 34 kHz for the next smaller device. PAGE 133 119 10 20 30 40 10-4 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 10 20 30 40 10-4 10-2 100 Uncertainty 10 20 30 40 0 50 100 150 Freq [kHz]Phase(EAFR) [Deg] 10 20 30 40 2 4 6 Freq [kHz] Data Theory Figure 5-10: Device 1A-6-5, electrically actu ated center deflection for the device with a radius of 1500 m and no central mass, packaged in the quarter-wave resonator package. Frequency response is shown in the left column with the corresponding uncertainty gi ven in the right column. With the addition of a central mass to th is size device, with outer radius of 1500 m the resonant frequency drops from 25 kHz to 10.5 kHz. This is shown in Figure 5-11 for Device 1A-10-6 and corroborates that the addition of the mass does not increase the stiffness enough to cancel out the lowe ring effect on the resonance frequency. The largest device is shown in Fi gure 5-12, with a radius of 1800 m without a central mass. A resonant frequency of around 20 kHz was measured for this device, as compared to 23.5 kHz for the next smaller device. Also note the smaller resonances at around 26 kHz and 45 kHz that are most likely effects from higher order modes. PAGE 134 120 5 10 15 10-4 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 5 10 15 10-4 10-2 100 Uncertainty 5 10 15 -100 0 100 Freq [kHz]Phase(EAFR) [Deg] 5 10 15 2 4 6 8 Freq [kHz] Data Theory Figure 5-11: Device 1A-10-6, el ectrically actuated center de flection for a device with a radius of 1500 m and a central mass, pack aged in the quarter-wave resonator package. Frequency response is shown in the left column with the corresponding uncertainty gi ven in the right column. Overall, the electrically act uated frequency response test s demonstrated the general trends that were expected, namely that of a decrease in resonant frequency with each increase in outer radius, as well as a decrease in resonant frequency due to the addition of central pillar in the center of the diaphragm that acts like a point mass. Shown in Figure 5-13 is a summary graph of the low frequency sensitivity as measured well below each individual resona nce. Note the general trend towards increasing sensitivity with increasing radius as expected from a physical standpoint, although the variation in sensitivity among id entical devices is on the same order of PAGE 135 121 magnitude. These results as well as the meas ured resonant frequenc ies and sensitivities at resonance are summarized in Table 5-3. 10 20 30 40 10-4 10-2 100 Frequency ResponseMag(EAFR) [ m/V] 10 20 30 40 10-4 10-2 100 Uncertainty 10 20 30 40 -300 -200 -100 0 100 Freq [kHz]Phase(EAFR) [Deg] 10 20 30 40 2 4 6 8 10 12 Freq [kHz] Data Theory Figure 5-12: Device 1A-3-7, electrically actu ated center deflection for the device with a radius of 1800 m and no central mass, packaged in the sealed-cavity package. Frequency response is sh own in the left column with the corresponding uncertainty gi ven in the right column. A graph of the measured resonant frequenc ies under electrical excitation is shown in Figure 5-14. Two series of data are show n, both with and without a central mass. The central mass was measured to have the effect of reducing the resonant frequency as was expected. These results show that, as desired, the effect of the addi tion of mass due to the central pillar was dominant over any decrea se in compliance that resulted from the presence of the pillar. Also, as the radi us is increased, the resonance frequency decreases. Overall, the range of re sonant frequencies varies from 6.73 kHz to 60.34 kHz PAGE 136 122 800 1000 1200 1400 1600 1800 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Outer Radius (R2) [ m]Low Frequency Sensitivity [ m/V] Open Cavity w/ central mass Open Cavity w/o central mass Sealed Cavity w/ central mass Sealed Cavity w/o central mass Theory Figure 5-13: Electrically actuated sensitivity at low frequency (well below resonance) ( 120.95 RR ). 800 1000 1200 1400 1600 1800 10 20 30 40 50 60 70 Outer Radius (R2) [ m]Mechanical Resonant Frequency [kHz] Exp. w/o mass Exp. w/ mass Theory w/o mass Theory w/ mass Figure 5-14: Summary of electrically actuated resonant frequencies. PAGE 137 123 The effect of the central mass is illustrated more explicitly in Figure 5-15, where the negative shift in resonant frequency due to the additional mass is plotted versus the original resonant frequency. The values ar e expressed in percenta ge change from the original values, where a positive number indicates a drop in resonant frequency. The mean percentage change was experimentally found to be 58.93 %, as compared to a predicted shift of 57.76 %. Also the shift wa s found to be more prominent for the larger devices with the lower resonant frequencies 20 30 40 50 60 70 0 20 40 60 80 100 Resonant Frequency [kHz]Frequency Shift [%] Theoretical Experimental Figure 5-15: Drop in resonant frequency due to the addition of the central mass. Overall, significant numerical differences were seen between theory and experiment, as evidenced in Table 5-3. Qual itatively, however the device behavior did generally follow the expected trends, particular ly for the resonant frequencies, as seen in Figure 5-14 and Figure 5-15. On the othe r hand, while the low frequency sensitivity did generally increase with radius, as exp ected, the experimental behavior was not as PAGE 138 124 predictable and varied greatly in response to packaging a nd the presence of the central mass. The measured behavior was most consis tent with the theory in the sealed cavity packages, suggesting that unwanted acoustical interactions related to the package geometry, may be altering the low frequency sensitivity of the open cavity packaged devices. Table 5-3: Summary of elect rically actuated frequency response measurements. Experimental Theoretical %Difference Device # 2R m mR m mresf kHz lowEASens mV resEASens mV mresf kHz lowEASens mV %mresf %lowEASens 1A-5-1 900 0 n/a n/a n/a 65.68 0.011 n/a n/a 1A-6-1 900 0 50.8 0.005 0.30 65.68 0.011 22.54 54.55 1A-6-2 900 180 23.30 0.015 6.66 28.08 0.011 17.02 -36.36 1A-10-2 900 180 25.20 0.007 0.47 28.08 0.011 10.26 36.36 1A-11-3 1200 0 34.25 0.025 0.45 48.14 0.013 28.85 -92.31 1A-5-4 1200 240 16.69 0.005 1.54 20.37 0.013 18.07 61.54 1A-10-4 1200 240 14.12 0.030 19.70 20.37 0.013 30.68 -130.7 1A-6-5 1500 0 25.2 0.015 0.88 38.01 0.014 33.70 -7.143 1A-6-6 1500 300 9.53 0.050 8.18 15.98 0.014 40.36 -257.1 1A-10-6 1500 300 10.56 0.010 6.60 15.98 0.014 33.92 28.57 1A-3-7 1800 0 19.84 0.008 4.20 31.41 0.015 36.84 46.67 1A-6-7 1800 0 22.72 0.020 0.62 31.41 0.015 27.67 -33.33 1A-3-8 1800 360 6.74 0.078 12.20 13.15 0.015 48.75 -420.0 Linearity Once the frequency response was measured for each device, experiments to characterize the linearity of the response were performed. The measurements were made at the resonant frequency of each device, in order to maximize deflection. A zero DC offset sinusoidal voltage signal was used to dr ive the diaphragm while the velocity at the center of the diaphragm was measured using th e laser vibrometer. The center deflection was then derived from this measurement a nd plotted versus the driving voltage. The resulting plot is shown in Figure 5-16 for all of the tested devices. In the figure, most of the measured devices exhibited a linear response up to about 50 mV of input voltage. PAGE 139 125 Increasing the voltage beyond that did not lead to a corresponding increase in deflection for these devices. From these results, the ma ximum input voltage fo r a linear response appears to be independent of the device radi us. This suggests that it is not due to a mechanical factor that would increase when the device is scaled. Instead, it is most likely due to a depoling of the piezoelectric materi al, leading to a reduc tion in piezoelectric properties and the re sulting deflection. 