DEVELOPMENT OF AN ULTRASONIC PIEZOELECTRIC MEMS-BASED RADIATOR FOR NONLINEAR ACOUSTIC APPLICATIONS
By BENJAMIN ANDREW GRIFFIN
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
2009
1 °c 2009 Benjamin Andrew Griffin
2 To my wife, Elizabeth, with much love and appreciation.
Isaiah 40:31 ...but those who hope in the Lord will renew their strength.
They will soar on wings like eagles; they will run and not grow weary, they will walk and not be faint.
3 ACKNOWLEDGMENTS Financial support for this work has been provided by graduate fellowships from the National Science Foundation and the University of Florida. I thank my advisors, Mark
Sheplak and Louis N. Cattafesta III, for their many helpful technical discussions, as well as their career and personal advice. I am also grateful to my committee members, Havana V. Sanka, David Arnold, and Nab Ho Kim, for their expertise and assistance in the success of this project. I am especially grateful to my many colleagues in the Interdisciplinary Microsystems Group. I would like to thank my predecessors, Venkataraman Chandrasekaran and Guiqin
Wang, for establishing a firm foundation upon which this work was built. Former colleagues David Martin and Stephen Horowitz are greatly appreciated for their mentorship and train- ing during our concurrent association with the Interdisciplinary Microsystems Group. I have much gratitude for contemporaries Brian Homeijer and Vijay Chandrasekharan as we have “come of age” as graduate students together. Their engaging technical discussions, friend- ship, and comradery have been a sustaining force in my graduate career. I am also indebted to Matthew Williams, who I have worked closely with on this project. Without his addition to the Interdisciplinary Microsystems Group, this undertaking would not have been as suc- cessful. In addition, I am grateful to Chase Coffman, Dylan Alexander, and John Griffin for their assistance with experimental setups, package fabrication, and data acquisition. I would also like to acknowledge all of the Interdisciplinary Microsystems group whose contributions are too numerous to list. I am particularly thankful to Avago Technologies Limited for the access to their fab- rication facilities. Special thanks goes to David Martin and Osvaldo Buccafusca at Avago for their special attention and personal time devoted to this project. I also acknowledge Dynatex International for their skill in wafer separation. I am especially grateful for the excellent machining work performed by Ken Reed at TMR engineering. The Mechanical and Aerospace Engineering departmental staff is thanked for their kind assistance.
4 I thank my parents, Mike and Barbara Griffin, for instilling in me a good work ethic and perseverance; without which, this project would not have been as successful. I would also like to express appreciation to my brother John and sister-in-law Karen who never hesitated to provide any assistance and support they could give. Above all, I am grateful to my wife, Elizabeth, for her patience, encouragement, support, and love.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 4
LIST OF TABLES ...... 9
LIST OF FIGURES ...... 11
ABSTRACT ...... 17
CHAPTER 1 Introduction ...... 19 1.1 Parametric Arrays ...... 19 1.2 Transducer Issues ...... 22 1.2.1 Current Limitations ...... 23 1.2.2 Potential Transducer Solution ...... 24 1.3 Research Objectives ...... 24 1.4 Dissertation Overview ...... 24 2 Nonlinear Acoustics ...... 25 2.1 Finite Perturbation Acoustic Theory ...... 25 2.2 Parametric Array ...... 30 2.2.1 Model Equations of Nonlinear Acoustic Theory ...... 31 2.2.2 Sound Beam Solutions ...... 34 2.2.3 Existing Implementations ...... 38 2.2.4 MEMS Parametric Arrays ...... 47 2.3 Conclusion ...... 50
3 Air-Coupled MEMS Ultrasonic Transducers ...... 51 3.1 Principles of Transmitter Operation ...... 51 3.2 Acoustic Sources ...... 55 3.2.1 Planar Radiation ...... 55 3.2.2 Array of Sources ...... 61 3.2.3 Acoustic Attenuation ...... 63 3.2.4 Summary ...... 66 3.3 MEMS Actuators ...... 66 3.3.1 Electrostatic Transduction ...... 67 3.3.2 Piezoelectric Transduction ...... 80 3.3.3 Thermoelastic Actuation ...... 95 3.4 Conclusion ...... 100
6 4 Ultrasonic Radiator Design ...... 102 4.1 Avago’s FBAR Process ...... 102 4.2 Fabrication ...... 104 4.3 Package ...... 105 4.4 Conclusion ...... 108
5 Modeling ...... 109
5.1 Equivalent Circuit ...... 109 5.1.1 Acoustical Domain ...... 112 5.1.2 Electrical Domain ...... 118 5.1.3 Transduction ...... 118 5.1.4 Equivalent Circuit ...... 119 5.1.5 Approximate Performance ...... 121 5.1.6 Example Device ...... 122 5.2 Nonlinear Acoustic Modeling ...... 130 5.3 Conclusion ...... 132
6 Design Optimization ...... 133 6.1 Methodology ...... 133 6.2 Radiator Optimization ...... 134 6.2.1 Limitations-Constraints ...... 135 6.2.2 Problem Formulation ...... 137 6.3 Results ...... 138 6.4 Alternate Designs ...... 142 6.5 Conclusions ...... 143 7 Experimental Setup and Results ...... 145 7.1 Fabrication Results ...... 145 7.2 Electrical Characterization ...... 148 7.2.1 Setup ...... 148 7.2.2 Results ...... 149 7.3 Device Topography ...... 151 7.4 Electromechanical Characterization ...... 153 7.4.1 Setup ...... 155 7.4.2 Frequency Response Function ...... 157 7.4.3 Diaphragm Resonance ...... 159 7.4.4 Linearity ...... 160 7.4.5 Variable Back Cavity ...... 163 7.4.6 Vacuum Experiments ...... 167 7.5 Electroacoustic Characterization ...... 171 7.5.1 Setup ...... 171 7.5.2 Results ...... 174 7.6 Performance as a Parametric Array ...... 176
7 7.7 Conclusion ...... 178 8 Conclusion and Future Work ...... 180
8.1 Conclusions ...... 180 8.2 Recommendations for Future Work ...... 182 8.3 Recommendations for Future Design ...... 183
APPENDIX
A NONLINEAR ACOUSTIC MODELING ...... 186 A.1 Westervelt Parametric Array Solution ...... 186 A.2 Berktay Solution ...... 189 B PLATE MODEL ...... 191
B.1 Basic Assumptions ...... 192 B.2 Static Equilibrium ...... 192 B.3 Constitutive Equations ...... 194 B.4 Governing Differential Equations ...... 196 B.5 General Solution ...... 197 B.6 Boundary and Matching Conditions ...... 198 B.7 Incremental Plate Deflection ...... 200
C UNCERTAINTY ANALYSIS ...... 201 C.1 Electrical Characterization Uncertainties ...... 201 C.1.1 Electrical Impedance ...... 201 C.1.2 Element Extraction ...... 202 C.2 Electromechanical Characterization Uncertainties ...... 206 C.2.1 Velocity ...... 206 C.2.2 Volume Velocity ...... 207 C.2.3 Resonant Frequency ...... 208 C.2.4 Damping Coefficient Estimation ...... 208 C.2.5 Variable Back Cavity ...... 210 REFERENCES ...... 213
BIOGRAPHICAL SKETCH ...... 227
8 LIST OF TABLES Table page 2-1 CMUT transducer array specifications and performance presented by Wygant et al. [15]...... 48
2-2 Transducers used in parametric array experiments...... 49 3-1 Air-coupled cMUT characteristics...... 81
3-2 Piezoelectric film properties...... 87
3-3 Air-coupled pMUT characteristics...... 96 3-4 Thermoelastic MEMS characteristics...... 100 4-1 Typical epoxy dispense and cure parameters used for die attachment. PCB boards were pre-heated at 145 ◦C for 5 min. before epoxy application...... 106 4-2 Wirebonder settings...... 106 5-1 Geometry of example device...... 123 5-2 Material properties of AlN and Mo...... 123
5-3 Comparison of electrical element fit to full model...... 129 6-1 Avago Technologies Limited process options where j = 1, 2 refers to the inner and annular plate sections, respectively...... 135 6-2 Design optimization results...... 140
6-3 Alternate optimization results...... 143 7-1 Film stress target and realized values for the wafer provided by Avago...... 146 7-2 Objective function of device designs in dB referenced to the target design, UFPA08. Blanks refer to devices that violate modeling constraints...... 147 7-3 The extract impedance parameters from the experimental impedance measurement.151 7-4 Damping and quality factor values for the different devices at vacuum and STP. 169
7-5 Performance of the die. The center deflection after back cavity tuning is shown in parentheses...... 179 8-1 Comparison of air-coupled MEMS radiator performances...... 182
C-1 Frequency bounds on system fit...... 210 C-2 Uncertainties in the resonant frequency and response as a function of measure- ment uncertainties...... 211
9 C-3 Screw depth measurement with uncertainties...... 212
10 LIST OF FIGURES Figure page 1-1 Observer of a museum display is listening to commentary while disinterested pedestrians pass by in quiet (adapted from Holosonics [3])...... 20
1-2 Sound fields of a conventionally sized radiator at audio and ultrasonic frequencies (adapted from Holosonics [3])...... 21
2-1 Wave steepening due to nonlinear wave propagation. Adapted from Blackstock [8]...... 29 2-2 The frequency distribution of a nonlinearly propagating wave...... 30 2-3 Parametric array radiation of sound (adapted from Holosonics) [3]...... 31
2-4 The sound field and length scales associated with the sound beam solutions. Note that this figure is highly schematic...... 35 2-5 ATC bimorph piezoelectric actuator with attached cone for enhanced volume velocity [11]...... 44 3-1 Generic acoustic transmitter [49]...... 51 3-2 Bulk acoustic transmitter...... 52 3-3 Generic bending mode acoustic transmitter [49]...... 52
3-4 Frequency response of an ultrasonic transmitter...... 53 3-5 Comparison of the ideal and physical transducer sensitivities versus the forcing voltage for a fixed frequency...... 54 3-6 Speaker modeled as a baffled piston...... 56
3-7 Arbitrarily shaped baffled piston (adapted from Blackstock [8])...... 57 3-8 Circular piston in an infinite baffle (adapted from Blackstock [8])...... 57 3-9 Comparison of the directivity of a compact and non-compact source...... 59
3-10 Radiation impedance of a baffled, circular piston...... 60
3-11 Line array of N ideal monopoles (adapted from Blackstock [8])...... 62
3-12 Array directivity for 3 and 7 monopole arrays at kd = 1...... 63 3-13 Sound absorption coefficient of air at 70% relative humidity (after Zuckerwar [54]). 65
3-14 One-dimensional electrostatic transducer...... 68
11 3-15 Pull-in point where the change in the electrostatic force is equal to the change in the mechanical force...... 70
3-16 Plot of the mechanical force, FM , and the electrostatic force, FE, at different voltages versus the gap distance x (Note that all values are non-dimensional). . 71 3-17 Device formed by combining macro- and micromachining (adapted from Higuchi [59])...... 72
3-18 Capacitive cMUT of the E.L. Ginzton Laboratory (adapted from Haller [66]). . 73
3-19 Nitride diaphragm cMUT with vacuum sealed back cavity (adapted from Jin et al. [71])...... 74
3-20 Capacitive transducer with a polysilicon diaphragm and doped bottom electrode (adapted from Eccardt et al. [78])...... 75
3-21 Micromachined capacitive device with coupled Helmholtz resonator formed by the resonant cavity and throat (adapted from Parviz et al. [81])...... 76 3-22 A cMUT whose cavities are formed by an anisotropic etch (adapted from Torndahl et al. [82])...... 77 3-23 Capacitive transducer with polysilicon moveable membranes (adapted from Buhrdorf et al. [83])...... 77 3-24 A cMUT fabricated using MUMPS (adapted from Oppenheim et al. [84]). ... 77 3-25 A cMUT fabricated using an SOI wafer bonded to a patterned substrate (adapted from Huang et al. [85,86])...... 78 3-26 A cMUT with nitride diaphragm (adapted from Kim et al. [90])...... 79 3-27 Isometric view of an ideal perovskite structure. [91]...... 82
3-28 Isometric views of ideal wurtzite and perovskite structure. [91]...... 83
3-29 One-dimensional piezoelectric transducer (adapted from [16])...... 84
3-30 Effect of the d31 coefficient in piezoelectric thin films...... 85 3-31 Hysteresis loops of piezoelectric materials (adapted from [91]) ...... 87
3-32 Two side-by-side pMUTs formed by a nitride diaphragm and ZnO annular ring (adapted from Percin et al. [115])...... 88
3-33 Dome shaped pMUT with nitride diaphragm and ZnO piezoelectric layer (adapted from Cheol-Hyun et al. [100])...... 88 3-34 Square diaphragm pMUT using PZT (adapted from Mohamed et al. [102,120]). 89
12 3-35 Square diaphragm pMUT using PZT (adapted from Lee et al. [121])...... 90 3-36 PZT microspeaker (adapted from Zhu et al. [124])...... 90
3-37 PZT diaphragm (adapted from Zhu et al. [131])...... 91
3-38 PZT microspeaker (adapted from Zhu et al. [53])...... 92 3-39 Square diaphragm pMUT utilizing a P(VDF-TrFE) film for actuation (adapted from Lam [101])...... 93 3-40 Oxide diaphragm formed by the BOX layer of an SOI wafer and actuating by a PZT film (adapted from [103])...... 93
3-41 A pMUT used in an energy harvester (adapted from [46])...... 94 3-42 An AlN audio microspeaker pMUT (adapted from [135])...... 94
3-43 Proximity sensor that utilizes thermoelastic actuation and piezoresistive sensing (adapted from Brand [138])...... 98
3-44 Thermoelastic actuator with a buckled diaphragm (adapted from Popescu et al. [147]...... 98 3-45 Thermoelastic/piezoresistive proximity sensor that uses the device layer of an SOI wafer to form a circular diaphragm (adapted from Chandrasekaran et al. [148]). 99 3-46 Thermoelastic proximity sensor using polysilicon for the heater and piezoresistors (adapted from Rufer et al. [149])...... 99
4-1 Cross-section of the structural layers and back cavity possible using the Avago FBAR process...... 103 4-2 MEMS- based ultrasonic radiator for parametric array applications...... 104 4-3 PCB board for device package...... 105 4-4 The aluminum block containing variable back cavity, dowels, screw holes for align- ment to the PCB, and an anchor screw hole...... 107
4-5 Packaged device mounted to variable back cavity...... 107 5-1 Diagram of incremental diaphragm deflection...... 110
5-2 Equivalent circuit model...... 111 5-3 Rigid duct model...... 115
5-4 The back cavity of the transducer with the die, pcb, and aluminum sections. . 116 5-5 Circuit representation of the back cavity using transfer matrices...... 116
13 5-6 Edge of the diaphragm showing the front to back cavity vent (not to scale). .. 118 5-7 Electrical elements of the equivalent circuit...... 118
5-8 Two-port representation of electro-acoustic transduction...... 120
5-9 Equivalent acoustic circuit model...... 120 5-10 Volume velocity frequency response calculated using the full sensitivity equivalent circuit and the high frequency approximation...... 124 5-11 Total impedance of the vent and cavity in parallel versus just the cavity impedance...... 124
5-12 Diaphragm impedance comparison to the radiation impedance...... 125 5-13 Contributions to the overall acoustic impedance...... 125
5-14 Input and the electrical impedance comparison...... 126 5-15 Γ factor of the input impedance...... 126
5-16 Radiation impedance comparison of the full model and LEM...... 127 5-17 Back cavity impedance comparison of the full model and LEM...... 127 5-18 Input impedance comparison of the full model and LEM...... 128 5-19 Full model and curve fit comparison assuming a LEM of the input impedance. 129
5-20 Diagram of transducer array showing radius definitions (Not drawn to scale). . 130 5-21 Output of the example parametric array at 1 m...... 132 6-1 Output of the parametric array at 1 m...... 138
6-2 Sensitivity of the normalized objective function to the normalized design vari- ables...... 140 6-3 Sensitivity of the objective function due to changes in the individual design vari- ables show active constraints and bounds...... 141
6-4 Sensitivity of the objective function to the nonlinear deflection constraint. ... 142 6-5 Sensitivity of the objective function to the minimum resonant frequency con- straint...... 142
7-1 Reticle labeling convention...... 148 7-2 The front panel connections on the HP 4294A Impedance Analyzer where red is the high connection, black is the low connection, and green is ground...... 149
14 7-3 The real and imaginary part of the electrical impedance displaying both the experimental data for all transducers and the theoretical model...... 150
7-4 Initial diaphragm deflections...... 152
7-5 Scanning laser vibrometer system used for electromechanical characterization. . 154
7-6 The scan grid overlayed with the microscope picture of the device...... 157
7-7 Velocity frequency response function for the center grid point, where the uncer- tainty is the 95% confidence in the mean estimate...... 158 7-8 Displacement frequency response function for the center grid point, where the uncertainty is the 95% confidence in the mean estimate...... 158 7-9 Volume velocity sensitivity, where the uncertainty is the 95% confidence in the mean estimate...... 159 7-10 Displacement sensitivity cross-sections...... 160 7-11 Resonant mode shapes...... 161 7-12 The velocity sensitivity versus excitation voltage...... 162
7-13 Velocity and displacement response versus resonant tone excitation amplitude. . 163 7-14 Measurements of the back cavity...... 166 7-15 Ideal piston and back cavity...... 166 7-16 Imaginary parts of the diaphragm impedance and cavity impedances of varying back cavity depth...... 167 7-17 Vacuum chamber diagram...... 168 7-18 Device comparison of the frequency response function between rough vacuum and STP conditions...... 171 7-19 Resonant performance reference to STP versus increase vacuum chamber pressure.172
7-20 Setup for the electroacoustic characterization...... 172 7-21 Directivity measurements along with the predicted directivities found by extrap- olating the LV measurements using Rayleigh’s integral...... 175 7-22 On axis pressure response...... 177
7-23 Parametric array output at 1 m of a an array of 4,500 radiators...... 178 8-1 Stress fields inside a deflected clamped plate...... 184
8-2 Alternate design for improved performance...... 184
15 B-1 Cross sections that show the correspondence between the plate model and the device...... 191
B-2 Isometric view of an infinitesimal plate element. Three layers are shown for illustration but the plate element is considered to have an arbitrary number of layers in the derivation of the governing equations [161]...... 193
B-3 Diagram of incremental diaphragm deflection...... 200
C-1 Random and bias errors in the real and imaginary parts of the impedance. ... 202
C-2 Impedance fit in comparison to the experimental data...... 205 C-3 Schematic of the uncertainty in the resonant frequency calculation...... 208 C-4 Damping coefficient curve fit...... 209
16 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 AN ULTRASONIC PIEZOELECTRIC MEMS-BASED RADIATOR FOR NONLINEAR ACOUSTIC APPLICATIONS By Benjamin Andrew Griffin May 2009 Chair: Mark Sheplak Cochair: Louis N. Cattafesta III Major: Mechanical Engineering
The development of a piezoelectric micromachined ultrasonic transmitter is presented. The transducer is evaluated as an ultrasonic source for parametric arrays. A parametric array is an acoustic technology that leverages the nonlinear demodulation of ultrasound to create a highly directional beam of audible sound analogous to a flashlight.
The transmitter was formed by a circular composite diaphragm that was radially non- uniform. The diaphragm consisted of molybdenum annular electrodes and an aluminum
nitride diaphragm. The composite diaphragm was released using a combination of deep reactive ion and oxide etching. Transduction of the diaphragm occurred when an electric field was induced across the annular piezoelectric layer of aluminum nitride. The electric field caused mechanical strain within the piezoelectric layer through the piezoelectric effect. The strain coupled into force and moment resultants that generated diaphragm deflection. By supplying an ac voltage be- tween the electrodes, an oscillating electric field caused diaphragm vibration. The vibrating diaphragm in turn generated acoustic waves. The overall device model was formed using composite plate mechanics, lumped element
modeling (LEM), and acoustic theory. Through LEM an equivalent circuit of the device was formed that incorporated electrical, mechanical, and acoustic components. Nonlinear
17 acoustic theory was used to predict ultrasonic demodulation. The device performance model was used in a geometric constrained optimization scheme.
The optimal design based on the LEM was used to form the primary design. A series of secondary designs were formed by constraining the device layer thicknesses and performing optimization using radial geometry as design variables and considering deviations in stress. The device was fabricated at Avago Technologies Limited’s foundry. A unique package with back cavity depth control was designed and fabricated. This was followed by experimental characterization. Electrical characterization consisted of measurement of device impedance. Mechanical characterization included mode shape and resonant frequency measurements.
Acoustic characterization encompassed farfield acoustic measurements of a single device.
The characterization results showed significant mismatch between devices as well as the equivalent circuit model. Four out of the six devices tested had resonant frequencies near 60 kHz. The remaining two devices had resonant frequencies of 31 kHz and 44 kHz. The equivalent circuit predicted a resonance of 39 kHz. The variation between the results was attributed to stress variations across the wafer that occurred during fabrication. A variable back cavity was used to tune the devices and maximize sensitivity at resonance. A 220% improvement in the resonant deflection sensitivity of the 31 kHz resonant device was found by tuning the back cavity. A nonlinear acoustic calculation was used to project the performance of a 150 mm diameter device array as an acoustic source for a parametric array. The sound pressure level of a 5 kHz audible tone was 42 dB at 1 m. The low projected audible output does not show good promise of the application of this design to a conventional audible parametric array. Recommendations for future work focus on a more robust device design and packaging improvements for other ultrasonic applications.
18 CHAPTER 1 INTRODUCTION
The goal of this research effort was to develop a microelectromechanical systems (MEMS) based ultrasonic transmitter that was designed to serve as an element of a parametric acous- tic array sound source transducer. The parametric array is an acoustic technology analogous to a flashlight. In a parametric array, the distortion of high amplitude ultrasound creates a highly directional beam of audible sound, or an “Audio Spotlight” [1]. The sound beam is similar to a flashlight in that objects are only “illuminated” with sound if they are in the beam path.
This chapter begins with an introduction to parametric arrays. Next, general issues and limitations associated with existing parametric array sound source transducers are discussed, leading to the motivation for the development of a MEMS-based transducer. Finally, the research objectives and overall dissertation organization are given. 1.1 Parametric Arrays
The objective of conventional audio systems is to create an audio environment where all listeners, regardless of position with respect to the sound source, experience the same audio quality. An example is an automobile sound system which, when balanced, will fill an entire vehicle with sound, giving each passenger an essentially equivalent audio experience. Typical audio systems, however, have little ability to control the distribution of sound [2].
The control of sound distribution is useful in environments where audible communi- cation is needed but disturbance of bystanders is undesirable. An example is a museum display similar to Figure 1-1 where an observer of an exhibit listens to a recording on the finer points of the display while disinterested pedestrians pass by undisturbed [3]. The fol- lowing are other possible applications of parametric arrays: long distance communication similar to bullhorns [4], human-humanoid communication [5], drive through communica- tions perceptible only to the customer, nondestructive evaluation [6], active noise control systems [7], virtual headphones that provide a private audio experience without physical ear
19 attachments, automobile speakers where passenger and driver can enjoy different audio [3], military communications and weapons applications [3], etc.
Figure 1-1. Observer of a museum display is listening to commentary while disinterested pedestrians pass by in quiet (adapted from Holosonics [3]).
The ability of a single source to control the distribution of sound depends on the size of the source in comparison to the wavelength of the projected sound [8]. A source whose size is small relative to the projected wavelength will create an omnidirectional sound field that can be heard equally in all directions as shown in Figure 1-2. For example, a conventional audio system uses a single sub-woofer to generate bass frequencies on the order of 100 Hz or lower where the corresponding wavelengths are on the order of meters or larger [9]. Since the wavelengths are much larger than the sub-woofer, an omnidirectional sound field is generated.
In contrast, a conventional audio system uses multiple tweeters to generate frequencies on the order of 1 kHz or higher where the wavelengths are on the order of 10 centimeters or less [9]. At these frequencies the tweeter is no longer much smaller than the wavelengths, resulting in a sound field that is no longer omnidirectional. Thus, conventional audio systems use multiple tweeters that combine to create a more uniform sound distribution [9].
20 A source that is much larger than its projected wavelength, however, creates a highly directional sound field. A sub-woofer would have to be on the order of meters to project directional, low frequency audio wavelengths. For example, a sub-woofer with a diameter of
2.8 m is required to create a 1 kHz beam with a -6 dB beamwidth of 10 degrees (beamwidth is defined in Chapter 3). On the other hand, the projection of ultrasound in air with wavelengths measured in millimeters by a conventionally sized source can produce a beam. Ultrasound is defined as sound whose frequency range is greater than the upper limit of the human hearing range, 20 kHz [9]. For instance, a speaker 70 mm in diameter that generates ultrasound at approximately 40 kHz has a -6 dB beamwidth of 10 degrees.
Audio Ultrasound
Radiator
Figure 1-2. Sound fields of a conventionally sized radiator at audio and ultrasonic frequencies (adapted from Holosonics [3]).
Ultrasound, though, is not perceptible to the human ear, but it is possible to create audible sound from the distortion of ultrasound. High intensity ultrasound distorts as it
propagates due to convective and local heating effects [8]. The distorted wave can be de-
scribed by a Fourier series that consists of the summation of multiple harmonic components.
21 When two tones within 20 kHz of each other are projected by the source, the waveform dis- tortion produces audible frequencies. This process is predictable. In this manner, the beam of ultrasound acts as a source for the audible frequencies. Since the originating ultrasound is projected as a beam, the audio frequencies also propagate as a beam. A parametric ar- ray uses this nonlinear acoustic phenomenon to produce a controlled distribution of audible sound by projecting a beam of ultrasound. More details on parametric array operation can be found in Chapter 2. 1.2 Transducer Issues
A parametric array is usually not formed by a single source, but an array of many tiny sources [1]. The sources are small such that their fundamental resonance occurs at ultrasonic frequencies. A small single transducer, however, cannot produce enough sound for adequate parametric conversion. Therefore, the transducers are arrayed to increase the sound output. The projected sound of an array of small transducers, however, will be a beam of sound if the size of the array, or array aperture, is large with respect to the ultrasonic wavelengths [8]. This is due to constructive and destructive interference between the sound fields of the individual transducers [8].
Parametric array transducers usually operate in the lower ultrasonic range because sound absorption in air increases with frequency [1, 10–12]. Operating at lower ultrasonic frequencies maximizes the amount of power that is converted from the source to the audio frequencies. However, it is important for parametric arrays to stay away from the upper limit of human hearing frequency range as a safety precaution. Most parametric arrays operate around 40 kHz as a tradeoff between performance and safety [1,10]. There are specific desirable attributes of air-coupled ultrasonic transducers for para- metric array sources. The most important attribute is to have a large sound output at the fundamental tones. A large sound output increases the amount of energy available for con- version to the audible difference tone. In addition, a large number of radiators increases the overall sound output of the array. More transducers per area, known as packing density,
22 allows the array size to remain manageable. Matched device characteristics such as resonant frequency and phase are important to ensure that all elements are achieving their maximum output at the fundamental ultrasonic tones and that their sound fields add constructively
along the beam axis of the array. The efficiency of the conversion of energy from the ultra- sonic tones to the audible difference tone is very low as will be discussed in Chapter 2 [13]. Thus, it is important to have high emission efficiency with respect to the input electrical energy to ensure that the majority of the energy lost is in the ultrasonic to sonic conversion. A transducer with an adjustable resonant frequency is desirable so that the same transducer can achieve maximum sound output at multiple ultrasonic tones. For instance, half of the transducers in an array could emit at one resonant tone while the other half emit at another resonant tone. Also, it is difficult to fabricate and package a device with a resonant frequency that exactly matches the model prediction. Thus, it is beneficial to have the ability to adjust the transducer’s resonant frequency to match originally designed performance. It is ideal for the transducer to have the full human hearing range bandwidth about its resonant peak.
1.2.1 Current Limitations
At the present time, parametric arrays are constructed from conventional off-the-shelf
components such as piezoelectric disks or radical designs presented without modeling [1, 11, 14, 15]. There have been no optimal designs of transducers for parametric arrays in the literature presented in Chapter 2 at this junction. Such a design would have to meet unique challenges:
• Resonant frequency of approximately 40 kHz as a tradeoff between absorption of ul- trasonic frequencies and safety of human hearing
• Overcome low conversion efficiency between the amplitude of fundamental tones to the audible difference tone
• Large packing density
• Adequate ultrasonic sound pressure level bandwidth around its resonant peak to achieve audio bandwidth in the distorted signal
23 • Matched resonant frequency and phase characteristics 1.2.2 Potential Transducer Solution
One possible solution is a MEMS transducer that can leverage the substantial batch fabrication technology of the integrated circuits industry. Commonly, MEMS are formed from silicon wafers with both electrical and mechanical components. They range from 1- 1000’s of microns in size, allowing for large packing densities. Batch fabrication is a favorable technology when trying to match device characteristics. The fabrication technology also leads to reasonable costs since each wafer contains many devices and the wafers are typically fabricated in lots of multiple. Therefore, the cost of production of a single wafer is divided among the many devices it produces [16]. 1.3 Research Objectives
The goal of this dissertation is the optimal design, fabrication, and characterization of a first generation ultrasonic transmitter for parametric array applications. Experimental characterization results such as mechanical mode shapes and acoustic output are analyzed to determine viability of the transducer for a parametric array source. 1.4 Dissertation Overview
This dissertation is organized into 8 chapters. Chapter 1 presents the background, motivation, and goals of the project. Chapter 2 discusses nonlinear acoustics including theory and implementations of parametric arrays. Chapter 3 reviews different types of electroacoustic transducers with specific attention paid to ultrasonic, air-coupled MEMS. Chapter 4 gives an overview of the device design and fabrication. Chapter 5 presents the device modeling. Chapter 6 summarizes the optimization methodology. The experimental setup and results are presented in Chapter 7. Chapter 8 discusses the conclusions and future work.
24 CHAPTER 2 NONLINEAR ACOUSTICS
The field of fluid dynamics describes fluid particle motion and its interaction with solid bodies [17]. A complete fluid dynamic field description includes three-dimensional, thermo- viscous, and unsteady effects. A linear acoustic description is derived by perturbing the fluid field variables by an infinitesimal amount and linearizing the governing equations of compressible fluid dynamics. Linear acoustics is successful in describing common acoustic phenomena where the small-signal assumption is valid [8]. The science of nonlinear acoustics forms a bridge between the full viscous compressible flow mechanics description and linear acoustics. Although the study of nonlinear acoustics is the analysis and measurement of finite acoustic perturbations, nonlinear acoustic theory neglects insignificant effects of a full
fluid mechanics model when appropriate.
This chapter presents fundamentals of nonlinear acoustic theory. An example of finite wave propagation is demonstrated using plane acoustic waves. The nonlinear acoustic equa- tions most applicable to the analysis of parametric acoustic arrays are introduced. Solutions to these equations for sound beams are presented. Finally, parametric array implementations found in the literature are summarized.
2.1 Finite Perturbation Acoustic Theory
To describe motion of a thermoviscous fluid, four governing equations are used: continu- ity, momentum, energy, and thermodynamic state [18]. The following equations are derived assuming a homogeneous fluid medium with viscosity and heat conduction coefficients that are independent of the introduced disturbance [18].
The continuity equation is given by [17]
Dρ + ρ∇~ · V~ = 0, (2–1) Dt where ρ is the mass density, V~ is the fluid velocity vector, and D ()/Dt is the material derivative given by D ()/Dt = ∂ ()/∂t + V~ · ∇~ ().
25 The momentum equation, ignoring body forces, is given as [18] µ ¶ DV~ 1 ³ ´ ρ + ∇~ P = µ∇2V~ + µ + µ ∇~ ∇~ · V~ , (2–2) Dt B 3
where P is the thermodynamic pressure, ∇2 is the Laplacian operator, µ is the shear viscos- ity, and µB is the bulk viscosity. The shear viscosity relates the velocity gradient to the shear stress and is a measure of a fluid’s ability to transfer momentum to particles perpendicular to the momentum direction. The bulk viscosity accounts for differences between the thermo- dynamic pressure and the sum of all normal stresses [18]. The momentum equation using the bulk viscosity formulation is an approximation that is valid at low frequencies where the time required to reach thermodynamic equilibrium is much smaller than the period of oscillation. At higher frequencies, the bulk viscosity model breaks down due to non-equilibrium effects. The changes in the thermodynamic state occur so rapidly that thermodynamic equilibrium is not reached during each cycle. The result is an energy loss called molecular relaxation in gases [17]. In air, molecular relaxation is dominated by the time it takes the vibration energy mode of nitrogen, 100 ms, and oxygen, 1 ms, to reach thermodynamic equilibrium [8]. As will be shown in Section 3.2.3 of Chapter 3, the acoustic attenuation of a 40 kHz tone is -1.29 dB per meter, of which thermoviscous effects account for -0.26 dB per meter, relaxation from nitrogen vibration accounts for -0.005 dB per meter, and relaxation from oxygen vibration
accounts for -1.03 dB per meter. If the bulk viscosity model holds, then the appropriate entropy formulation of the energy equation is [18] ³ ´ µ ³ ´¶ Ds 2 1 ↔ ρT = κ∇2T + µ ∇~ · V~ + µ ↔ε − δ ∇~ · V~ , (2–3) Dt B 3
↔ where s is specific entropy, ε is the strain-rate tensor given in indicial notation as εij =
³ ´ ↔ 1 ∂ui + ∂uj , κ is thermal conductivity, and δ is the Kronecker delta function. 2 ∂xj ∂xi The equation of state for a single-phase fluid in thermodynamic equilibrium states that
any thermodynamic variable can be described as a function of two other thermodynamic
26 variables [17]. It is common in linear acoustics to express thermodynamic pressure as a function of density and entropy,
P = P (ρ, s) . (2–4)
The linear, lossless acoustic wave equations for homogeneous, initially quiescent media are derived when the perturbations of the fluid are assumed to be very small with respect to the ambient conditions. Since the ambient pressure is many orders of magnitude greater than common acoustic levels, the change in the thermodynamic pressure is assumed to be
2 small in comparison to the isentropic bulk modulus, ρ0c0, such that
0± 2 p ρc0 ¿ 1, (2–5)
where c0 is the small-signal, isentropic sound speed defined by
p c0 = γRT0. (2–6)
The medium is assumed to be initially quiescent. Thus, the particle velocity magnitude is assumed small in comparison to the isentropic speed of sound, ° °. °~ 0° °V ° c0 ¿ 1. (2–7) ° °. °~ 0° The ratio of the velocity amplitude to the isentropic speed of sound, °V ° c0, is also known as the acoustic Mach number. In addition to the small perturbations assumptions, the speed of sound is assumed to be the small-signal, isentropic speed of sound,
c = c0. (2–8)
For linear, lossless propagation of acoustic waves in homogeneous media, the governing equation in terms of velocity perturbation is [8]
∂2V~ 0 − c2∇2V~ 0 = 0. (2–9) ∂t2 0
Further details on linear, lossless, acoustic motion can be found in the text by Blackstock [8].
27 For very intense sound pressure levels, assumptions 2–5, 2–7, and 2–8 are violated and the linear acoustic description breaks down. There are two physical effects that simultane- ously contribute to nonlinear acoustic propagation: convection and local heating [8].
The convective effect occurs at high sound pressure levels (SPL) when the particle veloc- ity becomes significant in comparison to the speed of sound. For example, the propagation speed of constant phase, cph, of a one dimensional wave becomes the sound speed, c (T ), plus the one-dimensional particle velocity, u0,
cph = c (T ) + u0, (2–10)
√ where c (T ) = γRT . The local heating effect is the result of the speed of sound’s dependence upon tem- perature. Large amplitude waves compress and expand the gas, creating a non-negligible temperature change and thus a change in the speed of sound. It can be shown that the first order approximation of the speed of sound for lossless, one-dimensional finite amplitude plane waves is [18]
ph 0 c = c0 + βu , (2–11)
where β is the coefficient of nonlinearity of a gas given by [18],
γ + 1 β = , (2–12) 2
and γ = cp is the ratio of specific heats. cv As observed in Equation 2–10, the propagation speed of a constant phase is dependent
upon the magnitude of the particle velocity. Given u0 as the amplitude of the velocity, the phase corresponding to the peak of the sine wave shown in Figure 2-1 will propagate
at c|peak = c0 + βu0, while the phase corresponding to the trough propagates at c|trough =
c0 − βu0. This will lead to steepening of the wave as shown in Figure 2-1. As evident in the figure, eventually the peak of the wave will catch up to the trough causing the wave to form
a shock as shown in case (d). Thermoviscous dissipation across the shock prevents the peak
28 from passing the trough of the wave. Once the shock is formed, dissipation across the shock will weaken the wave as it propagates [8].
c= c + β u peak 0 0
c= c zero 0 x x x x
(a) c= c − β u (b) (c) (d) trough 0 0
Figure 2-1. Wave steepening due to nonlinear wave propagation. Adapted from Blackstock [8].
