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- Permanent Link:
- https://ufdc.ufl.edu/UFE0015581/00001
## Material Information- Title:
- Nonlinear Dynamics of a Dual-Backplate Capacitive MEMS Microphone
- Creator:
- LIU, JIAN
- Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Diaphragm ( jstor )
Electric potential ( jstor ) Electrostatics ( jstor ) Mechanical forces ( jstor ) Mechanical shock ( jstor ) Microelectromechanical systems ( jstor ) Microphones ( jstor ) Nonlinearity ( jstor ) Stiffness ( jstor ) Velocity ( jstor ) City of Gainesville ( local )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Jian Liu. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 7/12/2007
- Resource Identifier:
- 659872404 ( OCLC )
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NONLINEAR DYNAMICS OF A DUAL-BACKPLATE CAPACITIVE MEMS MICROPHONE By JIAN LIU 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 2007 Copyright 2007 by Jian Liu To my parents and my beloved wife, Zhen ACKNOWLEDGMENTS Financial support for this research was provided in part by a National Science Foundation grant (ECS-0097636), and financial support from a NSF CAREER award (CMS-0348288) is also gratefully acknowledged. First, I would like to express my sincere gratitude to my advisor, Professor Mark Sheplak, for giving me the opportunity to work at the Interdisciplinary Microsystems Group. His guidance over the years has been invaluable and his desire for excellence has had a very positive influence on me. I would like to thank my chair, Professor Brian P. Mann, for many valuable insightful talks and encouragement during this research. His guidance and help will always be cherished. I would also like to thank Professors Toshikazu Nishida, Louis N. Cattafesta, and Bhavani V. Sankar for their help and discussions on different aspects of this multidisciplinary research and for serving on my committee. My acknowledgements go to all my colleagues at the Interdisciplinary Microsystems Group. Special thanks go to David Martin and Karthik Kadirvel for working together over the past five years and many valuable discussions on this research. In particular, I want to thank David Martin for helping me improve my English over the years . I would like to express my deep appreciation to my parents for their endless support, understanding and guidance throughout my life. Finally, I would like to thank my beloved wife, Zhen, for her love, patience and sacrifice, without which this dissertation would not have been done. I am forever grateful for her love. TABLE OF CONTENTS page ACKNOWLEDGMENT S .............. .................... iv LIST OF TABLES ............ ...... .__ ...............x.... LIST OF FIGURES .............. ....................xii AB STRAC T ................ .............. xvii CHAPTER 1 INTRODUCTION ................. ...............1.......... ...... Nonlinear Dynamics Issues .............. ...............2..... Obj ective and Approach ................. ...............5.......... ..... Research Contributions............... .............. Dissertation Organization ................ ...............6................. 2 BACKGROUND .............. ...............7..... Microphone Basics .............. ......... .. ............ Conventional versus Silicon Microphones ................. ............... ......... ...7 Metrics of Performance ................. ...............8................ Transduction Mechanisms .............. .... ... ...._.._ ...............11...... Introduction to Electromechanical Transducers ................. ........................11 Piezoelectric Microphones .............. ...............12.... Piezoresistive Microphones ................. ...............13................. Optical M icrophones ................. ...............13.......... ..... Electrostatic Microphones ................. ......... ...............14....... Electrostatic Microphones .............. ...............15.... Electret Microphones ................. ...............19................. Condenser Microphones ................. .......... ...............22....... Single-backplate condenser microphones .................. ................2 Dual-backplate condenser microphones............... ..............2 Maj or Previous Work on Capacitive Silicon Microphones ................. ................ ..30O Basics of Nonlinear Dynamics .............. ...............37.... N onlinearities .............. ............ ........ .... .......3 Steady State, Autonomous System and Fixed Point .............. ....................3 Stability, Basin of Attraction and Phase Portrait ................. .......................39 B ifurcation............... .. ..... .... ... ....... .......4 Previous Work on Nonlinear Dynamics of Electrostatic MEMS Devices ........._.....42 3 NONLINEAR DYNAMIC MODEL .....__.....___ ..........._ ............5 M icrophone Structure .................. .....____ ...............51..... Displacement Solutions of the Diaphragm ...._._._._ .......____ ......._.. .........5 Small Displacement Solution .............. .. ......... ..............5 Energy Method and Large Displacement Solution .............. ....................5 Procedure of energy method .............. ...............55.... Large displacement solution............... ...............55 Lumped Element Modeling of the Microphone ........._._._......____ .........._.....58 Lumped Parameters of Diaphragm ...._._._._ ..... ... .___ ....._.. ...........5 Lumped mass............... ....... .. ..........6 Lumped linear stiffness and compliance ...._._._._ ............ ........._.....61 Lumped area ............ ..... .._ ...............62... Lumped cubic stiffness............... ...............6 Lumped Damping Coefficient ....__ ......_____ .......___ .............6 Lumped Stiffness of the Cavity ................. ...............66........... ... Nonlinear Dynamic Model .............. ...............70.... Discussion of Nonlinearities............... ............7 Nonlinear Finite Element Analyses ................ ...............76................ Stiffnesses of the Diaphragm ................. ............. ...............77. .... Electrostatic Forces by CoSolveEM Simulations ................. ............ .........80 Summary ........._.. ..... ._ ...............83..... 4 APPROXIMATE SOLUTIONS OF NONLINEAR GOVERNING EQUATIONS..84 Introducti on ................. ..... ......_..._ ..... ._. ._ .... .. ..............8 Governing Equation for the Electrical Square Wave Excitation ..........._..._ ..............86 Approximate MTS Solution for the Electrical Square Wave Excitation....................89 Approximate Solution by the MTS Method ........._._.._......_.. ........_.._.....90 Discussion of the MTS Approximate Solution..........._.._.. ......._.._........._..92 Approximate solution with zero initial conditions ................. ................. .93 Approximate solutions in other applications ....._____ ....... ...__...........94 Validity Region of the Approximate MT S Solution ................. .....................94 Results of linear case ................. ...............95................ Results of weakly nonlinear case ................. ...............96........... ... Results of highly nonlinear case ................. ............ ............... 98..... Governing Equati on for the El ectri cal Sinu soi dal Excitati on .........._.... ........._.....100 Approximate HB Solution for the Electrical Sinusoidal Excitation ................... ......103 Approximate Solution by a HB Method............... ...............103 Validity Region of the Approximate HB Solution ................. ............. .......106 Results of small THD case ....._ .....___ ........__ ............0 Results of transition THD case ....._ .....___ ......... ............10 Results of large THD case .............. ... ...... .. ......._ ............1 Governing Equation for the Sinusoidal Acoustical Pressure Excitation ..................1 13 Approximate Solutions for the Sinusoidal Acoustical Pressure Excitation. .............116 HB Approximate Solution ................. ...............117........... ... MT S Approximate Solution .................. ...............118................ Validity Region of Approximate Solutions ................. ................. ...._ 120 Results of linear case ................. ...............121............... Results of weakly nonlinear case ................. ...............................122 Results of highly nonlinear case ................. ...............123........... ... Sum m ary ................. ...............125......... ...... 5 PULL-IN INST ABILITIE S ................. ......... ...............126 ..... Quasi-Static Pull-in due to an Applied DC Voltage ................. .......................126 Equilibrium Points and Local Stabilities ....._____ ..... ... .............._..128 Graphical Analysis ..................... ...............13 1 Critical Quasi-Static Pull-in Voltage ............... ...... ....___ .......