101 102 101 102 103 AC Driving Voltage [mV]Center Deflection [nm] Device 1A-6-1 Device 1A-10-2 Device 1A-5-4 Device 1A-6-5 Device 1A-10-6 Device 1A-3-7 Figure 5-16: Magnitude of the resonant center deflection versus driving voltage. Another convenient way of looking at th e nonlinear behavior is by calculating the mechanical sensitivity to voltage inputs as th e driving voltage is varied. The resulting plot is shown in Figure 5-17. From this plot it is easy to see where the sensitivity starts to drop for most of the devices, indica ting the onset of nonlinear behavior. PAGE 140 126 101 102 102 103 104 AC Driving Voltage [mV]Mechanical Sensitivity [nm/V] Device 1A-6-1 Device 1A-10-2 Device 1A-5-4 Device 1A-6-5 Device 1A-10-6 Device 1A-3-7 Figure 5-17: Mechanical sensitivity at resonance versus driving voltage. Acoustical Measurements Acoustically Actuated Frequency Response Plane-Wave Tube Sealed cavity package The devices that were packaged in the sealed cavity packages (Device 1A-10-2, Device 1A-5-4 and Device 1A-3-7) were tested in the plane-wave tube, with a nominally average incident acoustic pressure of 92.6 dB. The frequency response results shown below in terms of magnitude and phase ar e plotted in Figure 5-18 over the frequency range of 250 Hz to 6700 Hz for Device 1A-5-4, which has a radius of 1200 m and a central mass. The plots for the other devices were not shown to avoid needless repetition, PAGE 141 127 as they are all highly similar in behavior. The lower frequency limit was set to avoid speaker inefficiencies while the upper limit wa s due to the onset of higher order modes in the tube. The important thing to note from all of these results is the relatively flat magnitude over the frequency range, which is to be expected well below resonance. A slight dip in the low frequency region is mo st likely due to a vent resistance arising from a small leakage in the packaging. The other factor that will lead to a low-frequency rolloff is due to the p eB R C time constant of the piezoelect ric. For the device shown in Figure 5-18, p eB R C =0.0089, and therefore the relate d corner frequency, given by 1 p eB R C was found to be 112.7 Hz As can be seen in Figure 5-18, the corner is closer to 1792 Hz and therefore unrelated to the dielectric losses. Among all of the tested devices, a relative ly flat magnitude of the response was seen, with the numerical value of the ma gnitude as the only significant difference between devices. This value was found to in crease with increasing radius and occurs because the testable frequency range was we ll below all of the resonant frequencies, leading to a compliance dominated testing regi me. Once again, this frequency range was limited by the cut-on conditions for higher order modes in the plane wave tube. Quarter-wave resonator package The same experiment was then performed on the devices that were packaged in the quarter wave resonator package (Device 1A -5-1, Device 1A-11-3, Device 1A-10-4, Device 1A-6-5, Device 1A-10-6, Device 1A-3-7 and Device 1A-3-8). The results for Device 1A-10-4 with a radius of 1200 m and a central mass and Device 1A-6-5 with a radius of 1500 m and no central mass are shown in Figure 5-19 and Figure 5-20 in terms of magnitude and phase. Note that a dip occurs in the magnitude at around 3.694 PAGE 142 128 kHz instead of the flat responses of the sealed cavity packages. The dip arises due to a quarter wave anti-resonance that causes a maximum in the acoustic impedance of the vent channel, as seen by the diaphragm. At this frequency, the diaphragm sees a significantly higher impedance, and the resul ting load acts to reduce the deflection, thereby leading to a stiffening effect on the diaphragm. 1000 2000 3000 4000 5000 6000 10-6 10-5 Mag(AE FR) [V/Pa]Freq [Hz] 1000 2000 3000 4000 5000 6000 -100 -50 0 50 Phase(AE FR) [Deg]Freq [Hz] Data Theory Figure 5-18: Device 1A-5-4, magnitude and pha se of the acoustically actuated frequency response in a sealed cavity package fo r the device with a radius of 1200 m and a central mass. Using an analytical approach to calculate the impedance of a tube with an open end yields a theoretical maximum in impedance (and hence minimum in sensitivity) occurring at 3.882 kHz leading to a difference of 4.84% between the experimental and theoretical values. The radiation impedan ce from the open end of the tube was also considered, however it was found to have only a small impact on the response, by PAGE 143 129 extending the effective le ngth of the tube to efflll where l is the effect due to the radiation mass. 1000 2000 3000 4000 5000 6000 10-8 10-5 10-3 Mag(AE FR) [V/Pa]Freq [Hz] 1000 2000 3000 4000 5000 6000 0 50 100 150 200 Phase(AE FR) [Deg]Freq [Hz] Data Theory Figure 5-19: Device 1A-10-4 in a quarter-wave resonator package. Magnitude and phase of the acoustically actuated frequency re sponse for the device with a radius of 1200 m and a central mass. For all of these devices, the antiresonant frequency is the same while the overall magnitude increases with each larger device The constant antiresonant frequency is indicative of this packaging do minated effect, rather than a device dominated effect. The packaging is identical for all of these de vices, meanwhile the geometry of the device itself varies. Furthermore, the antiresonant frequency has a quarter wavelength that corresponds closely to the length of the backsi de hole in the resonator package. This quarter-wave resonator behavior was not initially intended, but became obvious in retrospect, considering the geometry of the package. PAGE 144 130 1000 2000 3000 4000 5000 6000 10-8 10-5 10-3 Mag(AE FR) [V/Pa]Freq [Hz] 1000 2000 3000 4000 5000 6000 -200 -100 0 100 200 Phase(AE FR) [Deg]Freq [Hz] Data Theory Figure 5-20: Device 1A-6-5 in a quarter-wave resonator package. Magnitude and phase of the acoustically actuated frequency re sponse for the device with a radius of 1500 m and no central mass. Acoustic Input Impedance Mea surements Plane-Wave Tube The results of the acoustical input impedance measuremen ts on the devices in the sealed cavity packages are shown in Figure 5-21 to Figure 5-23 in terms of normalized resistance and reactance. The normalization was performed with respect to the acoustic impedance of the plane wave tube, and can be found as inin oo tube tubeZZ j c Z A (5.5) where tubeA is the cross-sectional area of the tube. PAGE 145 131 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.94 0.96 0.98 1 Frequency [Hz]Mag(R) 2000 2500 3000 3500 4000 4500 5000 5500 6000 -20 -10 0 10 20 Frequency [Hz]Phase(R) [Deg] Theory Data Figure 5-21: Device 1A-10-2, Normalized acous tic impedance in a sealed cavity package for the device with a radius of 900 m and a central mass. The impedances were shown over the same range of 250 Hz to 6700 Hz. The reactance of these devices was generally negative, as would be expected below resonance, where the behavior is domina ted by the compliance of the diaphragm. Additionally, the resistance s hows some unusual and incons istent features below 2 kHz; however, due to the close proximity to unity of the reflection coefficient at these frequencies, small measurement errors and un certainties are magnified when calculating the acoustic impedance. PAGE 146 132 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.9 0.95 1 Frequency [Hz]Mag(R) 2000 2500 3000 3500 4000 4500 5000 5500 6000 -20 -10 0 10 20 Frequency [Hz]Phase(R) [Deg] Theory Data Figure 5-22: Device 1A-5-4, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1200 m and a central mass. 