The distorted waves in cases (b), (c), and (d) of Figure 2-1 can be described mathemat- ically with a Fourier series whose fundamental frequency is the originating frequency shown in case (a), " # X∞ jn2πf0t P = Re Pne . (2–13) n=1 Thus, as the wave propagates, power redistributes from the fundamental frequency to its harmonics. Figure 2-2 shows the power distribution between the frequencies at different distances along the propagation path. Notice that as the wave travels, the power of the harmonics has increased and, correspondingly, the power of the fundamental has decreased.
Thus, if a source produces a high amplitude tone, f0, the spectrum measured at a finite distance from the source will also include its harmonics (2f0, 3f0, 4f0, etc.) [18]. In practice, nonlinear effects are perceptible on a log scale at SPLs where assumptions 2–5 through 2–8 are still relatively valid. For example, waveforms in air of SPLs above 120 dB distort enough to produce noticeable harmonics. At 120 dB in air the acoustic ° °. °~ 0° ∼ Mach number is approximately °V ° c0 = 0.06 for a plane wave. Although the acoustic Mach number may seem small with respect to unity, wave distortion is significant enough to produce harmonics on a log scale. For instance, an experimental array that projected 128 dB
(distance not given) at 25 and 30 kHz produced 100 dB of the audible difference frequency,
5 kHz, at 4 m [19].
29 log(Pa 2/Hz) log(Pa 2/Hz)
f f f0 2 f0 3 f0 4 f0 5 f0 f0 (b) The power in the sinusoidal acoustic (a) The frequency distribution of the power wave redistributes to its harmonics as it at the origin. propagates.
Figure 2-2. The frequency distribution of a nonlinearly propagating wave.
2.2 Parametric Array
If the source were to generate two high SPL, ultrasonic tones, fa > fb, the harmonics of each, as well as sum and difference frequencies, fa + fb and fa − fb, respectively, are produced [20]. If the difference between the two initial tones is within the human hearing bandwidth, the difference frequency is audible. If the size of the source is large with respect to the ultrasonic wavelengths, then the ultrasonic sound field will be a directional beam [8]. The harmonic, sum, and difference frequencies will maintain the directivity of the originally generated ultrasonic tones [20]. Thus, a low frequency ”spotlight” [1] is produced. If the difference frequency had been produced linearly, it would have an omnidirectional field.
Figure 2-3 compares the ”spotlight” [1] produced by a parametric array with the directivity of a conventional transducer. Also, the frequency content of the parametric array at the source and at a finite distance from the source is shown. Notice that along the propagation path, harmonics are produced at the expense of the power of the fundamental frequencies. The generation of the difference tone is the fundamental mechanism behind parametric arrays.
30 SPL + fa f b
− f fa f b fb fa 2 fb 2 fa Ultrasound
Parametric Array Audible Sound
Virtual Sources Conventional Audio
Ultrasound Radiator
SPL
f fb fa Ultrasound
Figure 2-3. Parametric array radiation of sound (adapted from Holosonics) [3].
2.2.1 Model Equations of Nonlinear Acoustic Theory
This section outlines the major equations used in parametric array analysis. The farfield acoustic radiation of a linear acoustic speaker is analyzed using Rayleigh’s integral [8]. The radiation from a parametric array, however, requires more complex modeling. The following section presents the two most frequently used equations for analyzing parametric arrays, the Westervelt [20] and Khokhlov-Zabolotskaya-Kuznetsov (KZK) [21, 22] equations. Two analytical solutions to the KZK equation that are applicable to parametric arrays are given in Section 2.2.2.
2.2.1.1 Westervelt Equation
The concept of a parametric array was first introduced by Peter Westervelt [20]. In his paper, Westervelt discussed the mixing of two tones in a sound beam to produce a difference
31 tone. Westervelt’s original derivation of an inviscid, second order wave equation begins with Lighthill’s equation [23]:
2 2 2 ∂ ρ 2 ∂ ρ ∂ Tij 2 − c0 = , (2–14) ∂t ∂xi∂xi ∂xi∂xj
¡ 2 ¢ where Tij = p − c0ρ δij + ρuiuj + τij. (2–15)
This is Lighthill’s exact equation of fluid motion where the stress tensor in Equation 2–15 acts as a source term for the rest of the otherwise quiescent media [23]. The first term in Equation 2–15 accounts for deviations from isentropic behavior, the second term is the Reynold’s turbulent stress, and the third term is the viscous stress tensor. In Westervelt’s original derivation, the viscous stress tensor is ignored.
Manipulation of Equation 2–14 results in Westervelt’s second order wave equation. To arrive at the Westervelt equation, each dependent variable is perturbed about a nominal value, the second-order Taylor series expansion of pressure with respect to density is substi- tuted, and the linear acoustic equations are back-substituted to arrive at an equation with a single dependent variable. The resulting equation is [20]
2 02 2 0 β ∂ p ¤ p = − 4 2 , (2–16) ρ0c0 ∂t where the prime refers to the perturbation about the nominal value, β is the nonlinear
γ+1 2 ∂2 1 ∂2 coefficient equal to , and ¤ = 2 − 2 2 is the D’Alembert operator. When the 2 ∂xi c0 ∂t thermoviscous effects neglected by Westervelt in the original derivation of Equation 2–16 are included the resulting equation is [18],
3 0 2 02 2 0 δ ∂ p β ∂ p ¤ p + 4 3 = − 4 2 , (2–17) c0 ∂t ρ0c0 ∂t where δ is the diffusivity of sound given by µ ¶ µ ¶ 1 4 κ 1 1 δ = µ + µB + − . (2–18) ρ0 3 ρ0 cv cp
32 In Equation 2–18 cv and cp are the specific heats at constant volume and pressure, respec- tively.
The Westervelt equation is second order in the small parameterε ˜ that represents the
2 magnitude of both the acoustic Mach number, ε = u/c0 , and η = µω/ρ0c0. The small parameter η measures the balance between viscous shear stress and pressure fluctuations. Division of the numerator and denominator of η by ω2 results in
µ η = k2 (2–19) ρ0ω
where k = ω/c0 is the wavenumber. Equation 2–19 is the square of the ratio of an oscillatory diffusion length scale to the wavelength. The Westervelt equation also ignores the Lagrangian density defined by [18]
02 1 2 p ` = ρ0u − 2 . (2–20) 2 2ρ0c0
The Lagrangian density is the difference between the kinetic and potential energy densities of the wave and is zero for progressive plane waves in a quiescent medium [24]. The Westervelt Equation 2–17 continues to be valid to second order for non-plane waves
when the cumulative effects dominate the local effects. Cumulative effects are significant as the wave propagates, whereas local effects are ignored after a wavelength. The convective and local heating effects are examples of cumulative effects. Examples of local effects include linearizing a vibration to a fixed surface at a boundary condition or the substitution of linear acoustic relations into the nonlinear equations [18]. After a wavelength, the errors in these approximations are small in comparison to the effect of cumulative wave distortion. Local effects become important in compound wave fields such as standing waves and in the radiation pressure [18].
Further assumptions lead to a subset of the Westervelt equation, the KZK equation,
that is suited for directional sound beams.
33 2.2.1.2 KZK Equation
The KZK [21, 22] equation is the most frequently used model that describes the ef- fects of diffraction, absorption, and nonlinearity in sound beams [18]. It is derived from the Westervelt equation, 2–17, by transforming the spatial scales so that the effects of diffrac- tion, absorption, and nonlinearity are all of orderε ˜2 and all other higher order terms are disregarded. The following scales are introduced into the Westervelt equation:
1/2 1/2 x1 =ε ˜ x, y1 =ε ˜ y, z1 =εz, ˜ and τ = t − z/c0, (2–21) where x and y are in the plane of the acoustic source, z is along the propagation direction, and τ is the retarded time. The Laplacian in the D’Alembert operator in Equation 2–17 becomes µ 2 2 ¶ 2 2 2 2 ∂ ∂ 2 ∂ 2 ∂ 1 ∂ ∇ =ε ˜ 2 + 2 +ε ˜ 2 − ε˜ + 2 2 . (2–22) ∂x1 ∂y1 ∂z1 c0 ∂z1∂τ c0 ∂τ Introducing this and the new scales into the Westervelt Equation 2–17 while dropping the higher order terms gives µ ¶ ∂2 ∂2 2 ∂2p δ ∂3p β ∂2p2 ε˜ 2 + 2 p − ε˜ + 4 3 = − 4 2 . (2–23) ∂x1 ∂y1 c0 ∂z1∂τ c0 ∂τ ρ0c0 ∂τ
Rewriting the equation in terms of x, y, and τ, the resulting KZK equation is
∂2p c δ ∂3p β ∂2p2 − 0 ∇2 p − = , (2–24) ∂z∂τ 2 ⊥ 2c3 ∂τ 3 2ρ c3 ∂τ 2 | {z } | 0{z } | 0 {z0 } Diffraction Absorption Nonlinearity
2 ∂2 ∂2 where ∇⊥ = ∂x2 + ∂y2 in Cartesian coordinates. The diffraction term in Equation 2–24 accounts for changes in the plane perpendicular to the z-axis of propagation. Thus, to first order inε ˜, this equation models directional, quasi-planar beams and accounts for diffraction, absorption, and nonlinearity effects [18].
2.2.2 Sound Beam Solutions
In this section, two sound beam solutions that are commonly used in parametric array analysis are introduced. The Westervelt solution is the original sound beam solution based
34 on the Westervelt equation. The Berktay solution is a variation of the Westervelt solution that uses amplitude modulation as the source function.
L P 1/z roll-off S le b di u A
x Primary sound field
a 0 z
y ′ Lα R0 z “Sources” for difference frequency
Figure 2-4. The sound field and length scales associated with the sound beam solutions. Note that this figure is highly schematic.
Before discussing sound beam solutions for parametric arrays, it is important to discuss the scales used in these analysis to make assumptions that allow an analytical solution to be
formed. First, as shown in Figure 2-4 the primary sound field is assumed to be a collimated wave of radius a that experiences attenuation due to acoustic absorption in the absence of spherical spreading. The primary fields in the next two sound beam solutions take on the form
−α0z p1 (r, z) = p0H (a − r) e (2–25)
where p0 is the pressure given by p0 = ρ0c0u0, α0 is the absorption at the average primary frequency, and the Heaviside function is defined by 0 x < 0 H (x) = 1 x = 0 . (2–26) 2 1 x > 0
35 The validity of this assumption is based on the primary field being sufficiently attenuated by the transition from the nearfield to the farfield at approximately the Rayleigh distance,
1 2 R0 = 2 ka (for details on nearfield and farfield see Section 3.2.1), that there is no longer enough power to contribute significantly to the production of the difference frequency. The effective length of the parametric array, also known as the absorption length, is given by
1 La = . (2–27) 2α0
The consequence of this assumption is that spherical spreading can be ignored since nonlinear production in the farfield is negligible to that in the nearfield. If the primary sound field is still significant in the farfield, this assumption will be violated and the following solutions will over predict the nonlinear demodulated field since the primary sound fields in the farfield will not experience attenuation due to spherical spreading. Another assumption as shown in Figure 2-4 is that the distance of the measurement point from the source of the parametric array, z0, is much larger than the region of nonlinear production. This assumption is valid
0 as long as z À La. If this assumption is violated, the amplitude of the difference frequency at the point z0 would be over-predicted since both solutions consider the entire primary field out to infinity as the source of the difference frequency. Without going to numerical solutions of the KZK equation that require large resource allocation [18], the analytical sound beam solutions are generally used in this dissertation for first order approximations of the demodulated sound field.
2.2.2.1 Westervelt’s Parametric Array Solution
In Westervelt’s “Parametric Array” paper [20], Equation 2–16 is solved for the pressure at the difference frequency of two adjacent tones produced by a radiating circular piston. The assumed pressure field of the primary tones are
−αaz p1a (r, z) =p0aH (a − r) e (2–28)
−αbz and p1b (r, z) =p0bH (a − r) e . (2–29)
36 The basis for assuming a collimated primary sound field is on the assumption of large acoustic absorption to limit the nonlinear production to the nearfield of the sound source (radiator
nearfield will be defined in Chapter 3).
The resulting pressure of the difference frequency as a function of the distance along the axis normal to the radiator, z, and the angle with respect to the normal, θ, is
2 2 −α z p0ap0bβk a e 1 2 j(ω−τ− 2 k z tan θ) p (θ, z) ' − 2 Dw (θ) DA (θ) e , (2–30) 8ρ0c0α0 z
where the minus subscript indicates properties that correspond to the difference frequency,
p0a and p0b are the originating pressures of the two fundamental frequencies given by p0 =
ρ0c0u0, u0 is the piston velocity, a is the piston radius, α0 and α− are the absorption co- efficients of the original tones and the difference tone [8], respectively, and the Westervelt directivity and aperture factor are given by
1 DW (θ) = 2 (2–31) 1 + j (k /2α0) tan θ 2J (k a tan θ) and D (θ) = 1 , respectively. (2–32) A k a tan θ
2.2.2.2 Berktay Solution
Berktay [25] introduced an alternate sound beam solution that was similar to Wester- velt’s. In this case, however, the source is a modulated signal,
p (r, z = 0, t) = p0E (t) sin [ω0t + ϕ (t)] H (a − r) , (2–33)
where p0 is the pressure given by p0 = ρ0c0u0. The amplitude and phase modulation (E (t) and ϕ (t) , respectively) are assumed to be slowly varying functions of time in compar- ison to the carrier frequency, ω0. A derivation of Berktay’s solution is found in Appendix A. The audible pressure solution is [18]
2 2 2 2 βp0a d (E (τ)) p2 (0, z, τ) ' 4 2 , (2–34) 16α0ρ0c0z dτ
37 where a is the source radius and α0 is the absorption at the fundamental frequency ω0. If a sinusoidal amplitude modulating function, for example E (t) = 1 + sin (ωt), is used, then the pressure at the frequency ω is proportional to the frequency squared,
2 p2 ∝ ω . (2–35)
Thus, Berktay’s solution predicts a 40 dB per decade roll-off as lower modulating frequencies are approached. 2.2.3 Existing Implementations
Parametric arrays have been studied in depth over the past 50 years. Many of the early studies focused on operation in water for directive sonar [20, 26, 27]. Only recently has parametric operation in air begun to garner attention. The first implementation in air showed that wave distortion could produce a difference tone [10]. A research group in Japan associated with Nagoya University and the University of Electro-communications, Tokyo, dominated the 1980’s and early 1990’s with studies focused on eliminating harmonic
distortion and reducing power consumption [1,19,28–32]. The turn of the century marked the first commercialization efforts by Holosonics, headed by Joseph Pompei [2,12], and American Technology Corporation (ATC) [11], founded by Woody Norris [33]. Presented below are implementations of the parametric array in air found in the literature. The focus is on the transducer and array specifications as well as the production of a difference frequency. A
summary of the arrays is presented in Table 2.2.4. 2.2.3.1 Airborne Arrays
Bennett and Blackstock [10] experimentally demonstrated the first parametric array in air. They used an oil-filled emitter 64 mm in diameter originally designed for use as a hydrophone in water. It had two resonant frequencies of 18.6 and 23.6 kHz. The source was particularly weak (110 dB at 0.3 m of 18.6 and 23.6 kHz), resulting in poor parametric
conversion. Due to the limitations of the experimental space, all measurements were taken
in the nearfield. They were able to measure, however, an increase in the difference frequency
38 on axis culminating in a peak and then declining after the region of parametric conversion. This measurement demonstrated nonlinear conversion from the source frequencies to the difference frequency. After the peak, the absorption of the high frequency signals weakened the nonlinear conversion. Beyond the peak, the difference frequency was dominated by its own absorption. At a distance of 0.3 m from the transducer they were able to measure 50 dB of the 5 kHz difference frequency.
A Japanese research group associated with Nagoya University and the University of Electro-communications, Tokyo, headed by Yoneyama and Kamakura began the next major effort in 1983. Their first paper was also the first work where practical audio output was obtained [1]. This effort, aptly named the ”Audio Spotlight,” employed an array of 547 PZT bimorph emitters (overall array and emitter size omitted) arranged in a hexagon and operating with a resonance of 40 kHz and secondary peak just below 60 kHz. They were able to obtain a narrow beam with a difference frequency at 1 kHz of about 80 dB at 4 m using amplitude modulation of the source signal. They used Berktay’s solution to recognize the correlation between the audio signal and the modulating function. As predicted by Berktay’s solution, experiments showed the dependence of the audio signal amplitude on the audio frequency squared from about 200 Hz to 2 kHz. For parametric generation of audio frequencies above 2 kHz, the response of the transducer above 40 kHz began to decline such that the SPL output of the primary signals was much lower. They also recognized that modulating the carrier signal with a dc offset sinusoid generated harmonic distortion of the audio signal. This can easily be seen by employing a modulating function E (t) = 1+m·g (t) in the Bertkay solution, where m is the modulation index and g (t) is the audio signal. The resulting demodulated signal is
βp2a2m d2g (τ) βp2a2m2 d2g2 (τ) p (0, z, τ) ' 0 + 0 , (2–36) 2 8α ρ c4z dτ 2 16α ρ c4z dτ 2 | 0 0 0{z } | 0 0 0{z } ps pd where ps is the desired pressure signal and pd is harmonic distortion. As predicted by the Berktay solution, if g (t) is a harmonic function, then the audio output has a 40 dB/decade
39 slope with increasing frequency. This was the first work to propose the use of an equalizer to give the audio signal a -40 dB/decade frequency characteristic before implementing it in the amplitude modulation function, E (t).
The paper by Yoneyama et al. [1] spurred on further efforts in the Japanese community to eliminate the harmonic distortion of the audio signal produced when using the modulating function. Kamakura [28] presented results from a rectangular array of 581 PZT bimorph transducers operating at 40 kHz with an effective array radius of 15 cm. They were able to obtain approximately 68 dB of 1 kHz difference frequency at 9.5 m.
Aoki et al. [32] showed good agreement between numerical solutions of the KZK equation and experiments. The array consisted of 1410 piezoelectric transducers of 1 cm diameter and a resonance at 28 kHz. The array had a radius of 21 cm, and the primary tones were 27
and 30 kHz. Given a source level of 112 dB, they obtained approximately 65 dB of 3 kHz difference frequency at 3 m. At a source level of 133 dB, they obtain approximately 103 dB of the 3 kHz difference frequency at 3 m.
In the next paper by Kamakura et al. [30], the square root of the entire modulating function was taken to eliminate harmonic distortion of the audio signal. They also introduced a low frequency envelope function, e (t), in an effort to reduce the power output of the parametric array. The modulating function in relation to the Berktay solution [25] for this case is p E (t) = e (t) + m · s (t), (2–37)
where s (t) is the desired audio signal. The Fourier transform of Equation 2–37 consists of an infinite number of frequencies due to the square root function. Thus the transducer required a large bandwidth around its resonant peak to be able to accurately replicate function 2–37. They were able to show a decrease in the harmonic distortion of the audio signal as well as a 36% decrease in the power requirement with respect to double sided modulation while
maintaining tone quality. Their experimental implementation used a 44 by 50 cm2 array of
2000 PZT bimorphs with a 28 kHz resonance.
40 Aoki et al. [34] created a parametric array to improve speech articulation in reverberant tunnels. The array consisted of 91 piezoelectric bimorphs 1 cm in diameter with resonant frequencies of 40 kHz. The total array aperture was 11 cm. The primary frequencies used
were 38.5 and 41.5 kHz. They extrapolated the sound pressure level at the source from lower level forcing as 134.5 dB. The 3 kHz audio signal produced was approximately 88 dB at 1 m. They performed numerical solutions of the KZK equation that showed good agreement with experiments. Next, Kamakura et al. [19] conducted experiments on a rectangular array and compared their results to numerical simulations. Numerical simulations based on the KZK equation were conducted with both rectangular and circular source faces and both showed excellent agreement with the experimental data. The array consisted of 1102 piezoelectric bimorphs
in a 24 by 44 cm2 rectangle. They radiated at 25 and 30 kHz at 116 dB at 0.7 m from the source. They obtained about 80 dB of 5 kHz at 4 m with this arrangement. Radiating at 128 dB (position not given) of the primary frequencies they obtained approximately 103 dB at 4 m of the difference frequency.
Pompei [12] presented the next implementation of the parametric array in air. His paper discussed solutions for signal processing and different modulation techniques that emphasized the difference frequency in comparison to its harmonic distortion. The setup used an array (35 cm in diameter) of wideband devices generating with a carrier frequency of 60 kHz. Pompei recognized that to get a flat frequency response of the audio signal, the parametric
array must contend with the 40 db/decade slope with increasing frequency that the Berktay solution predicts. He claimed that using the square root of one plus the double integral of the desired audio signal, s ZZ E (t) = 1 + g (t) dt2, (2–38)
as the modulating function would counteract the 40 dB/decade slope. The modulating
function 2–38 had a problem similar to the equalizer proposed by Yoneyama et al. [1].
The transducer required a large bandwidth to produce enough frequency components to
41 give an accurate representation of Equation 2–38. Instead, Pompei utilized the roll-off of his transducers to compensate for the problem of the 40 dB/decade slope resulting in a fairly flat audio spectrum. Therefore, equalization by double integration was unnecessary in the preprocessing. The design of these transducers is not presented. The results showed significant reduction in total harmonic distortion with just the square root pre-processing. The study obtained about 77 dB of 1 kHz difference frequency at 3 m.
Havelock and Brammer [35] made a comparison of two directional sources using the same compression drivers; a perforated pipe and a parametric array. They used four 50 mm diameter piezoelectric disks in an array with an overall array diameter of 170 mm. The source level for the parametric array was 134 dB at 0.25 m and 28 kHz. At 4 m they were able to measure 50 dB at 1 kHz and 40 dB at 300 Hz. In the 2-5 kHz range of the difference frequency, they were able to show the frequency squared dependence predicted by Berktay’s solution.
In addition to sound reproduction applications, parametric arrays have also been used for nondestructive evaluation (NDE). In 2000, Kaduchak et al. used a parametric array to excite resonances of elastic, fluid filled containers to determine the fluid type [6]. The parametric array consisted of 48 off-the-shelf air-coupled piezoceramic transducers (AirMar model number AT200) 15.8 mm in diameter with a resonance around 200 kHz. The trans- ducers were arranged in a rectangular array with center to center spacing of 19 mm. The authors measured the carrier frequency of 217 kHz at 0.5 m to be 115 dB. They also reported a difference frequency measurement of 85 dB at 3 m, but did not report the actual difference frequency. The authors showed good agreement between a beamwidth measurement at 3.9 kHz and the Westervelt directivity (sound pressure levels are normalized). ATC produced a white paper that contained a literature review of parametric array implementations in air as well as a brief description of their own efforts at ATC [11]. Here they described some of their signal processing solutions as well as a monolithic film ultra- sonic transducer. They introduced single sideband (SSB) amplitude modulation as having
42 an advantage over double sideband (DSB) amplitude modulation. DSB amplitude modula- tion resulted in the source emitting a fundamental carrier tone plus two adjacent tones. The resulting audio signal had a second harmonic with the same amplitude of the primary audio tone [11]. SSB amplitude modulation emitted only two tones: the fundamental carrier tone and one adjacent tone. This is similar to the Westervelt solution where two adjacent tones served as the fundamental frequencies. Their claim was that single side band modulation requires no preprocessing to produce zero harmonics of a pure tone and thus only requires bandwidth equal to the difference tone. They gave an example of generating two audio fre- quencies simultaneously and the use of an iterative scheme to mitigate the distortion. The scheme introduced new frequencies into the spectrum generated by the transducer. These frequencies, however, were within the bandwidth of the larger audio signal and thus do not require more bandwidth from the transducer. They compared and contrasted using upper and lower SSB modulation. They claimed that lower sideband modulation was more bene- ficial since there was less absorption at these frequencies and transducers tended to behave more erratically at frequencies above their resonance. In the section on transducers, Croft et al. discussed designing transducers that counteract the frequency squared dependence predicted by the Berktay solution. The ATC piezoelectric bimorph actuator used to date is shown in Figure 2-5. A cone was incorporated with the piezoelectric bimorph to improve volume velocity by essentially creating a piston that vibrates at the center velocity. A quar- ter wavelength back cavity provided zero cavity impedance at the carrier frequency. They were able to get 123 dB from a Nicera AT40-12P transducer at 30 cm (frequency not listed). A key point of the paper was that an array with a larger area but smaller output per area can outperform a smaller array with larger output per area. The Berktay solution is directly proportional to area. To get the same amplitude as a large array, a considerable amount of power must be supplied to a smaller array. If the amplitudes of the fundamental tones are large enough, the waves will form shocks before sufficient parametric conversion can take place. Once the waves shock, dissipation across the shock dominates and absorbs the power
43 of the waves. In this case the array is saturated since an increase in the input power will no longer cause an increase in the power of the audio signal. This idea resulted in ATC designing large monolithic structures with the hope of getting better parametric conversion.
Their transducer was formed using a polyvinylidene diflouride (PVDF) film laid over holes drilled into a plate. To aide in the conversion of the contraction of the PVDF film into vertical deflections, vacuum was applied to the back of the plate causing the film to have an inward static deflection. Their initial device was 44.45 mm (1.75”) in diameter with 85 hexagonal close packed 3.57 mm (9/64”) diameter holes with center to center spacing of 4.06 mm (0.160”). This produced a structure with a resonance of 37.23 kHz. The primary frequency output was 136.5 dB (distance from source unreported).
Cone
Piezoelectric bimorph λ 4
Figure 2-5. ATC bimorph piezoelectric actuator with attached cone for enhanced volume velocity [11].
Moon et al. [36] constructed a flexural-type transducer utilizing piezoelectric actuation. The PZT layer had a diameter of about 9 mm. The primary frequencies used in the experi- ments were 42.24 and 43.25 kHz. They mounted their transducers at angles to each other in an attempt to avoid sidelobes. All experimental measurements were made at 120 mm from the array. All measured amplitudes given were normalized so an absolute measurement of the parametric conversion is not available [36].
Roh and Moon [14] gave results from an array of 60 thickness mode piezoceramic trans- ducers operating at 650 kHz. The actuators had a diameter of 30 mm with a PZT-4 thickness
44 of 3 mm. The total array diameter was 40 cm. By having such a high operating frequency, the transducers had enough bandwidth to supply the full human hearing range (20 kHz is a small percentage of 650 kHz). The impedance bandwidth of the individual transducer was
flat from 580-670 kHz within 1 dB. The array pattern was designed by treating the individual transducers as point sources. They were able to measure an average sound pressure level of 103 dB and a bandwidth of 17.5 kHz of the audio signal at 1 m. They showed a flat audio frequency response of the measured audio signal. These results seem counterintuitive since Berktay’s solution predicts the audio signal to have a frequency squared dependence given the flat transducer response they showed in paper. Absorption of the primary signals was extremely high at 650 kHz. This would severely limit the length of the parametric array and the interaction volume such that the amount of demodulation would be limited. It should be noted that the authors did not report what ultrasonic transducer they used to make their SPL measurements in air.
Another paper focused on pre-processing was written by Young and Sung [37]. They mitigated the audio signal distortion resulting from the square root of the equalized modu- lation signal by manipulating the phase and amplitude of the frequency components of the square rooted signals. They measured the SPL output and then changed the amplitude and phase until the THD was minimized. They operated at a carrier frequency of 40 kHz. The primary waves were about 120 dB at 2 m from the array. They claimed significant reduction in the THD with respect to double sided modulation. They did not provide quantitative
SPL levels of the difference frequency. A Russian group reported experimental results with the goal of using the parametric array in air for atmospheric sounding [38]. They gave results from multiple array types: a single capacitive membrane radiator, an array of bimorph piezoelectric radiators, and a focused array of radiators. The capacitive radiator was a large metallized polymer membrane stretched over a conducting surface that has been roughened. The roughness created air pockets over which the membrane vibrates. They were able to measure a demodulated signal
45 of 55 dB at 3 kHz and 60.5 dB at 4 kHz. The transducer for the array of radiators consisted of a bimorph piezoelectric plate 10.6 mm in diameter with a 7 mm diameter conical horn made of light metal attached. The SPL at 40 kHz was about 85 dB for a single transducer. They
arranged these transducers in a planar array (PA) of two rows of seven and a seven by seven array. Measurements were made at 3.3 m from the transducers (results were normalized). Another experiment used a focused array (FA) with groups of 90 small radiators in rings. The normalized directivity patterns of these arrays were given without reference to the SPL levels obtained.
Parametric arrays have also been evaluated as an instrument for an active noise control system [7]. Brooks et al. measured ultrasonic and demodulated signals from the ATC parametric array. At 1 m, they measured a 1 kHz demodulated signal of about 90 dB from
a 48 kHz carrier frequency. In 2006, Kamakura et al. presented a conference paper that focused on increasing the power efficiency of parametric arrays [39]. They reported a 12.5 by 25 cm2 array of 286 monomorph ceramic transducers. The transducers are Nipon Ceramic Co., Ltd, Type AT/R40-10 and are 10 mm in diameter. They obtained 120 dB of the 39.3 kHz carrier frequency at 3 m.
Other researchers have looked at using parametric arrays in cell phones to decrease diffusivity of audible sound [40]. In 2006, Nakashima et al. mounted 16 piezoelectric trans- ducers on a cell phone sized device. They were able to obtain 134 dB at 1 m of the 40 kHz
carrier frequency. This resulted in a 65 dB difference frequency of 1 kHz at 1 m.
In 2007, Chen et al. [41] presented a conference paper that identified the problem of the infinite bandwidth that was required for the transducer to produce a square root modulating function similar to Equation 2–37, where e(t) = 1 and s(t) was a simple sinusoid. They conducted experiments using first, second, and third order Maclaurin series approximations of the square root modulating function. The experiments showed a decrease in the total harmonic distortion for the second and third order series approximations. The array was
46 300 mm by 300 mm and the carrier frequency was 40 kHz. The modulating factor, m, was 0.7. At a distance of 2 m, they obtained a 2 kHz signal ranging from 70.58-72.65 dB.
Recently, Peifeng et al. [42] presented experimental results of a parametric array looking
at applications in virtual reality systems and home theaters. They conducted experiments where two ultrasonic sources, one emitting at 40 kHz and the other at 41 kHz, intersected orthogonally at 1 m. Each array consisted of 91 off-the-shelf piezoelectric transducers of 8 mm radius and 45 degree individual beam angles. The output of a single array was 140 dB of the 40 kHz carrier frequency at 1 m. At the intersection point, 40 dB of the 1 kHz difference frequency was measured.
Ying and Chen et al. [43] presented a similar conference paper to Chen et al. [41] that investigated truncated square root modulating functions with a different transducer. In this
case, a transducer made of PVDF (Polyvinylidene Fluoride) was used to produce a carrier frequency of 42 kHz. They reported 62 dB at 1 m.
2.2.4 MEMS Parametric Arrays
Although most parametric arrays have used conventional transducers, there have been a few recent studies on MEMS-based parametric arrays.
In 2006, Haksue et al. presented a micromachined parametric array that focused on rang- ing applications such as robotics [44,45]. The application required a very narrow beamwidth from a small actuator. Thus, the authors chose an ultrasonic difference frequency of 40 kHz and a piezoelectric micromachined ultrasonic transducer (pMUT) as the actuator. PMUTs are covered in more detail in Section 3.3.2. The pMUT was fabricated by depositing a 2.5 µm lead zirconate titanate (PZT) layer on a silicon-on-insulator (SOI) wafer similar to the work of Horowitz et al. [46] covered in Section 3.3.2.1. The authors fabricated two diaphragms with radii of 1200 µm and 1420 µm, both with 15 µm thick silicon layers. The resonant
frequencies of the two devices were 135 and 95 kHz, respectively. The two diaphragms were hexagonally arranged to form a device array 35 by 30 mm2. At 0.18 m, the SPL of the 40
kHz difference frequency was 85 dB. The excitation signal was 8.5 volts peak-to-peak.
47 Table 2-1. CMUT transducer array specifications and performance presented by Wygant et al. [15]. Design A B Membrane diameter (mm) 4 4 Membrane thickness (µm) 40 60 Cavity depth (µm) 36 16 AC excitation (V peak-to-peak) 200 200 DC bias (V) 380 350 Center frequency (kHz) 46 55 -3 dB bandwidth (kHz) 2.0 5.4 Pressure at 3 m (dB re 20 µP a) 115 107
In 2007, Wygant et al. presented a conference paper on a micromachined parametric array [15]. The array was formed by a wafer of capacitive micromachined ultrasonic trans- ducers (cMUT) with vacuum sealed back cavities. Capacitive transduction and cMUTs are covered in Section 3.3.1. The fabrication of these devices was similar to those presented by Ergun et al. [47], which is covered in Section 3.3.1.1. The device was formed by an electrically conductive diaphragm over a back cavity at vacuum. Two different transducer designs were fabricated. Both had 4 mm diameters, but designs A and B had diaphragm thicknesses of
40 µm and 60 µm, respectively. The A and B design properties are outlined in Table 2-1. The diaphragms had large static deflections since the back cavities were at vacuum. The static deflection was averaged over the area of the diaphragm and was reported as an aver- age of 2 µm. The transducer performance including the large static deflection was modeled using finite element analysis (FEA). It was not reported if a nonlinear analysis was used. The stiffness of the diaphragm was dominated by the static deflection due to atmospheric pressure across the diaphragm [15]. The resonant frequency of designs A and B were 46 and 55 kHz, respectively. An entire array of devices was 8 cm in diameter. The pressure at 3 m was 115 and 107 dB for the A and B arrays, respectively. The diaphragms were actuated with a 200 V peak-to-peak AC signal with a 350-380 V bias voltage. To demonstrate the parametric array effect, the design B array was excited at 52 kHz (100 dB) and 57 kHz (110 dB) resulting in a 58 dB, 5 kHz difference frequency at 3 m.
48 Table 2-2. Transducers used in parametric array experiments.
Author No. Type Primary Primary Diff. Freq. Diff. SPL Distance Freq. (kHz) SPL (dB) (kHz) (dB) (m) Bennett et al. 1 oil-filled hydrophone 18.6 & 23.6 110 5 50 0.3 1975 [10] Yoneyama et al. 547 PZT bimorph 40 1 80 4 1983 [1] Kamakura et al. 581 PZT bimorph 40 1 68 9.5 1984 [28] Aoki et al. 1410 φ1 cm piezoelectric 27 & 30 112 3 65 3 1991 [32] transducer; 133 103 φ42 cm array Kamakura et al. 2000 PZT bimporph; 44×50 28 1991 [30] cm2 array Aoki et al. 91 φ1 cm piezoelectric 38.5 & 41.5 134.5 3 88 1 1994 [34] bimorph; φ11 cm array Kamakura et al. 1102 PZT bimporph 24 by 44 25 & 30 116 5 79 1994 [19] cm2 array 128 100 Pompei et al. φ35 cm array 60 1 77 3 1999 [12] Havelock et al. 5 φ5 cm piezoelectric plates 134 1 50 0.25 2000 [35] Kaduchak et al. 48 φ1.58 cm piezoceramic 217 115∗ 85 3 2000 [6] Croft et al. 85 φ4.5 cm PVDF 37.23 136.5 2001 [11] transducer w/ φ0.36 cm cavities at vacuum Moon et al. φ0.9 cm flexural 42.24 & 2002 [36] piezoelectric disk 43.25 Roh et al. 60 φ3.0 cm piezoceramic; 650 17.5† 103 2002 [14] φ40 cm array Kim et al. 40 120 2 2002 [48] Vinogradov et al. PA:49; φ1.06 cm piezoelectric 40 85‡ 2005 [38] FA:90 bimorph w/ φ0.7 cm cone Kamukura et al. 286 φ1 cm monomorph 39.3 120 3 2006 [39] ceramic transducers; 12.5 by 25 cm2 array Nakashima et al. 16 piezoelectric transducers 40 134 1 65 0.5 2006 [40] Haksue et al. 16 pMUT 95 & 135 40 85 0.18 2006 [44] Chen et al. 300 by 300 mm2 array 40 2 70.58– 1 2007 [41] 72.65 Peifeng et al. 91 φ8 mm piezoelectric 40 140 1 40§ 1 2007 [42] transducers Ying et al. PVDF transducer 42 62 1 2007 [43] Wygant et al. φ0.4 cm cMUT; 52 & 57 100 & 110 5 58 3 2007 [15] φ8 cm array ∗ Measured at 0.5 m. † Bandwidth of the difference frequency. ‡ Single transducer. § Measured at the intersection of the two beams.
49 2.2.4.1 Summary
Up until recently, most parametric array implementations used piezoelectric bimorph transducers as listed in Table 2.2.4. The bimorph transducers were off the shelf components and were not designed specifically as parametric array sources. Recent work focused on inno- vative transducer designs that were specific to parametric arrays such as the micromachined transducer arrays of Haksue et al. [45] and Wygant et al. [15]. Thus far, the published pMUT designs of Haksue et al. have focused on ranging applications where the demodulated signals were ultrasonic and not applicable to sound reproduction. Wygant et al. were the first to produce an audible tone using a micromachined parametric array. The primary sound levels, however, were weak in comparison to previous works. This led to a low audio output of 58 dB even at a difference frequency of 5 kHz. Also, the capacitive transduction scheme used by Wygant et al. required large ac voltages in addition to a large dc bias where the peak voltage signal could be almost 600 V [15]. The voltage requirement precluded the cMUT array from general commercial applications. In all cases, an optimal transducer design for parametric array applications is not found in the current literature.