___..........3 Quasi-Static Pull-in by a Sub critical Pitchfork Bifurcation .............. ................134 Potential Advantage of Geometric Nonlinearity .............. .....................136 Compact Quasi-Static Stable Operation Range ................. .......................138 Quasi-Static Pull-in due to an Applied Acoustical Pressure ................. .................1 40 Equilibrium Points and Local Stabilities ................. .............................141 Critical Quasi-Static Pull-in Pressure ................. ............. ......... .......14 Dynamic Pull-in due to a Mechanical Shock Load .............. ....................14 Problem Formulation.......................... .........4 Equilibrium Points and Local Stabilities ................. .............................153 Phase Portrait and Basins of Attraction ................. ........... ............... ....155 Dynamic Pull-in due to a Mechanical Shock Load ................ ............... .....158 Potential Advantage of Geometric Nonlinearity .............. .....................162 Dynamic Pull-in due to an Acoustic Shock Load ........................... ...............164 Problem Formulation............... ..............16 Numerical Simulation Results ................. ...............167................ Simulated Dynamic Pull-in Results ................. ...............169............... Effect of Damping on Dynamic Pull-in ................... .......... ................ ...172 Effect of Geometric Nonlinearity on Dynamic Pull-in .............. ................... 173 Summary ................. ...............175................ 6 SYSTEM IDENTIFICATION BY PRELIMINARY EXPERIMENTS ..................177 Experiment Setup and Procedures ................ ...............177............... Experiment Setup .............. ...............177.... Experiment Procedures ................. ....... ...... ...............179...... Procedures for the electrical square wave excitation ................. ...............179 Procedures for the electrical sinusoidal excitation ................. .................1 80 Results of Electrical Square Wave Excitation ................. ................ ......... .181 Results of Bottom Backplate Excitation............... ...............18 Results of Top Backplate Excitation ................. .........___. ....... 185......... Results of Electrical Sinusoidal Excitation............... ...............18 Results of Bottom Backplate Excitation............... ...............18 Results of Top Backplate Excitation ....._.__._ ..... ..._. .. ...._.__........19 Di scussion of Analysi s Results ........._._._ ...._._ ...............191. Summary ........._.___..... ._ __ ...............193.... 7 CONCLUSIONS AND FUTURE WORK ....._.__._ ..... ... .__. ....._._..........19 Conclusions............... .... ....................19 Recommendations for Future Work ......__....._.__._ ......._._. ...........19 APPENDIX A LARGE DISPLACEMENT ENERGY SOLUTION OF A CIRCULAR DIAPHRAGM ................. ...............199......... ...... B APPROXIMATE SOLUTION FOR A GENERAL NONLINEAR SECOND ORDER SY STEM ............ ..... ._ ...............204... Introducti on ............... .. ..........__...... ... ... .... ....... ..........20 Approximate Solution by the Multiple Time Scales Method .............. ................205 C COEFFICIENTS OF THE APPROXIMATE HARMONIC BALANCE SOLUTION .............. ...............213.... D APPROXIMATE SOLUTIONS FOR A SINUSOIDAL ACOUSTICAL PRESSURE EXCITATION .............. ...............215.... HB Approximate Solution ................ ...............217................ MT S Approximate Solution .............. ...............220.... E UNCERTAINTY ANALYSIS .............. ...............227.... Uncertainty Analysis Methods .............. ...............227.... Uncertainty Sources ................. .... .......... ...............229...... Uncertainty in the Experimental Data ................ ........... ...... .... ......... ......229 Errors of Approximate Solutions and Nonlinear Least-Squares Algorithms....230 Uncertainties Caused by the Fabrication Process ................. ......................231 Preliminary Uncertainty Analysis Results ................. ............. ......... .......231 LIST OF REFERENCES ................. ...............237................ BIOGRAPHICAL SKETCH .............. ...............248.... LIST OF TABLES Table pg 2-1 Maj or previous work in capacitive silicon microphones. ........._._... ........_........3 1 3-1 Material properties and physical parameters of the 2000Pa microphone in metric units (material: polysilicon). ............. ...............67..... 3-2 Maj or specifications of the diaphragm mesh with converged displacement results. ............. ...............78..... 3-3 Comparison of nonlinear FEA and LEM results ......___ ... .....