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.9 0.95 1 Frequency [Hz]Mag(R) 2000 2500 3000 3500 4000 4500 5000 5500 6000 -30 -20 -10 0 10 20 Frequency [Hz]Phase(R) [Deg] Theory Data Figure 5-23: Device 1A-3-7, Normalized acoustic impedance in a sealed cavity package for the device with a radius of 1800 m and no central mass. PAGE 147 133 Quarter-Wave Resonator Package The same measurements were then performe d on the devices that were packaged in the quarter-wave resonator package (Device 1A-6-1, Device 1A-11-3, Device 1A-10-4, Device 1A-6-5, Device 1A-10-6, Device 1A-3-7 and Device 1A-3-8). The results of some of these measurements are shown in Figure 5-24 and Figure 5-25, for Device 1A6-1 with a radius of 900 m and no central mass and Device 1A-11-3 with a radius of 1200 m and also no central mass, respectively. The plots for the other devices were not shown to avoid needless repetition, as they are all highly similar in behavior. Note how the reactance is no longer solely negative, due to the presence of the quarter wave resonator. Furthermore, a peak was consis tently observed in the resistance near 1500 Hz. 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.8 0.85 0.9 0.95 1 Frequency [Hz]Mag(R) 2000 2500 3000 3500 4000 4500 5000 5500 6000 -20 0 20 Frequency [Hz]Phase(R) [Deg] Theory Data Figure 5-24: Device 1A-6-1, Normalized acoustic impedance in a quarter-wave resonator package for the device with a radius of 900 m and no central mass. PAGE 148 134 The important thing to note from Figure 5-24 and Figure 5-25 is the decidedly different character of the impedance as compar ed to the previous experiments with sealed cavity devices. With the sealed cavity devi ces, the reactance was primarily negative and increased more or less monotonically with fr equency, however the introduction of the quarter-wave resonator package alters the impedance such th at positive reactances occur more frequently and the reactance in general ha s a more variable natu re. The significant changes in the impedance highlight the role of the packaging in the overall behavior. 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.8 0.9 1 Frequency [Hz]Mag(R) 2000 2500 3000 3500 4000 4500 5000 5500 6000 -20 -10 0 10 20 Frequency [Hz]Phase(R) [Deg] Theory Data Figure 5-25: Device 1A-11-3, Normalized acoustic impedance in a quarter-wave resonator package for the device with a radius of 1200 m and no central mass. Initial Deflection Measurements Wyko Optica l Profilometer Using the Wyko Optical Profilometer, the in itial static deflection was measured for four devices. The initial deflection that aris es due to residual stresses in the composite PAGE 149 135 structure is plotted versus radi us in Figure 5-26 to Figure 5-29. Note that a small step is evident in the region near the clamped edge, resulting from the height difference between the piezoelectric ring and diaphragm surface. Also note that the two even numbered devices, Figure 5-27 and Figure 5-29, have a fl atter central region due to the presence of the central pillar. Figure 5-26: Device 1A-11-3, Ini tial static deflection resulting from residual stresses for a device with a radius of 1200 m and no central mass. There are several key points of note from the initial static deflection plots. First of all, the static deflection was found to be si gnificant and on the same order of magnitude as the diaphragm thickness. The models that were used to predict the device behavior are linear models that depend on the assumption of small deflections. The large initial deflections violate this assumption and leads to a device that behaves differently than predicted using a linear theory. PAGE 150 136 Figure 5-27: Device 1A-6-6, Initial static defl ection resulting from residual stresses for a device with a radius of 1500 m and a central mass. Figure 5-28: Device 1A-3-7, Initial static defl ection resulting from residual stresses for a device with a radius of 1800 m and no central mass. Knowledge of the deflection curv e is critical to accurate prediction of the device, as many of the lumped element parameters, such as the piezoelectric coefficient, Ad, are fundamentally derived from the curve. Because of the nonlinear nature of the deflection, the models cannot accurately predict the device behavior. Therefore, in order to compare PAGE 151 137 the experimental data against a more r easonable theoretical prediction, measured deflection curves were used to compute new values for the piezoelectric coefficient, Ad. Additionally, the measured dielectric loss resistance, pR, and damping factor,m R were included into the model, are shown in Table 54 In the theoretical curves, previously shown in Chapter 2, these were not include d. The energy harvesting measurement section utilizes these new predictions for comparis on to the measurements. Notice from Table 5-4 that the measured Ad values are much lower than the predicted values of 131.9310 and 122.4210 for Device 1A-4-4 and Device 1A-3-8 respectively. The explanation behind this phenomenon will be discussed later in this chapter. Table 5-4: Revised lumped element parameters used for comparing experimental data to theory. New and revised columns are marked with an asterisk. Device k Ad* aDC aD M aDmassM ebC p R *m R Pa V V Pa 3m V 3m Pa 4kg m 4kg m nF k 4kg ms 1A-4-4 -9.29 -7.16E-6 0.0083.08E-1 42.07E-142602.5 11052 26.9 414.63.24E7 1A-3-8 -16.92 -4.71E-5 0.0282.15E-1 41.43E-131009.8 4912.2 51.3 233.55.09E7 Figure 5-29: Device 1A-3-8, Initial static defl ection resulting from residual stresses for a device with a radius of 1800 m and a central mass. PAGE 152 138 Energy Harvesting Measurements The primary application of the work presente d in this dissertation is as a harvester of acoustic energy. For this application, it is useful to look at the device performance at resonance as a function of the applied acousti c pressure. In order to accomplish this, a load is placed across the output of the resona tor and the voltage is measured across that load as the pressure is varied. Before va rying the pressure, however, it is useful to optimize the load in order to maximize the energy transfer. Recall from Eq. (2.12), that under the constraint of a purely resistive load, the power transfer to the load is maximized when the load resistance is equal to the magnitude of the output impedance. This optimal value was determined experimentally by measur ing the power while the load is varied. The resulting value was then compared against the measured output impedance of the device, in order to check for c onsistency with Eq. (2.12). Optimal Resistance Using a Stanford Research Systems SRS 785 Dynamic Signal Analyzer, the optimal resistance for maximum power transfer wa s determined. First however, the exact resonant frequency values had to be determ ined, therefore the SRS785 was first set to record 500 averages of 800 frequency bins over a frequency span of 25.6 kHz. For Device 1A-4-4, no window was needed as a ch irp input was used, while for Device 1A-38, a Hanning window was used in conjuction with a band-limited white noise signal input. This measurement resulted in m easured resonant frequencies of 13.568 kHz and 5.232 kHz for Device 1A-4-4 and Device 1A-3-8, respectively. Once the resonant frequencies were determined, the source was changed to a sinusoidal signal at each of these frequencie s for the respective devices. The output voltage was then measured while the load resistance was varied from 46.4 to 1.003 PAGE 153 139 M The output power was then calculated according to Eq. (2.117) and the results were then plotted as shown in Figure 5-30. The figure is overlaid with the theoretical values for comparison. As can be seen, ther e is less than one order of magnitude between data and theory, which overpredicted the pow er for Device 1A-4-4 and underpredicted the power for Device 1A-3-8. The experimentally determined, optimal resistance for both of these devices was found to be 982.9 Optimal Energy Harvesting Using the optimal frequency and load re sistance measured pr eviously, the input signal was steadily increased in amplitude wh ile the output voltage was measured. The power was then calculated based on the volta ge and resistance measurements, again according to Eq. (2.117). The resulting output voltage for each device is shown in Figure 5-31 and is overlaid with the theoretical predictions for the output voltage. 