2.3 Conclusion
This dissertation aims at realizing a micromachined transducer optimally designed for parametric array applications using an electroacoustic model. The following chapter presents basics of electroacoustic actuators and review the relevant micromachined devices. In Chap- ter 4, the transducer design is presented. In Chapters 5 and 6, the nonlinear acoustic theory of the current chapter is combined with the transducer basics of Chapter 3 to form an electroacoustic model and an objective function for optimization. Finally, the experimental results of Chapter 7 are evaluated for applicability to parametric arrays using the nonlinear acoustic theory of the current chapter.
50 CHAPTER 3 AIR-COUPLED MEMS ULTRASONIC TRANSDUCERS
This chapter presents an overview of air-coupled MEMS ultrasonic transmitters. The key characteristics of air-coupled transmitters are described. Next, the major sensing and transmitting mechanisms used in micromachined ultrasonic transducers (MUTs) are pre- sented. Finally, existing air-coupled MUTs are reviewed. 3.1 Principles of Transmitter Operation
An acoustic transmitter, or radiator, is a transducer that converts an electrical signal into an acoustic signal. Figure 3-1 displays a representation of a generic acoustic transmitter. The transmitter acts as an electroacoustic transfer function that shapes the frequency and phase content of the radiated acoustic wave. An electrical signal is applied to the acoustic transmitter. The transduction mechanism causes a mechanical element to deflect. The mechanical element imparts motion to neighboring fluid particles. This fluid particle motion consists of two parts: hydrodynamic fluctuation and gas compression. The energy of the compressed gas propagates into the fluid medium in the form of an acoustic wave.
Electrical Radiated Transmitter Input Acoustic Wave
Figure 3-1. Generic acoustic transmitter [49].
Acoustic radiation in ultrasonic transmitters is normally accomplished by either longi- tudinal bulk resonators or bending mode devices. The vibration of a bulk material shown in Figure 3-2 is predominately used in macro-sized transmitters [50].
Diaphragm deflection is used predominantly in MUTs. The transduction mechanism causes bending of the diaphragm that leads to out of plane motion as shown in Figure 3-3.
51 Acoustic Field
Bulk Vibrations
Figure 3-2. Bulk acoustic transmitter. Acoustic Field
Diaphragm
Figure 3-3. Generic bending mode acoustic transmitter [49].
If the transmitter is linear, then the pressure output at a given distance, Pout (ω), is
related to the input electrical signal, Vin (ω), as
Pout (ω) = Htrans (ω) Vin (ω) , (3–1)
where Htrans (ω) is a function of the acoustic frequency response function of the transmitter and the acoustic propagation to the point of interest. The frequency response function is separated into its magnitude and phase,
j∠Htrans(ω) Htrans (ω) = |Htrans (ω)| e . (3–2)
The magnitude of the frequency response function, |Htrans (ω)|, is a measure of the trans- mission sensitivity. The phase, ∠Htrans (ω), is the delay of acoustic propagation plus that of the transmitter with respect to the input electrical signal.
52 0
−6 dB
−5
bandwidth −10 Amplitude (dB re Peak) −15
−20 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 Normalized Frequency (f/f ) peak
Figure 3-4. Frequency response of an ultrasonic transmitter.
Transmitters are designed to have large sound pressure level outputs at a given distance. In order to maximize the sound pressure level output, ultrasonic transmitters typically op- erate around their fundamental resonant frequency. The resonant performance is dominated by damping. A representative frequency response function of an under-damped ultrasonic transmitter is shown in Figure 3-4. The bandwidth is a measure of the width of the funda- mental resonant peak. It is often defined as the frequency difference -6 dB, or quarter power, below the peak [50]. The quality factor, Q, is a measure of the sharpness of the resonant peak. The quality factor is defined as [51]
H (ω ) Q = r , (3–3) H (0) where ωr is the damped resonant frequency and H (0) is the frequency response function at dc. The quality factor of an under-damped, second order system where ζ2 < 1/2 is given
53 by [51] 1 Q = p , (3–4) 2ζ 1 − ζ2 where ζ is the damping ratio. As the damping decreases, the quality factor increases and the peak becomes sharper. Although the peak sensitivity increases, there is a tradeoff in
loss of bandwidth. The half-power bandwidth (-3 dB), Br, of an under-damped second order system, where ζ ≤ 0.1, is approximately [51]
ζω B ≈ r . (3–5) r π
The damping ratio of many macro ultrasonic transducers is designed to mediate the tradeoff between sensitivity and bandwidth to meet the needs of the application.
ity tiv si n e Ideal S t u tp u Physical O d n u o S Voltage
Figure 3-5. Comparison of the ideal and physical transducer sensitivities versus the forcing voltage for a fixed frequency.
An ideal transducer’s sound output sensitivity at a given distance remains constant re- gardless of the input electrical power. However, a physical device’s sensitivity decreases at
large forcing voltages due to nonlinearities as shown in Figure 3-5. Material stiffening at large deflections and acoustic harmonic generation (discussed in Chapter 2) are examples of sources of nonlinearities. The nonlinearities cause distortion in the pressure output of the
transmitter. The distortion is measured in terms of the harmonics generated in reference to
the fundamental signal. In parametric arrays, acoustically generated harmonics are desirable
54 for the production of audible frequencies. Harmonics produced by diaphragm stiffening, how- ever, are undesirable because they bleed energy from the fundamental frequencies, lowering the amount of energy that can be transferred from the mechanical to acoustic domain.
Transmitters are designed to have large acoustic output, but at the cost of increasing excitation power requirements. Thus, the transmitters are designed to have a large trans- mission sensitivity to minimize power requirements. The power transmission efficiency from the mechanical to acoustic domains has a large impact on the overall device efficiency. Sec- tion 3.2 on acoustic sources gives specific details on how power efficiency of acoustic sources varies depending on their size in comparison to acoustic wavelengths.
Transmitters are designed to have specific sound fields. For instance, conventional speak- ers ideally fill an entire room with sound. Ultrasonic transmitters, however, are usually designed to produce a confined sound field such as a beam. The measure of an acoustic transducer’s ability to distribute sound is the directivity pattern. Examples of fundamental acoustic sources and their directivity patterns are given in the following section as examples.
3.2 Acoustic Sources
This section uses examples of acoustic sources to present the concepts of radiation impedance, nearfield and farfield, directivity, volume velocity, and attenuation. One example is a planar radiator called a piston in an infinite baffle. Another example is a linear array of monopole sources that demonstrates how the acoustic fields of many sources combine in the farfield. The concept of sound absorption due to thermoviscous and relaxation effects is also introduced [8].
3.2.1 Planar Radiation
Many acoustic sources are modeled as planar radiators where out-of-plane structural velocity acts as a boundary source for acoustic waves in a fluid. A baffled loudspeaker is an example of a planar radiator. The acoustic analysis of a speaker can be very complex depending upon its deflection shape. The speaker in Figure 3-6 can be approximated as a baffled piston with a reasonable degree of precision [8].
55 ∞
∞ (a) Speaker. (b) Baffled piston.
Figure 3-6. Speaker modeled as a baffled piston.
A baffled piston is a rigid plane where all points are fixed except for an area that vibrates normal to the plane. In Figure 3-7, the white rigid baffle remains stationary while the gray
piston deflects in the z direction. The analytical approximation of the acoustic farfield due to baffled piston vibration is called Ralyeigh’s integral [8]. For an arbitrarily shaped piston as shown in Figure 3-7, Rayleigh’s integral gives the pressure in the farfield at point L (x, y, z) as [8]
Z 0 0 u˙ p (x , y ; t − R/c0) p (x, y, z; t) = ρ0 dS, (3–6) S 2πR q 0 2 0 2 2 where R = (x − x ) + (y − y ) + z , ρ0 is the ambient density, andu ˙ p is the piston acceleration. Many speaker systems can be approximated by an axisymmetric circular piston where
all points within the circle are vibrating harmonically with the same velocity amplitude, u0. Note that the sign convention of harmonic vibrations is a positive exponent, i.e. ejωt. The pressure at the point (r, θ) shown in Figure 3-8, where r is the distance from the piston center and θ is the angle made with the z-axis, is [8]
R 2J (ka sin θ) p (r, θ; t) = jP 0 1 ej(ωt−kr), (3–7) 0 r ka sin (θ)
56 x L( x, yz , ) dS( x′, y ′ ,0 )
Rigid z baffle Piston
y
Figure 3-7. Arbitrarily shaped baffled piston (adapted from Blackstock [8]).
1 2 where J1 ( ) is the first order Bessel function of the first kind [52], R0 = 2 ka is the Rayleigh distance, and P0 = ρ0c0u0.
x (r,θ )
r
θ Rigid baffle z Piston
y
Figure 3-8. Circular piston in an infinite baffle (adapted from Blackstock [8]).
The Rayleigh distance is a length scale that estimates the transition from a radiator’s nearfield, where pressure and velocity have a phase difference and waves are collimated, to the radiator’s farfield, where pressure and velocity are in phase and the waves spread spherically [8]. The pressure P0 should not be confused with the pressure on the piston face.
57 Instead, P0 is the pressure at the piston face that would exist if the piston face extended to infinity and had velocity amplitude u0 [8]. Many times, radiators are designed to have a certain SPL at a given distance. Since this distance can vary depending on the application, the source level (SL) is introduced to compare the relative strength of acoustic sources [8]. The SL is the SPL measured on the radiation axis of a sound source that is extrapolated to a 1 m distance by assuming spherical spreading. For instance, if prms is the rms pressure on the radiation axis of a sound source at a distance r, the source level is given as
prms r SL = 20 log10 . (3–8) pref (1 m)
The directivity, D (θ), gives the pattern of the pressure in the farfield [8]. The directivity function is the farfield pressure divided by the maximum pressure at the same distance,
P (r, θ) D (θ) = , (3–9) P (r, θmax) where θmax is the angle of maximum farfield pressure. The directivity of the baffled circular piston is 2J (ka sin θ) D (θ) = 1 . (3–10) ka sin (θ) The Maclaurin series expansion of Equation 3–10 for small ka is [8]
1 1 D (ka sin θ) = 1 − (ka sin θ)2 + (ka sin θ)4 + ··· . (3–11) 8 192
For a compact source where ka ¿ 1, the directivity function approaches 1. Thus, a compact source has a uniform pressure in terms of θ in the farfield.
For a non-compact source where ka À 1, J1 (ka sin θ) will be zero at multiple angles as shown in Figure 3-9. These are termed nulls. In between the nulls are minor lobes called sidelobes. Note that the sidelobes decrease in amplitude further from the central lobe.
Also, each sidelobe is 180◦ out of phase with its neighboring sidelobes. Figure 3-9 is the directivity pattern of a piston in an infinite baffle. A more realistic acoustic source will have
58 a deflection mode that is a function of radius and that goes to zero at the edge leading to sidelobe suppression [8].
It is general practice to report the -6 dB beamwidth of the main lobe as a measure of the source’s directivity. The -6 dB beamwidth denotes the angle at which the signal is a quarter of its maximum power. In Figure 3-9, the beamwidth of the compact source is much larger than that of the non-compact source.
θ 6 dB
θ compact 2 −6 dB
θ non-compact 2 −6 dB
Non-compact Compact Source
Figure 3-9. Comparison of the directivity of a compact and non-compact source.
Equation 3–7 is the farfield pressure. The radiation impedance seen by the piston is defined as [8]
Pav Zp = , (3–12) u0 where Pav is the average pressure at the piston face. The resulting radiation impedance, shown in Figure 3-10, is [8] · ¸ J (2ka) K (2ka) Z = ρ c 1 − 1 + j 1 . (3–13) p 0 0 ka ka
59
1
0.8 0 c
0 Real
ρ 0.6 /
p Imaginary Z 0.4
0.2
0 0 2 4 6 8 10 ka
Figure 3-10. Radiation impedance of a baffled, circular piston.
The Maclaurin series expansion of the piston radiation impedance for small ka is " Ã !# (ka)2 (ka)4 4 2ka (2ka)3 Z = ρ c − + ··· + j − + ··· . (3–14) p 0 0 2 12 π 3 45
Thus, for a compact source where ka ¿ 1, the radiation impedance becomes " # (ka)2 8ka Z = ρ c + j . (3–15) p 0 0 2 3π
Clearly, for ka ¿ 1, the imaginary part of the radiation impedance will dominate. Equa- tion 3–13 is rewritten as (ka)2 8a Z = ρ c + jωρ . (3–16) p 0 0 2 0 3π The second term in Equation 3–16 is equivalent to a lumped mass. Thus, the dominant loading on a compact radiator is a cylindrical slug of fluid with the radius of the piston and
8a a height of 3π . The majority of the compact radiator’s energy that is transferred to the fluid is reversibly stored as kinetic energy of the slug of fluid instead of creating acoustic waves that propagate away from the radiator.
On the other hand, the radiation impedance of a non-compact transducer approaches the
characteristic specific acoustic impedance, ρ0c0, as ka becomes large as shown in Figure 3-10.
60 Thus, the majority of the non-compact radiator’s energy is dissipated in the form of acoustic waves that travel away from the radiator. The non-compact radiator is more efficient than the compact radiator.
The efficiency of compact versus non-compact transducers is contrasted by looking at the power emitted by the baffled piston, µ ¶ πa2u2ρ c 2J (2ka) W = 0 0 0 1 − 1 . (3–17) 2 2ka
The power emitted by compact and non-compact radiators reduces to
πa2u2ρ c (ka)2 W = 0 0 0 (3–18) c 2 2 πa2u2ρ c and W = 0 0 0 , (3–19) nc 2 respectively. The power emitted by the compact radiator where ka ¿ 1 is dependent upon (ka)2 while the non-compact radiator is independent of frequency.
3.2.2 Array of Sources
As discussed in Chapter 2, a parametric array commonly uses many smaller sources to form its ultrasonic speaker. Thus far, baffled circular pistons have been introduced as acoustic transmitters. The acoustic field of many of these sources, called an array, is discussed here. As an example, a line array of ideal point sources, or monopoles [8], is discussed. A correction for non-ideal sources is addressed at the end of the section.
The pressure field of a single monopole is [8]
A p = ej(ωt−kr), (3–20) 0 r where A is the amplitude. Consider a linear array of N ideal, equally spaced monopoles separated by a distance d as shown in Figure 3-11. The pressure of the sum of the monopoles is [8] ¡ ¢ sin N kd sin θ ¡ 2 ¢ p = p0 kd , (3–21) sin 2 sin θ
61 θ
d p−3 p−2 p−1 p0 p1 p2 p3
Figure 3-11. Line array of N ideal monopoles (adapted from Blackstock [8]). where d is the spacing between monopoles. From Equation 3–21, the pressure on axis is p (θ = 0) = N · p0. Thus, the array directivity is ¡ ¢ sin N kd sin θ ¡ 2 ¢ Da (θ) = kd . (3–22) N sin 2 sin θ The directivity function is plotted in Figure 3-12 for arrays with different number of sources at an equal spacing of kd = 1. As shown, increasing the number of elements in the array decreases the beamwidth of the sound field. Note that in Equation 3–22, as kd approaches zero, the directivity function approaches one. As kd approaches zero the monopoles line up on top of each other forming a single source.
The acoustic field of a line array of directional sources is simply the monopole field multiplied by the directivity of the directional sources, De (θ) [8]. Thus, the directivity of an array of N non-ideal sources is [8]
D (θ) = Da (θ) De (θ) . (3–23)
The basic principles of the analysis of a line array can be extended to a two-dimensional array such as a parametric array source. As will be shown in Section 3.3, the size of a MEMS bending mode ultrasonic transmitter scales on the order of 1 mm for resonant frequencies in the low ultrasonic range around 40 kHz [53]. The directivity of a single of these transducer is
62
90 1 120 60 N=3 0.8 N=7
0.6 150 30 0.4
0.2 θ
180 0
210 330
240 300 270
Figure 3-12. Array directivity for 3 and 7 monopole arrays at kd = 1.
fairly omnidirectional with a ka ≈ 0.7. On the order of 103 devices [1] would form the source of the parametric array. Similar to the example of the linear array, the increase in number of transducers and subsequent increase in array diameter creates a narrow array directivity, resulting in the beam generally required for parametric array operation.
3.2.3 Acoustic Attenuation
Ultrasonic transducers, by definition, operate at very high frequencies where acoustic attenuation is a significant effect that must be taken into account. Acoustic attenuation, or absorption, is the description of energy loss of an acoustic wave. Viscosity, heat conduction, molecular relaxation, as well as other loss mechanisms result in acoustic attenuation [8].
To find an analytical solution of absorption that includes the contributions of attenu- ation, the physical mechanisms are individually included in the governing fluid mechanics equations. The equations are combined to form a wave equation [8]. Plane waves are con-
sidered here for the purpose of discussion. It can be shown that the solution to the wave
63 equation including dissipation mechanisms takes the familiar form [8]
j(ωt−kz) u = u0e . (3–24)
In Equation 3–24, k is allowed to be complex and assumes the form [8]
k = β − jα. (3–25)
Thus, a dispersion relationship where the phase velocity is dependent upon frequency is
introduced into Equation 3–24 [8] resulting in
−αz j(ωt−βz) u = u0e e , (3–26)
where α is called the absorption coefficient. The new phase speed, cph, is related to β by
ω c = . (3–27) ph β
The absorption coefficient for each loss mechanism is found from the corresponding solution to the wave equation. The absorption coefficients are then summed to obtain the total absorption. For example, to find the absorption coefficient that includes thermoviscous and relaxation effects, the sum of the individual absorption coefficients is taken,
X i α = αtv + αmr, (3–28) i
i where αmr is the relaxation absorption coefficient for each constituent of the gas. Although the absorption coefficients from Equation 3–28 are solved using plane wave solutions, the same coefficients are used in the analysis of spherical waves [8].
The thermoviscous absorption coefficient is proportional to frequency squared [8],
2 αtv ∝ f . (3–29)
64 The absorption coefficient due to molecular relaxation is dependent upon the frequency as [8] 2 i f αmr ∝ 2 2 , (3–30) f + fr,i where fr,i is the relaxation frequency of each of the molecular constituents of the gas. Acoustic absorption in air is dominated by thermoviscous absorption and relaxation associated with the vibrational modes of nitrogen and oxygen [8]:
αair = αtv + αO2 + αN2 . (3–31)
102
α = α + α + α 0 air N O tv 10 2 2
10−2
−4 α 10 air Sound absorption (1/m) α N 2 α O 10−6 2 α tv
101 102 103 104 105 106 Frequency (Hz)
Figure 3-13. Sound absorption coefficient of air at 70% relative humidity (after Zuckerwar [54]).
The relaxation frequencies are dependent upon the water vapor content. Thus, the absorption coefficient is substantially dependent upon humidity. Absorption at 70% relative humidity is given in Figure 3-13. At 40, 80, and 120 kHz, the absorption at 70% relative
65 humidity is -1.3, -3.0, and -4.7 dB/m. At 20% humidity, the absorption at 40 kHz drops to -0.8 db/m.
From the Bertkay solution from Section 2.2.2.2, the demodulated sound scales as the
inverse of the absorption coefficient of the fundamentals. Once the demodulated sound is created, it travels as a normal acoustic wave and experiences acoustic absorption. How- ever, since the frequencies of the demodulation sound are usually in the audible range, its attenuation is commonly neglected. 3.2.4 Summary
The basics of acoustic radiators have been covered in the previous sections. The prin- ciples introduced are used to evaluate the devices covered in the following literature review
of MEMS actuators and as the basis for much of the modeling in Chapter 5.
3.3 MEMS Actuators
A MEMS transducer could serve as an ultrasonic radiator while leveraging the sub- stantial batch fabrication technology of the integrated circuits industry. In recent years, a large body of work has emerged on MEMS designed for ultrasonic use in air. These devices, commonly referred to as micromachined ultrasonic transducers (MUTs), are usually planar radiators that use a variety of actuation and sensing mechanisms. Many MUTs are used as pulse/receive devices in which they transmit an acoustic signal and then receive acoustic reflections. The devices can use the same transduction mechanism, such as electrostatic, for both sensing and transmitting or a combination of different transduction methods. For a
general transducer, the electromechanical transduction equation in terms of power conjugate variables is given by [55] V ZEB TEM I = . (3–32) F TME ZMO U ¯ V ¯ where V is voltage, F is force, I is current, U is velocity, ZEB = I U=0 is the blocked ¯ ¯ F ¯ V ¯ electrical impedance, ZMO = U I=0 is the open-circuit mechanical impedance, TEM = U I=0
66 ¯ F ¯ is the open-circuit electromechanical transduction factor, and TME = I U=0 is the blocked electromechanical transduction factor. Note that the product of power conjugate variables, such as force and velocity or current and voltage, results in power.
The major sensing and transmitting mechanisms used in MUTs described in the lit- erature are presented. The major mechanisms can be split into two categories: reciprocal and non-reciprocal. Reciprocal mechanisms are used to both actuate and sense changes in a physical system. By definition, reciprocity indicates that the open-circuit and blocked elec- tromechanical transduction factors in Equation 3–32 are equal [55]. Reciprocal mechanisms prevalent in the MUT literature are electrostatic and piezoelectric. Electrostatic transduc- tion utilizes the capacitance-displacement relationship between two or more electrodes [56]. Piezoelectric transduction makes use of the piezoelectric effect, where direct polarization of a piezoelectric material gives rise to an elastic strain. Conversely, elastic stress of the material creates a change in electric potential [56].
Thermoelastic actuation is the major non-reciprocal actuating mechanism observed in the literature. Temperature gradients created within a structure cause non-uniform thermal expansion and thus a finite amount of displacement [57]. The literature review is organized according to the transduction method. A brief review is presented of electrostatic, piezoelectric, and thermoelastic transduction with each followed by a review of MEMS-based actuators. The performance and fabrication of the devices is presented. Finally, the tradeoffs among the different mechanisms are discussed in the conclusion.
3.3.1 Electrostatic Transduction
To introduce electrostatic transduction typically used in bending mode transducers, a simplified transducer consisting of two conducting parallel plates of area A separated by a gap x0 is shown in Figure 3-14. The bottom plate is fixed. The balance between the mechanical restoring force (represented by a spring), the electrostatic force between the two plates, and the gravitational force holds the top plate in place.
67 Mechanical spring V
= x x 0 x′
x = 0
Figure 3-14. One-dimensional electrostatic transducer.
Displacement of the top plate is denoted x0 (t) such that the distance between electrodes
0 is x0 − x (t). The capacitance between the plates is given by
CEB CE (t) = 0 , (3–33) 1 − x (t)/x0
where CEB = ε0A/x0 is the blocked capacitance [56]. The blocked impedance of an electro- static transducer is the ratio of the voltage to current while maintaining the gap. If a voltage is supplied across the plates, a charge is stored on the capacitor. The voltage is related to the charge by [56] µ ¶ Q (t) Q (t) x0 (t) V (t) = = 1 − . (3–34) CE (t) CEB x0 The total force acting on the plate is the sum of the electrostatic and mechanical forces (ignoring gravitational effects). To calculate the electrostatic force, the displacement deriva- tive of the electrical potential energy is taken [56]. The electric potential energy is given by Q2 (t) WE = . (3–35) 2CE (t) Thus, the electrostatic force is
2 dWE 1 Q (t) FE = − 0 = − . (3–36) dx 2 CEBx0
68 Assuming a linear stiffness relationship between the mechanical force and displacement,
0 the mechanical force is given by Fm (t) = x (t)/CM where CM is the mechanical compliance represented by the spring in Figure 3-14. The total force acting on the top plate is
x0 (t) 1 Q2 (t) F (t) = − . (3–37) CM 2 CEBx0
Equations 3–33, 3–34, and 3–37 form three coupled, nonlinear equations. It is common practice to supply a dc bias as well as an ac voltage across the plates
0 for linearization purposes, V (t) = V0 + V (t). The magnitude of the bias is adjusted to improve the linearity between the ac signal and the plate deflection. The potential across
0 the plates induces a charge that also consists of dc and ac components Q (t) = Q0 + Q (t), where Q0 = V0CEB. To find the linear transduction equations, it is assumed that the ac components of charge, voltage, and displacement are much smaller that the dc components. Thus, the linearized equations are µ ¶ x0 (t) CE (t) = CEB 1 − , (3–38) x0 Q0 (t) V V 0 (t) = − 0 x0 (t) , (3–39) CEB x0 x0 (t) V and F 0 (t) = − 0 Q0 (t) . (3–40) CM x0
Assuming time harmonic signals, the linearized transduction Equations 3–39 and 3–40 are written in matrix form as
0 1 V0 0 V jωC − jωx I (t) = EB 0 . (3–41) F 0 − V0 1 U 0 (t) jωx0 jωCM where I0 and U 0 are ac current and velocity. The upper right and lower left terms in Equa- tion 3–41 are known as the open and blocked electromechanical transduction factors, re- spectively [55]. Linear electrostatic transduction is reciprocal because these two terms are equal.
69 Equation 3–41 is a linearized transduction equation whose degree of robustness depends upon small deflections and small ac signals. When deflections become large, the linear assumption is violated and substantial harmonics are produced. The charge dependence of
Equation 3–36 is converted to a voltage dependence by substituting Equation 3–34:
x0 (t) 1 ε AV 2 (t) F (t) = − 0 . (3–42) 0 2 CM 2 (x0 − x (t))
The voltage squared dependence of Equation 3–42 is nullified by supplying a dc bias voltage across the plates as described above. However, the dependence of Equation 3–42 on the inverse of the gap squared remains. For small deflections, x0 (t), with respect to the nominal gap, x0, the denominator of Equation 3–42 can be linearized,
1 x0 (t) ' 1 + 2 . (3–43) 0 2 (1 − x (t)/x0) x0
On the other hand, if the dynamic deflections are substantial with respect to the gap, odd
harmonics are produced by the displacement dependence of the second term in Equation 3– 42 [58].
dF dF E< M dx dx = x x 0 x′ 2 dFE dF M x= x > 3 0 dx dx
x = 0
Figure 3-15. Pull-in point where the change in the electrostatic force is equal to the change in the mechanical force.
For the top plate to be in static equilibrium, the mechanical and electrical forces must
balance each other. For a small dc voltage, there exists a small displacement where the forces
balance and the net force, given by the static version of Equation 3–42, equals zero. At this
70 0 point the plate is stable since the gradient of the mechanical restoring force, dFM /dx , is
0 positive and the gradient of the electrical force, dFE/dx , is negative as shown in Figure 3-15. As the voltage increases, the gap decreases and eventually reaches an unstable point at which the gradient of the net force equals zero. Beyond this point, the electrostatic force grows faster than the mechanical restoring force as shown in Figure 3-15. This combination of gap and voltage is known as the static pull-in point. A plot of the mechanical restoring and electrical forces for increasing voltage are shown in Figure 3-16. To find the static pull-in voltage and gap, the net force and net force gradient are both set equal to zero and solved, resulting in
2 xPI = x0 (3–44) s3 3 8x0 and VPI = . (3–45) 27CM ε0A
Pull-in can be detrimental in capacitive transducers as the diaphragm will effectively crash
1 F M 0.8 F E ) M
/ C 0.6 V / V =1.0 0 PI x x / x =2/3 0 V / V =0.8 0.4 PI V / V =0.6 PI Force / ( 0.2 V / V =0.4 PI
0 1 0.8 0.6 0.4 0.2 0 Gap ( x / x ) 0
Figure 3-16. Plot of the mechanical force, FM , and the electrostatic force, FE, at different voltages versus the gap distance x (Note that all values are non-dimensional). into the back plate. Even if the top plate is undamaged, the voltage must be lowered below the pull-in voltage before it can be released since intermolecular cohesive forces act to keep
71 the plates together. Thus, pull-in limits the maximum deflection possible which is directly related to its ability to project high intensity sound.
The following section is a review of MEMS electrostatic actuators for air applications.
A comparison of the resonant frequencies and outputs of the devices to those of parametric array implementations to date (see Section 2.2.3) provide insight into the applicability of this technology to parametric arrays. The following device review also gives insight into the fabrication technologies used to create these devices. 3.3.1.1 MEMS Electrostatic Actuators
There is a large body of work on capacitive micromachined ultrasonic transducers (cMUTs). Many cMUTs are designed for proximity sensing at very short distances. Thus, the resonant frequencies are very large to improve the spatial resolution. The effect of absorption is not detrimental due to the short working distances. Most of the operating frequencies of the cMUTs are much higher than those considered suitable for parametric array operation. However, an overview of ultrasonic, electrostatic transducers is presented to explore the device types possible with MEMS technology.
Aluminum Polyester Cavity
Oxide Silicon
Figure 3-17. Device formed by combining macro- and micromachining (adapted from Higuchi [59]).
Efforts began with Higuchi et al.’s [59] presentation of a hybrid ultrasonic sensor/actuator shown in Figure 3-17 that utilizes both micro- and macro-machining. The cavity and back electrodes were formed by anisotropic etching of the silicon and subsequent aluminum depo- sition. The wafer was diced into 20 mm by 30 mm chips containing 32 elements. A 12 µm thick polyester film with a 50 nm Al layer was stretched over the cavities and anchored with machine screws. Three square cavities 80 µm, 40 µm, and 10 µm on a side were fabricated.
72 Their resonant frequencies were 140 kHz, 180 kHz, and 215 kHz, respectively. The 80 µm device array had an acoustic output of approximately 105 dB at 50 cm from the array at 140 kHz.
Suzuki et al. [60] continued the work of Higuchi et al. and presented a similar hybrid ultrasonic sensor/actuator. Micromachining was used to define pyramid shaped back cavi- ties 10-40 µm on a side and an aluminum back electrode. Then, the wafer was diced and packaged and a stretched, metallized polyester diaphragm was attached over the cavities. They achieved a transmit sensitivity of 119.1 dB re 1 µPa/V at 50 cm from the devices at 150 kHz and a receiving sensitivity of 0.47 mV/Pa from 10-130 kHz at a bias voltage of 30 V.
The transducers presented in by Schindel and Hutchins et al. [61–65] are similar to those
of Suzuki et al. [60]. A KOH etch was used to define cavities on the silicon wafer 40 µm on a side with 80 µm center-to-center spacing. The cavities were arranged in a rectangular array 25 mm2. A metallized, 5 µm thick Kapton film was stretched and mechanically attached over the cavities. The reported experimental results were normalized.
Silicon Nitride Oxide Etch holes Oxide posts Approximate Gold Silicon diaphragm size
(a) Cross-section. (b) Top-view.
Figure 3-18. Capacitive cMUT of the E.L. Ginzton Laboratory (adapted from Haller [66]).
The first air-coupled ultrasonic MEMS of the E. L. Ginzton Laboratory at Stanford University is shown in Figure 3-18 [66–69]. The device consisted of an array on a 1 cm square die of nitride membranes 750 nm thick with 50 nm thick gold top electrode and a
highly doped (100) silicon substrate back electrode (substrate also has a layer of gold on the
backside for electrical contact). The nitride and gold electrode layers were deposited over a
73 layer of 1 µm thick sacrificial oxide on a silicon wafer. Holes of 3 µm diameter were etched through the top gold electrode and the silicon nitride layer. The holes were etched at 100, 50, and 25 µm spacing to create three types of devices. A timed hydrofluoric acid oxide etch released the membrane and created posts (approximately 20 by 20 µm2) that supported the membranes as shown in [66]. Thus, the back cavity was common to all of the diaphragms. The 100 µm devices had a resonance of 1.8 MHz while the 50 µm devices peaked at 4.5 MHz. Using an optical interferometer, the peak center displacement of the 100 µm device was 23 nm/V with a 100 V bias voltage. The maximum ac voltage was 16 Vpp resulting in an approximately 300 nm center displacement.
A similar device was fabricated by Ladabaum et al. [70]. In this device the holes used for the previously described timed etched were filled with a low pressure chemical vapor
deposition nitride. The resulting nitride diaphragm thickness was 600 nm. The resonant frequencies were 1.8 MHz for a 100 µm diameter membrane and 12 MHz for a 12 µm diameter membrane. Diaphragm Nitride Aluminum electrode
Silicon Vacuum sealed cavity
Figure 3-19. Nitride diaphragm cMUT with vacuum sealed back cavity (adapted from Jin et al. [71]).
Jin et al. presented an array of cMUTs formed by a nitride diaphragm with an alu- minum top electrode as shown in Figure 3-19 [47,70–77]. The heavily doped silicon substrate formed the bottom electrode. A 200-1000 nm amorphous silicon layer was used as a sacri- ficial layer because it had excellent selectivity with respect to the nitride diaphragm. The
amorphous silicon was etched into hexagonally shaped islands that defined the cavities after
diaphragm release. Nitride was deposited over the islands and patterned for access to the
74 amorphous silicon for diaphragm release. After the sacrificial amorphous silicon etch, nitride deposition sealed the cavities. The diaphragm thickness varied from 0.5-2 µm. A thin layer of nitride existed between the substrate and back cavity that served as an etch stop during
the diaphragm release. Cell dimensions ranged from 20-100 µm. They reported 110 dB of dynamic range in air at a resonance of 2.3 MHz for a device with cells 100 µm wide and 0.5 µm thick with an air gap of 1.0 µm. A summary of the work on this transducer was presented by Ergun et al. [47].
diaphragm polysilicon oxide/nitride aluminum
field oxide back cavity doped silicon silicon bottom electrode
Figure 3-20. Capacitive transducer with a polysilicon diaphragm and doped bottom elec- trode (adapted from Eccardt et al. [78]).
Eccardt et al. [79, 80] presented the capacitive transducer illustrated in Figure 3-20. Hexagonal, polysilicon membranes with side lengths of 40 µm and thickness of 400 nm and cavities of 450 nm depth formed the transducer. They achieved approximately 10 nm of deflection around a resonance of 10 MHz. A new fabrication step was added in hopes of increasing the electrostatic force [78]. The process was a modified CMOS process that fixed the gap thickness. Bumps were introduced on the bottom of the diaphragm. When the diaphragm was collapsed due to pull-in, these bumps formed the edge of a smaller diaphragm with a gap thickness the size of the bumps. Of course, even though the electrostatic force was increased by reducing the gap thickness, pull-in restricted the deflection and thus the pressure amplitude. Thus, the maximum center deflection of the smaller membrane was the new gap thickness. Testing of these transducers was conducted in water.
75 ThroatDoped Silicon Throat
Doped Polysilicon Membrane Glass Substrate Resonant Chamber
Figure 3-21. Micromachined capacitive device with coupled Helmholtz resonator formed by the resonant cavity and throat (adapted from Parviz et al. [81]).
Parviz et al. [81] described the use of an electrostatic actuator coupled to a Helmholtz
resonator in an effort to produce acoustic streaming at high frequencies (≥ 100 kHz). Acous- tic streaming is the creation of a mean flow by a high frequency acoustic field [18]. Appli- cations of the device included propulsion, micro-cooling, and micro-pumping. A square, polysilicon membrane 1.36 µm thick and 1.2 mm on a side was suspended 3 µm below a perforated, boron doped single crystal silicon layer. Oxide and nitride layers served as pas- sivation of the membrane. On the opposite side of the membrane was a resonant chamber that led to a throat. By supplying an alternating voltage across the diaphragm, an acoustic field was established within the resonant chamber. If the acoustic signal was high enough, it generated acoustic streaming through the throat. Actuators covered an entire 4 inch wafer in 4 quadrants (992 total devices). The diaphragm was claimed to collapse when a voltage was applied. The measurement microphones used to characterize the devices were audio grade and therefore only had a flat bandwidth out to 20kHz. Thus, the measurements at higher frequencies were subject to the roll-off response of the microphone. They reported a peak at 96 kHz. They also noted the harmonics generated by the collapse mode actuation.
Torndahl et al. [82] presented a capacitive transducer with a pyramid shaped back cavity and sides of length 40 or 60 µm and diaphragm thicknesses of 8 or 12 µm. The diaphragm of the cMUT was formed by a metal on top of a polymeric membrane. The doped silicon
substrate served as the back plate. The resonances of the cMUTs varied from 400 kHz to 1
MHz. Light diffraction tomography was used to measure the acoustic field of the transducer.
76 Electrode Insulating Film
Silicon
Figure 3-22. A cMUT whose cavities are formed by an anisotropic etch (adapted from Torn- dahl et al. [82]).
Experiments on an array of the 40 µm sized cavities with the 8 µm thick diaphragm produced 137.5 dB at 450 kHz at 35 mm from the device surfaces. Aluminum Polysilicon LTO LTO
Oxide Electrodes Silicon
Figure 3-23. Capacitive transducer with polysilicon moveable membranes (adapted from Buhrdorf et al. [83]).
Buhrdorf et al. [83] detailed a cMUT that utilized a polysilicon diaphragm. A doped polysilicon region defined the upper electrode. The diaphragm was about 1 µm thick with a gap height between 300-400 nm. Line arrays of the devices had a width of 100 µm. The hexagonal diaphragms were packed in this region in rows of two or three devices. The devices obtained over 100 nm/V drive sensitivity at a resonance of approximately 5.7 MHz when a dc bias of 82 V was supplied across the diaphragm. Etch holes Nitride Diaphragm Polysilicon
Silicon Cavity
Figure 3-24. A cMUT fabricated using MUMPS (adapted from Oppenheim et al. [84]).