___ ................80 4-1 Given and extracted (via MTS solution) parameters for a linear case. ....................96 4-2 Given and extracted parameters (via MTS solution) for a weakly nonlinear case...97 4-3 Given and extracted (via MTS solution) parameters for a highly nonlinear case....98 4-4 Results of the maximum error and sum of residual squares for each test case......100 4-5 Given and extracted (via HB solution) parameters for the small THD case..........108 4-6 Given and extracted (via HB solution) parameters for the transition THD case. ..110 4-7 Given and extracted (via HB solution) parameters for the large THD case...........112 4-8 Results of the maximum error and sum of residual squares for each test case......1 12 4-9 Parameters used for the comparison of approximate and numerical solutions. .....121 5-1 Force parameters for a designed 2000Pa capacitive MEMS microphone .............128 5-2 Parameters for the numerical study of an N-wave excitation. ............. .................168 6-1 Re sults of sy stem parameters of the b ottom b ackplate excitati on ................... .......1 84 6-2 Results of system parameters of the top backplate excitation. ........._..... ..............187 6-3 Amplitudes and phase of the integrated averaged steady-state center displacement of the bottom backplate excitation. ................ ................. .... 189 6-4 Re sults of sy stem parameters of the b ottom b ackplate excitati on ................... .......1 90 6-5 Amplitudes and phase of the integrated averaged steady-state center displacement of the top backplate excitation. ................ ................ ......... 190 6-6 Results of system parameters of the top backplate excitation. .............. ..... ..........191 6-7 Theoretical mean values and uncertainties of system parameters for a given 95% confidence level............... ...............191. 6-8 Nominal values of system parameters of the microphone. ................ .................192 E-1 Uncertainties caused by the fabrication process for a 95% confidence level. .......231 E-2 Sensitivity coefficients used in the uncertainty analysis. ................... ...............23 E-3 Theoretical mean values and uncertainties of system parameters caused by fabrication for a given 95% confidence level............... ...............236. LIST OF FIGURES Figure pg 1-1 Schematic of a dual-backplate capacitive MEMS microphone............... ...............3 1-2 Schematic of an electrical model of the microphone. ................ .......................3 2-1 A typical frequency response plot with a defined sensitivity and bandwidth. ...........8 2-2 Typical noise power spectral density plot for a microphone. ............. ...................9 2-3 Time histories and power spectra of a pure and two distorted sinusoidal waves.....10 2-4 A simplified model of an electrostatic microphone. ............. ......................15 2-5 Schematic of a capacitive microphone with an electret diaphragm. ........................ 19 2-6 A simplified qausi-static model of an electret microphone............... ...............2 2-7 Illustration of the critical bias charge of an electret microphone. ................... .........21 2-8 Schematic of a single-backplate condenser microphone ................. ................ ...22 2-9 Illustration of mechanical and electrostatic forces for a single-backplate condenser microphone............... ...............2 2-10 Illustration of quasi-static pull-in of a single-backplate condenser microphone. ....25 2-11 Simplified circuit of a single-backplate condenser microphone with a pream plifier. ............. ...............26..... 2-12 Effect of cubic nonlinearity on the system frequency response ............... ...............38 2-13 Phase plane traj ectories around Eixed points of a dual-backplate capacitive MEMS microphone (sink points are indicated by blue crosses, and saddle points are indicated by blue circles)............... ...............40 2-14 Typical bifurcation diagrams for one-dimensional autonomous systems: (a) Saddle-node bifurcation; (b) Pitchfork bifurcation; (c) Transcritical bifurcation....41 3-1 3D cross-section view of the microphone structure (not to scale). ........._._.............52 3-2 Top-view photograph of the microphone ................. ...............52............... 3-3 Schematic of a clamped circular diaphragm under a transverse uniform pressure loading. .............. ...............53.... 3-4 Displacement components in the neutral plane of a circular diaphragm. ................56 3-5 Normalized mode shape for several pressure values (2000, 10000 and 100000 Pascal s) ................. ...............58................. 