101 102 103 104 105 106 10-16 10-15 10-14 10-13 10-12 10-11 10-10 Load Resistance [ ]Power Delivered to Load [W] Data-Device 1A-4-4 Data-Device 1A-3-8 Theory-Device 1A-4-4 Theory-Device 1A-3-8 PAGE 154 140 Figure 5-30: Measured power delivered to a lo ad as function of the load resistance for Device 1A-4-4 and Device 1A-3-8 as compared against theoretical values. The input acoustic pressure was 92 dB for these measurements. As can be seen in the figure, th e output voltage was linear up to 125 dB for Device 1A-4-4 and up to 133 dB for Device 1A-3-8. The output voltage was seen to range between 22 V and 4.6 mV. Using the voltage data, the output power was then calculated and is shown in Figure 5-32. 90 100 110 120 130 140 150 100 101 102 103 104 105 Input Acoustic Pressure [dB]Voltage across Load [ V] Data-Device 1A-4-4 Data-Device 1A-3-8 Theory-Device 1A-4-4 Theory-Device 1A-3-8 Figure 5-31: Measured output voltage across the load as a function of applied acoustic pressure and compared to theoretical values. The load resistance was fixed at the optimal value of 982.9 (f = 13.568 kHz and 5.232 kHz for Device 1A4-4 and Device 1A-3-8, respectively) The power density was then calculated ba sed on a square unit cell with lateral dimensions equal to the diameter of the di aphragm. The resulting values are shown in Figure 5-33 and are again overlaid with theoretical values. Note from the graph that the maximum power density measured was around 1 2Wcm at around 150dB which is PAGE 155 141 considerably lower than the avai lable acoustic power density of 100 2mWcm at the acoustic pressure of 150 dB. 90 100 110 120 130 140 150 10-14 10-12 10-10 10-8 10-6 Input Acoustic Pressure [dB]Power Delivered to Load [W] Data-Device 1A-4-4 Data-Device 1A-3-8 Theory-Device 1A-3-8 Theory-Device 1A-4-4 Figure 5-32: Measured power delivered to load as a function of applied acoustic pressure and compared to theoretical values. The load resistance was fixed at the optimal value of 982.9 (f = 13.568 kHz and 5.232 kHz for Device 1A-4-4 and Device 1A-3-8, respectively) This can be seen more clearly by looking at the efficiency, The efficiency was calculated according to Eq. (2. 118) for each of the devices, and is shown in Figure 5-34. As can be seen, the efficiency was fairly constant near 0.012 % for Device 1A-4-4 in the linear regime and 4410 % for Device 1A-3-8 in the linear regime. For comparison, the theoretical model predicts efficiencies of 0.056 % and4310 % respectively. PAGE 156 142 90 100 110 120 130 140 150 10-14 10-12 10-10 10-8 10-6 10-4 Acoustic Input Pressure [dB]Power Density [W/cm2] Theory-Device 1A-3-8 Theory-Device 1A-4-4 Data-Device 1A-4-4 Data-Device 1A-3-8 Figure 5-33: Measured power density delivered to load as a function of applied acoustic pressure and compared to theoretical values. (f = 13.568 kHz and 5.232 kHz for Device 1A-4-4 and Device 1A-3-8, respectively) 90 100 110 120 130 140 150 10-5 10-4 10-3 10-2 101 Input Acoustic Pressure [dB]Efficiency [%] Data-Device 1A-4-4 Theory-Device 1A-4-4 Data-Device 1A-3-8 Theory-Device 1A-3-8 Figure 5-34: Measured overall efficiency of each device overlaid with theoretical values for comparison. (f = 13.568 kHz and 5.232 kHz for Device 1A-4-4 and Device 1A-3-8, respectively). PAGE 157 143 A question then arises regarding the orig in of the low efficiency and whether improvements in either the manufacturing or design of the devices could improve the efficiency, and thereby the output power. Seve ral issues were faced during the fabrication process that limited the efficiency of the devices. First, the materials and processes led to a large residual stress in several layers of th e device, most notably the titanium dioxide. The effect of this large tensile stress, 2TiO is twofold: (1) it al ters the deflection mode shape of the devices and (2) it leads to a large, non-linear, initial deflection of the diaphragm. The end result is a modifica tion to the piezoelectric coefficient, Ad and ultimately the coupling coefficient, k. It is difficult to asce rtain the exact magnitude of this effect since the deflection lies in the nonlinear regime and the analytical model is only linear. Nevertheless, a 1st order estimate of the effect can be made by using the linear model and comparing the results for the existing high-stress case to an ideal zerostress case. Using this method, the power density decreased by a factor of 4.3 over the ideal zero-stress case. A second issue arising from the fabrica tion is the overha nging metal on the electrodes that created short circuits unde r high electric fields. The field limitation imposes a maximum poling voltage that can be a pplied. As the piezoelectric coefficient, Ad, is proportional to the polarization, the full potential poling was not realized. The effects of this issue, along with the unmode led (non-linear) portion of the stress induced effects, lead to a reduction in power dens ity by a factor of 24.6 below the ideal case. The final fabrication issue concerns the c hoice of the starting piezoelectric material and the process by which it is deposited on the wafer. The particular sol-gel technique that was employed in this research produces a typical 31d of -50 p CN, whereas other PAGE 158 144 variations of the sol-gel process as well as techniques such as sputtering can produce a film with a 31d of -120 p CN. The output power density is sm aller by a factor of 2.8 as a result of the lower value of 31d as compared to other reported PZT thin films. The final issue that reduces the output power density concerns the quality factor, Q, of the device. The measured quality fact or for Device 1A-3-8 is approximately 10, leading to a damping factor of 0.05. This damping accounts for radiation resistance and internal damping, as well as radiation to the supports. This damping factor is much higher than a more typical 0.02, leading to a reduction in power density by a factor of 2.5 Through improvements in the fabrication and design, leading to a reduction in the damping factor, the output power density would increase by a factor of 2.5. Additionally, provided that all of the othe r sources of inefficiency are addressed, including the PZT material quality, the poling capability, and th e residual stress, the overall power density would increase by a factor of 740. The resu lting power density curv e is shown in Figure 5-35, overlaid with the currently achievable results. With these improvements, at 150 dB, the output power densit y would be on the order 2500 Wcm. Additionally, the individual contributions to this improvement are illustra ted in Figure 5-36, clearly showing the prominent role of improving Ad to the overall results. Nonlinear behavior is eviden t in Figure 5-31 through Figur e 5-34. Physically, there are several possible sources for the nonlinea r behavior. The overall response may be affected by one or all of these sources. Fi rst of all, a mechani cal nonlinearity arises, when the deflection of a plate is on the order of the thickness of that plate. Under large deflections, the neutral axis of the plate st retches by a finite amount, requiring energy in addition to the energy required for bending of the plate. The resulting effect is an PAGE 159 145 increase in stiffness above that predicted by linear, small-deflection plate theory. The increase in stiffness leads to a lower sensitiv ity and an increased resonant frequency. 90 100 110 120 130 140 150 10-14 10-12 10-10 10-8 10-6 10-4 102 Acoustic Input Pressure [dB]Power Density [W/cm2] Realistically Optimal Current Results Figure 5-35: Currently achievable power dens ity for Device 8 and potentially achievable power density using an impr oved fabrication process. 