77 Oppenheim et al. [84] contributed a capacitive transducer fabricating using the multi- user MEMS process (MUMPS). The diaphragm and backside electrode were formed by two polysilicon layers as shown in Figure 3-24. The diaphragm thickness and back cavity were constrained by the process to be 2 µm. The device had a hexagon top plate that was 45 µm on each side. Etch holes 5 µm in diameter allowed the oxide layer to be sacrificed. The etch holes also served as vents so that the device did not respond to changes in ambient pressure. The resonant frequency was 3.47 MHz.
LTO Diaphragm
Cavity Aluminum Oxide
Silicon
Figure 3-25. A cMUT fabricated using an SOI wafer bonded to a patterned substrate (adapted from Huang et al. [85,86]).
Another paper by the Stanford group described cMUTs similar to those already pre- sented created using a new fabrication method [85,86]. They defined the cavity on one wafer using either a simple isotropic oxide etch or a combination of oxide etch and silicon etch depending on the cavity depth desired. An SOI wafer was bonded to the original wafer. The diaphragm was formed by the single crystal silicon layer of the SOI wafer by grinding the bulk silicon of the SOI and sacrificing the BOX layer. Square membranes up to 750 µm in size were fabricated. For a 650 µm, 4.2 µm thick device with a gap of 11.5 µm, they measured a resonant frequency of 310 kHz. This fabrication method avoided some of the difficulties of the previous methods such as the nitride fill-in to vacuum seal the back cavity. They conducted pulse/receive experiments in air where the transducer has a narrow band of about 278 kHz.
The Stanford research group also explored three different operating modes for their cMUT transducers [76,87–89] : conventional mode where the total of dc bias and ac voltage never exceed the pull-in voltage, collapse mode where the dc bias exceeds the pull-in voltage
78 such that the diaphragm is collapsed to the bottom electrode, and collapse-snap-through mode where the dc bias is less than the pull-in voltage but the total of the dc bias and ac voltage exceeds the pull-in threshold causing the diaphragm to dynamically collapse to the
bottom electrode and then snap back when the voltage drops below pull-in. In collapse mode, diaphragm vibration occurs between its anchor to the substrate and the region where it has collapsed to the bottom electrode. The enhanced operation of the collapse-mode transducers over the conventional transducers is due to the decreased gap thickness causing an overall increase in capacitance and electric field [88, 89]. Experimental results for collapse-snap- through mode were reported in water [87].
Aluminum electrode Diaphragm
Cavity Nitride
Silicon
Figure 3-26. A cMUT with nitride diaphragm (adapted from Kim et al. [90]).
Kim et al. [90] designed and fabricated a cMUT with a 0.4 µm thick nitride membrane 45 µm in diameter and 15 µm spacing. The top electrode was formed by 0.2 µm of aluminum as shown in Figure 3-26. The bottom electrode was formed by the silicon substrate with a metal back-plate. The air gap was 0.3 µm. Finite element analysis predicted a resonance of 7.4 MHz. With a 40 V dc bias and 5 V ac signal, 0.2 µm of deflection was obtained.
3.3.1.2 Summary
A summary of the cMUTs is contained in Table 3-1. All the devices have resonant frequencies from the 100’s of kHz up to the MHz range. These frequencies are most likely not suitable for parametric array transducers due to large attenuation at high frequencies and the longer working distance required of the parametric array. It is also extremely difficult
to make quantitative measurements of the sound field at high frequencies in air. Generally,
B&K serve as measurement standards for microphones. The B&K 4138 1/4-inch free field
79 microphone has an upper bandwidth limit of 100 kHz and the B&K 4138 1/8-inch pressure field microphone has an upper bandwidth limit of 140 kHz. A free field microphone is intended to be used to measure the free field sound pressure level whereas the pressure field
microphone is intended to measure uniform pressures at a sound hard surface. Measurements beyond the upper bandwidth limit of the microphone are attenuated by the roll-off in the microphone response. Also, at these high frequencies, the short wavelengths interact with the microphone to change the average pressure acting on the diaphragm resulting in erroneous pressure readings.
In addition, the backsides of many cMUTs’ diaphragms are vacuum sealed such that the atmospheric pressure will cause a static deflection. A static deflection will cause a change in stiffness of the diaphragm. Thus, a change in atmospheric conditions would cause a change
in the resonant performance of the device. The transduction mechanism of cMUTs is inherently nonlinear resulting in harmonic distortion produced by the diaphragm vibrations. Also, the nonlinear transduction mecha- nism results in a pull-in voltage that further constrains device performance.
3.3.2 Piezoelectric Transduction
Piezoelectric transduction is similar to electrostatic transduction in that it involves the application or measurement of an electric potential between two electrodes across a dielec- tric. In the case of electrostatic transduction, the dielectric is often times air or vacuum and the transduction involves the mechanical motion of one or both electrodes as described in the previous section. Since the dielectric properties of air and vacuum are similar, electro- static transduction is nominally dependent upon electrode geometry and electrode material properties. In contrast, piezoelectric transduction involves the relationship between the electric
field and the stress/strain in the dielectric. The dielectric in this case is a piezoelectric
material, such as quartz (SiO2) or lead zirconate titanate (Pb[ZrxTi1−x]O3 0 < x < 1), commonly known as PZT. Thus, piezoelectric transduction is not only dependent upon
80 Table 3-1. Air-coupled cMUT characteristics. Author Diaphragm Gap Transmit Output Resonant Dimensions Sensitivity Frequency Higuchi et al. 80 µm by 12 NA NA 105 dB at 50 140 kHz 1986 [59] µm cm Suzuki et al. 40 µm by 12 NA 19.1 dB re 1 NA 150 kHz 1989 [60] µm µbar/V at 50 cm Schindel et al. 40 µm by 5 NA NA NA NA 1995 [64] µm Eccardt et al. 40 µm by 400 450 nm NA 10 nm 10 MHz 1996 [79] nm Haller et al. ϕ100 µm by 1 µm 23 nm/V 300 nm 1.8 MHz 1996 [66] 750 nm Ladabaum et al. ϕ100 µm by 1 µm NA NA 1.8 MHz 1998 [70] 600 nm Jin et al. ϕ100 µm by 1 µm NA NA 2.3 MHz 1998 [72] 0.5 µm Parviz et al. 1.2mm by 3 µm NA NA 96 kHz 2000 [81] 1.36 µm Torndahl et al. 40 µm by 8 NA NA 137.5 dB at 450 kHz 2002 [82] µm 35 mm Huang et al. 650 µm by 11.5 µm NA NA 310 kHz 2003 [85] 4.2 µm Buhrdorf et al. 50 µm by 1 300-400 nm 100 nm/V NA 5.7 MHz 2003 [83] µm Oppenheim et al.45 µm by 2 2 µm NA NA 3.47 MHz 2003 [84] µm Kim et al. ϕ45 µm by 0.3 µm NA 0.2 µm 7.4 MHz 2005 [90] 0.6 µm electrode geometry and diaphragm mechanical properties, but also highly dependent upon the piezoelectric material properties. This section defines piezoelectricity and discusses its properties and materials commonly found in microsystems.
A piezoelectric material is most generally defined as a crystalline material with non- centrosymmetric structure, with the exception being the point group 43 piezoelectric material [91]. Examples of centrosymmetric and non-centrosymmetric crystalline structures are given in Figure 3-27. Piezoelectric materials exhibit a coupling between electrical and mechanical energy domains. Mechanical stress applied to a piezoelectric material induces an electric
field. This is known as the direct piezoelectric effect. On the other hand, the converse
81 piezoelectric effect is the generation of a mechanical strain by the application of an electrical field.
(a) Centrosymmetric crystal unit cell. (b) Non-centrosymmetric crystal unit cell.
Figure 3-27. Isometric view of an ideal perovskite structure. [91].
A subset of piezoelectric materials are classified as pyroelectric. These materials exhibit a change in polarization due to a temperature change [91]. Polar materials have a net charge distribution in the absence of an electric field [92]. One such distribution is a dipole which is the created by the separation of positive and negative charges [93]. Only piezoelectric materials that are polar are also pyroelectric. An example of a non-polar piezoelectric material is SiO2. Aluminum nitride (AlN) and zinc oxide (ZnO) are examples of pyroelectric (i.e., piezoelectric and polar) materials [91]. Both materials have wurtzite structures as shown in Figure 3-28a. The structure consists of two hexagonal crystal lattices that are tetrahedrally interconnected [91].
Some pyroelectric materials exhibit a property where the direction of their dipole can be changed by the temporary application of an electric field. These materials are termed ferroelectric [91]. An example of a pyroelectric material that is not ferroelectric is AlN. PZT is an example of a ferroelectric material. The crystalline structure of PZT is perovskite as shown in Figure 3-28b. Ferroelectric materials lose their piezoelectric property when
82 Al N Zr/Ti Pb O Zn O (b) Perovskite crystal structure of PZT. (a) Wurtzite crystal structure of AlN and ZnO.
Figure 3-28. Isometric views of ideal wurtzite and perovskite structure. [91]. the temperature is raised beyond the Curie temperature. At this point, the atoms relax into their centrosymmetric positions [93]. Within a single ferroelectric crystal, domains of opposite but equal polarization will form such that the net polarization is zero. On a macro- scale, the symmetry of the polarization is accentuated by the multi-crystalline structure of many piezoelectric materials [91]. Thus, on a macro-scale, the piezoelectric material is not piezoelectrically active. A ferroelectric material is made piezoelectrically active by applying an electric field that is strong enough to reverse the direction of the polarization of the crystal domains so that they are most closely aligned to the electric field. The field strength at which reversal begins is known as the coercive field [94]. This process is termed poling.
Many times, poling is achieved by heating beyond the Currie temperature and cooling under a weaker electric field [91].
83 Piezoelectric transduction is often approximated by the linear piezoelectric constitutive equations [94],
σ Di = ²ijEj + diqσq (3–46)
E and εp = Spqσq + djpEj, (3–47)
2 where Di is the electric displacement vector [C/m ], εp is the engineering strain vector, σp
is the mechanical stress vector [Pa], and Ej is the electric field vector [V/m], diq is the
σ piezoelectric constant tensor [C/N], ²ij is the permittivity tensor at constant stress [F/m],
E and Spq is the elastic compliance tensor at constant electric field [1/Pa]. The three tensors are material constants. The terms of the piezoelectric constant tensor in Equations 3–46 and 3–47 determine the level of coupling the between the electrical and mechanical domain.
V V V = 0 P P P
expansion contraction
Figure 3-29. One-dimensional piezoelectric transducer (adapted from [16]).
Consider the one-dimensional simplified case of a bulk vibrator shown in Figure 3-29 [56]. To describe the behavior, the constitutive equations, 3–46 and 3–47, are rewritten to express the electric displacement in terms of charge q, the strain in terms of displacement x, the stress in terms of mechanical force F , and the electric field in terms of voltage V ,
q =CEF · V + d · F (3–48)
and x =d · V + CMS · F, (3–49)
respectively, where d is the piezoelectric modulus, CMS is the short-circuit mechanical com- pliance, and CEF is the free electrical compliance. From Equation 3–48, the piezoelectric
84 material acts like a normal elastic material if the top and bottom electrodes are shorted such that V = 0. Similarly, from Equation 3–49, the piezoelectric material behaves like a normal dielectric material between capacitor plates when the plate is ”free” such that F = 0.
For a time harmonic system, the transduction equations are written in conjugate power
variable form as [56] I jωCEF jωd V = . (3–50) U jωd jωCMS F Note that Equation 3–50 is written in admittance (i.e., inverse of impedance) form since the effort variables are given as output. The off-diagonal terms are equal indicating that the
linear model of the piezoelectric transducer is reciprocal [56].
The piezoelectric modulus, d, for the thickness mode transducer shown in Figure 3-29 is directly related to the d33 term of the piezoelectric constant tensor that relates an electric field in the 3 direction to elastic strain in the same direction through the converse piezoelectric effect. In contrast, many MEMS devices that utilize piezoelectric transduction are bending mode structures with a piezoelectric thin film. These devices rely on the d31 coefficient for transduction. This coefficient relates an electric field in the 3 direction (transverse to the plane of the film) to the elastic strain in the 1 direction (along the plane of the thin film). As illustrated in Figure 3-30, strain within the thin film deposited on the membrane or beam
can cause out of plane deflections [16].
piezoelectric film electrodes 3
1
passive material
Figure 3-30. Effect of the d31 coefficient in piezoelectric thin films.
85 Equation 3–50 is written in the electrical and mechanical domains. Often, it is conve- nient to model the transduction in the electrical and acoustic domains. In this case, velocity and force become volume velocity, Q, and pressure, P , and the two-port electro-acoustic model is expressed as I jωCEF jωdA V = , (3–51) Q jωdA jωCAD P
where dA is the effective acoustic piezoelectric coefficient and CAD is the short circuit acoustic compliance of the diaphragm [95].
Although Equations 3–46 and 3–47 are frequently used to describe piezoelectric trans- duction, a common characteristic of piezoelectric materials is a nonlinear materials response known as hysteresis. This phenomenon is characterized by a phase lag in response to ex- ternal forcing, whether mechanical or electrical. Because of the lag in response, materials that demonstrate hysteresis are memory dependent where their current reaction is dependent upon their past response to forcing [96]. Hysteresis is commonly characterized by the “hys- teresis loops” shown in Figure 3-31 that shows the multi-valued dependence of a material’s response to external forcing. The cause of hysteresis is complex and in some definitions may include other nonlinearities. Hysteretic sources are classified as either intrinsic or extrinsic. Intrinsic properties are inherent to the unit cell of the material. Extrinsic properties are due to effects of non-uniformity such as defects, grain boundaries, and domain boundaries. For example, hysteresis in PZT is dominated by the movement of domain boundaries under the application of an electric field [91,97].
Piezoelectric materials commonly found in microsystems are AlN [98], zinc oxide (ZnO) [99,100], copolymer of vinylidene fluoride with vinyl trifluorethylene (P(VDF/TrFE)) [101], and PZT [46,102,103]. P(VDF/TrFE) is a ferroelectric polymer that can be poled [91]. PZT is not CMOS compatible due to lead contamination. In addition, high precision chemical
etches of PZT do not exist [91]. Since PZT is ferroelectric, it must be poled in the proper
86 t ty en si m en e d c e 0 la g sp ar i h D C
0 0 Voltage 0 Stress
Figure 3-31. Hysteresis loops of piezoelectric materials (adapted from [91])
Table 3-2. Piezoelectric film properties. Material Property AlN [111] PZT [112,113] ZnO [114] E1 (GPa) 283 96 98.6 ρ (103 kg/m3) 3.26 7.7 5.7 d31 (pm/V) -2.6 -130 -5.5 d33 (pm/V) 5.5 290 10.3 ²33,r 10.7 1300 8.5 orientation [104]. AlN and ZnO are not ferroelectric materials, and therefore cannot be poled by the application of an external field. As a result, their fabrication must specifically orient the crystals in the same direction to make the material piezoelectrically active. Although this is difficult for bulk piezoelectrics, it is possible for thin film deposition, such as RF- sputtering [105–108] or pulsed laser deposition [109]. Although AlN and ZnO have lower piezoelectric constants in comparison to PZT (see Table 3-2 for a comparison of material properties), they do not require poling and have the benefit of being lead free. In the case of AlN, stable films have been obtained [98,110] and it is compatible with integrated circuit (IC) fabrication. The following section gives an overview of piezoelectric micromachined ultrasonic trans- ducers (pMUTs). The resonant frequencies and outputs of the devices are similar to the ranges observed in the review of a parametric array implementations in Section 2.2.3. These
87 devices provide a review of methods for creating a pMUT that can be used in a parametric array.
3.3.2.1 MEMS Piezoelectric Transducers
The E.L. Ginzton Laboratory at Stanford University has also presented work in air- coupled pMUTs [99, 115–119]. The diaphragm is nitride with a ZnO ring at the outer edge
of the diaphragm as shown in Figure 3-32. The diaphragms are released using two different methods. In the first method the sacrificial oxide is etched via access holes in the silicon membrane. In the other method, DRIE is used to etch through the bulk silicon to access the sacrificial oxide. The diaphragms are 100 µm in diameter and are arranged with 150 µm spacing. A transmit sensitivity of 0.15 µm/V is observed at a resonant frequency of 2.85 MHz. ZnO Cr/Au Vent Ti/Au Nitride
Oxide Cavity
Silicon
Figure 3-32. Two side-by-side pMUTs formed by a nitride diaphragm and ZnO annular ring (adapted from Percin et al. [115]).
Parylene Aluminum Nitride
Silicon ZnO
Figure 3-33. Dome shaped pMUT with nitride diaphragm and ZnO piezoelectric layer (adapted from Cheol-Hyun et al. [100]).
88 Cheol-Hyun and Sok [100] presented fabrication details for a dome shaped ultrasonic transducer shown in Figure 3-33. The transducer featured a 1.5 µm thick nitride diaphragm 2 mm in radius with a 0.5 µm thick layer of deposited piezoelectric material (ZnO) to provide
transduction. The unique fabrication process included photolithography methods used to pattern electrodes onto the three-dimensional dome. The sound output from the domed transducer was about 113 dB at 140 kHz (measured 2 mm from the transducers with a Bruel & Kjaer 4135 1/4-inch free-field microphone with an upper frequency response of 100 kHz). Cheol-Hyun and Sok claimed that the major benefits of using a dome are a wrinkle and crack free diaphragm and an increased transduction of in plane strains into vertical deflections.
Oxide Electrodes PZT
Nitride Silicon
Figure 3-34. Square diaphragm pMUT using PZT (adapted from Mohamed et al. [102,120]).
A piezoelectric transducer was developed by Mohamed et al. for echo ranging ef- fects [102, 120]. The transducer consisted of a square diaphragm of PZT on nitride. The diaphragms, ranging from 100 µm to 1500 µm on a side and 2.5 µm thick, were released using DRIE. Echo ranging tests were conducted with a degree of success, although the authors pointed out that the transducer impedance was not matched to that of the pulse-receive controller used in the experiments. Beginning in 2004, Lee et al. presented a series of papers on a square pMUT formed by
depositing PZT on an SOI substrate [121–123]. Three variations of the square diaphragm
shown in Figure 3-35 were fabricated, each 700 µm on a side. The buried oxide (BOX) layer
89 Pt/Ti lower Pt upper electrode PZT electrode
SiO 2 SiliconSi
Figure 3-35. Square diaphragm pMUT using PZT (adapted from Lee et al. [121]).
of the SOI substrate was removed for two devices, resulting in total thicknesses of 4.65 and 3.86 µm and resonant frequencies of 90.8 and 87.6 kHz, respectively. The BOX layer was not etched for the third device resulting in a total thickness of 5.60 µm and a resonant frequency of 111 kHz. The authors noted substantial initial deflection of 4-8 µm in the diaphragm due to residual stresses. Actuation sensitivities were absent from the studies.
PZT Platinum Oxide
Silicon
Figure 3-36. PZT microspeaker (adapted from Zhu et al. [124]).
Zhu et al. introduced a sequence of papers from 2004 through 2007 covering their PZT
based microspeaker shown in Figure 3-36 [124–130]. The square diaphragm was composed
of a SiO2/Pt/PZT/Pt/Ti/SiO2 composite formed by an anisotropic etch from the back side
2 of the substrate. The diaphragm was 600 by 600 µm and 2.2 µm thick. Using a 4 Vpp excitation, 8.36 µm of displacement was obtained at a resonant frequency of 67.2 kHz. Two smaller peaks in their displacement spectrum at 26.9 and 44.7 kHz were noted and attributed
90 to the compressive stress in the bottom, thermal oxide layer. Acoustic measurements were
conducted on a single microspeaker as well as 2 by 2 and 3 by 3 arrays. For a 4 Vpp excitation signal at 40 kHz, Zhu et al. measured 58.1, 64.0, and 67.5 dB at 20 mm from the single
device, the 2 by 2 array, and the 3 by 3 array, respectively. A similar device with a 1000 by 1000 µm2 and 2.15 µm thick diaphragm was also fabricated [130]. The device experienced a static deflection due to residual stress. At the resonant frequency of 54 kHz, the measured acoustic signal was 96 dB at 5 mm for a 4 Vpp excitation signal.
PZT Pt
TiO 2 SiO 2 Si
Figure 3-37. PZT diaphragm (adapted from Zhu et al. [131]).
Additionally work in 2007 from Zhu et al. focused on an in-plane polarized PZT dia-
phragm with interdigitated electrodes as shown in Figure 3-37 for microspeakers and sen- sors [131, 132]. The membrane was 1000 by 1000 µm2 and 3 µm thick. A potential was established between alternating Pt electrodes so that the fringe of the electric field traveled through the PZT diaphragm. The PZT film was polarized in-plane at 100 kV/cm so that the electric field generates strain via the d33 piezoelectric coefficient, whose magnitude was over two times the d31 piezoelectric coefficient utilized in previously described pMUTs. Given an
91 excitation signal of 4 Vpp, 73.8 dB was measured at 100 mm at a resonant frequency of 43.8 kHz.
Polyimide Pt/Ti Pt/Ti
PZT
Oxide Nitride Silicon
Figure 3-38. PZT microspeaker (adapted from Zhu et al. [53]).
In 2005, Wang et al. presented the pMUT design shown in Figure 3-38 that utilized a thick PZT layer for actuation [53]. The diaphragm was formed by thick PZT and silicon layers as well as silicon dioxide, silicon nitride, and platinum/titanium layers. Devices with
separate dimensions were fabricated and characterized. The first diaphragm was 2 mm by 2 mm and utilized a 7 µm thick PZT layer with a 10 µm silicon layer. For an excitation signal consisting of 20 V dc and 30 Vpp, 107 dB was measured at 12 mm for a resonant frequency of 41.2 kHz. The second diaphragm was 1.5 mm by 1.5 mm with a 3.5 µm thick PZT layer. The silicon layer of the membrane was completely etched. For a 20 V dc plus a
25 Vpp excitation signal, a 120 dB acoustic signal was measured at a resonant frequency of 76 kHz.
Lam et al. [101] described a 2.3 mm square silicon/oxide composite diaphragm actuated
using a P(VDF-TrFE) film. The film was sandwiched between two aluminum electrodes. The electrodes and film were deposited after the diaphragm was released and then poled. The maximum displacement at a resonance of 40.8 kHz was 0.9 µm (drive of 7 V). Muralt et al. [103, 133] presented a piezoelectric transducer design that was adapted
for use in air. Bridges connected the diaphragm to the substrate to obtain a larger volume
92 Aluminum electrodes P(VDF-TrFE)
Oxide
Silicon
Figure 3-39. Square diaphragm pMUT utilizing a P(VDF-TrFE) film for actuation (adapted from Lam [101]). PZT Au/Cr
Pt/Ti
Oxide Silicon
Figure 3-40. Oxide diaphragm formed by the BOX layer of an SOI wafer and actuating by a PZT film (adapted from [103]).
velocity per applied voltage. The diaphragm was formed from the single-crystal silicon of an SOI wafer with the BOX forming the four bridges that anchor the diaphragm to the substrate. They formed a 1.1 mm diameter, 8.2 µm thick diaphragm that resonated at 49 kHz with an approximate amplitude of 4 µm.
An acoustic energy harvester was designed by Horowitz et al. using a MEMS piezo-
electric transducer [46]. Measurements of the transducer output characteristics were also conducted in the author’s dissertation [134]. The 3 µm thick, 1.2 mm radius, circular dia-
phragm was formed from the device silicon layer of an SOI waver. An annular ring (1.1 mm
93 Platinum Electrodes TiO 2 PZT
BOX Silicon
Figure 3-41. A pMUT used in an energy harvester (adapted from [46]).
inner radius) of PZT was formed at the edge of the diaphragm. For a resonant frequency of 34.25 kHz, the drive sensitivity was 450 nm/V.
Mo AlN
Nitride
SiliconSi
Figure 3-42. An AlN audio microspeaker pMUT (adapted from [135]).
Recently, Seo et al. designed and fabricated AlN microspeakers as shown in Figure 3-42 for audio applications [135]. A square diaphragm 4 mm by 4 mm and a circular diaphragm 4 mm in diameter were fabricated. The diaphragm consists of a 0.5 µm thick AlN layer between Mo layers on a 1 µm thick silicon nitride layer. The top Mo layer is split into 2 electrodes. The bottom electrode was grounded. The excitation signals applied to the top two electrodes were 180 degrees out of phase. For a 20 Vpp excitation signal, acoustic measurements at 3 mm with a B&K 4192 1/2-inch pressure-field microphone were 100 dB at a resonant frequency of 10 kHz for the circular diaphragm and 76 dB at a resonant frequency of 10.5 kHz for the square diaphragm.
3.3.2.2 Summary
A summary of the discussed pMUTs is contained in Table 3-3. The frequency ranges
of the devices are comparable to those of the parametric array implementations to date
94 reviewed in Section 2.2.3. In contrast to the cMUTs, a dc voltage is not necessary to linearize the piezoelectric transduction mechanism. Harmonic distortion of diaphragm deflection for pMUTs is limited to geometric and hysteresis effects. Piezoelectric transducers do, however, have an upper limit to the driving voltage. Large voltages can cause depoling of ferroelectric materials or even dielectric breakdown of general piezoelectric materials.
Similar to the sound measurements of the electrostatic transducers in Section 3.3.2.2, the sound output measurements outlined in Table 3-3 suffer from measurement difficulties. The B&K 4135 microphone used by Cheol-Hyun and Sok [100] to measure a 140 kHz signal has an upper bandwidth limit of 100 kHz. Unless the measurement is compensated, the microphone sensitivity will not be the same as the flat-band sensitivity due to the roll-off of the microphone response beyond resonance. Also, the wavelength at 140 kHz in air is
2.5 mm. Since the transducer is less than a wavelength from the microphone, the reflected waves from the microphone will load the diaphragm and change its response. The 10 and 10.5 kHz measurements by Seo et al. [135] at 3 mm suffer from a similar proximity issues between the transducer and microphone.
3.3.3 Thermoelastic Actuation
Thermoelastic actuation utilizes the coupled temperature-displacement characteristics of the material for actuation. A common heat source in thermoelastic MEMS is Joule heating created by the thermal dissipation in a resistor. The resulting non-uniform thermal gradients create thermal forces and moments that couple into structural deflection. Composite plates often serve as the thermoelastically actuated structure. The non-symmetrical material prop- erties further enhance the temperature gradients. Dynamic oscillations of the structure are obtained when the voltage across the resistor is varied. Joule heating is quadratically dependent upon the applied voltage. To create oscillations of the same input frequency, the input signal is offset by a dc bias. Thus, the resultant
95 Table 3-3. Air-coupled pMUT characteristics. Author Diaphragm Piezoelectric Deflection Output Resonant Dimensions Material Sensitivity Frequency Percin et al. ϕ100 µm by 0.3 µm ZnO 0.15 µm/V NA 2.85 MHz 1998 [115] Cheol-Hyun et al. φ4 mm by 1.5 µm ZnO NA 113 dB 140 kHz 1999 [100] at 2 mm Mohamed et al. 0.1-1.5 mm by PZT NA NA NA 2001 [102, 120] 2.5 µm Lee et al. 700 by 700 by PZT NA NA 90.8 kHz 2004 [121] 4.65 µm3 700 by 700 by 87.6 kHz 3.86 µm3 700 by 700 by 111 kHz 5.60 µm3 Zhu et al. 600 by 600 by PZT 4.18 µm/V 8.36 µm 67.2 kHz 2004 [124] 2.2 µm3 1000 by 1000 by 96 dB 54 kHz 2.15 µm3 Wang et al. 2 by 2 mm2 by PZT NA 107 dB 41 kHz 2005 [53] ∼20 µm at 12 mm 1.5 by 1.5 mm2 by 120 dB 76 kHz ∼13 µm Lam et al. 2.3 mm by NA P(VDF- NA 0.9 µm 40.8 kHz 2005 [101] TrFE) Muralt et al. ϕ1.1 mm by 8.2 µm PZT NA 4 µm 49 kHz 2005 [103] Horowitz et al. φ2.4 mm by 3 µm PZT 450 nm/V NA 34.25 kHz 2006 [46] Zhu et al. 1000 by 1000 by PZT NA 73.8 dB 43.8 kHz 2007 [131] 4.65 µm3 at 100 mm Seo et al. 4 by 4 mm AlN NA 100 dB 10 kHz 2007 [135] at 3 mm φ4 mm 76 dB 10.5 kHz at 3 mm spectrum of Joule heating contains a dc and ac component plus a single harmonic,
1 1 V 2 = (V + V sin (ωt))2 = V 2 + V 2 + V V sin (ωt) − V 2 cos (2ωt) . (3–52) dc ac dc 2 ac dc ac 2 ac
The magnitudes of the input voltages can be varied to emphasize the original frequency component. The static component, however, creates a static deflection of the structure, changing the compliance and resonant frequency of the device. Thermoelastic devices have
96 favorable scaling with decreased size because thermal inertia decreases with volume, resulting in faster response time [136].
MEMS thermoelastic actuators for air applications are reviewed in the following section.
3.3.3.1 MEMS Thermoelastic Actuators
In a series of papers, Brand, Baltes, Paul, Hornung, and company presented a thermally actuated proximity sensor that utilized piezoresistors for sensing [137–146]. Implanted p- type dopants formed the heater and piezoresistors. The device consisted of a 1 mm square diaphragm defined by a KOH etch. The diaphragm was composed of silicon with oxide and nitride insulating layers as shown in Figure 3-43. A compressive residual stress in the oxide layer was a result of fabrication and caused buckling of sufficiently thin diaphragms. Reactive ion etch varied the diaphragm thickness from 8-16 µm to study the effects of buckling including static deflection height, resonant frequency, and vibration amplitude. The resonant frequency reached a minimum when the thickness reached the critical value at which buckling occurs. At some thickness just below the critical thickness, Brand et al. achieved beneficial kinematic amplification while reducing the resonant frequency. Further reduction of the thickness resulted in buckling of the plate and an increase in the resonant frequency, as well as a decrease in the vibration amplitude due to stiffening. The critical buckling thickness of the device occurred at about 6.7 µm when the resonance was approximately 50 kHz with a deflection amplitude on the order of 900 nm. The authors were able to show good agreement between experiments, FEA, and analytical solutions up to the buckling point; afterwards the slope of the data shows agreement. A summary of the work on this proximity sensor was found in the paper by Brand et al. [138]. Also, Hornung and Brand [57] authored a book on the design of thermoelastic proximity sensors.
Another micromachined thermoelastic transducer with piezoresistive sensing was pre- sented by Popescu [147]. A silicon/oxide composite formed the diaphragm. The diaphragm
97 Nitride Doped Heater Doped Piezoresistor
Epitaxial Silicon
Oxide Epi Contact
Silicon
Figure 3-43. Proximity sensor that utilizes thermoelastic actuation and piezoresistive sensing (adapted from Brand [138]).
Oxide Aluminum
Polysilicon Heater
Silicon
Figure 3-44. Thermoelastic actuator with a buckled diaphragm (adapted from Popescu et al. [147].
also contained a polysilicon heater and an aluminum ring layer that served as a thermal con- duit. The membrane was 4 mm by 4 mm and had an initial buckled height of 20 µm. Static deflections of up to 50 µm were obtained with dissipated power of more than 1.3 Watts.
Chandrasekaran et al. [148] presented a thermoelastic proximity sensor similar to that of Brand et al. The diaphragm in this case was circular and was released using DRIE. The diameter was 1 mm and the thickness varied from 6-10 µm. The major innovation introduced in this work was the use of electronic-through-wafer interconnects (ETWIs) for electrical connections. ETWIs allowed electrical connections to the back of the wafer creating a more robust package. Chandrasekaran et al. fabricated three device designs and observed buckling phenomenon similar to that reported by Brand et al. A 9 µm thick device achieved almost 200 nm of deflection at a resonance of 55 kHz.
98 Aluminum Doped Piezoresistors
BOX Doped Heater Silicon Oxide Polysilicon Through-Wafer Interconnects
Figure 3-45. Thermoelastic/piezoresistive proximity sensor that uses the device layer of an SOI wafer to form a circular diaphragm (adapted from Chandrasekaran et al. [148]).
Oxide Polysilicon Heater Nitride
Thermopile Polysilicon Piezoresistors Silicon
Figure 3-46. Thermoelastic proximity sensor using polysilicon for the heater and piezoresis- tors (adapted from Rufer et al. [149]).
Another thermoelastic proximity sensor [149], shown in Figure 3-46, was formed using a 0.8 µm complementary metal-oxide-semiconductor (CMOS) process. The diaphragm was composed of oxide and nitride layers embedded with polysilicon resistors that served as four piezoresistors and a heater. Four thermopiles were also contained within the diaphragm to measure its temperature. The diaphragm thickness varied from 4.2 to 5.2 µm and its side length was 1.3 mm. For a single device, a 1/4” Bruel & Kjaer 4135 measured 48 dB at a resonant frequency of 41.3 kHz at a distance of 10 mm. 3.3.3.2 Summary
A summary of the thermoelastic actuators is contained in Table 3-4. The frequency ranges of the thermoelastic devices are also comparable to those of the parametric array implementations to date reviewed in Section 2.2.3. However, the transduction mechanism of
99 thermoelastic actuation is inherently quadratic. Thus, from Equation 3–52, the input elec- trical signal is converted into dc heating and harmonic deflection as well as the fundamental component. It would be problematic to use thermoelastic actuation to supply the multiple frequency components needed to produce an audio signal. 3.4 Conclusion
The actuation methods covered in this review are electrostatic, piezoelectric, and ther- moelastic. The benefits of all three should be considered when choosing a transduction mechanism for a micromachined ultrasonic transducer.
From Section 3.3.1, the electrostatic force is dependent upon the inverse of the gap to the second power. It would seem advantageous to minimize the gap. As discussed in the electrostatic section, however, reducing the gap reduces the deflection that the electrostatic transducer can achieve before pull-in occurs. Therefore, enhancing the transmit sensitivity reduces the deflection capabilities of an electrostatic speaker [55]. The electrostatic trans- duction mechanism is inherently quadratic and thus requires a bias voltage to “linearize” the transducer. Generally, the cMUTs in the literature do not reflect the frequency range or outputs established in the literature for parametric array transducers. Thus, electro- static transduction is not chosen as the transduction mechanism for the parametric array transducer.
Table 3-4. Thermoelastic MEMS characteristics. Author Diaphragm Heater Type Output Resonant Dimensions Frequency Brand et al. 1 mm by 6.7 µm P-type doped 900 nm 50 kHz 1997 [138] Popescu et al. 4 mm by 15 µm aluminum ring 50 µm (static) NA 1996 [147] Chandrasekaran et al.1 mm by 9 µm P-type doped 200 nm 55 kHz 2002 [148] Rufer et al. 1.3 mm by polysilicon 5 mPa at 10 mm 41.3 kHz 2006 [149] 4.2-5.2 µm
100 The thermoelastic devices in the literature favorably reflect the frequency ranges and outputs required of a parametric array transducer. The thermoelastic transduction mecha- nism, however, is also inherently quadratic. Similar to electrostatic transduction, providing the ac signal with a dc offset emphasizes the fundamental frequency. However, this intro- duces a static heating power that is dependent upon both the oscillating amplitude and the dc bias. The static heating affects the resonant frequency of the device, making the bandwidth dependent upon the forcing amplitude. A device whose resonant frequency is dependent upon the input signal increases the complexity of the control circuitry. Thus, thermoelastic actuation is not chosen as the transduction mechanism.
Piezoelectric transduction does not require a bias voltage for a linear input-output relationship. On the other hand, piezoelectric devices do exhibit hysteresis effects which can generate harmonics and degrade device performance. The literature on pMUTs, though, compares favorably with the performance required of a parametric array transducer. Thus, piezoelectric transduction is selected for the device design.
101 CHAPTER 4 ULTRASONIC RADIATOR DESIGN
This chapter describes the piezoelectric ultrasonic radiator designed for nonlinear acous-
tic applications. The devices were designed around Avago Technologies Limited’s high volume, film bulk acoustic resonator (FBAR) process. The design of structures using an established fabrication process is a common goal of MEMS design since it avoids process development and the associated non-recoverable engineering costs [16]. The tradeoff of this method is the constraint of the device design by the limitations and uncertainties associated with the process. Also, uncertainties within the process may arise that were not crucial to the original application of the fabrication process. Considering these tradeoffs, the opportu- nity to leverage an existing high volume commercial fabrication facility such as Avago’s was
favorable for the design of an ultrasonic radiator for nonlinear acoustic applications.
The structures available using Avago’s FBAR process is covered in Section 4.1 including an overview of fabrication steps. Afterwards, the implementation of the FBAR process to form the ultrasonic radiator structure is outlined in Section 4.2. Finally, the packaging design
and methods are summarized in Section 4.3. 4.1 Avago’s FBAR Process
An FBAR is a thickness mode piezoelectric device that is used as an electrical-mechanical- electrical filter, most commonly in the send and receive of cell phone signals [150, 151]. FBAR’s consist of a piezoelectric material between two metal electrodes. The piezoelectric material in Avago’s FBAR is aluminum nitride (AlN) while the metal electrodes are molyb- denum (Mo) [98]. Additional AlN layers above and below the electrodes serve as passivation and structure [98]. The FBAR fabrication was leveraged to form an AlN/Mo/AlN/Mo/AlN/AlN composite
diaphragm on a silicon substrate with a back cavity as shown in Figure 4-1 [151]. The fabrication begins with a 150 mm diameter, 675 µm thick silicon substrate. First, a shallow cavity that defines the diaphragm boundary is etched in the front of the silicon [98, 151].