3-6 Repetitive pattern of holes in the top backplate. ................ ...........................64 3-7 Simplified lumped element model of a dual-backplate capacitive microphone. .....68 3-8 A nonlinear dynamic model of a dual-backplate capacitive microphone. ...............70 3-9 Free body diagram of the nonlinear dynamic model ................. .......................71 3-10 Nonlinear vs. linearized mechanical and electrical forces of a single-backplate capacitive microphone............... ...............7 3-11 Nonlinear vs. linearized mechanical and electrical forces of a dual-backplate capacitive microphone............... ...............7 3-12 3D mesh of the diaphragm in CoventorWare 2003 ................. .......__. ........._.77 3-13 Transverse center deflections of the diaphragm under the uniform pressure. ..........78 3-14 Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thickness direction). ............. ...............79..... 3-15 Plot of simulated and modeled electrostatic forces for the top capacitor. ................81 3-16 Plot of simulated and modeled electrostatic forces for the bottom capacitor. .........82 4-1 Dynamic model for an electrical square wave excitation on the top backplate. ......86 4-2 Plot of electrostatic and approximate electrostatic forces. ................... ...............87 4-3 Comparison of simulated and MTS-based curve fitting center displacement results. ............. ...............96..... 4-4 Comparison of simulated and MTS-based curve fitting center displacement results. ............. ...............97..... 4-5 Comparison of simulated and MTS-based curve fitting center displacement results. ............. ...............99..... 4-6 Dynamic model for an electrical sinusoidal excitation on the bottom backplate...101 4-7 Simulated sinusoidal response of the diaphragm. ............. ......................0 4-8 Simulated power spectrum of the steady state displacement. ............. .................108 4-9 Simulated sinusoidal response of the diaphragm. ............. ......................0 4-10 Simulated power spectrum of the steady state displacement. ............. .................109 4-11 Simulated sinusoidal response of the diaphragm. ................ ........................11 1 4-12 Simulated power spectrum of the steady state displacement. ............. ..... ............11 1 4-13 Dynamic model for the sinusoidal acoustical pressure excitation. ................... .....113 4-14 Plot of net electrostatic and approximate net electrostatic forces. ................... ......1 14 4-15 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions for a linear case. ............. ...............122.... 4-16 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions for a weakly nonlinear case ................. ......._.._.. ......123 4-17 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions for a highly nonlinear case. ................... ...............12 5-1 Plot of the ND mechanical and net electrostatic forces ................. ................ ...13 1 5-2 Plot of the ND mechanical and net electrostatic forces ................. ................ ...133 5-3 A subcritical pitchfork bifurcation illustrating quasi-static pull-in due to an applied DC voltage. ................. ...............135._._.. ..... 5-4 A subcritical pitchfork bifurcation illustrating quasi-static pull-in due to an applied DC voltage (versus bias voltages). ............. ...............136.... 5-5 Plot of a non-dimensional net electrostatic force and different non-dimensional mechanical forces. .............. ...............137.... 5-6 Quasi-static stable operation range of the microphone in a 3D space.. ........._.......138 5-7 Quasi-static stable operation range of the microphone in a 3D space (versus DC voltages). ............. ...............139.... 5-8 Plot of the ND net restoring and electrostatic forces ................. ............. .......143 5-9 Plot of the ND net restoring and electrostatic forces ................. ............. .......145 5-10 Quasi-static pull-in due to varying ND parameters ................. .......................146 5-11 Quasi-static pull-in due to varying acoustic pressure and DC bias voltage. ..........147 5-12 Three commonly used nonlinear mechanical shock load models (impulse, half sine and triangle). ............. ...............152.... 5-13 Phase plane traj ectories around the equilibrium points ................. ................ ...155 5-14 Three basins of attraction for a DC bias of 25 V. ............. ......................156 5-15 Basins of attraction within the physical backplates for a DC bias of 25 V............157 5-16 Stable and unstable non-dimensional center displacement responses with two initial non-dimensional velocities. ............. ...............159.... 5-17 Phase plots of a stable response and a dynamic pull-in due to a large initial velocity imposed by a mechanical shock load. ............. ...............159.... 5-18 Dynamic pull-in due to a combination of DC bias voltage and a mechanical shock load. ........... ..... .._ ...............161... 5-19 Phase plane trajectories for an added geometric nonlinearity case. Sink points are indicated by blue crosses, and saddle points are indicated by blue circles. .....162 5-20 Expanded stable operation region of the microphone due to the added geometric nonlinearity ................. ...............163................ 5-21 A typical N-wave with an amplitude and a duration time ................. .................1 65 5-22 The Fourier transform of a typical N-wave ................. .............................166 5-23 Transient non-dimensional center displacement response of diaphragm due to an N-wave with an amplitude of 125 dB SPL and a duration time of 2 ms................1 68 5-24 Transient non-dimensional center velocity response of diaphragm due to an N- wave with an amplitude of 125 dB SPL and a duration time of 2 ms. ........._.._......169 5-25 Dynamic pull-in due to an N-wave with a ND amplitude of 1.29 and a ND duration time of 2.3. ............. ...............170.... 5-26 Threshold of dynamic pull-in due to an N-wave ................. ........................170 5-27 Threshold of dynamic pull-in due to an N-wave with a normalized pressure parameter ................. ...............172................ 5-28 Effect of damping on dynamic pull-in threshold. ............. ......................173 5-29 Effect of geometric nonlinearity on dynamic pull-in threshold. ................... .........174 6-1 Block diagram of the experiment setup ................. ...............178........... .. 6-2 Laser beam spot (red dot) impinges the diaphragm through the center hole of the top backplate. ............. ...............179.... 6-3 Simplified circuit to generate the high voltage signal ................. ......._.._.. ......181 6-4 Measured averaged center velocity response for an applied square wave with an amplitude of 5V ................. ...............182................ 6-5 Integrated center displacement response for an applied square wave with an amplitude of 5V ................. ...............183................ 6-6 Constructed phase plot for an applied square wave with an amplitude of 5V.......183 6-7 Comparison of integrated and curve-fit center displacements for an applied square wave with an amplitude of 5V ................. ...............184........... .. 6-8 Measured averaged center velocity response for an applied square wave with an amplitude of 18V ................. ...............185............... 6-9 Integrated center displacement response for an applied square wave with an amplitude of 18V ................. ...............186............... 6-10 Constructed phase plot after 12 Cls for an applied square wave with an amplitude of 18V ................. ...............186................ 6-11 Comparison of integrated and curve-fit center displacements for an applied square wave with an amplitude of 18V ................. ...............187............. 6-12 Measured averaged steady-state center velocity response (asterisk) for a sinusoidal excitation with an amplitude of 9V and a frequency of 1 14.4 k 6-13 Comparison of the integrated (red asterisk) and curve-fitting (blue solid line) steady-state center displacement results for a sinusoidal excitation with an amplitude of 9V and a frequency of 1 14.4 k 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 NONLINEAR DYNAMICS OF A DUAL-BACKPLATE CAPACITIVE MEMS MICROPHONE By Jian Liu May 2007 Chair: Brian P. Mann Cochair: Mark Sheplak Major Department: Mechanical and Aerospace Engineering This work presents an investigation of the electromechanical nonlinear dynamics of a dual-backplate capacitive MEMS (microelectromechanical systems) microphone. A large displacement solution via an energy method has been utilized to provide linear and cubic lumped stiffnesses of the circular diaphragm of the microphone. A nonlinear dynamic model of the microphone is developed using lumped element modeling. Theoretical lumped stiffnesses of the diaphragm are verified by nonlinear finite element analyses and the errors for the linear and cubic stiffnesses are approximately 1.3% and 5.0% respectively. The critical quasi-static