72% 13% 8% 7% dA TiO2 Ideal PZT Q Figure 5-36: Breakdown of the individual cont ributions to the overall improvement in power that is achievable under better fabrication conditions. PAGE 160 146 The effect on the resonant frequency was measured for one particular device that exhibited strong nonlinear behavior and is show n in Figure 5-37. The resonant frequency was seen to approximately shift linearly as the acoustic pressure was increased. Additionally, in the frequency domain, the nonlinearity generates harmonics at integer multiples of the fundamental signal frequency, thereby dumping some of the energy that would have otherwise resided at the fundamental frequency. 0 200 400 600 800 1000 10.15 10.2 10.25 10.3 10.35 10.4 Input Acoustic Pressure [mPa]Resonant Frequency [kHz] fres = 250*Pin + 10.15E3 [Hz] data 1 linear fit Figure 5-37: Resonant frequenc y vs. acoustic pressure, arising from nonlinear response. PAGE 161 147 CHAPTER 6 ALTERNATIVE APPLICATION PIEZOELECTRIC MICROPHONE The previous chapter on experimental characterization measured the overall characteristics of the device in terms of im pedance, acoustical response, and electrical response in a general sense, as well as an evaluation of the performance of the device as an energy harvester. The goal was to unde rstand the actual behavior and to try and explain the difference between the actual behavior and the predicted behavior. This chapter serves to enhance our u nderstanding of the device performance by looking at an alternative application of the piezoelectric composite diaphragm, by using it as a piezoelectric microphone. For the ener gy harvester devices, we are using the piezoelectrically generated charge as an energy source, whereas in a microphone application, this charge (or corresponding voltage) is a signa l for transmitting information about the acoustical environment. For this set of experiment s, a piezoelectric diaphragm with 1.8 mm diameter and no central mass was used. Experiments were performed both as a sens or (acoustically ac tuated) and as an actuator (electrically actuated) in order to obtain the performance specifications. The packaged device was mounted to the end of a 1 x 1 cross section plane wave tube (PWT) for obtaining the frequency response a nd linearity. A Bruel and Kjaer (B&K) 1/8 Type 4138 microphone was mounted on the wall of the PWT near the end of the tube to record the incident sound pressure level (SPL). The signal was provided by a PULSE Multi-Analyzer system to a BMS 2 coaxial co mpression driver, via a Techron amplifier. PAGE 162 148 The PULSE analyzer was also used to reco rd the signals from the B&K microphone and the piezoelectric microphone. First, the linearity was tested using a sinusoidal signal at 1 kHz. Measurements were taken with a binwidth of 1 Hz and the signal falling direc tly on the center of a bin to avoid leakage. Additionally, 100 averages were taken at each point. The measured linearity of the sensor at 1 kHz is shown in Figure 6-1. The device was experimentally found to be linear up to (at least) the ma ximum testable sound pressure level of 169 dB. The sensitivity was measured over th is range to be approximately 11.0 VPa Furthermore, the response flattens out below 45 dB as the signal level approaches the noise floor. The maximum testable level is limited here by the out put capacity of the compression driver. The measured sensitivity was found to be 0.75 VPa 75 100 125 150 175 10-8 10-7 10-6 10-5 10-4 10-3 10-2 Output Voltage [V]Input Acoustic Pressure [dB] Data Fit R2=0.9995 Figure 6-1: Linearity of the microphone device at 1 kHz. PAGE 163 149 The frequency response was measured us ing a periodic random noise acoustic signal of 94dB and 1000 spectral averages over a band width from 0 to 6.4 kHz with a 1 Hz bin width. Once again, the testable frequency range was limited on the upper end by the onset of higher order modes. The resu lting frequency response of the sensor is shown in Figure 6-2 in terms of magnitude and phase. 101 102 103 10-7 10-6 10-5 Frequency [Hz]Magnitude [V/Pa] 101 102 103 -50 0 50 100 Frequency [Hz]Phase [Deg] Figure 6-2: Frequency res ponse spectrum in terms of magnitude and phase. The response is flat over the testable frequency range up to 6.7 kHz, except for a low frequency rolloff at approximately 100 Hz. This is most likely due to the vent resistance; however, the magnitude of this c ontribution is unknown, as the value for the vent resistance due to packaging leaks has not been measured. Additionally, there is a low-frequency rolloff due to the parallel combination of the capacitance and resistance in PAGE 164 150 the piezoelectric material, however the corner frequency due to this combination is 6.7 Hz, well below that of the measurable corner frequency of 100 Hz. The noise floor spectrum of the sensor is shown in Figure 6-3 from 0 Hz to 12.8 kHz, along with the noise floor due to the experimental setup alone. The noise floor was found to generally decrease with increasing frequency. The noise spectrum of the experimental setup alone wa s obtained by shorting the inpu ts to the preamplifier and recording the resulting output voltage signal, using the same conditions and setup as the sensor noise floor measurement. This noise spectrum is characteristic of a resistance shunted by a capacitance. The resistance in this case is the sensor dielectric resistance, p R and the capacitance is the blocked electrical capacitance, ebC. The resistance and capacitance form a low-pass tran sfer function that spectrally shapes the thermal noise. When integrated over an infinite bandwidth, the resulting mean square noise voltage is given by ebkTC, where k is Boltzmanns constant and T is the ambient temperature. For this device, the low pass filter has a cutoff frequency of 6.7 Hz, determined by 12peb R C There are three characteristic regions in this spectrum. Below 6.7 Hz, there exists a relatively flat region where the noise is dominated by the resist ance of the sensor. Then, from 6.7 Hz up to approximately 2 kHz a 1 f rolloff in the noise is observed that is consistent with a capacitive dominated noise source. Above 2 kHz, the noise spectrum flattens out again as the sensor noise approaches the setup noise. For a 1 Hz bin width centered at 1 kHz, the output voltage with no acoustic signal applied is 3.69 nV that corresponds to an equivale nt acoustic pressure of 4.93 mPa or 47.8 dB. PAGE 165 151 100 102 104 45 50 55 60 65 70 75 80 85 Freq [Hz]Magnitude [dB] Sensor Setup Figure 6-3: Noise floor spectrum of output voltage when no acous tic signal is applied, as well as noise floor due to measurement setup alone. Because the device is a linear, reciprocal electromechanical transducer, it can operate in both an electrically driven and ac oustically driven modes and the electrically driven frequency response can be used to gain additional information about the sensor, such as the bandwidth. In particular, as the acoustically actuated frequency response could only be measured up to 6700 Hz due to testing limitations of the plane wave tube, the bandwidth can be approximated via us e of the electrically actuated frequency response. The electrically actuated frequency res ponse is shown in Figur e 6-4 in terms of magnitude and phase. The resona nt frequency was found to be 50.8 kHz and provides an estimate of the usable bandwidth as a microphone, as the electrically and acoustically PAGE 166 152 actuated resonance frequencie s are related. A summar y of the overall performance specifications of the microphone is given in Table 6-1. 