102 Next, a sacrificial material is deposited. The sacrificial material is thinned back to the substrate such that the shallow cavity remains filled.
AlN Mo Si
Back Cavity
Figure 4-1. Cross-section of the structural layers and back cavity possible using the Avago FBAR process.
Once the substrate has been planarized, AlN and Mo layer depositions begin. Although the exact deposition process for these layers is proprietary, sputtering of AlN and Mo films is common in the literature [105–108,110]. First, the structural layer (referred to in this disser- tation as “scaffolding”) of AlN is deposited. Next, a thin AlN layer is deposited (not shown in Figure 4-1). Thin AlN layers are commonly used in AlN/Mo composites as seed layers to orient the subsequently deposited Mo and AlN films [106]. The bottom Mo electrode is deposited over the AlN seed layer. Next, the active piezoelectric AlN layer is deposited followed by the top Mo electrode. Finally, an AlN passivation layer is deposited [98]. Since AlN is a dielectric, this layer electrically isolates the top Mo electrode from the environment. All Mo and AlN layers are residually stressed in either tension or compression. More dis- cussion on layer stresses is contained in the following paragraph. All layers other than the AlN scaffolding and seed layers can be patterned according to designer defined geometry. Once the layer depositions are complete, the back cavity is formed using a deep reactive ion etch (DRIE) [152] from the back of the substrate to the sacrificial layer. The resulting cavity is straight walled. The sacrificial layer is then etched from the backside to release the AlN/Mo/AlN/Mo/AlN/AlN composite as shown in Figure 4-1.
As noted previously, each of the AlN and Mo layers emerge from the fabrication pro- cess with residual stresses. Avago attempts manipulation of the stress in each layer by
103 changing the deposition parameters to achieve stress design targets between -100 and 100 MPa. Selection of stress targets is covered in Chapter 6. Measure of stresses are reported in
Chapter 7.
4.2 Fabrication
By patterning and etching select layers in the AlN/Mo stack, the ultrasonic radiator shown in Figure 4-2 was produced. The structure in Figure 4-2(a) possesses top and bottom Mo annular electrodes. The AlN passivation layer is also annular. The active piezoelectric AlN and AlN passivation films are continuous across the diaphragm.
Gold (Au) bond pads and a front side vent are also options in Avago’s process as seen in the top view image of the device in Figure 4-2(b). The vent allows static pressure equalization between the front of the diaphragm preventing a static deflection of the diaphragm due to atmospheric pressure changes.
Piezoelectric Ring Diaphragm Vent
Si Cavity Au Bond Substrate Pad
(a) Device cross-section. Front-side Vent
(b) Top view of the device showing the Au bond pads and front side vent.
Figure 4-2. MEMS- based ultrasonic radiator for parametric array applications.
Once batch fabrication was complete, the wafers were singulated into die by Dynatex International using a scribe and ”smart break” process [153]. The separated die are approx- imately 3 mm by 3 mm.
104 4.3 Package
The ultrasonic device was incomplete without a package to provide a supporting struc- ture and conduit for electrical excitation. The package also provided a unique variable back cavity length. This section describes the package and its effect on device performance.
Back cavity Die recess extension Dowel Beveled hole screw hole
Epoxy reservior
BNC feed- throughs
Figure 4-3. PCB board for device package.
The device was mounted in the printed circuit board (PCB) board as shown in Figure 4- 3. The PCB board was formed of FR4 material with gold plated bond pads. A recess was
machined in the PCB board so that the die could be flush mounted. The circular holes at the four corners of the recess were wells for epoxy to hold the die in place. The wells gave the epoxy room to expand while curing to prevent imparting stress to the die. The back cavity of the device was extended via a hole machined through the PCB board in the middle of the recess. Dowel and beveled machine screw holes were added to align and mount the
PCB board to the aluminum block shown in Figure 4-4, which contains the variable back cavity. Machining of the PCB boards was accomplished at either TMR Engineering or in house using a Sherline 2000-series CNC mill.
The attachment of die to a supporting structure has the potential to affect device per-
formance. It is possible for an epoxy to conduct heat and/or electricity or act as an insulator.
105 Table 4-1. Typical epoxy dispense and cure parameters used for die attachment. PCB boards were pre-heated at 145 ◦C for 5 min. before epoxy application. Dual Bond Katiobond Pressure (psi) 45 45 Vacuum (mmHg) 29.5 29.5 Dispense Time (s) 2 2 Tip (gauge) 30 30 Cure Type Heat/Optical Optical Oven 135 C, 3 min. NA mW Lamp (405 nm) ∼ 0.5 cm2 at 0.3 m 10 min. 10 min.
Epoxies may be cured in a variety of ways, including both thermal and optical cures or mix- ing with a catalyst. Stress can be transferred to the device due to thermal mismatch with the epoxy or pcb board at elevated curing temperatures and/or contraction/expansion of the epoxy during curing. To mitigate the stress imparted to the device by the package, epoxy wells were created that allowed expansion during the curing process. Two different epoxies were used in this work to attach the die to the PCB board: Cyberbond’s Dualbond 707 and Katiobond 45952. Both are one-part, modified resin epoxies that cure optically using visible wavelengths from 400 to 550 nm. Dualbond 707 has the added benefit of curing ther- mally. The epoxies were dispensed using EFD’s Ultra 2400 Series Dispensing Workstation, a pneumatic epoxy dispenser. Dispense and cure parameters are contained in Table 4-1.
Table 4-2. Wirebonder settings. Search Force Time Power Ball Bond* 6.5 3 5 3 Wedge Bond* 6.5 7 5 4.5 * Loop height: 6.5, ball size: 4, temperature of work holder: 128 ◦C.
Once the die was attached to the PCB, electrical connections were made via Au wire bonds using a Kulicke & Soffa 4124 Series Manual Ball Bonding System. The first bond was a ball-bond and the second a wedge bond. To augment successful wirebonding, the PCB was placed on a heated work holder. Wirebonding parameters and settings are contained in
Table 4-2.
106 Threaded Back A Secure screw holes cavity hole Variable A-A back cavity
A
Dowel Threaded holes (b) Cross-section of Al block showing the variable screw holes back cavity. (a) Isometric view of the Al block on which the PCB is mounted.
Figure 4-4. The aluminum block contains variable back cavity, dowels and screw holes for alignment to the PCB, and an anchor screw hole.
(a) 3-D rendered image. (b) Photograph.
Figure 4-5. Packaged device mounted to variable back cavity.
Once the connections between the die and the PCB board were complete, the PCB was aligned and mounted to the aluminum block shown in Figure 4-4 via two dowels and four beveled machine screws. The PCB board was aligned with the aluminum block such that the through hole in the PCB was centered on the variable back cavity hole in Figure 4-4(a). A cross section of the aluminum block is shown in Figure 4-4(b). A #0 machine screw was used to change the depth of the back cavity. By turning the screw, the depth of the back cavity was adjusted. A single turn constitutes a change in depth of 320 µm. Quarter turn
107 marks were made in the aluminum block giving a depth resolution of approximately 80 µm. Experimental results for a changing back cavity depths including theoretical comparisons are given in Chapter 7. The fully assembled package, including BNC electrical connection, is show in Figure 4-5. 4.4 Conclusion
The ultrasonic device structure and package based on the FBAR process of Avago Technologies Limited is presented in this chapter. The device makes use of piezoelectric transduction and consists of a radially non-uniform diaphragm consisting of AlN and Mo
layers. The major fabrication steps are covered. The device packaging, including a variable
back cavity, is outlined. Next, Chapter 5 presents the model of the packaged device.
108 CHAPTER 5 MODELING
The design of an ultrasonic radiator for maximum pressure output required detailed un- derstanding of the entire system. The device behavior was determined by geometry, material properties, fabrication induced effects such as residual stress, and acoustic interactions of the package. A comprehensive design model was necessary to elicit the relationships between design parameters and device performance so optimization could be performed.
This chapter presents a detailed model of the linear performance of the ultrasonic ra- diator. The model incorporates the interaction between the electrical, mechanical, and acoustical domains. The interaction between the domains is captured using an equivalent circuit. In the equivalent circuit, the domains are represented using combinations of circuit elements. The circuit elements are formed using models discussed in this chapter and in Appendix B. Section 5.1 presents the equivalent circuit model of the ultrasonic device. Derivation of the acoustical, mechanical, and electrical circuit elements are included. Section 5.2 outlines a nonlinear acoustic model for an array of MEMS transmitters. The source condition for the numerical solution is solved from the lumped element model. The final section presents modeling results from a non-optimal example of a design that does not incorporate in-plane stress.
5.1 Equivalent Circuit
In this section, the linear behavior of the ultrasonic resonator is explored using an equivalent circuit derived using conjugate power variables, such as voltage and current or pressure and volume velocity. A schematic of the device is shown in Figure 5-1. The device components are represented by lumped elements when the device dimensions are smaller than the wavelengths of interest. For example, the diaphragm is modeled as a second order system with mass, compliance, and resistance. The circuit representation of a second order system is a inductor, capacitor, and resistor in series. Resistors represent dissipative elements.
109 Inductors symbolize kinetic energy storage elements, and capacitors depict potential energy storage elements. Transformers and gyrators represent transduction from one energy domain to another. The conjugate power variables in the domains represent “efforts” and “flows.”
For example, in the electrical impedance domain, efforts and flows are voltages and currents, respectively. On the other hand, in the acoustical impedance domain, efforts and flows are pressure and volume velocity, respectively. For an impedance analogy, elements that share a common effort are connected in parallel while elements that share a common flow are connected in series. When the wavelengths of interest are on the order of the device dimensions, the accuracy of lumped elements deteriorate and distributed system models such as transfer matrices are added to the equivalent circuit.
Piezoelectric Diaphragm Vent R Ring Rrad AV
Cavity MAD, C AD , R AD Z AC ddie Sdie
S pcb d pcb
Sscrew dscrew
Figure 5-1. Diagram of incremental diaphragm deflection.
The equivalent circuit that models the device performance is formed using models from both the acoustic and electrical domains as shown in Figure 5-2. A transformer is used to connect the two energy domains. The diaphragm is modeled as a piston with an acoustic
mass, MAD, compliance, CAD, and resistance, RAD. The equivalent resistance combines the dissipation effects in the diaphragm due to thermoelastic dissipation, acoustic radiation into the supports, and other effects. The diaphragm also is loaded by a radiation impedance, Zrad, due to loading of the air. The radiation impedance is complex. The reactance represents
110 energy storage in the forms of potential and kinetic energy. The resistance accounts for energy transferred to an acoustic wave that travels away from the diaphragm. The back cavity consists of three sections: silicon wafer, pcb, and threaded aluminum channel. Due to back cavity tuning, the acoustic wavelength is on the order of the total back cavity depth. Therefore, transfer matrices are used to find an equivalent back cavity impedance by modeling the back cavity as three pipes of differing cross-sectional area. Dissipation in the back cavity is also included in the model. Finally, the side vent equilibrates the static pressure between the back cavity and the atmosphere to prevent static diaphragm deflection. The side vent is best represented by an acoustic resistance and mass due to the short length of the vent in comparison to the acoustic wavelengths. The mass represents the storage of kinetic energy of the air as it moves in and out of the vent. The resistance represents dissipation due to viscous losses at the vent wall. Acoustic Domain Electrical Domain Q I φ Zrad A :1 + M AD RAD CAD RES
Z AC RAV CEB REP V -
Figure 5-2. Equivalent circuit model.
The electrical domain is modeled as a resistance in series with a capacitance and resis- tance in parallel. The capacitance is formed between the top and bottom electrodes across the piezoelectric AlN layer. The parallel resistance models dissipation due to leakage through the dielectric formed by the AlN piezoelectric layer. The series resistance models dissipation due to the electrical leads, wirebond contacts, and interconnects between the device pads and the molybdenum electrodes.
111 The following sections outline the derivation of the elements used in the equivalent circuit. Also, the transduction equations of the transformer that connects the two domains is covered. Finally, the result elements are combined to form the equivalent circuit.
5.1.1 Acoustical Domain
The diaphragm, radiation impedance, back cavity and vent are all modeled in the acous- tic domain. The following sections outline the formulation of each impedance. 5.1.1.1 Diaphragm Impedance
Lumped elements are used to model the diaphragm impedance when the wavelengths of interest are larger than the diaphragm radius. The wavelengths of interest for the diaphragm are the acoustic wavelength [8], λA, and bending wavelength [154], λBW . The acoustic wavelength is the ratio of the speed of sound in air to the frequency,
c λ = . (5–1) A f
When the acoustic wavelength is much larger than the diaphragm radius, the pressure is assumed uniform over the diaphragm. The definition of the bending wavelength of this specific structure is complex due to the radial non-uniformity of the diaphragm. The scaling of the bending wavelength is more easily illustrated by considering a uniform, isotropic plate with clamped boundary conditions. The bending wavelength scales as follows, s
1/4 D λBW ∝ 2 . (5–2) ρAω
When the bending wavelength is larger than the diaphragm radius, the diaphragm is assumed to vibrate in the fundamental mode such that the entire deflection mode shape is in phase. Thus, the diaphragm can be approximated as a lumped mass and compliance. Dissipation in the form of a resistance is introduced so that the diaphragm response at resonance is finite.
112 The acoustic compliance of the diaphragm, CAD, is estimated as the volume displace- ment per applied pressure when the top and bottom electrodes are shorted, ¯ ¯ ∆–V ¯ CAD = ¯ . (5–3) P V =0
Thus, CAD, is known as the short circuit acoustic compliance. The volume deflection is found from the incremental deflection of the diaphragm, winc, as Z
∆–V = winc (r)2πrdr. (5–4)
The incremental deflection of the plate is found from the mechanical model in Appendix B.
The acoustic mass of the diaphragm, MAD, is [56] Z µ ¯ ¶ ¯ 2 winc (r)¯ MAD = 2π ρA ¯ rdr, (5–5) ∆–V V =0 where ρA is the areal density given by Z
ρA (r) = ρ (r) dz. (5–6)
Note that the areal density is a function of radius due to the non-radially uniform composite diaphragm. By inserting Equation 5–3 into Equation 5–5, the acoustic mass is rewritten in terms of the acoustic compliance as Z µ ¯ ¶ ¯ 2 2π winc (r)¯ MAD = 2 ρA (r) ¯ rdr. (5–7) CAD P V =0
A resistance, RAD, is added to the diaphragm to account for dissipation. An accurate model that incorporates all of the dissipation mechanisms within the diaphragm is beyond the scope of this work. The resistance value is based on previous experimental results of piezoelectric diaphragm transducers [134]. The resistance is found from the damping coefficient by [51] r MAD RAD = 2 ζ. (5–8) CAD
113 5.1.1.2 Radiation Resistance
The impedance of the air as seen by the diaphragm is found from the solution of a piston
in an infinite baffle as given in Chapter 3. The expression, repeated here for completeness, is · ¸ ρ0c0 J1 (2kaeff ) K1 (2kaeff ) Zrad = 2 1 − + j . (5–9) πaeff kaeff kaeff Note that the piston radius is not the outer radius of the diaphragm but the effective piston radius found by equating the volume displacement of the diaphragm to a piston with an
effective radius, aeff . The effective area, Aeff , is given by [56] ¯ ¯ ∆–V ¯ Aeff = ¯ . (5–10) winc (r = 0) V =0 p Thus, the piston radius is aeff = Aeff /π. 5.1.1.3 Back Cavity
As seen in Figure 5-1, the back cavity consists of three sections of differing cross-sections and materials. The back cavity is modeled as three sound hard rigid ducts of varying cross- sectional area. Dissipation in the back cavity is accounted for in the form of acoustic absorp-
tion due to thermoviscous dissipation and molecular relaxation as well as boundary layer absorption. Acoustic absorption due to molecular relaxation and thermoviscous dissipation
in air, αair, is covered in Section 3.2.3. Dissipation due to boundary layer absorption occurs due to two phenomenon. First, there is a no-slip boundary condition at the back cavity walls due to the fluid viscosity. The transition region between the wall and the bulk acoustic os- cillation is known as the acoustic boundary layer [8]. The second boundary layer absorption phenomenon is the thermal boundary layer. At the wall’s edge, the expansion/contraction of the fluid that is normally assumed adiabatic in free-space is now isothermal since the boundary ideally acts as an infinite heat sink. The absorption of both boundary layers can be accounted for in a combined boundary layer absorption coefficient [8], r µ ¶ 1 ωµ γ − 1 √ αBL = 2 1 + , (5–11) a 2ρ0c0 P r
114 where µ is the dynamic viscosity of air and P r is the Prandtl number, which is the ratio of the viscous diffusion rate to the thermal diffusion rate. The acoustic and boundary layer absorption are combined into an overall absorption coefficient for a rigid walled duct,
α = αair + αBL. (5–12)
x
0
Figure 5-3. Rigid duct model.
To derive the back cavity impedance, first consider plane waves traveling along a straight walled duct of circular cross-section, as in Figure 5-3. The pressure and volume velocity in
the duct are given by [8]
P = Ae−jkxˆ + Bejkxˆ (5–13) AS BS and Q = e−jkxˆ − ejkxˆ , (5–14) Z0 Z0
respectively, where S is the cross-sectional area of the duct and kˆ is the complex wave number, [8] kˆ = k − jα. (5–15)
Now consider a finite duct of length `. The pressure and volume velocity at the duct entrance (x = 0) are
P0 = A + B (5–16) S S and Q0 = A − B , (5–17) Z0 Z0
respectively.
115 At length `, the pressure and volume velocity are
−jk`ˆ jk`ˆ P` = Ae + Be (5–18)
S −jk`ˆ S jk`ˆ and Q` = A e − B e . (5–19) Z0 Z0
Equations 5–16 and 5–18 are combined to yield P0 P` = [T ] , (5–20) Q0 Q` where [T ] is the transfer matrix that relates the pressure and volume velocity at the end of the duct to that at the beginning of the duct. The transfer matrix is given by [8] ³ ´ ³ ´ ˆ Z0 ˆ cos k` j S sin k` [T ] = ³ ´ ³ ´ . (5–21) j S sin k`ˆ cos k`ˆ Z0
PCB Aluminum Die
x SSi S pcb Sφ Screw
Figure 5-4. The back cavity of the transducer with the die, pcb, and aluminum sections.
Q0 + Blocked p0 TSi Tpcb TAl -
Figure 5-5. Circuit representation of the back cavity using transfer matrices.
116 A cross section of the back cavity showing the three sections is shown in Figure 5-4. The sections are estimated as rigid ducts of differing cross section and length, each with their own transfer matrix. The roughness of threads in the aluminum section are assumed negligible since their pitch is much smaller than the acoustic wavelengths of interest. The back-cavity terminates at the aluminum screw, which is assumed to be a sound hard boundary condition. The total back cavity transfer matrix is given by the multiplication of the transfer matrices of each section,
[TBC ] = [TSi][Tpcb][TAl] . (5–22)
The circuit representation of the back cavity is shown in Figure 5-5. Note that the sound hard boundary termination dictates an open circuit after the last transfer matrix. The impedance of the back cavity is found from the ratio of the pressure and the volume velocity at the back cavity entrance, given as P0 P` = [TBC ] , (5–23) Q0 0 where the ’0’ and ’`’ subscripts refer to the cavity entrance and termination, respectively. Thus, the cavity impedance is given by
P0 TBC 11 ZAC = = . (5–24) Q0 TBC 21
5.1.1.4 Vent
The back cavity is vented to the front surface of the die through a small, short duct underneath the edge of the diaphragm as shown in Figure 5-6. The vent is a 2 µm thick channel that is 50 µm in length and 25 µm in width. The vent is modeled assuming fully- developed pressure driven flow [17]. The resistance of the channel is [16]
12µL R = , (5–25) AV W h3 where h is the height, W is the width, and L is the length.
117 Cross Section Vent Top View
20 µm 20 µm Opening m m µ µ 5 0 Vent 2 2 50 µm m µ 2 50 µm
Figure 5-6. Edge of the diaphragm showing the front to back cavity vent (not to scale).
5.1.2 Electrical Domain
The electrical components of the lumped element model include the capacitance formed between the Mo electrodes, the parallel resistance due to leakage through the AlN dielectric layer, and the series resistance due to wirebond contacts and line resistance. These elements are shown in Figure 5-7. The resistances are found from experimental curve fits of the electrical impedance in Chapter 7. The free capacitance, CEF , is given as
²Ap CEF = (5–26) hp where ² is the absolute permittivity, Ap is the planform area of the electrodes, and hp is the thickness of the piezoelectric aluminum nitride layer.
+ RES CEF REP V _
Figure 5-7. Electrical elements of the equivalent circuit.
5.1.3 Transduction
Recall that piezoelectric transduction, being reciprocal, can be used for both actuation and sensing. These two modes of transduction are termed the direct and converse piezo- electric effects, respectively [91]. The harmonic transduction equations of a piezoelectric
118 transducer from Chapter 3, repeated here for convenience [95], are I jωCEF jωd V = . U jωd jωCMS F
The actual device is modeled in the electric and acoustic domains. Thus the displace- ments and forces become volume velocities and pressure, respectively. The two-port, electro-
acoustic model can be expressed as I jωCEF jωdA V = , (5–27) Q jωdA jωCAD P
where Q is the volume velocity and dA is the effective acoustic piezoelectric coefficient found as [56], ¯ ¯ ∆–V ¯ dA = ¯ . (5–28) V P =0 The two port model, Equation 5–27, can be represented using a transformer with a parallel shunt capacitance on the electrical side and a series compliance on the mechanical side as
shown below in Figure 5-8. The transformer turns ratio φA is the electro-acoustic transduc- tion coefficient,
dA φA = − . (5–29) CAD The blocked electrical compliance is
¡ 2¢ CEB = 1 − k CEF , (5–30)
where k2 is the impedance-coupling factor given as
d2 k2 = A . (5–31) CEF CAD
5.1.4 Equivalent Circuit
The elements derived in Section 5.1.1 through Section 5.1.3 are combined to form equiv-
alent circuit as shown previously in Figure 5-2. The electrical side of the transformer is
119 q I φ :1 + A +
CAD p CEB V _ _
Figure 5-8. Two-port representation of electro-acoustic transduction. converted into the acoustic domain using the electroacoustic transduction coefficient. The electrical elements in the acoustic domain are defined as
CEB CEBA = 2 , (5–32) φA 2 REPA = REP φA, (5–33)
2 and RESA = RESφA. (5–34)
The resulting equivalent acoustic circuit is given in Figure 5-9, where acoustic equivalent input voltage is p2 = φAV . The impedances in Figure 5-9 are
Z AD Q
Zrad + M AD RAD CAD RESA
Z AC RAV CEBA REPA p2
Z AV Z EBA -
Figure 5-9. Equivalent acoustic circuit model.
1 ZAD = RAD + sMAD + , (5–35) sCAD ˜ ZAD = ZAD + Zrad, (5–36)
ZAV = RAV , (5–37)
REPA and ZEBA = . (5–38) 1 + sCEBAREPA
120 The sensitivity of the equivalent radiator circuit is given in terms of the volume velocity output of the diaphragm per voltage input,
Q Q φA S = = φA = ³ ´ ³ ´. (5–39) V p2 RESA ˜ ZAC ZAV ZEBA − 1 + ZEBA + ZAD + ZEBA ZAC +ZAV
As is subsequently discussed in Section 5.2, the volume velocity sensitivity is a useful metric when forming an array of radiators.
Another important performance parameter is the input electrical impedance that is driven by the voltage source,
REP 1 Zinput = RES + , (5–40) 1 + jωCEBREP 1 + Γ where ³ ´ ZAC ZEBA Z + 1 Γ = ³ AV ´ (5–41) ˜ ZAC ZAD + 1 + ZAC ZAV
5.1.5 Approximate Performance
By applying a couple assumptions, Equations 5–39 and 5–40 are simplified and insight into the importance of particular device elements is gained. First, the series electrical resis- tance is expected to be much smaller than the total impedance of the blocked capacitance and parallel electrical resistance, R ES ¿ 1. (5–42) ZEB This is confirmed experimentally in Chapter 7. The result of this assumption is that the acoustic equivalent input voltage is applied directly across the acoustic elements on the left hand side of Figure 5-9.
Also, the vent impedance is only important at low frequencies where the back cav- ity impedance is large enough to cause a current divider. Since the device operates near resonance, the effect of the vent in parallel to the back cavity impedance is negligibly small,
Z AC ¿ 1. (5–43) ZAV
121 These approximations are validated using an example device in Section 5.1.6. Applying these assumptions, the sensitivity simplifies to
φ S = − A . (5–44) ˜ ZAC + ZAD
Thus, the high frequency performance of the device is mainly dependent on the diaphragm and back cavity impedances. The resonant performance is dominated by the resistance of
the diaphragm, radiation, and cavity impedances. Given Equation 5–43, the input electrical
impedance, Zinput, simplifies to
REP 1 Zinput = RES + , (5–45) 1 + jωCEBREP 1 + Γ
where Z Γ = EBA . (5–46) ˜ ZAD + ZAC The following section gives the predicted performance of an example device. 5.1.6 Example Device
In this section, the performance of an example device of moderate dimensions is given to illustrate the results of the equivalent circuit. The geometry of the diaphragm is given in Table 5-1. For this example, the stress in the diaphragm layers is assumed to be neutral. Also,
the back cavity is assumed to be exactly a quarter of the diaphragm’s resonant wavelength
such that at resonance the back cavity impedance, ZAC , is zero. Dissipation in the back cavity is also neglected in the example. The damping ratio is assumed to be 0.03 based on previous experience [46]. The parallel electrical resistance is based on a resistivity extracted from
initial experimental impedance results and is given in Table 5-2 along with other material properties. The series electrical resistance is assumed to be 40 Ω based on initial experimental impedance results.
The equivalent circuit results in Figure 5-10 show a resonant frequency of 45.2 kHz and a
volume velocity radiation sensitivity of 34.2 mm3/s/V at resonance. Further insight into the device model is gained by making relevant comparisons of device performance and impedance.
122 Table 5-1. Geometry of example device. Scaffolding AlN 2 µm Bottom Mo Electrode 0.2 µm Piezoelectric AlN 0.8 µm Top Mo Electrode 0.2 µm Top AlN Passivation 0.05 µm
Inner Radius (R1) 350 µm
Outer Radius (R2) 500 µm
Table 5-2. Material properties of AlN and Mo. Property AlN Mo‡ E (GP a) 283† 329 ν 0.29† 0.31 ρ (kg/m3) 3250* 1029 † d31 (pm/V ) −2.6 - † ²33,r 10.7 - * ρp (MΩ·m) 22.5 - † Tsubouchi et al. [111]. * Avago supplied. ‡ [155].
In Figure 5-10, the full sensitivity result, Equation 5–39, and its approximation, Equation 5– 44, are plotted. As observed in Figure 5-10, the approximate sensitivity estimation almost exactly reproduces the full sensitivity calculation over the frequency range of interest. This justifies the use of the high frequency estimation of the sensitivity in the optimization scheme in Chapter 6. A plot of the total impedance of the back cavity in parallel with the vent resistance is shown in Figure 5-11. Clearly, the vent impedance is large enough in comparison to the cavity impedance that it can be neglected. Thus, the high frequency assumption that
ZAC /ZAV ¿ 1 is further justified. It is also informative to note the relative contribution of the radiation impedance in comparison to the diaphragm impedance in Figure 5-12. The radiation impedance adds a significant contribution to the real component. It also contributes to the reactive compo- nent, leading to a slightly different resonant frequency. The shift in resonance leads to a
123 102
101 /s/V) 3 100
| (mm | −1 S
| 10 Full High Freq. −2 10 90
45 ) ° (
S 0 ∠ −45
−90 100 101 Frequency (kHz)
Figure 5-10. Volume velocity frequency response calculated using the full sensitivity equiv- alent circuit and the high frequency approximation.
101
0
/s)) 10 3
10−1
−2 |Z| (Pa/(mm Z || Z 10 AV AC Z AC −3 10 100 101 Frequency (kHz) Figure 5-11. Total impedance of the vent and cavity in parallel versus just the cavity impedance. large percentage relative error near resonance between the imaginary part of the diaphragm impedance, ZAD, and the total imaginary impedance, ZAD + Zrad. Based on these results, the radiation impedance is included in the model used in the optimization.
Next, the contributions of the back cavity impedance and the diaphragm impedance to the overall acoustic impedance are compared in Figure 5-13 where ZAM is the total impedance of the acoustic elements given by
e ZAV ZAC ZAM = ZAD + . (5–47) ZAV + ZAC
124 100 0.4 Z + Z
/s)) AD rad 3 75 Z 0.35 AD 50 ) (Pa/(mm ) Z 0.3 25 % Relative Error Re( 0
0 100 /s))
3 −50 75 −100 50 −150 ) (Pa/(mm ) Z 25
−200 % Relative Error Im( −250 0 100 101 100 101 Frequency (kHz) Frequency (kHz)
Figure 5-12. Diaphragm impedance comparison to the radiation impedance.
102 1 /s))
3 10 100 10−1 | (Pa/(mm |
Z −2 | 10 10−3
90 Z AM 45 Z +Z AD rad ) °
( Z AC
Z 0 ∠ −45
−90
100 101 Frequency (kHz)
Figure 5-13. Contributions to the overall acoustic impedance.
It is important to note that in this analysis, the back cavity depth is tuned to match the quarter wavelength of the resonant frequency of the diaphragm. Thus, although the con- tribution of the cavity impedance seems insignificant in comparison to the diaphragm, it should be included in the device model since the physically realized cavity depth does not exactly match the quarter wavelength.
The input electrical impedance, Equation 5–40, is plotted in Figure 5-14 together with
2 the uncoupled electrical impedance, RES + ZEB, where ZEB = ZEBA/φA. As observed in
125 the plot, the uncoupled electrical impedance almost exactly mimics Zinput except for around resonance, where the resistive component has a local peak at the resonant frequency. The difference between the input impedance and the uncoupled electrical impedance is the factor
that multiplies ZEB, 1/ (1 + Γ). The frequency dependence of Γ is shown in Figure 5-15 for both Equation 5–41 and the high frequency simplification, Equation 5–46. Clearly, the high frequency estimate of Γ is accurate over the frequency range of interest. The peak in Γ is also observed to occur at the resonant frequency. This is easily confirmed since the denominator of Γ is the same as that of the volume velocity sensitivity in Equation 5–44.
−1 7 10 10 Z Full 6 input 10 −2 HF
) R + Z 10 5 ES EB Ω
10 | ) ( ) Γ | Z 4 10 −3
Re( 10 103
2 −4 10 10 45 5 −10 0 )
Ω −45
Γ ) (
6
Z −10
∠ −90 Im( −135 7 −10 −180 100 101 100 101 Frequency (kHz) Frequency (kHz)
Figure 5-14. Input and the electrical Figure 5-15. Γ factor of the input impedance comparison. impedance.
In Chapter 7, the capacitance, parallel resistance, and series resistance are extracted from experimental measurements of the input impedance. As is shown in Figure 5-14, the acoustic and electrical domains are coupled, resulting in the peak in the resistive component of the input impedance. Equation 5–45 contains models of the radiation and back cavity impedances that are not lumped elements. Thus, a frequency response function estimation of the experimentally measured input impedance to Equation 5–45 using basic circuits analysis is not possible. A low order Taylor series approximation, or LEM, of the radiation and cavity
impedances results in a Zinput that can be fit to a frequency response function. First, the
126 approximation of the radiation impedance was given in Equation 3–16 and is repeated here for convenience,
2 (kaeff ) 8aeff Zrad ≈ Rrad + jωMrad = ρ0c0 + jωρ0 . 2Aeff 3πAeff
A comparison of the full radiation impedance given by Equation 5–9 to the lumped element model is given in Figure 5-16. The error is shown to be less than 5% in the real and imaginary parts over the frequency range of interest.
5 −1 Full 10 4 LEM ) /s)) 3 3 rad Z 10−3 2 Re( (Pa/(mm
% Relative Error 1 −5 10 0 5 4 ) /s)) 3 3 rad Z −2 2
Im( 10 (Pa/(mm
% Relative Error 1 0 100 101 100 101 Frequency (kHz) Frequency (kHz)
Figure 5-16. Radiation impedance comparison of the full model and LEM.
101
/s)) 0 3 10
10−1 | (Pa/(mm |
AC −2
Z 10
| Full LEM −3 10 100 101 Frequency (kHz)
Figure 5-17. Back cavity impedance comparison of the full model and LEM.
127 Second, the back cavity impedance is estimated as a capacitance in series with a mass given by 2 1 ρoc0 ρ0–V ZAC ≈ + jωMAC = + jω 2 , (5–48) jωCAC jω–V 3S where –V and S are the cavity volume and cross-sectional area, respectively. A comparison of the full cavity impedance model to the LEM is shown in Figure 5-17. The results show a significant error around the resonant frequency where LEM breaks down since the acoustic wavelength is no longer much larger than the cavity depth.
106 20 Full
) 5 10 15 Ω LEM ) ( 4 10 10 input Z 3
Re( 10 5 % Relative Error 2 10 0 −104 20
) 15 Ω −105 ) ( 10 input Z −106
Im( 5 % Relative Error −107 0 100 101 100 101 Frequency (kHz) Frequency (kHz)
Figure 5-18. Input impedance comparison of the full model and LEM.
A comparison of the total input impedance based on the full model versus the lumped element model is given in Figure 5-18. The error in the resistive component rises to 20%. The error in the extraction of the electrical components from experimental data is simulated by fitting the LEM input impedance model to the distributed impedance model. The transfer function used to fit the impedance is
3 2 s a0 + s a1 + sa2 + a3 Zinput = 3 2 . (5–49) s b0 + s b1 + sb2 + b3
128 The distributed impedance model and the curve fit using Equation 5–49 is shown in Figure 5- 19. The electrical impedances are found from the following equations
a0 RES = (5–50) b0 a0 REP = a3 − (5–51) b0 2 b0 and CEB = . (5–52) a1b0 − b1a0
The extracted electrical impedances are given in Table 5-3. Clearly, the use of the LEM to fit the electrical elements in Chapter 7 is justified.
106 1 Full ) 105
Ω Fit ) ( 4 10 0.5 input Z 3
Re( 10 % Relative Error 2 10 0 1 −105 ) Ω ) ( −106 0.5 input Z Im( −107 % Relative Error 0 100 101 100 101 Frequency (kHz) Frequency (kHz)
Figure 5-19. Full model and curve fit comparison assuming a LEM of the input impedance.
Table 5-3. Comparison of electrical element fit to full model. Actual Fit % Difference
CEB (pF) 48.572 48.574 0.0035 %
RES (Ω) 55.21 55.14 0.12 %
REP (MΩ) 43.90 43.90 0.000074 %
In the following section, the results of the equivalent circuit are used to estimate the
parametric array performance of an array of transducers.
129 5.2 Nonlinear Acoustic Modeling
The Berktay parametric array solution presented in Section 2.2.2.2 is used to estimate the nonlinear acoustic output of an array of transducers [25]. The source condition of the Berktay solution assumes a single piston. The implementation of the MEMS-based para- metric array utilizes an array of many sources, each with a volume velocity predicted by the equivalent circuit of the Section 5.1.4. Thus, the single piston volume velocity is taken to be the summation of the volume velocity of an array of N radiators. When forming the array pattern, hexagonal packing of the devices is assumed with a center to center spacing of (1 + η) dtrans, where dtrans is the diameter of the transducer and η is a positive number that accounts for the space between the diaphragm edges. For hexagonally packed circles, the total surface area of the transducers plus their spacing accounts for 90% [156] of the total array area as illustrated in Figure 5-20. Thus, the array diameter, d, is related to the transducer diameter by,
2 2 d × 90% = N ((1 + η) dtrans) . (5–53)
d
dtrans ( +η ) 1 dtrans
Figure 5-20. Diagram of transducer array showing radius definitions (Not drawn to scale).
130 The source waveform is µ ¶ d p (r, z = 0, t) = p E (t) sin (ω t) H − r , (5–54) 0 0 2
where the modulating function is given by
³ω ´ E (t) = sin a t , (5–55) 2
where ωa = 2πfa and fa is the audible tone of interest. Thus, the source condition is rewritten as µ ¶ hp ³ ω ´ p ³ ω ´ i d p (r, z = 0, t) = 0 cos ω − a t − 0 cos ω + a t H − r . (5–56) 2 0 2 2 0 2 2
Note that the Bertkay solution assumes that the source amplitudes at both of the origi- nating frequencies are the same. Although the amplitude of the volume velocity frequency response of the transducer varies around resonance, the same volume velocity amplitude can be achieved using two different voltage excitation signals. The upper limit of the volume ve-
locity amplitude is determined by its linear limit. As is shown experimentally in Section 7.4.4
that upper limit of linearity is dictated by the velocity amplitude. Thus, by compensating
the two voltage excitation signals, the same velocity is achieved at each frequency. Thus, p0 is found as 2ρ c Q p = 0 0 0 . (5–57) 0 π(d/2)2
where Q0 is the volume velocity of half of the transducers. The equivalent circuit results of the example in Section 5.1.6 show a resonant frequency of 45.2 kHz and a volume velocity radiation sensitivity of 34.2 mm3/s/V at resonance. Con- sider an example of a parametric array 141 mm (5.6 in) in diameter. Eight-thousand trans- ducers fit within the array diameter with a center to center spacing of 1.5 mm. A conserva-
tive estimate based on experimental results in Section 7.4.4 is a 1 V excitation at resonance.
Given the parametric array analysis based on Berktay’s solution [25] outlined in Section 5.2,
the audible frequency response at 1 m is given in Figure 5-21. Note that there is 20 db per
131 decade roll-off in the response at lower frequencies as is the characteristic of parametric array response [25]. The output at 5 kHz and 1 m is 41 dB re 20 µPa. If the working distance is reduced to 10 cm, the SPL is 61 dB.
60 40 dB/decade Pa µ 40
SPL re 20 20
0 1 2 5 10 20 Difference Frequency (kHz)
Figure 5-21. Output of the example parametric array at 1 m.
5.3 Conclusion
This chapter detailed the equivalent circuit used to model a MEMS-based ultrasonic radiator for parametric array applications. An equivalent circuit of the device was derived to evaluate the dynamic performance. A source condition was introduced for a nonlinear acoustic calculation based on the Berktay parametric array solution [25]. Through an ex- ample of a non-optimal device, the importance of individual elements was explored. The
models derived in this chapter are used in Chapter 6 in an optimization scheme.
132 CHAPTER 6 DESIGN OPTIMIZATION
It is difficult to intuitively choose device geometry to create an ultrasonic radiator de- sign that meets or exceeds the application requirements. Therefore, an optimization scheme is used intelligently determine device dimensions. In this chapter, the objective of the opti- mization and the constraints on the design are given. Optimization results including design sensitivity are given. The optimization will also provide insight into the sensitivity of the device behavior with respect to individual design variables. 6.1 Methodology
The objective of the optimization is to determine a transducer design that produces a maximum SPL of the difference tone at a distance of 1 m from the source. The 1 m distance was chosen as a proof of concept value to compare to results from the literature [10]. As discussed in Chapter 5, the array of transducers can be modeled by a single piston by equating volume velocities. As shown in Chapter 3, the on-axis output of an array of transducers is related to the output of a single transducer by a constant. Thus, the optimization of the output of an array of many transducers can be reduced to the optimization of the output of a single transducer. As discussed in the introduction of parametric arrays in Chapter 1 and Chapter 2, the pressures of the fundamental tones act as sources for the difference frequency. The pressures of the fundamental tones are assumed to be found from Rayleigh’s integral for a baffled piston, discussed in Chapter 3, because there is weak nonlinear conversion. The pressure is related to the diaphragm velocity as given in Chapter 3. Maximizing the velocity or volume velocity of a single transducer, however, does not take into account acoustic attenuation due to molecular relaxation as the acoustic waves propagate away from the transducer.
A more appropriate optimization scheme is based on the difference pressure result of the Berktay solution outlined in Chapter 2. The Berktay solution accounts for acoustic
133 attenuation. It is also based on the square of the transducer output. Thus, the difference tone pressure obtained from the Berktay solution served as the objective function.
MATLAB’s fmincon function will serve as the optimization tool in this methodology.
This function uses a general class of gradient-based local optimization called sequential quadratic programming with nonlinear constraints [157]. Details on fmincon can be found in MATLAB’s help file.
6.2 Radiator Optimization
The objective function is the difference tone pressure given by the Berktay solution
considering modulation by the audio frequency, fa, [158]
2 2 βρ0u0 (fc) Aπfa p¯2 (r = 0, z = 1m) ' 2 , (6–1) 4c0 [1m] α0 (fc)
where β, c0, and ρ0 are properties of the surrounding air. The array area, A, and the audio
frequency, fa = 5kHz, are fixed for the proof of concept application. The velocity amplitude,
u0, is directly related to the average velocity of the diaphragm at resonance by a constant.
The attenuation is dependent upon the carrier frequency, α0 (fc). The carrier frequency,
fc, is also the resonant frequency of the device. Thus, the optimization will maximize the following ratio from Equation 6–1:
2 u0 (fc) Fobj = . (6–2) α0 (fc)
Note that the resonant frequency, fc, and the average velocity at resonance, u0, are a result of the mechanics of the diaphragm coupled with its radiation impedance.
The volume velocity sensitivity to applied voltage, Equation 5–39, derived from the
equivalent circuit is converted to velocity sensitivity by division of the effective area, Aeff .
The applied voltage is limited by the breakdown voltage, VBD, of the AlN piezoelectric layer.
The breakdown voltage is found by multiplying the dielectric field strength of AlN, Em [159], by the thickness of the AlN piezoelectric thickness, hp. Note that the breakdown voltage is most likely overly generous as nonlinear effects will likely become appreciable at lower
134 excitation voltages. In the absence of a solution for the maximum voltage before the onset of nonlinearities, the breakdown voltage is applied as a scaling factor to ensure that the units of the squared term in Equation 6–2 are in terms of velocity. Thus, the objective function in Equation 6–2 becomes 2 (Em · hp · S (fc)) Fobj = . (6–3) α0 (fc) Since the dielectric strength is a material property fixed by using the Avago FBAR process, it can be removed. The final objective function is
2 (hp · S (fc)) Fobj = . (6–4) α0 (fc)
6.2.1 Limitations-Constraints
Optimization constraints are classified into four types: physical bounds, manufacturing limits, modeling limits, and operational requirements [160]. Avago Technologies dictated constraints on device geometry. The thin AlN seed layer thickness is restricted by the fabrication and is therefore not considered a design variable. The thickness ranges for the rest of the AlN and Mo layers are given in Table 6-1.
Table 6-1. Avago Technologies Limited process options where j = 1, 2 refers to the inner and annular plate sections, respectively. Layer Lower bound Upper bound (j) Scaffolding AlN H1 500 nm 2,000 nm (j) Bottom Mo Electrode H3 200 nm 400 nm (j) Piezoelectric AlN H4 300 nm 800 nm (j) Top Mo Electrode H5 200 nm 700 nm (j) Top AlN Passivation H6 50 nm 300 nm
The outer radius is also constrained such that,
250 µm ≤ R2 ≤ 750 µm. (6–5)
135 The lower bound is dictated by Avago to ensure the proper etch chemistry during the fabri- cation. The upper bound of 750 µm was suggested by Avago as the largest diaphragm they were comfortable releasing in their facility.
Several modeling constraints are also necessary. First, it is crucial to ensure that the diaphragm sections are accurately modeled as plates. The aspect ratio is thus constrained according to [161]
R1 P (1) ≥20, (6–6) Hi i R2 − R1 and P (2) ≥20. (6–7) Hi i This constraint could be relaxed by implementing Midlin plate theory that accounts for shear deformations to give accurate results for thicker plates.
The effects of residual stresses and radial non-uniformity of the diaphragm lead to initial deflection that is difficult to predict using the linear plate theory presented in Appendix B. For the linear plate model to accurately represent the diaphragm mass and compliance used in the equivalent circuit of Chapter 5, the initial deflection must remain linear. In addition, constraining the initial deflections will also ensure that the plate is pre-buckled. Buckled diaphragms of similar resonant frequency and sensitivity to pre-buckled diaphragms have shown a decrease in linear range [162]. Also, higher order buckling modes can have areas of the diaphragm that vibrate out of phase, reducing the efficiency of the device as a radiator. Thus, it is advantageous to constrain the device to be pre-buckled. To formulate the constraint on the initial deflection, the deflection mode shapes found from the linear and nonlinear plate models in Appendix B must be compared. Instead of comparing the mode shapes point by point, the total static volume deflections, –V , are compared between the linear and nonlinear solutions. The constraint on the initial device deflection is written as ¯ ¯ ¯ ¯ ¯–Vlin − –Vnl ¯ ¯ ¯ × 100% ≤ 10%. (6–8) –Vlin
136 A linearity constraint on the incremental plate deflection is not added. Any harmonic distortion of the diaphragm deflections are suspected to be weak in amplitude [58]. Therefore, harmonics of the diaphragm deflection will not be efficient radiators. The generation of vibrational harmonics, however, extracts energy from the fundamental vibration mode. No constraint is placed on the amplitude of the dynamic deflection of the diaphragm. This approach is based on the belief that if substantial harmonics are produced, the transducer will successfully be achieving large excursions. An operational constraint is applied to the carrier frequency to ensure that the devices operate safely. The carrier frequency must be high enough to guarantee that the amplitude modulation will not shift one of the originating tones into the human hearing range where the amplitude would be above the human threshold for pain. Although this constraint could be lowered depending on the modulation type, in the literature [11] the carrier frequency is usually constrained to be 35 kHz or higher,
fc ≥ 35 kHz. (6–9)
A sensitivity study of each of the constraints is provided in the optimization results. 6.2.2 Problem Formulation
Optimization schemes are written to minimize an objective function, fobj ([x]). The mathematical representation of the optimization is [160]
h Minimize i fobj ([x]) (j) (j) (j) (j) (j) [x]= R1,R2−R1,H1 ,H3 ,H4 ,H5 ,H6 such that LB ≤ [x] ≤ UB
R1 1 − P (1) /20 ≤ 0 Hi i (6–10)
R2−R1 1 − P (2) /20 ≤ 0 Hi i 1 − fc ≤ 0 ¯ ¯ 35kHz ¯ ¯ ¯–Vlin−–Vnl ¯ /0.1 − 1 ≤ 0, –Vlin
137 where fobj ([x]) is given by the negative of Equation 6–4. The design variables are the inner radius, the difference between the inner and outer radius, and the thicknesses of the bottom Mo electrode, piezoelectric AlN, top Mo electrode, and top AlN passivation layers. The
stress in all of the layers is assumed to be −20 MPa compressive stress with the exception of the AlN scaffolding layer which is assumed to be stress neutral.
6.3 Results
The optimal geometry and performance are given in Table 6-2. The volume velocity sensitivity is 68.2 mm3/s at a resonant frequency of 35 kHz. Using the same nonlinear acous- tic calculation outlined in Sections 5.2 and 5.1.6, the parametric array frequency response
at 1 m is given in Figure 6-1. Note that the array diameter is slightly larger at 154 mm (6 in) diameter due to the larger transducer diameter. The SPL is 54.5 dB re 20 µPa at 5 kHz. There is a 10.5 dB improvement in comparison to the non-optimal example with zero residual stresses of Section 5.1.6.
20
Pa 80 µ
60 SPL re 20 40
1 2 5 10 20 Difference Frequency (kHz)
Figure 6-1. Output of the parametric array at 1 m.
Insight into the importance of the design parameters is gained by noting the sensitivity of the objective function to changes in the design variables. The normalized change in the objective function versus the normalized design variables, represented by X, is shown in
Figure 6-2. The design variables and objective function are normalized by their optimal
values, Xopt and fobj (Xopt), respectively. The range in the design variables is ±20%. From
138 Figure 6-2, the objective function is most sensitive to the AlN scaffolding layer thickness, H1, and the inner radius of the annular piezoelectric ring, R1. Next, the annular ring thickness,
R2 − R1, and piezoelectric AlN thickness, H4, also affect the design performance. The objective function is least sensitive to the molybdenum electrode thicknesses, H3 and H5, and the upper passivation layer thickness of AlN, H6. It would seem that a decrease in the AlN scaffolding layer thickness or an increase in the inner radius of the annular piezoelectric ring would lead to enhanced performance of the device. As shown in Figure 6-3, however, this would lead to constraint violations.
The objective function and active constraint sensitivities for the AlN scaffolding layer
thickness, H1, the inner radius of the annular piezoelectric ring, R1, the annular ring thick-
ness, R2 − R1, and piezoelectric AlN thickness, H4, are given in Figure 6-3. The normalized constraints are plotted such that they are active when they exceed a value of one. The resonant frequency and nonlinear initial deflection constraints are active. The design space regions where the constraints are active are shaded in color. The upper or lower bound of
each design variable is violated in the hatched region. For example, in Figure 6-3(b), the AlN piezoelectric thickness is at its upper bound. An increase in the AlN piezoelectric thickness would result in a higher objective function but it is constrained by its upper bound. A de- crease in the AlN piezoelectric thickness would result in a decrease in the objective function as well as a violation of both nonlinear constraints.
The optimization is also sensitivity to the heuristic nonlinear constraints, specifically
the percent nonlinearity in the static deflection and the minimum resonant frequency. The optimization was repeated using different values for the heuristic constraints. In Figure 6-4, the relative sensitivity of the objective function to the choice of the percent difference between the linear and nonlinear static deflection is shown. The objective function is reduced by almost 30% if a more conservative constraint of 5% deviation is chosen. A less conservative
choice of constraint of 15% results in an increase in the objective function of almost 20%.
139 Table 6-2. Design optimization results.
(i) † Scaffolding AlN H1 2,000 nm (i) * Bottom Mo Electrode H3 200 nm (i) † Piezoelectric AlN H4 800 nm (i) * Top Mo Electrode H5 200 nm (i) * Top AlN Passivation H6 50 nm Inner radius R1 399 µm
Outer thickness R2 − R1 145 µm 3 Sensitivity S (fres) 68.2 mm /s
Resonant frequency fres 35 kHz
Parametric array output f− = 5 kHz 51.5 dB * Lower bound. † Upper Bound.
R 4 1 R −R 3.5 2 1 H
) 6
opt 3 H 5 (X H obj 2.5 4 H (X)/f 2 3
obj H f 1 1.5
1
0.8 0.9 1 1.1 X/X opt
Figure 6-2. Sensitivity of the normalized objective function to the normalized design vari- ables.
Thus, the optimization is more sensitive to a reduction in the constraint value rather than a relaxation. The relative change in the objective function with respect to the minimum resonant frequency constraint is shown in Figure 6-5. An increase in the minimum resonant frequency constraint to a more conservative value of 40 kHz results in a 10% decrease in the objective
140 (a) (c)
(b) (d)
Figure 6-3. Sensitivity of the objective function due to changes in the individual design variables show active constraints and bounds.
141 function. A relaxation of the resonant frequency constraint causes a similar increase the objective function.
An increase in the objective function on the order of 10-30% could be achieved by a
relaxation of either of the nonlinear constraints. The audible output of the parametric array, however, is reported in decibels. Therefore, the increase in SPL is not significant enough to warrant a revision of the nonlinear constraints.
1.1 1.2 1.05 (main) (main) 1 1 obj obj F F / /
obj 0.95 obj F F 0.8 0.9
5 10 15 30 35 40 f (min) (kHz) Relative Error (%) res Figure 6-4. Sensitivity of the objective Figure 6-5. Sensitivity of the objective function to the nonlinear de- function to the minimum res- flection constraint. onant frequency constraint.
6.4 Alternate Designs
The design found in Table 6-2 was created using fmincon by assuming that all of the layers had a −20 MPa compressive stress with the exception of the AlN scaffolding layer which is assumed to be stress neutral. This was the stress target of Avago Technologies Limited for the production run. There is evidence of large stress variations from prior AlN/Mo structures, both with reference to the stress targets and across the wafer. Taking this into account, a set of optimizations were performed considering variation in the stress of
±200%. The design variables for the secondary designs were the inner and outer radii of the annular ring since the thicknesses are fixed by the primary optimal design. This redundant design method provided a degree of robustness that ensured that some percentage of the final devices did not severely under-perform the original optimal design or violate constraints. The
142 sixteen device designs are given in Table 6-3 along with the objective function referenced to that of the optimal design, UFPA08.
Table 6-3. Alternate optimization results. Device Label Stress Inner Radius Outer Radius Objective Function (MPa) (R1 µm) (R2 µm) (dB re UFPA08) UFPA01 15 512 728 -12.4 UFPA02 10 483 700 -10.5 UFPA03 5 455 672 -8.5 UFPA04 1 398 611 -7.1 UFPA05 -5 403 618 -4.3 UFPA06 -10 379 665 -2.3 UFPA07 -15 360 572 -0.3 UFPA08 -20 399 544 0.0 UFPA09 -25 390 507 -1.3 UFPA10 -30 356 463 -2.9 UFPA11 -35 330 429 -4.2 UFPA12 -40 308 401 -5.2 UFPA13 -45 291 378 -6.1 UFPA14 -50 276 359 -6.8 UFPA15 -55 263 342 -7.5 UFPA16 -60 252 327 -8.1
6.5 Conclusions
The optimization methodology and design results are presented in this chapter. The results of the equivalent circuit in Chapter 5 together with Berktay’s nonlinear acoustic solution are used to form an objective function. The layer thicknesses and topographical geometry of the diaphragm are the design variables. A primary design is found using the fmincon optimization tool in Matlab. A sensitivity analysis of the objective function shows that the AlN scaffolding layer thickness and the inner radius of the annular piezoelectric ring are the most crucial design variables. Active constraints include upper and lower bounds on the layer thicknesses as well as the resonant frequency and nonlinear initial deflection constraints. A secondary optimization is performed on the topographical geometry while constraining the layer thicknesses and considering alternative in-plane stress. The result is
143 16 different device designs. The following chapter discusses the fabrication results and the device characterization.
144 CHAPTER 7 EXPERIMENTAL SETUP AND RESULTS
This chapter describes the experimental setups and results used to perform characteri-
zation. First, initial results of the fabrication are discussed along with a method for picking the best case design. Next, electrical experimental setup and characterization results are outlined. Third, the diaphragm initial deflection is given. Next, the electromechanical char- acterization results using a scanning laser vibrometer are presented. Finally, ultrasound measurements of the device that define its directivity and on axis response are given in the electroacoustic characterization results section. The measurement results are used to estimate the difference frequency production of an array of devices.
7.1 Fabrication Results
As discussed in Chapter 6, sixteen different device geometries were fabricated. From the measurement of stress during the fabrication process, the performance of each device design was projected using the equivalent circuit from Chapter 5. The most promising device was chosen based on the predicted performance.
First, the stress measurement results are given. Stress existed in each of the AlN and Mo layers due to fabrication effects as discussed in Chapter 4. Film stress causes wafer bow due to a mismatch in stress between the film and the substrate. The wafer bow caused by a film deposition was found by measuring the wafer profile before and after film deposition. The film stress was found from the wafer bow measurement using Stoney’s formula [152]. The film stress measurements were conducted at Avago using a Tencor Flexus FLX 5400. The
measurement results along with the stress targets are given in Table 7-1. Uncertainties in the stress measurements were estimated by Avago at ±20 MPa for stresses whose magnitude was less than 60 MPa and ±30% for larger stresses. It was also important to note that Avago claims that the stress varied across the wafer, with the inner portion of the wafer being more tensile than the outer portion. This was confirmed by a visual inspection of the wafer. The diaphragms along the outer portion of the wafer appeared to have large initial deflections
145 indicating an overall compressive stress. The diaphragms along the inner portion of the wafer appeared flat indicating that they are more tensile. The stress measurement using the Tencor Flexus FLX 5400 was not capable of determining localized stress variations. Thus, the stress measurements reflect integrated stress affects from across the wafer and possess an unknown uncertainty that is a function of die location on the wafer. Since the averaged stress through the layer thicknesses indicated in Table 7-1 was tensile while the outer part of the wafer was clearly under compressive stress, it was inferred that the inner reticles are under a higher tensile stress than that measured by Avago. The ramification was that the device predicted given the stresses in Table 7-1 was more compliant that the actual devices tested. Thus, it was probable that the theoretical model would under predict the resonant frequency and over predict the device sensitivity. This hypothesis was confirmed from the comparison of the experimental and theoretical results of the following sections.
Table 7-1. Film stress target and realized values for the wafer provided by Avago. Layer Target As fabricated (MPa) (MPa) Scaffolding AlN (2 µm) 0 17 Bottom Mo Electrode (0.2 µm) -20 90.1 Piezoelectric AlN (0.8 µm) -20 -20 Top Mo Electrode (0.2 µm) -20 -31 Top AlN Passivation (0.05 µm) -20 -365.1
Based on the stress results from Table 7-1, the equivalent circuit from Chapter 5 was used to predict the performance of the fabricated devices. The predicted objective function of the fabricated devices in given in Table 7-2 in dB referenced to the target design, UFPA08. For example, the performance of the fabricated UFPA08 device in the second column shows a predicted loss in the objective function of -12.9 dB with respect to the target UFPA08 device in the first column. The predicted performance of the fabricated UFPA01 device, on the other hand, shows a loss in the objective function of -11.8 dB with respect to the target
146 UFPA08 device. Given the slightly enhanced performance with respect to the target design, the UFPA01 device was chosen as the device under test (DUT).
Table 7-2. Objective function of device designs in dB referenced to the target design, UFPA08. Blanks refer to devices that violate modeling constraints. Device Targeted As fabricated UFPA01 -11.8 UFPA02 -11.9 UFPA03 -12.0 UFPA04 -12.4 UFPA05 -12.3 UFPA06 -13.4 UFPA07 -12.8 UFPA08 0.0 -12.9 UFPA09 -3.6 -14.0 UFPA10 -6.2 -14.5 UFPA11 -8.0 -15.0 UFPA12 -9.4 -15.5 UFPA13 -10.7 -16.1 UFPA14 -11.7 -16.7 UFPA15 -12.7 -17.3 UFPA16 -13.7 -18.0
As described in Section 4.1, projection lithography was used to form the planform geometry of the devices. Projection photolithography steps a single mask pattern across a wafer in a grid to form multiple copies of the same mask geometry on a single wafer [163]. Each copy of the mask is known as a reticle. The result of the fabrication was a grid of reticles that each contained a copy of the UFPA01 device. The devices are labeled according to their reticle row (RR) and reticle column (RC) as shown in Figure 7-1. The devices characterized were RR1RC3, RR2RC3, RR2RC4, RR3RC5, RR5RC5, and RR5RC2, which are indicated in Figure 7-1. All of the die with the exception of RR5RC2 were mounted to a pcb board that allowed for a variable back cavity, as described in Chapter 4. The exception was the RR5RC2 device which was mounted to a pcb board with no back cavity through hole. Thus, the back cavity depth of RR5RC2 was limited to the silicon thickness.
147 RR1RC1 RR1RC2 RR1RC3 RR1RC4 RR1RC5 RR1RC6
RR2RC1 RR2RC2 RR2RC3 RR2RC4 RR2RC5 RR2RC6
RR3RC1 RR3RC2 RR3RC3 RR3RC4 RR3RC5 RR3RC6
RR4RC1 RR4RC2 RR4RC3 RR4RC4 RR4RC5 RR4RC6
RR5RC1 RR5RC2 RR5RC3 RR5RC4 RR5RC5 RR5RC6
RR6RC1 RR6RC2 RR6RC3 RR6RC4 RR6RC5 RR6RC6
Figure 7-1. Reticle labeling convention.
7.2 Electrical Characterization
Electrical characterization consisted of impedance measurements using an HP 4294A
Impedance Analyzer. The blocked capacitance, CEB, parallel loss resistance, RP , and series loss resistance, RS, were extracted from the impedance measurements. 7.2.1 Setup
The HP 4294A Impedance Analyzer has 4 ports used to connect to the DUT. To make an impedance measurement, the low current and low potential terminals were tied together as are the high current and high potential terminals as shown in Figure 7-2. The resulting low and high terminals are connected across the DUT for impedance testing. The HP 4294A impedance analyzer uses an I-V method to measure impedance where
V VHp−Lp Zsensor = = . (7–1) I IHc−Lc
The impedance analyzer was set to perform a linear frequency sweep from 1 to 200 kHz with 801 points with a 100 mV source. The bandwidth settings range from 1 to 5, where 5 corresponds to the narrowest notch filter the impedance analyzer used to measure the impedance at a frequency bin. The tradeoff is an increase in time of the measurement.
Point averaging and averaging of sweeps are turned off. The measurement was repeated 30
148 L CUR L POT H POT H CUR
LH DUT
Figure 7-2. The front panel connections on the HP 4294A Impedance Analyzer where red is the high connection, black is the low connection, and green is ground.
times and averaging was conducted during post-processing to obtain estimates of the random error in the measurement. The estimation of the bias error was taken from the operation manual [164]. The contribution of the random and bias errors are indicated in the following
results. A comparison of the random to bias errors is presented in Appendix C.
Impedance measurements were conducted pre- and post-packaging to identify packaging effects on the impedance. The pre-packaged impedance measurements were conducted using a Wentworth Labs probe station and Signatone Model S-725 probes. Non-repeatability of the short calibration and drift of the probe station led to erroneous impedance measurements. Thus, the pre-packaged impedance measurements are not presented in this work.
7.2.2 Results
The impedance measurements are compared to theoretical predictions of Zinput from
Section 5.1.4 in Figure 7-3. The series resistance, RES, and parallel resistance, REP , used in
the theoretical prediction of Zinput in Equation 5–40 were based on an average of the values extracted from the experiment as 39 Ω and 31 MΩ, respectively. As observed, there is a good match between the imaginary part of the measured input impedance of all the devices
as well as the theoretical model. As was shown in Section 5.1.6, the blocked capacitance
dominates the imaginary component of the impedance. Since the devices come from the same wafer and were formed using projection lithography, which indicates that the planform geometry should be well matched, this result is expected.
149 The measured resistive components are qualitatively similar with an average slope of approximately −22 dB per decade. In comparison to the measurements, the theoretical model on the resistive component of the input impedance shows signification error. First,
the local maximum peaks that were shown in Section 5.1.6 to coincide with the resonant frequency occurred at different frequencies. This was evidence that the resonant frequency
varies between devices and is confirmed in Section 7.4. Second, the theoretical model begins to asymptote to the series resistance value of 39 Ω after the peak in the resistance. Before the peak, the theoretical resistance is dropping at approximately −40 dB per decade. The implication of the approximately 20 dB per decade difference in the resistance roll-off between the experiments and the theory is that there is a term that multiplies the resistance that increases linearly with frequency that has not been accounted for in the theory. The direct resultant of the increase in the resistive component of the impedance is a reduction in the volumetric flow rate sensitivity of the device.
Mean 95% Confidence 105 105
104 104 ) Ω 103 103 R ( 102 102
101 101
4 −10 6 RR5RC5 10 RR1RC3 105 RR2RC3
) 5 4 RR2RC4 Ω 10 −10 RR3RC5
X ( 3 10 RR5RC2 Theory 102 −106 1 10 100 101 102 100 101 102 Frequency (kHz) Frequency (kHz)
Figure 7-3. The real and imaginary part of the electrical impedance displaying both the experimental data for all transducers and the theoretical model.
150 The complex impedance data in air was fit using a function in Matlab, invfreqs, which implements a damped Gauss-Newton method to fit an analog transfer function to complex data that is a function of frequency [165]. The data was fit to the impedance model derived in Chapter 5 which is repeated here for convenience,
REP 1 Zinput = RES + , 1 + jωCEBREP 1 + Γ
Z where Γ = EBA . ˜ ZAD + ZAC
The impedance fitting algorithm was not able to capture the local maximum in the resistive component. The resulting extracted electrical elements are given in Table 7-3. The error between the curve fit and the experimental data over the frequency range of the fit is less than 10% for the resistive component other than at resonance and less than 5% for the reactive component over all frequencies. A comparison of the curve fits to the experimental data is given in Appendix C. As observed, there is a wide variation in the series and parallel resistances. This is probably due to a low sensitivity of the curve fit to these components.
Table 7-3. The extract impedance parameters from the experimental impedance mea- surement.
CEB (pF) REP (MΩ) RES (Ω) RR1RC3 93 81 13 RR5RC5 94 33 63 RR2RC3 94 12 57 RR2RC4 94 18 41 RR3RC5 94 26 39 RR5RC2 93 18 21 Theory 101 31 39
7.3 Device Topography
Film stress in non-radially uniform devices causes initial diaphragm deflection. Large compressive stresses can even lead to buckling of the diaphragm, resulting in large initial deflections and\or multi-mode deflections [166]. Initial deflection and/or buckling of the
151 diaphragm changes its stiffness resulting in a shift in device performance such as resonant frequency. In addition, differences in initial deflection across a wafer for devices of the same geometry is an indication of non-uniform stress over the wafer. A further addition
of stress can come from the packaging. Steps were taken to mitigate the packaged induced stress using the epoxy wells as discussed in Chapter 4. An indication of the effectiveness of the epoxy wells would be a comparison of static topography before and after packaging. The pre-packaged topography measurements were corrupted by multiple layer reflections. Unfortunately, the devices were already packaged before this affect was discovered such that these measurements are not represented in the current work.
Measurements of the static diaphragm deflections were made with a Zygo NewView
7200 non-contact scanning white light interferometer. A 5X objective combined with a 0.5X
field zoom lens to give a 2.83 × 2.12 mm2 field of view. The vertical resolution of the system is less than 0.1 nm. This measurement presented a challenge due to the multiple thin film layers of the AlN and Mo. Focus of the optical system resulted at multiple layer interfaces resulting in ambiguity as to the zero reference of each measurement. Thus, the measurements were all referenced to the top Mo electrode since focusing on this layer was repeatable in between measurements.
1
m) RR1RC3
µ 0 RR2RC3 RR2RC4 −1 RR3RC5 Inner RR5RC2 Annular Region Annular RR5RC5 −2 Region Region Initial Deflection (
−3 −600 −400 −200 0 200 400 600 Radius (µm)
Figure 7-4. Initial diaphragm deflections.
152 Initial deflections of the diaphragms are given in Figure 7-4. The vertical step towards the outer edge of the diaphragms marks the beginning of the annular region. Note that the vertical heights are referenced to the surrounding substrate. As is shown in the figure, all of the devices with the exception of RR5RC5 had similar initial deflection topography. There was a slight waviness on the order of 100 nm in the central region of devices RR2RC3, RR2RC4, and RR3RC5. Clearly, the initial deflection of the RR5RC5 device stood out from the other topographies. The center deflection was -1.3 µm down from the surrounding substrate. Since the center deflection was on the order of the diaphragm thickness, the diaphragm was likely buckled. This was an indication that the performance of RR5RC5 varies significantly with respect to the other devices, as confirmed in the following sections.
7.4 Electromechanical Characterization
The electromechanical characterization involved measuring the velocity frequency re- sponse, diaphragm resonance, first resonant mode shape, and linearity at resonance using the converse piezoelectric effect (see Section 3.3.2 for a discussion of the direct versus converse piezoelectric effect). Measurements were performed using optical interferometric techniques. An ac voltage excitation signal was applied to the DUT. The vibration of the diaphragm was measured using a Polytec MSV 300 scanning laser vibrometer shown in Figure 7-5. The laser vibrometer is a non-contact measurement system that utilizes optical interferometry to measure surface vibration velocities. A helium neon laser is scattered off the diaphragm and into the interferometer along with the reference laser beam. The reflected beam experiences a Doppler shift due to the velocity of the diaphragm [167]. Comparison of the frequency dif- ferences measured between the reflected and reference signals give the instantaneous velocity of the diaphragm. By coupling the vibrometer to a microscope and piezoelectric motors, the laser was scanned across the diaphragm surface to capture phase-locked mode shapes. The resonant frequency was found from the frequency response of the diaphragm. The resonant mode shape was extracted from the phase-locked scan of the diaphragm at resonance. The linearity
153 Scanning motors
LV microscope Optical signal attachments (to fiber interferometer)
Microscope
DUT
Excitation signal
Figure 7-5. Scanning laser vibrometer system used for electromechanical characterization. of the device at resonance was found by measuring the velocity amplitude of the device at a fixed frequency under increasing excitation. The laser vibrometer was also used to determine the impedance loading effect of the variable back cavity. The frequency response of the DUT was recorded for a series of back cavity depths. The dependence of the resonant frequency and sensitivity on the back cavity depth was investigated.
In addition to measuring the device response in air, vibrometer measurements were also made in a vacuum chamber. By measuring the device response while under vacuum, the effects of the radiation, back cavity, and vent impedances were mitigated. By comparison of the measurement in air and under vacuum, the effect of damping within the diaphragm, such as thermoelastic dissipation or anchor loss, was isolated from the radiation damping, back cavity dissipation, and vent resistance. A convergence study was done to investigate whether the vacuum level reached was low enough to negate the effects of air on device performance.
154 7.4.1 Setup
In the FFT measurement mode, the maximum (100) complex averages was selected. Two channels were measured, the vibrometer with units of velocity and the reference exci- tation signal with units of voltage. Both were ac coupled with a cut-on of approximately 200 Hz. The range was set according to the signal level given the sensitivity setting of the velocity decoder, which is discussed subsequently, to avoid clipping while maximizing dy- namic range. Since the excitation signal was periodic, no filters or windowing functions were applied. The frequency settings determined the sampling frequency and frequency resolution of the FFT. The sampling frequency was set to 2.56 times the upper limit of the bandwidth to avoid aliasing. The bandwidth was chosen from 0-50 or 0-100 kHz, depending upon the resonant frequency of the transducer and the desired spectra measured. The number of FFT lines was set to the maximum of 6400. Since the excitation signals were always periodic, no overlap was used. A trigger was unnecessary since the reference signal was measured and was periodic. For signal enhancement of the vibrometer signal, speckle tracking was used along with the fast setting. The speckle tracking setting made the position of the laser move randomly about the scan point each iteration to reduce the effects of the speckled nature of the back scattered light. The fast setting kept the number of averages at the same number as set in the measurement mode settings. There are 6 velocity settings for the vibrometer, 3 for each velocity decoder. The low frequency velocity settings of 1, 5, 25 mm/s / V correspond to maximum frequencies of 20, 50 and 50 kHz, respectively. The high frequency velocity settings of 10, 25, and 125 mm/s / V correspond to maximum frequencies of 0.2, 1, and 1.5 MHz, respectively. The tracking filter was set to slow to reduce speckle effects. The signals measured by the vibrometer include both the reference excitation signal,
Vref , and the surface velocity, Uvib, at a scan point whose location is denoted (x, y). The data extracted from the vibrometer was in the frequency domain. For frequency domain data,
155 the Fast Fourier Transform (FFT) of the reference and vibrometer signals were calculated,
jφref (f) V ref (f) =|V ref (f)|e , (7–2)
jφvib(f) and U vib(f) =|U vib(f)|e . (7–3)
The phase of the vibrometer signal was referred to the reference signal such that the FFT’s extracted from the vibrometer were actually
Vref (f) =|V ref (f)|, (7–4)
j(φvib(f)−φref (f)) and Uvib(f) =|U vib(f)|e . (7–5)
The vibrometer software also reported the auto-spectrums and cross-spectrums,
2 £ ¤ G = E V ∗ V , (7–6) VV T ref ref 2 G = E [U ∗ U ] , (7–7) UU T vib vib 2 and G = E [V ∗ U ] . (7–8) VU T vib ref
From the auto- and cross-spectrums, the vibrometer software calculates the coherence as ∗ 2 GVU GVU γVU = . (7–9) GVV GUU The auto- and cross-spectrums are also used to estimate the frequency response function (also known as velocity sensitivity) at a measurement point using the “optimal” Wiener
Filter to minimize dependence on the output noise of the measurement,
GVU HU = . (7–10) GVV
A discussion of the mean estimation and uncertainty in the vibrometer data is discussed in Appendix C.
156 7.4.2 Frequency Response Function
The frequency response function was measured by conducting a phase-locked scan of the diaphragm vibration under electrical excitation. The excitation signal was a periodic chirp waveform with zero offset which gives excitation at all frequency bins defined in the frequency settings of the vibrometer. The amplitude of the excitation signal was tuned to provide a maximum signal to noise ratio. The diaphragm and scanning grid are shown in Figure 7-6.
Figure 7-6. The scan grid overlayed with the microscope picture of the device.
The velocity frequency response function at the center of the diaphragm is given in Figure 7-7 with the 95% confidence of the uncertainty in the mean estimate of the frequency response. Similarly, the calculated center displacement sensitivity is given in Figure 7-8. Figure 7-7 is only the velocity sensitivity at a single grid point. As is seen in Rayleigh’s integral in Chapter 3, if the entire diaphragm is not deflecting in phase, then the total sound
output is diminished. Therefore, the volume velocity sensitivity is a more telling performance characteristic. Also, the volume velocity sensitivity can be readily compared to the result of the equivalent circuit in Chapter 5 since volume velocity is the flow variable of the acoustic domain in LEM. In addition, the volume velocity can be used to solve for an equivalent piston velocity of an array of many radiators for nonlinear acoustic calculations.
The volume velocity sensitivity shown in Figure 7-9 was found by numerically integrating
the mean complex frequency response function, Hb, over the cartesian grid given in Figure 7-6
157 Mean 95% Confidence 103 103 102 102
101 101 100 100 | ((mm/s)/V) U −1 −1 H | 10 10 10−2 10−2
−90 40 RR5RC5 RR1RC3 −135 30 )
° RR2RC3
) ( RR2RC4 U −180 20 H
( RR3RC5 ∠ −225 10 RR5RC2 Theory −270 0 101 102 101 102 Frequency (kHz) Frequency (kHz)
Figure 7-7. Velocity frequency response function for the center grid point, where the uncer- tainty is the 95% confidence in the mean estimate.