10 20 30 40 50 60 70 80 90 -20 -18 -16 -14 Magnitude [dB re 1 V/Pa]Freq [kHz] 10 20 30 40 50 60 70 80 90 -150 -100 -50 0 Freq [kHz]Phase [Deg] Figure 6-4: Electrically actuated frequency respons e of microphone device. Table 6-1: Summar y of experimental results of microphone. Specification Measured Value Diaphragm Size 1.8 mm diameter, 3 m thick Die Size 5 5 mmmm Measured Resonant Frequency 50.8 kHz Sensitivity 0.75 VPa Dynamic Range 47.8169 dB PAGE 167 153 CHAPTER 7 CONCLUSION AND FUTURE WORK The acoustic energy harvesting technique th at was developed in this dissertation provides a unique and innovative method for extracting energy from an otherwise inaccessible source. For the development of acoustic energy harvesters, piezoelectric composite diaphragms were designed based on analytical models of circular piezocomposite structures. Lumped element mode ling was then used to further understand the dynamics of the system. The modeling served as a basis around which the geometry and dimensions of numerous diaphragms were determined. The lumped element model developed in this dissertation serves as a design basis for other types of piezoelectric composite structures as well as energy harv esting devices employing other transduction methods. Furthermore, it is directly applicab le to those interested in the scaling and design of piezoelectric composite diaphragms. Following the design stage, a fabrication pr ocess flow and mask set was developed to synthesize the devices. Fabrication was then performed as prescribed in the process flow. Following this, the devices were packag ed into two different packages for testing purposes. Finally, a number of experiments were set up and performed on the devices to better understand the real behavior as compar ed to the theory. A graphical overview of this chapter is provided by th e roadmap of Figure 7-1. The main contributions of this disser tation included the development of the acoustic energy harvesting concept. While prev ious researchers had looked at harvesting PAGE 168 154 energy from vibration, solar, chemical and thermal energy among others, a thorough investigation into the harv esting of acoustic energy had not been performed. Additionally, in order to deve lop the concept, a comprehensive lumped element model (LEM) of the acoustic energy harvester was ut ilized. This model allowed for a single representation for the dynamics of the system as a whole, as well as the individual components. Development Issues Future Work Figure 7-1: Roadmap to Chapter 7. The implementation of the acoustic energy harvester led to the first reported integration of thin-film PZT with SOI-based MEMS. While numerous researchers had previously looked at thin-f ilm PZT, or MEMS using SOI wafers, none had sought to combine them. The high transduction coeffici ent of PZT along with a simpler fabrication route to a diaphragm via SOI wafers led to the decision to combin e these technologies. Finally, although the circular piezoelectric membrane was initially intended as a harvester of acoustic energy, it was discovered that it also served as fairly capable microphone. This is the first reported inst ance of an aeroacoustic capable piezoelectric microphone. Development Issues Several issues arose during the development of these de vices. First of all, the initial process flow that was developed cont ained more steps and masks than the process PAGE 169 155 flow that was presented here Initially, gold bond pads and an insulating oxide were incorporated to enable on-chip arrays of devices; however, this process flow was later simplified prior to fabrication to reduce cost, time and fabrication complexity. The next issue arose during fabrication of the devices, but was not discovered until packaging of the devices was attempted. Th e problem involved a thin strip of platinum that overhung the piezoelectric ring from the top electrode. Normally, for a metal liftoff step, a image reversal mask is used on positive photoresist to obtain a negative sloping sidewall prior to metal deposition. The negative slope permits a clean break in the metal for improved liftoff results. However, due to processing constraints, we had to use a negative photoresist for this pattern, resulti ng in a positive sloping sidewall. Then during the metal liftoff, a clean break was not ach ieved, resulting in a few microns of overhanging metal around the e dges of the ring. This was not noticed until electrical testing was performed on the devices after fabrication wa s completed. The testing showed that four of the six wafers had el ectrical short circuits between the top and bottom electrodes. Furthermore, the resistance of the short circuits was proportional to the perimeter of the ring, indicating it was due to an edge effect rather than a surface effect. The two remaining wafers still f unctioned correctly, though, despite the same overlapping metal issues, and were solely us ed for the experiments presented here. Another fabrication issue that arose concerne d the residual stress in the devices. It was discovered that the measured resi dual stress of the titanium dioxide (TiO2), at 1800 GPa was significantly higher than anticipated. This led to a large initial static deflection of the diaphragms. Additi onally, the buried oxide laye r on the backside of the PAGE 170 156 diaphragms proved difficult to remove. This residual stress of this layer further exacerbated the initial static deflection. Unfortunately, the large initial static defl ection placed the devices into a nonlinear regime of plate mechanics and led to a stiffe r device than was anticipated. Furthermore, the models that were developed for the design of the diaphragms were no longer applicable under this nonlinear condition. This caused numerous difficulties in fully understanding the overall behavior of the devices from a quantitative standpoint. Qualitatively, however, the devices did behave as expected, following anticipated trends and dependencies. Ultimately, however, the increased stiffening opened up a new application for the devices as a microphone. Because the stiffer devices had a higher resonance frequency, the usable bandwidth was extended be yond our expectations. Experimental characterization indicates a sensitivity of 11.0 VPa with a dynamic range from 44.1 dB to 153 20dBrePa and a resonant frequency of 22.72 kHz, suggesting a potential suitability as a microphone. Future Work A number of improvements can be made in the design of these devices. The fabrication process flow need s to be modified to avoid the platinum liftoff issue and resulting short circuits. Furthermore, more analysis and optimization of the residual stress in the various layers is needed, in particular for the TiO2. A reduction in this residual stress would improve performance by reducing the diaphragm stiffness, lowering the resonant frequency to a more testable range and ultimately allowing for comparison to a well understood linear model. Alterna tively, a modified process allowing for the PAGE 171 157 removal of most of the TiO2 would achieve the same results Additional work is also needed on developing a nonlinear ex tension to the model to facilitate the analysis of the existing structures. Other future work involves a more thor ough investigation of this device as a microphone, including a rigorous microphone calibration. Additionally, other applications need to be explored furthe r, including the use of the devices as a piezoelectric actuator and speaker, as well as the potential for active acoustic impedance control. PAGE 172 158 APPENDIX A MATLAB CODE lem.m clear all ; close all ; eh_setup; %Helmholtz Resonator Ma=(rho*Leff)/(pi*r_neck^2); %acoustic mass of neck Ca=volume/(rho*c^2); %acoustic compliance of cavity Ra=(40*8*mu*L)/(pi*r_neck^4); %acoustic resistance of neck % Readinput; silicon_sweep; %Reading shim properties platinum; %Reading platinum properties tio2; %Reading silicon doxide properties piezo_sweep; %Reading piezo props from console R21= R1/R2; % Give the values PP = 1; VV = 1; h_ca = 2e-4; density_air = 1.165; sound_velocity = 344; initialise; %Initialising sampling etc. N_in = .001; %Initial in-plane stresses in the inner region N_out = sigma02*t02+sigmap*tp; %Initial in-plane stresses in the outer region % Calculate the total deflection for the pressure loading totaldeflection_forP; % Calculate the total deflection for the voltage loading totaldeflection_forV; PAGE 173 159 % Calculate the initial deflection initialdeflection; % Incremental deflection for pressure loading Wi = W1p