Mean 95% Confidence 101 101 0 0 10 10 10−1
m/V) −1
µ 10 −2
| ( 10 W
| −2 10 10−3 −4 −3 10 10 40 180 RR5RC5 RR1RC3 30
) RR2RC3 °
) ( 90 20 RR2RC4 W ( RR3RC5 ∠ 10 RR5RC2 0 Theory 0 101 102 101 102 Frequency (kHz) Frequency (kHz)
Figure 7-8. Displacement frequency response function for the center grid point, where the uncertainty is the 95% confidence in the mean estimate.
158 Mean 95% Confidence 103 103 2 102 10 1 1 10 /s)/V)
3 10 100 0 10 10−1 | ((mm −2
Q −1 | 10 10 −3 10−2 10 40 −90 RR5RC5 30 RR1RC3 )
° RR2RC3
) ( 90 20 RR2RC4 Q (
∠ RR3RC5 10 RR5RC2 Theory 0 0 100 101 102 100 101 102 Frequency (kHz) Frequency (kHz)
Figure 7-9. Volume velocity sensitivity, where the uncertainty is the 95% confidence in the mean estimate. for each frequency component, ZZ Qb(f) = Hb(x, y; f)dxdy. (7–11)
The cartesian coordinate system of grid points was utilized to simplify the two-dimensional numerical integration. The tradeoff was the addition of noise from points not on the dia- phragm. As shown in Figure 7-9, the 95% confidence in the amplitude of the mean estimate of the volume velocity is at least two orders of magnitude below the mean estimate. 7.4.3 Diaphragm Resonance
As seen in Figure 7-9, the frequency response of devices varies, although four are well matched with resonant frequencies around 60 kHz. The device performance characteristics are summarized in Table 7-5. The RR5RC5 device gave the highest sensitivity at resonance and the lowest resonant frequency. It was expected that this device would behave drasti- cally different due to its large initial deflection. The RR3RC5 device also showed a higher sensitivity and lower resonant frequency than the 60 kHz devices. The RR3RC5 device did
159 not have a large initial deflection. However, the RR3RC5 device was closest to the RR5RC5 device on the wafer. Thus, the RR3RC5 device may have been located on the edge of a localized stress gradient on the wafer where the stress level is not yet high enough to create a large initial deflection. Displacement sensitivity cross-sections at resonance are shown in Figure 7-10(a) with error bars indicating 95% confidence. Also, the three-dimensional resonant mode shapes are contained in Figure 7-11. In Figure 7-10(b), the displacement sensitivity cross-sections are normalized by their peak value to give resonant mode shape cross-sections. The mode shapes are similar indicating that the modal mass of the diaphragms are also similar. This indicates that the shift in resonant frequency between devices was due to a change in stiffness between devices. This was an expected result since, as previously noted in Section 7.1, the stress was known to vary across the wafer. Shifts in stress between devices with the same geometry will create a shift in stiffness resulting in variations in sensitivity and resonant frequency.
0.5 RR5RC5 1 RR1RC3 0.4 RR2RC3 0.8 RR2RC4 m)/V)
µ 0.3 RR3RC5 0.6 RR5RC2 0.2 Deflection 0.4 Sensitivity
Sensitivity (( 0.1
Normalized Deflection 0.2
0 −500 0 500 0 −500 0 500 µm µm (a) Resonant displacement sensitivity cross-sections (b) Mode shape cross-sections. with 95% confidence intervals.
Figure 7-10. Displacement sensitivity cross-sections.
7.4.4 Linearity
The linearity of the diaphragm response at resonance was measured. A single excitation tone near the resonant frequency was applied to each diaphragm. The voltage amplitude was varied in steps of 50 mV or 100 mV. The diaphragm velocity was measured at a single
160 (a) RR5RC5 (d) RR2RC4
(b) RR1RC3 (e) RR3RC5
(c) RR2RC3 (f) RR5RC2
Figure 7-11. Resonant mode shapes.
161 120 RR5RC5 100 RR1RC3 RR2RC3 80 RR2RC4 RR3RC5 60 RR5RC2 40
20 Velocity Sensitivity ((mm/s)/V) 0 0 1 2 3 4 Voltage (V)
Figure 7-12. The velocity sensitivity versus excitation voltage. point at the diaphragm center using the laser vibrometer. The onset of nonlinearity was determined by monitoring the velocity sensitivity. A plot of the velocity sensitivity as a function of increasing voltage excitation is shown in Figure 7-12. The onset of non-linearity for each device is marked by the sudden drop in sensitivity. The maximum linear range of the device can also be witnessed through measurements of the displacement and velocity as functions of increasing excitation voltage. As shown in
Figure 7-13, the onset of nonlinearity marks a sudden decrease in the overall velocity and displacement. It is interesting to note that the displacement at the onset of nonlinearity is different for all devices as shown in Figure 7-13(a). This indicates that the onset of nonlin- earity is not a geometric affect as is normally considered in the nonlinear static deflection of plates. In static deflection, the onset of nonlinearity normally occurs when the displace- ment becomes a certain percentage of the diaphragm thickness. In contrast, Figure 7-13(b) shows that the velocity amplitudes at the onset of nonlinearity are nearly the same for all devices. This indicates that the limit of the linear range is likely due to a velocity threshold phenomenon.
Although the study of nonlinearities in actuators is an important subject matter, it is beyond the scope of this work. Here, it is sufficient to require that the excitation was kept
162 RR5RC5 800 150 RR1RC3 RR2RC3 600 RR2RC4 RR3RC5 100 RR5RC2 400
50 200 Center Velocity (mm/s) Center Displacement (nm)
0 0 0 2 4 0 2 4 Voltage (V) Voltage (V) (a) Displacement (b) Velocity
Figure 7-13. Velocity and displacement response versus resonant tone excitation amplitude. small enough in all other tests that the overall velocity response was below the linear velocity threshold indicated in Figure 7-13(b).
7.4.5 Variable Back Cavity
As was explained in Chapter 4, the devices were packaged with a variable back cavity (with the exception of RR5RC2). Testing was conducted at varying cavity depths to inves- tigate the effect of the back cavity on device performance. Qualitative comparisons were made with theoretical predictions.
7.4.5.1 Setup
The variable back cavity experiment was conducted using single point frequency re- sponse measurements and the pseudo-random excitation signal. For each measurement, the back cavity was adjusted by a quarter turn of the screw that marks the termination of back cavity. This involved the removal of the packaged device from under the microscope, adjust- ment of the back cavity, placement of the device back under the microscope, refocusing the microscope on the devices, and realignment of the laser with the center of the device. The uncertainties associated with the experimental process are discussed and analyzed further in Appendix C.
163 7.4.5.2 Results
The results of the back cavity tuning in terms of resonant frequency and resonant
frequency response are given in Figure 7-14(a) through Figure 7-14(f). The experimental results show that there is an optimal back cavity depth for performance. For instance, the center velocity sensitivity at resonance of the RR5RC5 device increases by a factor of 2.3
when the screw depth is tuned. Note that in Figure 7-14(c), the negative back cavity depth means that the screw was position up into the pcb board, decreasing the overall back cavity depth. Similar increases in the center velocity after tuning are seen for all devices. If the
cavity is extend beyond the depth for maximum performance, the performance decreases marking an increase in the back cavity impedance. If the depth is extend much beyond the optimal range, as in Figure 7-14(b), the performance reaches a minimum. At this point, the back cavity impedance is a maximum.
This is similar to the ideal case of a vibrating diaphragm at the end of a perfectly rigid back cavity as shown in Figure 7-15. The impedance of the diaphragm and back cavity are given by
1 ZAD = jωMAD + (7–12) jωCAD Z and Z = −j 0 cot (kd), respectively. (7–13) AC πa2
The vibration will reach a maximum when the back cavity and diaphragm have the same resonant frequency. This case is marked by the red “+” in Figure 7-16 where the imaginary part of the back cavity and diaphragm impedance both go to zero. This occurs when the depth is one-quarter the wavelength of the resonant frequency of the diaphragm. For the ideal case that lacks dissipation, the back cavity provides zero impedance at the diaphragm’s resonant frequency. If the depth is extended beyond the quarter wavelength, the impedance of the back cavity increases corresponding to a decrease in the performance and overall reso-
nant frequency. If the back cavity depth is extended to a half-wavelength of the diaphragm’s
resonant frequency, an ideal back cavity acts as an infinite impedance. Beyond, the half
164 32 44 30 42 (kHz) (kHz) res res f f 28 40 26 100 200
80 150
60 )| ((mm/s)/V) )| ((mm/s)/V) )| 100 r=0 r=0 ( (
U 40 U | | 50 1 1.5 2 2.5 3 3.5 1 2 3 4 Depth (mm) Depth (mm)
(a) RR3RC5 (c) RR5RC5
65 62 60 60 58
(kHz) (kHz) 56 res res f 55 f 54 52 50 80 60
50 60 40 )| ((mm/s)/V) )| ((mm/s)/V) )| 40 30 r=0 r=0 ( ( U U | | 20 20 1 2 3 4 1 2 3 Depth (mm) Depth (mm)
(b) RR1RC3 (d) RR2RC4
Figure 7-14. Measurements of the back cavity.
165 66 40 64 38 62 (kHz) (kHz) res res f f 60 36
58 34 60 600
500 40 400 )| ((mm/s)/V) )| 20 ((mm/s)/V) )|
r=0 r=0 300 ( ( U U | | 0 200 1 1.5 2 2.5 3 2 3 4 Depth (mm) Depth (mm)
(e) RR2RC3 (f) Theory
Figure 7-14. Measurements of the back cavity. wavelength, the back cavity impedance becomes periodic. Thus, there is a discontinuity in the resonant frequency as the cavity impedance acts as if it is less than a quarter wavelength. The diaphragm vibration will again reach a maximum, this time at three-quarters the wave- length. As the cavity depth continues to decrease, the behavior described will repeat.
Z AD a Z AC
d
Figure 7-15. Ideal piston and back cavity.
The equivalent circuit of Chapter 5 is used to model the effect of the back cavity for comparison with the experimental results. The result of the theory in Figure 7-14(f) shows a matching trend to the experimental results. As the screw depth is increased, the resonant response increases slightly. If the resonant frequency of the diaphragm was lower, the back cavity would provide even more stiffening to the device performance at small cavity depths and there would be a more pronounced increased in resonant response. This type of trend
166 1.5
Z 1 AD ) 0 Z (d/λ=0.15) AC 0.5 Z (d/λ=0.2) AC Z (d/λ=0.25) AC 0 λ ↑ d/ Z (d/λ=0.3)
Imaginary(Z/Z AC −0.5 Z (d/λ=0.35) AC
−1 0 1 2 3 f/f res Figure 7-16. Imaginary parts of the diaphragm impedance and cavity impedances of varying back cavity depth. is viewed in Figure 7-14(c) where the resonant frequencies of device RR5RC5 are closer to
30 kHz with a quarter wavelength near 2.9 mm as compared to the theoretical resonance prediction which is closer to 40 kHz and a quarter wavelength near 2.2 mm.
The theoretical result based on the equivalent circuit in Figure 7-14(f) also shows a decrease in the resonant frequency with an increase in screw depth. There is a discontinuity in the resonant frequency and resonant response. At this point, an anti-resonance of the back cavity moves through the frequency response. The frequency of the anti-resonance corresponds closely to the half wavelength equalling the cavity depth which is similar to the ideal back cavity discussed above. 7.4.6 Vacuum Experiments
As shown previously, the three energy domains must be accounted for in all measure- ments. For example, the effect of the acoustic and mechanical energy domains cannot be ignored when measuring the electrical impedance. The interaction of the acoustical energy domain is purely a result of the environment in which the device is placed. By placing the device in a vacuum, the effects of the acoustic elements in the equivalent circuit are elimi- nated isolating the electrical and mechanical energy domains. Thus, a direct comparison of
167 the mechanical and electrical domain between the equivalent circuit and experiments can be made.
7.4.6.1 Setup
A vacuum chamber was fabricated to facilitate these experiments. The vacuum chamber in Figure 7-17 is a 150 by 150 by 40 mm3 rectangular steel box capable of achieving a rough vacuum of around 100 mTorr. It is equipped with seven 1-1/3” CF flanged ports that support four single-ended BNC feedthroughs, a vacuum port, a pressure detection port, and a nitrogen supply port. A JB Industries’ Platinum Series LAV-3 vacuum pump is attached to the vacuum port. The pressure in the chamber is monitored by a KJLC Convectron Equivalent Gauge, rated to 0.1 mTorr. LV measurements are conducted through an optical port fitted with a Thor Labs BK7 broadband window that is 25.4 mm diameter and 5 mm thick. The port provides a 10 mm diameter viewing area. The focal plane using a 5x objective can penetrate up to 10.5 mm below the inner wall of the vacuum chamber.
Vacuum Line Optical Nitrogen Viewport Purge Line
Pressure Gauge BNC Feedthrough (x4)
Figure 7-17. Vacuum chamber diagram.
Single point LV measurements at the center of the diaphragm were made through the viewing window of the vacuum chamber. The measurements were made at different levels of vacuum to make sure the performance had asymptoted to the point where acoustic effects had been eliminated. A leak in the vacuum chamber prevented the chamber from achieving a vacuum level lower than 100 mTorr. Also, the leak prevented phase locked diaphragm
168 scans. Note that a shim was placed between the pcb and aluminum block to ensure that the back cavity was exposed to vacuum.
7.4.6.2 Results
The deflection sensitivities at the center point are given in Figure 7-18(a) through Fig- ure 7-18 for both standard temperature and pressure (STP) and rough vacuum conditions. The devices show a significant decrease in damping ratio and subsequent increase in the quality factor as shown in Table 7-4. The damping ratio, ζ, is calculated by fitting a second order system transfer function to the deflection sensitivity. Details of the fit are contained in Appendix C. The quality factor is calculated as [51]
1 QF = p . (7–14) 2ζ 1 − ζ2
Note that the damping coefficient decreased by at least an order of magnitude for all of the devices tested. This indicates that the dominant damping mechanisms for the device are due to fluid and acoustic interaction. A comparison of the experimental results with the theoretical prediction based on the equivalent circuit model shows an under-prediction of the damping ratio at STP. As covered in Chapter 5, the full equivalent model at STP accounts for damping from the diaphragm, back cavity, radiation, and vent impedances. At vacuum, the dominant damping mechanism from the theory is the damping ratio of the diaphragm, which is assumed to be 0.001 based on the results of RR2RC4 which has a similar resonant frequency predicted by the theory.
Table 7-4. Damping and quality factor values for the different devices at vacuum and STP.
fres(kHz) fres(kHz) γ γ QF QF (STP) (vacuum) (STP) (vacuum) (STP) (vacuum) RR5RC5 30.47 29.02 0.021 0.0022 24 224 RR1RC3 57.77 60.33 0.014 0.0003 35 1945 RR2RC3 59.48 61.22 0.015 0.0002 34 2001 RR2RC4 56.67 61.20 0.018 0.0010 27 485 RR3RC5 43.73 44.88 0.014 0.0004 36 1397 Equivalent Circuit 38.80 38.79 0.008 0.0010 63 499
169 1 1 10 10 Vacuum Vacuum 0 STP STP 10 100 /V) m/V) µ
µ −1 | (
| ( 10 −1
W/V 10 |
W/V −2 | 10
−3 −2 10 10 180 180 ) ) ° ° ) ( ) ( 90 90 W/V W/V ( ( ∠ ∠
0 0 20 40 60 80 10 20 30 40 50 Frequency (kHz) Frequency (kHz) (a) RR3RC5 (c) RR5RC5
1 1 10 Vacuum 10 Vacuum 0 STP 0 STP 10 10 /V) /V) µ −1 µ −1 | ( 10 | ( 10 W/V W/V | 10−2 | 10−2
−3 −3 10 10 180 180 ) ) ° ° ) ( ) ( 90 90 W/V W/V ( ( ∠ ∠
0 0 20 40 60 80 100 20 40 60 80 100 Frequency (kHz) Frequency (kHz) (b) RR1RC3 (d) RR2RC4
Figure 7-18. Device comparison of the frequency response function between rough vacuum and STP conditions.
170 1 1 10 Vacuum 10 Vacuum 0 STP STP 10 100 /V) m/V) µ
−1 µ
| ( −1
10 | ( 10 W/V |
−2 W/V −2 10 | 10
−3 −3 10 10 180 180 ) ) ° ° ) ( ) ( 90 90 W/V W/V ( ( ∠ ∠
0 0 20 40 60 80 100 10 30 50 70 Frequency (kHz) Frequency (kHz) (e) RR2RC3 (f) Equivalent Circuit
Figure 7-18. Device comparison of the frequency response function between rough vacuum and STP conditions.
Measurements were made at different levels of vacuum to test whether the fluid and/or acoustic effects had been completely mitigated. The damping ratio versus pressure is plotted
in Figure 7-19. As shown in the figure, complete convergence of the damping ratio was not obtainable due to limitations of the vacuum chamber. The drastic drop in damping from the STP conditions, however, is a good indication that the acoustic/fluidic effects have been
effectively if not completely isolated. 7.5 Electroacoustic Characterization
Measurements of the ultrasound production of each device were made to compare with the extrapolation of the vibrometer measurements via Rayleigh’s integral.
7.5.1 Setup
Ultrasound measurements were made with a 1/4” B&K 4939 free field microphone. The microphone was mounted on an optical table directly in front of a sheet of Sonex foam to min- imize reflections from behind the microphone as shown in Figure 7-20(b). The microphone mount is attached to the optical table using a micropositioner that is used for alignment.
171 10−2
RR5RC5 RR1RC3 −3
ζ 10 RR2RC3 RR2RC4 RR3RC5
10−4 200 400 600 800 Vacuum Pressure (mTorr) Figure 7-19. Resonant performance reference to STP versus increase vacuum chamber pres- sure.
The ultrasonic radiator was mounted to a baffle constructed using a Spectrum Z415 rapid prototyping machine as shown in Figure 7-20(a). The baffle is 200 mm in diameter. The baffle is mounted directly to a manual radial traverse. The baffle and radial traverse are then mounted to a 3 axis micropositioner attached to a uni-axis motorized traverse controlled by a Velmex VXM Stepping Motor Controller and Vexta Model PK264-03A-P1. A cross-haired laser alignment tool and the microphone and radiator micropositioners were used to align the microphone with the ultrasonic radiator.
Acoustic Foam
Radiator Microphone
Optical Table Baffle
Radial Traverse Traverse
Micropositioner Micropositioner
(a) Baffled ultrasonic radiator. (b) Microphone mount.
Figure 7-20. Setup for the electroacoustic characterization.
172 The motorized traverse was used to step the ultrasonic radiator away from the mi- crophone to measure the on axis response of the radiator. The traverse calibration was 6.4 µm/step. Note that the radiator and microphone were initially offset by a distance of 7.6 mm for each measurement. At a fixed distance from the microphone, the ultrasonic radiator was swept through a series of angles using the radial traverse to determine the directivity pat- tern. At each angle, the microphone measured the SPL. The radial traverse was stepped from −60◦ to +60◦ in steps of 4◦. The radial traverse had a minimum precision of 2◦. The estimated bias error of the zero degree reference is ±2◦. The estimated random error in each degree measurement is ±1◦.
The excitation signal was supplied by an Agilent 33120A function generator. The B&K 4939 free field microphone was powered by a B&K 2804 battery microphone power supply.
The microphone signal was amplified by a Stanford Research Systems SR560 low-noise am- plifier. The level of amplification varied depending on the signal level. A bandpass filter was applied by the amplifier with a cut-on frequency of 10 kHz and a cut-off frequency of 100 kHz. The roll-off of the amplifier is -6 dB/decade. The excitation and amplified microphone signals were measured using a National Instruments PXI-522 100 MHz, 100 MS/s, 14-Bit digitizer mounted in a National Instruments PXI-1045 General-Purpose 18-Slot Chassis for PXI. The signals were sampled at 500 kS/s and triggered off the sync signal from the Agilent function generator. One-hundred records of 5,000 samples were taken for each measurement. Com- plex averaging of the fourier transform of each record was performed. LabVIEW was used to control the data acquisition and motorized on-axis traverse.
The B&K 4939 free field microphone was calibrated using a B&K 4228 pistonphone. For the calibration, the amplification of the low-noise amplifier was set to 1 and the filter was set to high-pass and dc. The pistonphone emits a nominal value of 124 dB at 250 kHz. The microphone signal was sampled at 10 kS/s and 800 samples were taken per record length.
The Fourier transforms of 1,000 records were averaged. The result of the calibration is a
5.72 mV/Pa sensitivity.
173 7.5.2 Results
The experimental results are compared to the theoretical extrapolation of the vibrometer
measurements via Rayleigh’s integral (see Section 3.2.1 for more information on Rayleigh’s integral). The on axis response is found from Z H˙ (x0, y0; f) e−(jk+α)R P (x = 0, y = 0, z; f) = ρ0 dS, (7–15) S 2πR
where x0 and y0 refer to coordinates on the transducer face, H˙ is the acceleration sensitivity p measured by the vibrometer at each point x0 and y0 and frequency f, R = x02 + y02 + z2, α is the acoustic attenuation coefficient in air at frequency f (see Section 3.2.3 for more information on acoustic attenuation), and z is the distance from the transducer to the mi- crophone. Numerical integration over the grid of scan points was used to evaluate the double
integral in Equation 7–16. The results of the on axis measurements are compared to the computation in Figure 7-22. The directivity response was found using Rayleigh’s integral by considering the mea-
surement point to be at a fixed distance r and an angle θ with respect to the axis of the transducer. For these calculations, the directivity was found in the x − z plane. First, the pressure response at a fixed distance r for angles θ was found, Z H˙ (x0, y0; f) e−(jk+α)R P (r, θ; f) = ρ0 dS, (7–16) S 2πR q where now R = (x0 − r sin θ)2 + y02 + (r cos θ)2. The directivity, D (θ), was normalized by its θ = 0 response P (r, θ) D (θ) = . (7–17) P (r, θ = 0) The directivity measurement results and computations based on the projection of LV data
using Rayleigh’s integral are shown in Figure 7-21. Note that the results are reported in dB scale. The projection of the LV data using Rayleigh’s integral is fairly omnidirectional. This
is to be expected since the ka of the devices range from approximately 0.4 at 30 kHz to 0.8
at 60 kHz. At these values of ka, the radiators are fairly compact. In addition, the deflection
174
90 10 90 10 120 60 120 60 0 0
150 −10 30 150 −10 30 −20 −20
180 0 180 0
210 330 210 330
240 300 240 300 Experiment 270 f = 30 kHz Experiment 270 f = 56 kHz LV Projection LV Projection
(a) RR5RC5 (d) RR2RC4
90 10 90 10 120 60 120 60 0 0
150 −10 30 150 −10 30 −20 −20
180 0 180 0
210 330 210 330
240 300 240 300 Experiment 270 f = 57 kHz Experiment 270 f = 43 kHz LV Projection LV Projection
(b) RR1RC3 (e) RR3RC5
90 10 90 10 120 60 120 60 0 0
150 −10 30 150 −10 30 −20 −20
180 0 180 0
210 330 210 330
240 300 240 300 Experiment 270 f = 59 kHz Experiment 270 f = 56 kHz LV Projection LV Projection
(c) RR2RC3 (f) RR5RC2
Figure 7-21. Directivity measurements along with the predicted directivities found by ex- trapolating the LV measurements using Rayleigh’s integral.
175 mode shapes are a function of radius that go to zero at the edge of the diaphragm which causes suppression of sidelobes [8]. Although the directivity of the devices is omnidirectional, the sound field of an array of the devices will take on the array directivity pattern (see
Section 3.2.2 for more information on array directivity patterns). The experimental results showed a qualitative match to the projection of the LV data. The experimental results show some variation, but all the directivity patterns are well within ±6 dB. Sources of errors and non-uniformities in the measurement come from the finiteness of the baffle, a lack of ideal free-space surrounding the microphone, and misalignment of the transducer to the microphone. As seen in Figure 7-22, the 1/r roll-off in the farfield predicted by Rayleigh’s integral is well captured by the experimental results. The magnitude of the response, however, is not well matched for most of the devices. The 95% confidence random errors are so small that they could not be represented in the figures. Single point frequency response measurements before and after the on-axis experiment showed drift in the device response. For example, the experimentally measured on-axis pressure response in Figure 7-22(a) has a positive offset from the extrapolated results using Rayleigh’s integral. The center velocity sensitivity at 27 kHz, however, was shown to drift from 26.2 mm/s / V to 30.5 mm/s / V over the course of the measurement. This lead to discrepancies between the projected and measured ultrasound field.
7.6 Performance as a Parametric Array
The volume velocity calculations from the laser vibrometer measurements in Section 7.4.2
were used to simulate an equivalent source of an array of transducers for a parametric ar- ray. The nonlinear acoustic calculations outlined in Sections 5.2 and 5.1.6 are repeated for arrays of the DUTs. Consider an array diameter of 154 mm (6.1 in) in diameter. Forty-five
hundred transducers fit within the array diameter with a center to center spacing of 2.2 mm.
The transducers are driven at 80% of their maximum center velocity found in Section 7.4.4
and are assumed to vibrate in phase such that their volume velocities are perfectly summed.
176 10−1 10−1 f = 27 kHz f = 54 kHz
10−2 10−2
Pressure Sensitivity (Pa/V) 10−3 Pressure Sensitivity (Pa/V) 102 102 Distance (mm) Distance (mm) (a) RR5RC5 (d) RR2RC4
−1 10−1 10 f = 55 kHz f = 39 kHz
10−2 10−2 Pressure Sensitivity (Pa/V) Pressure Sensitivity (Pa/V) 102 102 Distance (mm) Distance (mm) (b) RR1RC3 (e) RR3RC5
10−1 f = 54 kHz f = 50 kHz
10−2 10−2 Pressure Sensitivity (Pa/V) Pressure Sensitivity (Pa/V) 102 102 Distance (mm) Distance (mm) (c) RR2RC3 (f) RR5RC2
Figure 7-22. On axis pressure response.
177 Given the parametric array analysis based on Berktay’s solution [25] outlined in Section 5.2, the audible frequency response at 1 m is given in Figure 7-23 for each transducer. The maximum output is based on an array of the RR5RC5 transducers, with a 41.9 dB signal at 5 kHz and 1 m. Note that the calculation based on the non-buckled RR5RC2 device was not far behind with a 40.6 dB tone at 5 kHz and 1 m. Lowering the distance requirement to 10 cm at 5 kHz results in a 61.9 dB signal for the RR5RC5 array and 60.6 dB for the RR5RC2 device.
60
Pa) RR1RC3 µ 40 RR2RC3 RR2RC4 RR3RC5 20 RR5RC2 RR5RC5 SPL (dB re 20 0 0.5 1 2 5 10 20 Difference Frequency (kHz)
Figure 7-23. Parametric array output at 1 m of a an array of 4,500 radiators.
7.7 Conclusion
The performance of the DUTs are summarized in Table 7-5. The realized layer stresses did not match the targets from Chapter 6. Also, the large uncertainty of the stress measure- ments plus the non-uniformity across the wafer led to poor theoretical match between the experimental and theoretical results. The deflection sensitivities and resonant frequencies are very comparable to the devices summarized in Chapter 3. It was shown that the deflection sensitivity could be improved by back cavity tuning. The center deflection sensitivity of 961.7 µm/V of the RR5RC5 device is remarkable. This device, however, was buckled and it is unlikely to be reproduced with a good degree of confidence. The performance of the devices are also hampered by the early onset of non-linearity at relatively low excitation voltages.
178 Table 7-5. Performance of the die. The center deflection after back cavity tuning is shown in parentheses.
3 fres(kHz) S (mm /s / V) W/V (nm / V) ± 95 % confidence +0.02 RR5RC5 30.47 −0.09 38.589 ± 0.008 442.0 ± 0.2 ( 961.7 ) +0.05 RR1RC3 57.77 −0.05 37.512 ± 0.008 181.0 ± 0.3 ( 211.6 ) +0.06 RR2RC3 59.48 −0.09 27.438 ± 0.009 113.3 ± 0.5 ( 147.4 ) +0.06 RR2RC4 56.67 −0.05 27.588 ± 0.009 144.8 ± 1.0 ( 150.4 ) +0.02 RR3RC5 43.73 −0.03 43.386 ± 0.008 285.3 ± 0.5 ( 359.7 ) +0.03 RR5RC2 57.00 −0.05 46.451 ± 0.012 224.5 ± 0.5
The nonlinear acoustic calculation shows a low amplitude at audible tones. The pro- jected parametric array output of 42 dB of a 5 kHz tone at 1 m is comparable to the first realization of a parametric array in air by Bennett et al. [10] which obtained a 50 dB, 5 kHz tone at 0.3 m (SL = 40 dB). However, an SPL of 40 dB is about the level of a whisper to the human ear [9]. A nominal array diameter of 154 mm was used in the calculations. The array diameter could be increased to match some of the examples from Chapter 2 and increase the overall sound output. This would lead to an increase in the number of devices beyond the 4,500 used in the calculation and may push the limits of practicality. In addition, it is problematic that this calculation is most likely a generous estimate given the lack of phase matching of the realized devices of similar resonant frequencies. Mismatched phase leads to an overall decrease in the total volume velocity output of the array and thus a decrease in parametric conversion. Given the results, it is unlikely that a practical parametric array can be realized with this device design. In Chapter 8, a summary is provided along with recommendations for future work.
179 CHAPTER 8 CONCLUSION AND FUTURE WORK
This chapter presents a summary of research goals, objectives, and key results. Recom- mendations for future work and designs are also presented. 8.1 Conclusions
Parametric arrays are a promising technology for the control of sound distribution. Possible benefits include lower noise pollution levels and enhanced theater systems. Much of the research on parametric arrays has been focused on signal processing. The development of ultrasonic transducers for parametric array sources has only recently begun to garner attention. The requirements of parametric array transducers are to provide a large sound output at ultrasonic frequencies and be phase matched at the excitation frequencies.
At the beginning of this work, there was no evidence of the application of MEMS technology to parametric arrays. In 2006, however, Haksue et al. [44, 45] showed results for an inaudible parametric array of pMUTs. The difference tone was 40 kHz and thus not appropriate for conventional audio applications of parametric arrays. Another example of a MEMS-based parametric array that arose during the course of this study was a cMUT transducer array presented by Wygant et al. in 2007 [15]. They measured 58 dB of a difference frequency of 5 kHz at 3 m from the array. This required an excitation signal composed of a 200 V peak-to-peak AC component in addition to a 350-380 V bias voltage. The high voltage requirements degrade the appeal of this technology. Neither study utilized formal optimization methods when designing the transducers. In contrast to previous parametric array works, the aim of this study was to design an ultrasonic transducer for a MEMS-based parametric array by developing a comprehensive systems-level model of an AlN-based ultrasonic resonator and optimization tools for paramet- ric array transducers. The device model used LEM to form the electromechanical coupling and diaphragm model. Included in the model were packaging effects such as a variable back cavity whose equivalent impedance was formed using a set of transfer matrices. In terms of
180 optimization tool development for parametric array transducers, a unique objective function was formed using an analytical nonlinear acoustic solution for the demodulation of sound beams. Transducer designs developed using these tools were fabricated and characterized.
The device characterization showed a quantitative difference between the prediction of the equivalent circuit model and the measured device performance. The deviation was likely the result of non-uniform film stress across the wafer and large uncertainties in the mean stress measurements. The stress measurements indicated an average effective tensile stress across the wafer with estimated uncertainties of greater than 30%. By visual inspection, however, the diaphragms along the outer portion of the wafer appeared to have large initial deflections indicating an overall compressive stress. Along the inner portion of the wafer, however, the diaphragms appeared flat. The implication was that there were large stress gradients across the wafer. This was confirmed by the large variation in resonant frequencies between devices ranging from approximately 30 to 60 kHz. The stress gradients along with the large film stress uncertainties led to quantitative discrepancies between the theoretical prediction of the equivalent circuit and the experimental results.
The device characterization showed a qualitative experimental verification of the systems model of the back cavity. As predicted using the comprehensive systems model, improvement in device response by back cavity tuning was experimentally confirmed. The overall trend of resonant frequency versus back cavity depth was also qualitatively matched. This gave credence to the mantra of comprehensive design that includes both device and package.
Another notable result of the experimental characterization was the decrease in the damping ratio of an order of magnitude when the devices where placed in a rough vacuum environment. This provided evidence that the fluid environment was the source of the dominant damping mechanism.
The performance of the AlN radiators in this work are compared to previously reported air-coupled MEMS radiators for vibration measurements in Table 8-1. The AlN radiators show comparable performance in terms of volume velocity and resonant frequency. For
181 Table 8-1. Comparison of air-coupled MEMS radiator performances. Author Type Diaphragm Volume Velocity Resonant Area ¶ Sensitivity Frequency (mm3/s / V) (kHz) RR5RC5 AlN φ1.46 mm 39 30 RR1RC3 AlN φ1.46 mm 38 58 RR2RC3 AlN φ1.46 mm 27 59 RR2RC4 AlN φ1.46 mm 28 57 RR3RC5 AlN φ1.46 mm 43 44 RR5RC2 AlN φ1.46 mm 46 57 Zhu et al. 2004 [124] PZT 0.6 by 0.6 mm2 101† 67 Horowitz et al. 2006 [46] PZT φ2.4 mm 23* 34 Chandrasekaran et al. 2002 [148] Thermoelastic φ1 mm 3 mm3/s * ‡ 55 Brand et al. 1997 [138] Thermoelastic 1 by 1 mm2 45 mm3/s † § 50 * Calculated based on the fundamental mode of a circular plate [154]. † Calculated based on the fundamental mode of a square plate [154]. ‡ Volume velocity amplitude given 7 V ac, 7 V dc excitation. § Volume velocity amplitude given 2.8 V ac, 4 V dc excitation. ¶ φ = diameter . example, the AlN RR5RC2 device has a volume velocity of 46 mm3/s / V at a resonant frequency of 57 kHz. In contrast, the PZT device presented by Zhu et al. [128] gives a volume velocity of 101 mm3/s / V at a resonant frequency of 67 kHz. The AlN device is slightly more than 6 dB down in performance, even though the d31 constant of PZT is almost two orders of magnitude larger than that of AlN (see Table 3-2). Although the projected performance of these devices as a conventional, audio parametric array do not show good promise, the device comparison shows good overall performance of the AlN ultrasonic radiators in comparison to the current literature. Future work can focus on more robust designs and packaging improvements for other ultrasonic applications.
8.2 Recommendations for Future Work
There are several areas in which this work could be further developed. The most notable effects that lead to disagreement between the device model and characterization were the large uncertainties associated with the layer stresses. The most direct path for improvement in this area is to mitigate device sensitivity to the layer stresses in the FBAR process. One method used to minimize stress sensitivity is to add a structural layer that is stress neutral
182 or of a known and consistent stress. The thickness of the extra structural layer would have to be such that the sensitivity of averaged stress through the diaphragm thickness would be low. One design path of such a transducer would be design optimization with uncertainties.
A stochastic design optimization will sacrifice device performance to arrive at a robust design that can be consistently fabricated given the uncertainties in the fabrication process.
Another improvement that could lead to better designs would be a theoretical model that could predict the onset of dynamic nonlinearity. The current nonlinear plate model is static and does not incorporate the influence of the package. The experimental results showed that the onset of nonlinearity around resonance was not dictated by the magnitude of the deflection, but instead by the magnitude of the velocity. A maximum excitation voltage for linear operation that is a function of device geometry could be incorporated in the objective function. Thus, the objective function would give the best possible linear performance of the device.
Beyond improvements in the fabrication and modeling of the device, another area of improvement is the testing of devices in vacuum. Vacuum testing mitigates the effects of air coupling, allowing a direct comparison between experiment and the electromechanical model. The experimental results showed that the damping ratio had not converged to a constant value at the lowest vacuum levels achievable with the current setup. A reduction in vacuum level to the point of damping ratio convergence would give better confidence that all fluid/acoustics effects had been eliminated. Also, a leak in the chamber prevented scans of the diaphragm while under a constant level of vacuum. Elimination of the leak would allow scans of the diaphragm and more in depth studies of the electromechanics.
8.3 Recommendations for Future Design
The characterization of the devices has lead to the conclusion that these devices should be abandoned for audible parametric arrays. The transducer numbers required are compa- rable to some of the large high frequency cMUT arrays. Lower ultrasonic frequency devices
183 such as the ones presented in this study are much larger and require too much wafer to- pography, making the application to parametric arrays too expensive. This technology may, however, find an application in ultrasonic difference frequency parametric arrays similar to that of Haksue et al. [44,45] after a redesign for higher frequencies. In Section 8.2, the concept of decreasing device sensitivity to stress variations was pro- posed. This could be accomplished by eliminating the AlN scaffolding layer and performing the standard FBAR deposition on a silicon-on-insulator (SOI) wafer. The diaphragm release would then proceed exactly the same as the wafer release step used to form the devices in this dissertation with DRIE from the backside followed by an oxide etch to release the diaphragm. An example of a device that utilizes SOI technology is given in Figure 8-2.