W0vp; formmatrix_CMFa; % Voltage loading only Wiv = W1v W0vp; frommatrix_dA; k = sqrt((dA^2)/(Cefp*CaD)); %Calculate the value of 1-k^2 PHI =-dA/CaD; %Electro-acoustic Xduction coefficient PHIprime =-dA/Cefp; Ceb =Cefp*(1-k^2); Rrad_RES=(FREQ*2*pi*R2/c)^2*rho*c/(2*Area); %rad resistance at resonance RloadA=Rrad_RES; %Equivalent Circuit Zrad=tf([0 Rrad_RES],1); %radiation impedance ZaD=tf([MaD 0],1)+tf(1,[CaD 0]); %impedance of diaphragm Zeb=tf(1,[Ceb 0]); %blocked electrical impedance Zload1=tf(RloadA+Ra,1); ZL=tf([0 0],1); ZRopen=tf(10^9,1); ZRshort=tf(1,1); Zload2=tf(RloadA,1); Zneck=tf([Ma Ra],1); %impedance of neck Zcav=tf(1,[Ca 0]); %impedance of cavity Ztube=rho*c/Atube; wnHR=1/sqrt(Ma*Ca); fHR=wnHR/(2*pi); wnD=1/sqrt(MaD*CaD); fD=wnD/(2*pi); % alpha=Ma/MaD; Pmax=2.3*Esi/(R2/tsi)^4 PmaxdBSPL=20*log10(Pmax/20e-6) P=1; Ze1=Zeb*Zload1/(Zeb+Zload1); Ze1short=Zeb*ZRshort/(Zeb+ZRshort); PAGE 174 160 Ze2=Zeb*Zload2/(Zeb+Zload2); Zin1=Zneck+Zcav*(ZaD+Zrad+PHI^2*Ze1)/(Zcav+ZaD+Zrad+PHI^2*Ze1); Zin2=ZaD+Zrad+PHI^2*Ze2; Zin2short=ZaD+Zrad+PHI^2*Ze1short; Zout1=Zeb*((ZaD+Zrad+Zcav*Zneck/(Zcav+Zneck))/PHI^2)/(Zeb+(ZaD+Zrad+Zca v*Zneck/(Zcav+Zneck))/PHI^2); Zout2=Zeb*((ZaD+Zrad)/PHI^2)/(Zeb+(ZaD+Zrad)/PHI^2); Powin1=P^2/Zin1; Powin2=P^2/Zin2; %Frequency response Vout1=PHI*Ze1*P/(((Zneck/Zcav)+1)*(ZaD+Zrad+PHI^2*Ze1)+Zneck); %voltage generated by diaphragm on HR Vout2=PHI*Ze2*P/(ZaD+Zrad+PHI^2*Ze2); %voltage generated by diaphragm alone Voutmax1=PHI*Ze1*Pmax/(((Zneck/Zcav)+1)*(ZaD+Zrad+PHI^2*Ze1)+Zneck); %max voltage generated by diaphragm on HR Voutmax2=PHI*Ze2*Pmax/(ZaD+Zrad+PHI^2*Ze2); %max voltage generated by diaphragm alone Sens1Open=PHI*Ze1/(((Zneck/Zcav)+1)*(ZaD+Zrad+PHI^2*Ze1)+Zneck); %sensitivity of diaphragm on HR Sens2Open=PHI*Ze1/(ZaD+Zrad+PHI^2*Ze1); %sensitivity of diaphragm alone Sens1Short=PHI*Ze1short/(((Zneck/Zcav)+1)*(ZaD+Zrad+PHI^2*Ze1short)+Zne ck); %sensitivity of diaphragm on HR Sens2Short=PHI*Ze1short/(ZaD+Zrad+PHI^2*Ze1short); %sensitivity of diaphragm alone PA1=Zcav/(Zneck+Zcav); Pow1=Vout1^2/Ze1; Pow2=Vout2^2/Ze2; PowAvail=P^2/Ztube; Eff1=Pow1/Powin1; Eff2=Pow2/Powin2; PowMax1=Voutmax1^2/Ze1; PowMax2=Voutmax2^2/Ze1; bode(Vout1,Vout2); F=linspace(0,7000,100); W=2*pi*F; V1=freqresp(Vout1,W); V2=freqresp(Vout2,W); VMAX1=freqresp(Voutmax1,W); VMAX2=freqresp(Voutmax2,W); P1=freqresp(Pow1,W); P2=freqresp(Pow2,W); POWMAX1=freqresp(PowMax1,W); POWMAX2=freqresp(PowMax2,W); PowIn1=freqresp(Powin1,W); PowIn2=freqresp(Powin2,W); PAGE 175 161 ZIn1=freqresp(Zin1,W); ZIn2=freqresp(Zin2,W); ZIn2short=freqresp(Zin2short,W); PAmp1=freqresp(PA1,W); SENS1OPEN=freqresp(Sens1Open,W); SENS2OPEN=freqresp(Sens2Open,W); SENS1SHORT=freqresp(Sens1Short,W); SENS2SHORT=freqresp(Sens2Short,W); EFF1=freqresp(Eff1,W); EFF2=freqresp(Eff2,W); PHIarray=ones(length(F),1)*PHI; PHIprimearray=ones(length(F),1)*PHIprime; ZOUT1=freqresp(Zout1,W); ZOUT2=freqresp(Zout2,W); figure(2); subplot(2,1,1) semilogy(F,abs(V1(1,:)),F,abs(V2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Mag(V_o_u_t) [V]' ); subplot(2,1,2); plot(F,angle(V1(1,:))*180/pi,F,angle(V2(1,:))*180/pi); xlabel( 'Frequency [Hz]' ); ylabel( 'Phase(V_o_u_t) [Deg]' ); legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ); figure(3); subplot(2,1,1) semilogy(F,abs(P1(1,:)),F,abs(P2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Mag(\pi_o_u_t) [W]' ); subplot(2,1,2); plot(F,angle(P1(1,:))*180/pi,F,angle(P2(1,:))*180/pi); xlabel( 'Frequency [Hz]' ); ylabel( 'Phase(\pi_o_u_t) [Deg]' ); legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ); hold on ; figure(4); subplot(2,1,1); semilogy(F,real(ZOUT1(1,:)),F,real(ZOUT2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Re(Z_o_u_t) [Ohms]' ); subplot(2,1,2); plot(F,imag(ZOUT1(1,:)),F,imag(ZOUT2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Im(Z_o_u_t) [Ohms]' ); legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ); figure(5) subplot(2,1,1); semilogy(F,abs(PowIn1(1,:)),F,abs(PowIn2(1,:))); PAGE 176 162 axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Mag(\pi_i_n) [W]' ); subplot(2,1,2); plot(F,angle(PowIn1(1,:))*180/pi,F,angle(PowIn2(1,:))*180/pi); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Phase(\pi_i_n) [Deg]' ); legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ); figure(6); subplot(2,1,1); semilogy(F,abs(EFF1(1,:)),F,abs(EFF2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Mag(Efficiency)' ); subplot(2,1,2); plot(F,angle(EFF1(1,:))*180/pi,F,angle(EFF2(1,:))*180/pi); axis tight ; xlabel( 'Frequency [Hz]' ); ylabel( 'Phase(Efficiency) [Deg]' ); legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ); figure(7) subplot(2,1,1) semilogy(F,real(ZIn1(1,:)),F,real(ZIn2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ) ylabel( 'Input Impedance Real [kg/m^4*s]' ) legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ) subplot(2,1,2); plot(F,imag(ZIn1(1,:)),F,imag(ZIn2(1,:))); axis tight ; xlabel( 'Frequency [Hz]' ) ylabel( 'Input Impedance imag [kg/m^4*s]' ) legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ) figure(8) subplot(2,1,1); semilogy(F,abs(PAmp1(1,:))); xlabel( 'Frequency [Hz]' ); ylabel( 'Magnitude [dB]' ); subplot(2,1,2); plot(F,angle(PAmp1(1,:))*180/pi); xlabel( 'Frequency [Hz]' ); ylabel( 'Phase [deg]' ); figure(9); semilogy(F,abs(SENS2OPEN(1,:)),F,abs(SENS2SHORT(1,:)),F,1./abs(PHIarray ),F,abs(PHIprimearray)); figure(10) semilogy(F,imag(ZOUT1(1,:)),F,imag(ZOUT2(1,:))); axis tight ; xlabel('Frequency [Hz]' ) PAGE 177 163 ylabel( 'Output Impedance [kg/m^4*s]' ) legend( 'with Helmholtz resonator' 'without Helmholtz resonator' ) eh_setup.m clear all ; %Constants rho=1.165; %density of air mu=1.789E-5; %viscosity of air c=344; %speed of sound in air eps0=8.85E-12; %free-space permittivity %Vent/Neck L=3.18e-3; r_neck=(4.72E-3)/2; Leff=L+1.6*r_neck; %effective neck/vent length %Resonator Cavity r_cav=5E-3; depth=2E-3; area_cav=pi*r_cav^2; volume=1950E-9; %Piezo-Diaphragm tsi = 3E-6; %thickness of silicon R2 = 1.95E-3; % radius of silicon %R2=3e-3; tp = 0.6E-6; %thickness of piezo %tp = 10e-6; R1 = .95*R2; %inner radius of piezo %Impedance tube Atube=(1*2.54e-2)^2; silicon_sweep.m Esi = 169e9; %youngs modulus of shim vsi = 0.3; %poissons ratio of shim mu = 1; %const of proportionality of shear stress vs shear strain densitysi = 2500; %Density of the shim platinum.m %all distances in um %http://www.webelements.com/webelements/elements/text/Ag/phys.html Epl=170E9; vpl=0.38; tpl=170e-9; densitypl=21440; tio2.m % tio2 %all distances in um PAGE 178 164 %http://www.webelements.com/webelements/elements/text/Ag/phys.html E02=283E9; v02=0.28; t02=.1e-6; density02=2150; sigma02=1200e6; piezo_sweep.m %Material Properties of the piezo %all distances in um Ep = 30e9; %Youngs modulus of the piezo vp = 0.3; %Poissons ratio of the piezo material d31 = -50e-12; %electromechanical transduction const of the piezo densityp = 7600; %Density of the piezo epsilon0 = 8.85E-12; %permitivity of free space in F/m dielectricconstant = 1000; %relative permitivity of the piezo epsilon=dielectricconstant*epsilon0; %absolute permitivity of the piezo sigmap = 100E6; initialise.m num1 = 125; r = linspace(0,1,num1+1); deltar= r(2)-r(1); sr=size(r); theta=zeros(sr); totaldeflection_forP.m %%%% totaldeflection_forP.m %%%%%%%%%% P = -PP; V = 0; Ef = V/tp; %pressure loading only abdpiezo; %Calculate the stiffness matrix A B D constants1234; %Obtain the constants c1 c2 c3 c5 solver1; W1p = w; totaldeflection_forV.m %%%%%%%% totaldeflection_forV.m %%%%%%%%%% V = VV; %electric field in V applied in direction 3(Z) PAGE 179 165 Ef = V/tp; P = 0; %voltage loading only abdpiezo; %Calculate the stiffness matrix A B D constants1234; %Obtain the constants c1 c2 c3 c5 solver1; W1v = w; initialdeflection.m P = 0; V = 0; Ef = V/tp; %no pressure and voltage abdpiezo; %Calculate the stiffness matrix A B D constants1234; %Obtain the constants c1 c2 c3 c5 solver1; W0vp = w; formmatrix_CMFa.m %%%%%% formmatrix_CMFa.m %%%%%%%% % Intergrate the deflection to obtain energy, WW and total deflection jj=floor(R21*num1)+1; for i=1:num1+1 rad1=r(i)*R2; if (i PAGE 180 166 MaD = 2*pi*wwtot/(wtotal^2); % acoustical mass FREQ = sqrt(1/(CaD*MaD))/2/pi; % resonant frequency %R1byR2(index) = R21; % for the graphs Areap = pi*(R2^2-R1^2); % area of the pzt layer Cefp = epsilon*pi*(R2^2-R1^2)/tp; frommatrix_dA.m %%%%%%% frommatrix_dA.m %%%%%%%%%% % Intergrate the deflection to obtain energy, WW and total deflection jj=floor(R21*num1)+1; for i=1:num1+1 rad1=r(i)*R2; if (i PAGE 181 167 zout6 = t02 + tpl + tp + tpl + tsi/2; % A B D Matrixs for inner and outer A_in = Qsi.*(zin2-zin1) + Q02.*(zin3-zin2); B_in = Qsi.*((zin2^2-zin1^2)/2) + Q02.*((zin3^2-zin2^2)/2); D_in = Qsi.*((zin2^3-zin1^3)/3) + Q02.*((zin3^3-zin2^3)/3); A_out = Qsi.*(zout2-zout1) + Q02.*(zout3-zout2) + Qpl.*(zout4-zout3) + Qp.*(zout5-zout4) +Qpl.*(zout6zout5); B_out = Qsi.*((zout2^2-zout1^2)/2) + Q02.*((zout3^2-zout2^2)/2) + Qpl.*((zout4^2-zout3^2)/2)+ Qp.*((zout5^2-zout4^2)/2)+Qpl.*((zout6^2zout5^2)/2); D_out = Qsi.*((zout2^3-zout1^3)/3) + Q02.*((zout3^3-zout2^3)/3) + Qpl.*((zout4^3-zout3^3)/3)+ Qp.*((zout5^3-zout4^3)/3)+Qpl.*((zout6^3zout5^3)/3); % Computing D Mark(determinant of matrix mapping defined variables y1,y2 to U0 theta Dstar_in = D_in(1,1)-(B_in(1,1)^2)/A_in(1,1); Dstar_out = D_out(1,1)-(B_out(1,1)^2)/A_out(1,1); % Computing fictitious forces due to piezo Mp_in = 0; Np_in = 0; Mp_out =-Ef* (Ep/(1-vp)) d31 (zout5^2-zout4^2)/2; Np_out =-Ef* (Ep/(1-vp)) d31 (zout5-zout4); constants1234.m %%%%% constants1235.m %%%%%%% k_in = sqrt(N_in*(R1^2)/Dstar_in); k_out = sqrt(N_out*(R2^2)/Dstar_out); a_R = R1/R2; b_jy = besseli(1,k_out)/besselk(1,k_out); Gamma_out = B_out(1,1)/A_out(1,1); Gamma_in = B_in(1,1)/A_in(1,1); AB11_in =-(Gamma_in*A_in(1,1)-B_in(1,1))*k_in/2/R1; AB11_out =-(Gamma_out*A_out(1,1)-B_out(1,1))*k_out/2/R2; AB12_in =-(Gamma_in*A_in(1,2)-B_in(1,2))/R1; AB12_out =-(Gamma_out*A_out(1,2)-B_out(1,2))/R1; BD11_in =-(Gamma_in*B_in(1,1)-D_in(1,1))*k_in/2/R1; BD11_out =-(Gamma_out*B_out(1,1)-D_out(1,1))*k_out/2/R2; BD12_in =-(Gamma_in*B_in(1,2)-D_in(1,2))/R1; BD12_out =-(Gamma_out*B_out(1,2)-D_out(1,2))/R1; x_p = P*R2/2/N_out/besselk(1,k_out); % Compute the matrix A11 = besselj(1,k_in); A12 = 0; A13 = b_jy*besselk(1,k_out*a_R) besseli(1,k_out*a_R); A14 = 0; PAGE 182 168 A21 =-Gamma_in*besselj(1,k_in) ; A22 = R1; A23 = (-b_jy*besselk(1,k_out*a_R) + besseli(1,k_out*a_R))*Gamma_out; A24 = R1*(1/(a_R^2)-1); A31 = AB11_in*(besselj(0,k_in) besselj(2,k_in)) + AB12_in*besselj(1,k_in); A32 = A_in(1,1)+A_in(1,2); A33 = AB11_out*((besseli(0,k_out*a_R)+besseli(2,k_out*a_R))+b_jy*(besselk(0,k _out*a_R)+besselk(2,k_out*a_R))) ... AB12_out*(besseli(1,k_out*a_R)-b_jy*besselk(1,k_out*a_R)) ; A34 = A_out(1,2)*(1/(a_R^2)-1) A_out(1,1)*(1+1/(a_R^2)); A41 = BD11_in*(besselj(0,k_in)-besselj(2,k_in)) + BD12_in*besselj(1,k_in); A42 = B_in(1,1) + B_in(1,2) ; A43 = BD11_out*((besseli(0,k_out*a_R)+besseli(2,k_out*a_R))+b_jy*(besselk(0,k _out*a_R)+besselk(2,k_out*a_R))) ... BD12_out*(besseli(1,k_out*a_R)-b_jy*besselk(1,k_out*a_R)) ; A44 = B_out(1,2)*(1/(a_R^2)-1) B_out(1,1)*(1/(a_R^2) + 1); b1 =-P*R2*besselk(1,k_out*a_R)/2/N_out/besselk(1,k_out) + P*R1/N_out/2 + P*R1/2/N_in; b2 = (P*R2*Gamma_out/2/N_out)*(besselk(1,k_out*a_R)/besselk(1,k_out)1/a_R); b3 = x_p*AB11_out*(besselk(0,k_out*a_R)+besselk(2,k_out*a_R)) x_p*AB12_out*besselk(1,k_out*a_R) + (P/2/N_out)*(B_out(1,2)+ B_out(1,1) ... +(Gamma_out/(a_R^2))*(A_out(1,1)-A_out(1,2))) + (P/2/N_in)*(B_in(1,1)+B_in(1,2)) Np_out + Np_in + (N_out); b4 = x_p*BD11_out*(besselk(0,k_out*a_R)+besselk(2,k_out*a_R)) x_p*BD12_out*besselk(1,k_out*a_R) + (P/2/N_out)*(D_out(1,2)+D_out(1,1) + ... (Gamma_out/(a_R^2))*(B_out(1,1)-B_out(1,2))) +(P/2/N_in)*(D_in(1,1)+D_in(1,2)) Mp_out + Mp_in + (N_out)*(tp+tsi)/2; A = [A11 A12 A13 A14; A21 A22 A23 A24; A31 A32 A33 A34; A41 A42 A43 A44]; b = [b1 b2 b3 b4]'; c1234 = inv(A)*b; c1=c1234(1); c2=c1234(2); c3=c1234(3); c4=c1234(4); solver1.m % Composite plates; % Computing deflections and Forces at the interface by superposition % Finding Deflection in the central region and the annular region sr=size(r); PAGE 183 169 w=zeros(sr); jj=floor(R21*num1)+1; for i=1:num1+1 rad1=r(i)*R2; if (i PAGE 184 170 APPENDIX B DETAILED PROCESS FLOW Energy Harvester Process Traveler Table B-1: Process Traveler Wafer 4" n-type <100> SOI 6 wafers Step # Lab/Equip Process Description 0 Start with SOI wafer with 4000 oxide + 12 um of silicon overlayer. 1.1 Sandia National Labs (SNL) Reduce silicon overlayer using KOH (~1.7 minm ) and HNA (~2.6 minm ) etches. KOH: (Temp ~ 85 C) 3 wafers 320 s etch expected thickness of Si = 2.933 m 2 wafers 270 s etch expected thickness of Si = 4.350 m 3 wafers 212 s etch expected thickness of Si = 5.993 m HNA: Recipe 160 ml acetic acid, 60 ml nitric acid, 20 ml hydrofluoric acid 1 wafer 211 s etch expected thickness of Si = 2.86 m 1 wafer 176 s etch expected thickness of Si = 4.37 m 1.3 SNL Deposit 100 nm Ti Temescal BJD-1800 Evaporator (modified) Conditions: 772.1102.310atmPTorr Ti deposition rate = 3 sec Total thickness = 1000 1.4 SNL Oxidize in tube furnace at 650 C for 30 mins with dry 2O ambient to form 2TiO 2.1 SNL spin positive photoresist (Clariant AZP4330-RS)on back surface (1 m) positive resist and pattern backside alignment marks, (BS_Metal) Mask DARK FIELD evaporated HMDS onto wafer for adhesion spin @ 4000 RPM for 30 s baked @ 90 C for 90 s on hot plate yields 3.75 m thick resist 2.2 SNL Sputter deposit 15 nm of Cr on backside followed by 40 nm of Pt Conditions: 76.310atmPTorr Deposition rate = 3 sec PAGE 185 171 2.3 SNL Strip resist in Acetone for 2 min to liftoff Cr Ash resist residue in Oxygen plasma stripper 2.4 SNL spin negative photoresist (JSR) on front surface (1 m) negative resist and pattern bottom electrode, (ElectrodeBot) Mask DARK FIELD spin @ 3000 RPM for 30 s bake @ 120 C for 1 min pattern and expose ( Front to back alignment ) using Karl Suss MA6/BA6 Backside Aligner develop in AZ400K for 90 s bake @ 100 C for 45 s descum in Oxygen plasma (PDS/PDE-301) for 5 min 2.5 SNL Evaporate 30 nm of Ti in Temescal BJD-1800 Evaporator for adhesion of Pt to TiO2 @ 3 sec for 100 s 2.6 SNL Evaporate 170 nm of Pt in Temescal BJD-1800 @ 2.5 sec for 680 s 2.7 SNL Strip resist to liftoff Ti/Pt in acetone 3.1 SNL Spin coat 52/48 PZT solution @ 3000 RPM for 30 sec 3.2 SNL Pyrolize @ 300 C for 5 min on a hot-plate in an air ambient 3.3 SNL Repeat spin/pyrolize 4 times to achieve 400 nm thick PZT 3.4 SNL Furnace anneal at 650 C for 30 min 4.1 SNL spin negative photoresist (JSR) on front surface (1 m) negative resist and pattern top electrode, (ElectrodeTop) Mask DARK FIELD spin @ 3000 RPM for 30 s bake @ 120 C for 1 min pattern and expose ( Front to back alignment ) using Karl Suss MA6/BA6 Backside Aligner develop in AZ400K for 90 s bake @ 100 C for 45 s descum in Oxygen plasma (PDS/PDE-301) for 5 min 4.2 SNL Deposit 180 nm Pt in Temescal BJD-1800 @ 2.5 sec for 720 s. 4.3 SNL Strip resist to liftoff Pt in acetone 5.1 Army Research Lab (ARL) Etch PZT in 3:1:1 ammonium biflouride/hydrochloric acid/DI water uses Pt top electrode as etch mask 5.2 ARL Etch PZT residues left behind by previous etch dilute nitric acid/hydrogen peroxide etchant 6.1 UF Spin thick photoresist on backside positive resist and pattern cavity, (Cavity) Mask DARK FIELD 6.2 UF Plasma etch (DRIE) 500 um of silicon from backside PAGE 186 172 stops on BOX layer (STS MESC Multiplex ICP) 6.3 UF Strip resist in acetone. 6.4 UF Etch oxide (BOE) on backside to remove oxide mask and BOX Graphical Representati on of Process Flow 1000A BOX Figure B-1: Step 1: Deposit Ti on SOI Wafer and oxidize to TiO2. 1800A Figure B-2: Step 2: Deposit Ti/Pla tinum(30nm/170nm) -Lift-Off w/ Mask (ElectrodeBot). 4000A Figure B-3: Step 3: Spin PZT 6 times to achieve desired thickness. 1800A Figure B-4: Step 4: Depos it Platinum(180nm)-Lift-off w/ Mask (ElectrodeTop). 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