The current transducer is actuated by inducing an in-plane stress in the annular region of the diaphragm. There is no actuation at the center of the diaphragm. An alternate piezoelectric transducer design similar to that of [135] is proposed that utilizes actuation at both the diaphragm edge and center. To motivate the design of such a transducer, the stress
field induced in a clamped plate when it deflects is shown in Figure 8-1. As you can see, the stress field at the top of the plate is 180◦’s out of phase when comparing the center and edge of the plate. If the opposite phase stress could be induced at the center of the plate in addition to in the annular region, then additional performance could be realized.
VV-V Tension Tension Compression Device silicon Bulk silicon BOX Figure 8-1. Stress fields inside a deflected clamped plate. Figure 8-2. Alternate design for improved performance.
The alternate design would have two top electrodes, one outer annular electrode and one inner circular electrode. The bottom electrode should be continuous across the bottom of the diaphragm. The bottom electrode should be grounded. The center and annular electrodes would be excited with signals of opposite phase to insure that the piezoelectric layer is tensile
184 in the center region when it is compressive in the annular region, and vice versa. This would provide enhanced transducer performance and amplify the actuator response.
185 APPENDIX A NONLINEAR ACOUSTIC MODELING
A.1 Westervelt Parametric Array Solution
The inviscid form of Equation 2–17 is solved for a piston radiating at two adjacent tones for the difference frequency produced by their interactions [20]. First, axisymmetric waves
2 2 2 −1 are assumed such that ∇⊥ = ∂ /∂r + r ∂/∂r. The pressure is assumed to be composed of two parts:
p = p1 + p2, (A–1) where p1 is the solution to the KZK equation minus the nonlinear term and p2 is a small correction to p1 at the first harmonic and any sum and difference frequencies generated by p1. Time harmonic solutions are assumed,
1 £ ¤ p (r, z, τ) = q (r, z) ejωaτ + q (r, z) ejωbτ + c.c., (A–2) 1 2j 1a 1b and
1 £ ¤ p (r, z, τ) = q (r, z) ej2ωaτ + q (r, z) ej2ωbτ + q (r, z) ejω+τ + q (r, z) ejω−τ + c.c., 2 2j 2a 2b + − (A–3) where the subscripts + and - refer to tones formed by the summation and subtraction of the originating tones, respectively. The resulting linear equations for p1 are
∂q j 1a + ∇2 q + α q =0 (A–4) ∂z 2k ⊥ 1a 1a 1a ∂q j and 1b + ∇2 q + α q =0, (A–5) ∂z 2k ⊥ 1b 1b 1b ± 2 3 where α1a,b = δωa,b 2c0. Equations for the harmonic tones and sum and difference frequencies can be written using the solutions to Equation A–4 and Equation A–5 as sources. The output of the parametric array is the difference frequency component so it is the focus here. The equation governing of the difference frequency correction to 1a and 1b is
∂q− j 2 βk− (∗) + ∇⊥q− (r, z) + α−q− (r, z) = − 2 q1a (r, z) q1b (r, z) . (A–6) ∂z 2k− 2ρ0c0
186 The source term on the right hand side of Equation A–6 is the linear or primary tone solution. Thus, the primary waves along the z-axis are seen as virtual sources that ”pump energy resonantly and therefore most efficiently into the difference-frequency sound that propagates in the same direction [18].” Equations A–4 through A–6 are solved using Green’s functions combined with a Hankel transform that removes the r dependence. The Green’s function is " # 2 02 µ ¶ jka,b,−(r +r ) 0 −α (z−z0)− jk k rr a,b,− 2π z−z0 G (r, z|r0, z0) = a,b,− J a,b,− e ( ) , (A–7) 1a,b,− 2π (z − z0) 0 z − z0
where subscripts a, b, and − denote the Green’s functions for Equation A–4, Equation A–5, and Equation A–6, respectively. The solution of Equations A–4 and A–5 is given by Z∞ 0 0 0 0 q1a,b (r, z) = 2π q1a,b (r , 0) G1a,b (r, z|r , 0) r dr , (A–8) 0
0 where q1a,b (r , 0) is the boundary condition at z = 0. The solution for the difference frequency is Zz Z∞ πβk± 0 0 (∗) 0 0 0 0 0 0 0 q± (r, z) = ± 2 q1a (r , z ) q1b (r , z ) G± (r, z|r , z ) r dr dz . (A–9) ρ0c0 0 0 In Westervelt’s solution, the parametric array is assumed be emitting at two neighbor- ∼ ing tones ωa = ωb. It is assumed that absorption is strong enough to limit the nonlinear interaction of the neighboring tones to the acoustic nearfield of the source. This assumptions allows the nonlinear interaction to occur while the waves are still plane and collimated and
do not suffer from geometric spreading. The assumption is based upon α0z0 ≥ 1, where α0
and z0 are based upon the average radiated frequency [18]. The primary fields are assumed to be perfectly collimated such that
−αaz q1a (r, z) = p0aH (a − r) e (A–10)
−αbz and q1b (r, z) = p0bH (a − r) e , (A–11)
where a is the radius of the source.
187 Equations A–10 and A–11 are substituted into Equation A–9 and the following assump- tions are made:
0 0 • The nonlinear interaction is assumed to occur in z < z0 such that z − z can be
0 −1 0 0 replaced by z in the (z − z ) J0 (k rr /z − z ) term in Equation A–7 resulting in
−1 0 (z) J0 (k rr /z). • The upper limit in the integration over z0 in Equation A–9 becomes ∞ since the contributions to the integral are negligible for large z. With the above assumptions and substituting r = z tan θ, the integral becomes [18]
∞ a 2 −α z Z Z jk z2 tan2 θ jk r02 jp p βk e −α z0− − − − 0a 0b − 0 T 2(z−z0) 2(z−z0) 0 0 0 q (θ, z) ' − 2 J0 (k r tan θ) e r dr dz , (A–12) 2ρ0c0 z 0 0
where αT = αa + αb − α is the combined attenuation coefficient. The effective or absorption length of the parametric array is defined by the combined attenuation coefficient as
1 La = . (A–13) αT
This length establishes the length of the region of nonlinear interaction of the parametric array. More assumptions have to be made to arrive at an analytic solution.
• Integration in r0 is restricted to r0 < a such that the r02 term in the exponential are
1 2 negligible with respect to z − La À 2 k a . • The term containing tan2 θ in the exponential is expanded in a geometric series for
0 1 0 2 small z /z resulting in − 2 jk (z + z ) tan θ. The result of integration is
2 2 −α z jp0ap0bβk a e 1 2 − 2 jk z tan θ q (θ, z) ' − 2 Dw (θ) DA (θ) e , (A–14) 4ρ0c0αT z
where the Westervelt directivity and aperture factor are given by
1 DW (θ) = 2 (A–15) 1 + j (k /2αT ) tan θ
188 and 2J (k a tan θ) D (θ) = 1 , (A–16) A k a tan θ respectively. The main assumptions lead to the following restriction on the validity of Equa- tion A–14:
• Absorption is restricted to the nearfield such that La < z0. • The field points of measurement must be far from the nonlinear interaction region such
that z À La. A.2 Berktay Solution
The Berktay solution is derived from the KZK equation given by Equation 2–24. The
total pressure is assumed to be made up of a primary signal, p1, and a small correction, p2 similar to Equation A–1. The primary signal is assumed to be a collimated beam of sound of the form
−α(τ)z p1 (r, z, τ) ' p0e E (τ) sin (ω0τ + ϕ (τ)) H (a − r) , (A–17)
where α (τ) is dependent on the instantaneous angular frequency, Ω (t) = ω0 + ∂ϕ/∂t. The attenuation is assumed to be large enough to restrict the nonlinear generation to the nearfield.
The equation that governs the small correction, p2, is the time integrated KZK equation (note that the thermoviscous dissipation term, δ, has been set to zero),
τ Z 2 ∂p2 c0 2 β ∂p1 − ∇⊥p2dτ = 3 . (A–18) ∂z 2 2ρ0c0 ∂τ −∞
Using Green’s functions, the solution to Equation A–18 is written as
z ∞ 2 Z Z · 02 ¸ 0 β ∂ 2 0 0 r 0 0 dz p2 (0, z, τ) = 4 2 p1 r , z , τ − 0 r dr 0 . (A–19) 2ρ0c0 ∂τ 2c0 (z − z ) z − z 0 0 In the squaring of Equation A–17, the frequency content contains both a low and high frequency component. The high frequency component is ignored since it will be absorbed at
a much faster rate. 1 p2 (r, z, τ) ' p2e−2α(τ)zE2 (τ) H (a − r) . (A–20) 1 2 0
189 The measurement point, z, is assumed to be large enough to make the following as- sumptions before integrating Equation A–19:
r02 • Ignore the phase term 0 2c0(z−z ) dz0 dz0 • z−z0 is replaced by z • The upper limit on the z0 integral is changed to ∞
The result of integration is
2 2 2 µ 2 ¶ βp0a d E (τ) p2 (0, z, τ) ' 4 2 . (A–21) 16ρ0c0z dτ α (τ)
Note that the absorption of the low frequency component is weak as long as modulations are slowly varying. If the primary signal has constant phase, the attenuation coefficient reduces to α (τ) = α0 and Equation A–21 becomes Berktay’s solution [25]
2 2 2 2 βp0a d (E (τ)) p2 (0, z, τ) ' 4 2 . (A–22) 16α0ρ0c0z dτ
Berktay’s solution predicts a 40 dB per decade roll-off as lower frequencies are approached in the square of modulation signal.
190 APPENDIX B PLATE MODEL
A linear composite plate model originally developed by Wang et al. [168] was used
to calculated the lumped mass and compliance of the diaphragm. The derivation of the linear composite plate model is given in this appendix. In addition, a numerical nonlinear composite plate model developed by Williams et al. [169] was used to check the performance of device designs derived from the linear model. The implementation of the nonlinear model
as a constraint in the optimization scheme is covered in Chapter 6. The nonlinear numerical model is not presented here but can be found in the unpublished dissertation proposal of Williams et al. [169] and in future publications..
Piezoelectric (1) (2) Ring Diaphragm 5 Vent 2 1 4 3 2 1 Cavity r Si R1 R Substrate z 2
(a) Device cross section. (b) Plate model cross section.
Figure B-1. Cross sections that show the correspondence between the plate model and the device.
A cartoon cross-section of the device is shown in Figure B-1. The diaphragm is formed by a composite plate and an annular ring that contains the piezoelectric layer. The layers have different material properties and fabrication induced in-plane stresses that can be either
tensile or compressive. The layers are not uniform over the radius of the diaphragm. Two radii are defined as shown in Figure B-1(b). The plate is subject to voltage applied across the piezoelectric layer (3 of the annular section in Figure B-1(b)) and transverse pressure loading. To find an analytical solution, the plate is separated into inner (1) and annular (2) sections. The plate’s transverse and radial deflection can be expressed as functions of
pressure, voltage, in-plane stress, material properties, and geometry.
191 In this section, the governing equations for plate deflection are derived. The basic assumptions as well as the static equilibrium, kinematic, and constitutive equations are presented. These equations are combined to form the governing equations of the plate deflection. B.1 Basic Assumptions
The analytical model assumes an axisymmetric diaphragm that is transversely isotropic in each of its two sections. A transversely isotropic structure has constant properties through planes of the diaphragm. The axisymmetric assumption is justified since the diaphragm will be operated around its fundamental resonance where its axisymmetric mode dominates.
Kirchhoff’s hypothesis is used in the following derivation of the governing equations of the circular plate under voltage and pressure loading [161]. Kirchhoff’s hypothesis states that transverse normals remain straight, are perpendicular to the plate’s mid-plane, and do not elongate after deformation [161]. Kirchhoff’s hypothesis fails when large shearing forces cause deformation of the transverse normals (important in small, thick plates), compression of the plate thickness is non-negligible (i.e. the pressure gradient across the diaphragm thickness is large), and\or deflections are large.
B.2 Static Equilibrium
The static equilibrium equations are derived by considering the forces and moments acting on an infinitesimal piece of an axisymmetric, composite plate with an arbitrary number of layers, as shown in Figure B-2. Summing the forces in the radial direction results in the radial equilibrium equation, dN N − N r + r θ = 0. (B–1) dr r Summing the forces in the axial direction results in the transverse equilibrium equation,
r dw Q + p + 0 N = 0, (B–2) r 2 dr r
192 where w0 is the deflection in the transverse (z) direction. The moment equilibrium equation is dM M − M r + r θ = Q . (B–3) dr r r
dθ Qr
M r Nr Mθ Nθ
h
p Mθ N θ ∂M M+ r dr r ∂r
z = 0 ∂ r Qr ∂ + Nr Qr dr + ∂ Nr dr θ r ∂r z Figure B-2. Isometric view of an infinitesimal plate element. Three layers are shown for illustration but the plate element is considered to have an arbitrary number of layers in the derivation of the governing equations [161].
B.2.0.1 Kinematic Equations
Kirchhoff hypothesis suggests the following displacement field for an axisymmetric cir- cular plate [161]:
uz (r, z) =w0 (r) (B–4) ∂w and u (r, z) =u (r) − z 0 , (B–5) r 0 ∂r
where the subscript ”0” refers to the z = 0 reference axis shown in Figure B-2. The linear, axisymmetric strain-displacement relationships are ∂ur εr = ∂r . (B–6) ur εθ r
193 The displacement field relations Equation B–4 and Equation B–5 are substituted into the strain-displacement relationships resulting in 0 εr εr κr = + z (B–7) 0 εθ εθ κθ
where the strains in the reference plane are given by 0 du0 εr = dr , (B–8) 0 u0 εθ r
and the curvatures are given by 2 d w0 κr − 2 = dr . (B–9) 1 dw0 κθ − r dr
B.3 Constitutive Equations
The stress in the nth layer is related to the strain and the piezoelectric induced stress as [168] σn σn ε0 κ dn r 0 n r r n 31 = + [Q ] + z − Ef , (B–10) n n 0 n σθ σ0 εθ κθ d31
n n n n where [Q ], Ef , σ0 , and d31 are the stiffness matrix, electric field, fabrication induced stress, and piezoelectric constant of the nth layer, respectively. It is assumed that the fabrication induced stress is approximately uniform in the plane of the thin film [163]. The stiffness matrix is n 1 ν n E n [Q ] = 2 , (B–11) 1 − νn νn 1
n where E and νn are the Young’s modulus and Poisson’s ratio of the nth layer.
194 The forces resultants are found by integrating the stress through the thickness of the diaphragm, z Z T n Nr σr = dz, (B–12) N σn θ zB θ where zB and zT represent the bottom and top of the diaphragm, respectively. Substituting equation Equation B–10 yields [168] 0 Nr N0 εr κr Np = + [A] + [B] + , (B–13) 0 Nθ N0 εθ κθ Np where [A] and [B] are extensional and flexural-extensional stiffness matrixes given by
ZzT [A] = [Qn] dz (B–14)
zB ZzT and [B] = [Qn] zdz, (B–15)
zB respectively. The fabrication induced force per unit length, N0, is given by
ZzT n N0 = σ0 dz. (B–16)
zB The final term in Equation B–13 is the in-plane force per length due to the piezoelectric effect [168], ZzT n n n n NP = − Ef d31 (Q11 + Q12) dz. (B–17)
zB The moments per unit length are found by integrating the stress times its moment arm, z, through the thickness of the diaphragm, z Z T n Mr σr = zdz. (B–18) M σn θ zB θ
195 Carrying out the integration of Equation B–18 results in 0 Mr M0 εr κr Mp = + [B] + [D] + , (B–19) 0 Mθ M0 εθ κθ Mp where [D] is the flexural matrix given by
ZzT [D] = [Q] z2dz. (B–20)
zB
The final term in Equation B–19 is the moment resultant due to the piezoelectric effect [168],
ZzT n n n n Mp = − Ef d31 (Q11 + Q12) zdz. (B–21)
zB The fabrication induced moment is given by
ZzT n M0 = σ0 zdz. (B–22)
zB
B.4 Governing Differential Equations
The governing differential equation is written in terms of the deflection slope, defined as dw ϕ = − 0 . (B–23) dr First, Equations B–2 and B–3 are combined to form
pr dM M − M − ϕN + + r + r θ = 0. (B–24) r 2 dr r
Next, both Equations B–13 and B–19 are substituted into Equation B–24 and Equation B–1. The resulting equations are linearized and combined to form [168] µ ¶ d2ϕ 1 dϕ N 1 pr + − 0 + ϕ = (B–25) dr2 r dr D∗ r2 2D∗
196 and 2 ³ ´ d u0 1 du0 u0 B11 pr 2 + − 2 = ∗ + (N0) ϕ . (B–26) dr r dr r D A11 2 Note that the equation for the deflection slope, Equation B–25, is decoupled from the equation for the radial displacement, Equation B–26. B.5 General Solution
The solutions to Equations B–25 and B–26 are split into the three cases of in-plane
force per length: tension (N0 > 0), zero (N0 = 0), and compression (N0 < 0). The solutions to Equations B–25 and B–26 are [168] ³ ´ ³ ´ r r pr c1I1 + c2K1 − ,N0 > 0, λm λm 2|N0| 3 ϕ = pr + c1 r + c2 ,N = 0, (B–27) 16D∗ 2 r 0 ³ ´ ³ ´ r r pr c1J1 + c2Y1 + N0 < 0, λm λm 2|N0|
and h ³ ´ ³ ´i 1 B11 r r c3r + c4 + c1I1 + c2K1 ,N0 > 0, r A11 λm λm 3 B11 pr c3 c4 u0 = ∗ + r + ,N0 = 0, (B–28) A11 16D 2 r h ³ ´ ³ ´i 1 B11 r r c3r + c4 + c1J1 + c2Y1 N0 < 0, r A11 λm λm q D∗ where λm = . The deflection is found by integrating the deflection slope resulting |N0| in [168] h ³ ´ ³ ´i r r pr2 λm c1I0 − c2K0 − + c5,N0 > 0, λm λm 4|N0| pr4 c 2 w0 = + 1 r + c ln r + c ,N = 0, (B–29) 64D∗ 4 2 5 0 h ³ ´ ³ ´i r r pr2 −λm c1J0 + c2Y0 + + c5 N0 < 0. λm λm 4|N0|
197 The radial force and moment equations are found from Equation B–13 and Equation B– 19 as [168] ³ ³ ´ ³ ´ ´ γA12−B12 r r I1 c1 + K1 c2 + (A11 + A12) c3 r λm λm N0 > 0, A12−A11 p + 2 c4 + (B11 + B12) + Np r 2(N0) B11+B12 B11−B12 A11+A12 A11−A12 − 2 c1 + r2 c2 + 2 c3 − r2 c4 Nr (r) = N0 = 0, (B–30) pr2 p + ∗ (γA12 − B12) − N 16D r ³ ³ ´ ³ ´ ´ γA12−B12 r r J1 c1 + Y1 c2 + (A11 + A12) c3 r λm λm N0 < 0, A12−A11 p + 2 c4 + (B11 + B12) + Np r 2(N0)
and h ³ ´ ³ ³ ´ ³ ´´i γB12−D12 r γB11−D11 r r I1 + I0 + I2 c1 r λm 2λm λm λm h ³ ´ ³ ³ ´ ³ ´´i γB12−D12 r γB11−D11 r r + K1 − K0 + K2 c2 N0 > 0, r λm 2λm λm λm P B12−B11 p(D11+D12) + (B12 + B11) c3 + M0 + M + 2 c4 + r r 2(N0) P D11+D12 D11−D12 B11+B12 M0 + Mr − 2 c1 + r2 c2 + 2 c3 Mr = N0 = 0, 2 B11−B12 pr − 2 c4 + ∗ (3γB11 − 3D11 + γB12 − D12) r 16D h ³ ´ ³ ³ ´ ³ ´´i γB12−D12 r γB11−D11 r r J1 + J0 + J2 c1 r λm 2λm λm λm h ³ ´ ³ ³ ´ ³ ´´i γB12−D12 r γB11−D11 r r + Y1 − Y0 + Y2 c2 N0 < 0. r λm 2λm λm λm P B12−B11 p(D11+D12) + (B12 + B11) c3 + M0 + M + 2 c4 + r r 2(N0) (B–31)
The constants ci for i = 1 ... 5 are solved for all three cases from the boundary and/or matching conditions for each of the two plate sections. In the next section, the constants
(j) are referred to as ci , where j corresponds to the plate section. The details of the bound- ary/matching conditions are provided below. B.6 Boundary and Matching Conditions
The solutions for each plate section have five unknown constants. Therefore, ten bound-
ary/matching conditions are needed to solve for these constants. The plate is assumed to be
198 clamped at its edge. The boundary condition for a clamped plate is
(2) (2) (2) w0 (r = R2) = u0 (r = R2) = ϕ (r = R2) = 0. (B–32)
Also, the radial displacement and slope remain finite at the center of the plate
(1) (1) u0 (r = 0) and ϕ (r = 0) < ∞. (B–33)
There is a discontinuity in the plate at R1. The matching conditions between the sections are [168]
(1) (2) ϕ (R1) =ϕ (R1) , (B–34)
(1) (2) u0 (R1) =u0 (R1) , (B–35)
(1) (2) w (R1) =w (R1) , (B–36)
(1) (2) Nr (R1) =Nr (R1) , (B–37)
(1) (2) and Mr (R1) =Mr (R1) . (B–38)
(2) (2) Applying the finite boundary conditions, B–33, constants c2 and c4 are zero. Eight more integration constants are required. The remaining conditions are the three edge conditions, B–32, and five matching conditions, B–34 through B–38. It is convenient to solve for these coefficients using a matrix formulation. The conditions are written as general matching conditions as follows © ª © ª C(1) {c} + f (1) = C(2) {c} + f (2) (B–39) £ ¤ where C(i) and f (i) are the matrix of coefficients of the integration constants and the vector of free terms of the ith section, respectively. All of the conditions are combined to form a single equation as follows © ª © ª [C] {c} = f (1) − f (2) , (B–40) 8x8 8x1 8x1 8x1
199 where [C] and {c} are formed by the coefficients of the of first and second sections of the plate as follows: © ª (1) · £ ¤ £ ¤ ¸ c [C] {c} = − C(1) C(2) 3×1 . (B–41) © ª 8×8 8×1 8×3 8×5 c(2) 5×1 £ ¤ £ ¤ © ª © ª The coefficients C(1) and C(2) as well as f (1) and f (2) will depend on the © ª © ª corresponding in-plane loading case. Once [C], f (1) , and f (2) are selected, Equation B– 41 is solved for the integration constants {c}. The integration constants are then substituted back into equations (B–27)-(B–29). This completes the linear plate model solution. B.7 Incremental Plate Deflection
There is an initial deflection of the plate due to discontinuity in the fabrication induced force and moment resultants between the annular and inner plate sections. The stresses in the plate will adjust to ensure that the forces and moments match at the section interface. The stress adjustment leads to an initial deflection of the plate that occurs even when the pressure and voltage load are zero. The incremental deflection shown in Figure B-3 is calculated by subtracting the initial deflection. The incremental deflection is used in all integrals of the
lumped element modeling in Section 5.1.
winc
Figure B-3. Diagram of incremental diaphragm deflection.
200 APPENDIX C UNCERTAINTY ANALYSIS
This chapter contains details on the analysis used to find the uncertainty estimates in the experiments presented in Chapter 7. C.1 Electrical Characterization Uncertainties
This section contains details on the uncertainties in the impedance measurement.
C.1.1 Electrical Impedance
The impedance measurement setup and results are outlined in Section 7.2. Thirty impedance measurements were taken and averaged. Each impedance measurement is given in the following form
Zk = Rk + jXk, (C–1) where R is the real part of the impedance, X is the imaginary part of the impedance, and k stands for the kth measurement. The mean real and imaginary impedance are calculated as
£ ¤ 1 XN µˆ =E R¯ = R (C–2) R N k k=1 £ ¤ 1 XN andµ ˆ =E X¯ = X . (C–3) X N k k=1 The standard deviations are found using an unbiased estimator as v u u 1 XN ¡ ¢ σˆ =E [σ ¯ ] = t R − R¯ 2 (C–4) R R N − 1 k k=1 v u u 1 XN ¡ ¢ andσ ˆ =E [σ ¯ ] = t X − X¯ 2. (C–5) X X N − 1 k k=1
The normalized random error is given by
σˆR εˆr (R) = √ (C–6) µˆR N σˆX andε ˆr (X) = √ . (C–7) µˆX N
201 In addition to the random error in the measurements, the HP 4294A Impedance Analyzer has a bias error. The estimation of the bias error was taken from the operation manual [164].
A comparison of the random and bias error estimates is given in Figure C-1. For these measurements, the bias error reported in the manual dominates over the random error in the measurement.
100 εˆr RR5RC5 −1 10 εˆr RR1RC3 (R)
ε εˆr RR2RC3 10−2 εˆr RR2RC4 εˆ RR3RC5 10−3 r εˆr RR5RC2
εˆb RR5RC5 10−2 εˆb RR1RC3
εˆb RR2RC3 10−3
(X) εˆb RR2RC4 ε −4 10 εˆb RR3RC5
εˆb RR5RC2 10−5
Frequency (kHz)
Figure C-1. Random and bias errors in the real and imaginary parts of the impedance.
C.1.2 Element Extraction
The blocked electrical capacitance, CEB, parallel electrical resistance, REP , and series electrical resistance, RES, were extracted from the experimental results in Section 7.2. The extraction was achieved by fitting the complex impedance data using a function in Matlab, invfreqs, which implements a damped Gauss-Newton method to fit an analog transfer func- tion to complex data that is a function of frequency [165]. The data was fit to the impedance model given by Equation 5–49. The following plots show the real and imaginary parts of the impedance fit and experimental data along with the percent relative error.
202
103 30
) 20 Ω R ( 10 2 Experimental 10 % Relative Error Model 0 −104 5
4
) 3 Ω
X ( 2 −105 % Relative Error 1
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(a) RR1RC3
104 6
) 4 3 Ω 10 R ( 2 Experimental % Relative Error Model 2 10 1 −104
) −105 Ω 0.5 X (
−106 % Relative Error
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(b) RR2RC3
Figure C-2. Impedance fit in comparison to the experimental data.
203 4 10 10 8 ) 3 6 Ω 10
R ( 4 Experimental % Relative Error 2 Model 2 10 −104 1 )
Ω 0.5 X (
−105 % Relative Error
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(c) RR2RC4
104 20
15 ) 3 Ω 10 10 R (
Experimental 5 % Relative Error Model 2 10 −104 0.4
0.3 )
Ω 0.2 X (
−105 0.1 % Relative Error
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(d) RR3RC5
Figure C-2. Impedance fit in comparison to the experimental data.
204 104 15 ) 3 Ω 10 10 R (
Experimental 5 % Relative Error Model 2 10 5 −104 4
) 3 Ω
X ( −105 2
% Relative Error 1
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(e) RR5RC2
40 3 10 30 ) Ω
R ( 20
Experimental 10 % Relative Error Model 2 10 −104 5
4
) 3 Ω
X ( 2 −105 % Relative Error 1
0 101 102 101 102 Frequency (kHz) Frequency (kHz)
(f) RR5RC5
Figure C-2. Impedance fit in comparison to the experimental data.
205 C.2 Electromechanical Characterization Uncertainties
This section contains details on the analysis used to find the uncertainty estimates in the LV measurements. C.2.1 Velocity
The random uncertainty in the amplitude and phase of the velocity sensitivity, spectral quantities can be found using the coherence function. First, it is important to delineate the difference between the standard deviation in the measurement, σ, and the standard deviation in the mean estimate, σb. The standard deviation of the measurement is the
standard deviation of the a set of nd measurements. The standard deviation of the mean is
the standard deviation in the estimation of the mean expected if the nd measurements were re-conducted and the mean recalculated. The confidence that our mean estimate is correct is found from the standard deviation of the mean. The standard deviation of the mean is
related to the standard deviation of the measurement as
σ σb = √ . (C–8) nd
The confidence interval of the mean is given as
µ ± tnd;ασ,b (C–9)
where tnd;α is the value of the t distribution for nd averages and a confidence of (1 − 2α) × 100%. The confidence interval estimates given for the LV measurements in Section 7.4 were for 100 averages and 95% confidence in the mean estimation for which the t distribution value is approximately 2.
The random normalized uncertainty, ε, is defined as the standard deviation of the mean normalized by the mean, µ, σb ε = . (C–10) µ
206 The normalized random error of the magnitude of the velocity sensitivity is given as p 1 − γ2 VU ε|HU | = √ . (C–11) |γVU | 2nd
The standard deviation of the mean of the phase of the frequency response function is p 1 − γ2 σb = √ VU , (C–12) φHU |γVU | 2nd
where φH is the phase of the frequency response function. C.2.2 Volume Velocity
The LV software reports the averaged velocity sensitivity and the coherence between the measured velocity and excitation voltage. The mean estimate of the volume velocity sensitivity is based on an integration of the mean estimate of the velocity sensitivity as
given in Equation 7–11. Uncertainty estimates of the volume velocity sensitivity are based on the uncertainties in the velocity sensitivity of the individual scan points using a Monte Carlo simulation. The amplitude and phase of the velocity sensitivity at each point is randomly varied using a Gaussian random number generator. A distribution of complex volume velocities at each frequency f is calculated as ZZ ³ ´ b i i b i j(∠H(x,y;f)+´zασ∠H (x,y;f)) Q (f) = |H(x, y; f)| + zασH (x, y; f) e dxdy, (C–13)
where zα andz ´α are random numbers of normalized distribution whose standard deviation is 1. From the distribution, the standard deviation of the amplitude and phase of the volume velocity is calculated, v u u 1 XN σ (f) =t [|Qi(f)| − |Qb(f)|]2 (C–14) |Q| N − 1 i=1 v u u 1 XN and σ (f) =t [∠Qi(f) − ∠Qb(f)]2. (C–15) ∠|Q| N − 1 i=1
The volume velocity sensitivity including uncertainty bounds is shown in Figure 7-9.
207 C.2.3 Resonant Frequency
This section reports the method for determining the uncertainties in the resonant fre- quency reported in Table 7-5. A schematic of magnitude of the volume velocity sensitivity versus frequency around resonance is illustrated in Figure C-3 along with the 95% confi- dence intervals at each frequency bin. The resonant frequency was determined to occur at the maximum magnitude of the volume velocity sensitivity. The random uncertainty in the resonant frequency was found by finding the bandwidth around resonance where
Q (f) + 2σbQ (f) > Q (fres) − 2σbQ (fres) . (C–16)
Thus, as shown in Figure C-3, the resonant frequency with 95% confidence intervals was estimated as + +∆fres fres − . (C–17) −∆fres Note that the half-width of the frequency bin was also included in the uncertainties in Equation C–17.
y it c y lo it e v V ti e si m en lu S o ∆ − ∆ + V fres fres
f Frequency res
Figure C-3. Schematic of the uncertainty in the resonant frequency calculation.
C.2.4 Damping Coefficient Estimation
In Section 7.4.6, the damping coefficient is calculated by fitting a second order system transfer function to the deflection sensitivity. The fit was weighted using a triangular weight function that equal 1 at the resonant frequency and 0.25 at the prescribed frequency bounds.
The frequency bounds are given in Table C-1. The deflection sensitivity and second order system curve fits are given in Figure C-4.
208
Vacuum Vacuum Vacuum Fit Vacuum Fit −6 STP −6 STP 10 STP Fit 10 STP Fit m/V) m/V) µ µ | ( | (
W/V −8 W/V | 10 | 10−8
20 40 60 80 100 20 40 60 80 100 Frequency (kHz) Frequency (kHz) (a) RR1RC3 (b) RR2RC3
Vacuum Vacuum Vacuum Fit Vacuum Fit 10−6 STP STP STP Fit −6 STP Fit 10 m/V) m/V) µ µ | ( | ( W/V W/V | | 10−8 10−8
20 40 60 80 100 20 40 60 80 Frequency (kHz) Frequency (kHz) (c) RR2RC4 (d) RR3RC5
Vacuum Vacuum Fit STP 10−6 STP Fit m/V) µ | ( W/V | 10−8
10 20 30 40 50 60 Frequency (kHz) (e) RR5RC5
Figure C-4. Damping coefficient curve fit.
209 Table C-1. Frequency bounds on system fit. Device Condition Lower f bound (kHz) Upper f bound (kHz) RR1RC3 STP 23.4375 81.2500 RR1RC3 Vacuum 23.4375 81.2500 RR2RC3 STP 23.4375 85.9375 RR2RC3 Vacuum 23.4375 85.9375 RR2RC4 STP 23.4375 85.9375 RR2RC4 Vacuum 23.4375 85.9375 RR3RC5 STP 23.4375 54.6875 RR3RC5 Vacuum 23.4375 54.6875 RR5RC5 STP 7.8125 39.0625 RR5RC5 Vacuum 7.8125 40.625
C.2.5 Variable Back Cavity
Uncertainties in the back cavity experiment from Section 7.4.5 were introduced via the experimental setup. First, the back cavity was hand tuned to a back cavity depth by aligned to quarter turn marks introducing a random error in terms of back cavity depth. Once the back cavity depth had been set, the device was place on the microscope stage and aligned to the LV again introducing a random error. In addition, the resonant frequency and response also varied from measurement to measurement. In this section, the effect of these errors are isolated and compared.
C.2.5.1 Resonant Response
To compare the magnitude of the random error introduced by alignment and back cavity depth, the resonant response of a single device, RR5RC5, was recorded for each of three cases. In the first case, labeled “Re-measure” in Table C-2, the device was not moved on the microscope stage and the measurement was repeated 30 times. In the second case, the alignment of the device to the LV was disturbed without removing the device from the microscope stage. The resonant response was then measured after realigning the device to the microscope. The process was repeated thirty times. This case is labeled “Alignment” in Table C-2. In the final case, labeled “Back Cavity Tune” in Table C-2, the device was removed from the microscope stage and the back cavity depth was perturbed before realigning
210 it to the nominal cavity depth. The device was then placed back on the microscope stage and realigned to the LV. The resonant performance was recorded 30 times.
Table C-2. Uncertainties in the resonant frequency and response as a function of mea- surement uncertainties.
H(fres) (mm/s / 95% Confidence fres (kHz) 95% Confidence V) (mm/s / V) (kHz) Re-measure 98.51 0.03 29.5 0.01 Alignment 99.55 0.16 29.7 0.04 Back Cavity Tune 101.59 0.95 30.0 0.09
It is clear from Table C-2, the error introduced by setting the back cavity depth and aligning to the LV overrides the random error in the experiment. The 95% confidence increases by almost an order of magnitude in both the resonant velocity sensitivity and resonant frequency. Even with the error introduced by the setting the back cavity depth, the magnitude of the 95% confidence is in the third digit of the mean quantity.
Note that the measurements in this section were performed for a single device at the nominal back cavity depth. To represent the true uncertainty in the resonant velocity sen- sitivity and resonant frequency results presented in Section 7.4.5, the measurement would have to be repeated at each back cavity depth multiple time for each device. For example, the RR5RC5 device response was measured at 40 different screw depths. If the measurement at single back cavity depth was repeated 31 times to build up a 95% confidence similar to that presented in Table C-2, 1240 measurements would have to be made. Considering the 5 devices that have variable back cavities, the total measurements would exceed 6,000, which clearly surpasses practical limits. Instead, measurement of the back cavity depth and the random error incurred by resetting it at each quarter turn can give some understanding to the overall measurement uncertainty. This measurement is conducted in the following section.
211 C.2.5.2 Back Cavity Depth Measurement
A Keyence LK G-32 laser displacement sensor was used to measure the back cavity depth with respect to the aluminum block at quarter turns of the back cavity screw. The back cavity depth was perturbed and then reset 31 times to build up an uncertainty distribution.
The back cavity depths measured at quarter turns of the screw are given in Table C-3. Note that the negative depth at the nominal screw position of 0 refers to the end of the screw being above the aluminum block. Note that the standard deviation introduced by resetting the screw depth is approximately 4-5 µm.
Table C-3. Screw depth measurement with uncertainties. Screw Turns 0 0.25 0.5 0.75 1 Depth (µm) -45 33 113 196 274 σ (µm) 5 4 5 4 4
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226 BIOGRAPHICAL SKETCH Benjamin Andrew Griffin was born in 1980, in Orange Park, Florida. He attended Or- ange Park High School in Orange Park, Florida, graduating in 1999. He then enrolled at the
University of Florida where he received his bachelor’s degree in aerospace engineering from the University of Florida in 2003. During the summer of 2001, Benjamin joined the Interdis- ciplinary Microsystems Group where he worked on a wind tunnel strain gauge balance. In August 2003, Benjamin began his graduate studies at the University of Florida as a National Science Foundation Research Fellow. In May 2006, he earned a Master of Science degree in aerospace engineering. Benjamin is currently completing his doctoral degree in mechanical engineering at the University of Florida. His research interests include air-coupled ultrasonic microelectromechanical systems (MEMS), parametric arrays, proximity sensors, aeroacous- tic microphone design, thermoacoustic imaging, shear stress sensor, laser micromachining, MEMS design optimization, and electroacoustic transducer characterization.
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