pull-in voltage of the microphone is found to be approximately 41V both analytically and numerically. The phenomenon of qausi-static pull-in due to an applied DC voltage is illustrated by a subcritical pitchfork bifurcation. By using a phase portrait and basin of attraction, a mechanical shock load is related to dynamic pull-in. Further study shows that dynamic pull-in could potentially take place below the critical quasi-static pull-in voltage when the microphone is subj ect to a large mechanical shock load. The dynamic pull-in due to an acoustical pulse, in the form of an N-wave, has been investigated by using numerical simulation. A dynamic pull-in threshold curve has been obtained in terms of the duration time and amplitude of the N- wave for a given DC bias voltage. Studies of dynamic pull-in also show that several nonlinearities (geometric, electrostatic and mechanical/acoustical shock) compete with each other. An increased electrostatic nonlinearity and/or an increased mechanical/acoustical shock load destabilize the system while an increased geometric nonlinearity helps to stabilize the microphone and expands the stable operational range. The multiple time scales and harmonic balance methods are applied to obtain approximate solutions of the nonlinear governing equations under the electrical square, electrical sinusoidal and sinusoidal acoustical excitations. Based on the two approximate solutions for the electrical excitations and a nonlinear least-squares curve-fitting technique, system parameters are extracted from two types of experimental data. The preliminary uncertainty analysis, which includes only the uncertainties caused by fabrication, shows that the experimentally extracted linear natural frequency, damping ratio and nonlinear stiffness parameter fall within their conservative theoretical ranges for a 95% confidence level. XV111 CHAPTER 1 INTTRODUCTION During the past three decades, the demand for reducing noise pollution, especially in communities surrounding airports, has increased. To abate the aircraft noise, the generation and propagation of noise sources must be characterized. Aeroacoustic measurement tools are required to both investigate mechanisms of noise generation and validate methods of noise reduction [1]. To enable aeroacoustic measurements, measurement microphones with the instrumentation grade, cost efficiency, and small size must be developed. Currently, commercial traditional microphones such as B&K condenser microphones are widely used in the field of aeroacoustic measurements; however, those microphones are costly and not suitable for miniaturization. With the recent advancements in microelectromechanical systems (MEMS) technology, batch fabrication of microphones with smaller sizes and lower prices is now possible. A variety of transduction schemes, such as piezoelectric, piezoresistive, capacitive and optical, have been used in MEMS microphones [2]. Capacitive MEMS microphones have shown the potential to provide a dynamic range of 160 dB and a bandwidth of 90 k However, the development of capacitive MEMS microphones comes with several issues. The electrostatic force between the diaphragm and backplate of the microphone is inherently nonlinear. Pull-in instability [3, 4], a phenomenon in which the diaphragm collapses to the backplate or vice versa, occurs due to a large applied voltage and/or a large displacement. Since pull-in results in structural failure of the device when the process is not reversible, the microphone needs to be carefully designed to operate in a stable domain. In addition, the mechanical restoring force of the diaphragm becomes nonlinear for large displacements. Both mechanical and electrostatic nonlinearities interact with each other, which adds more nonlinear distortion and lowers the fidelity of the microphone. Moreover, an increased electrostatic nonlinearity could potentially destabilize the microphone system while a larger mechanical geometric nonlinearity could help to stabilize the microphone and expands the stable operation range. Therefore, a thorough understanding of aforementioned nonlinear issues becomes vital for the success of the capacitive MEMS microphone. This dissertation is part of a larger effort to develop a dual-backplate capacitive MEMS microphone for aeroacoustic applications. The focus of this dissertation is to investigate the electromechanical nonlinear dynamics of the microphone through theoretical analysis, numerical simulation and preliminary experimental characterization. In summary, the study of this dissertation will help improve the overall performances of the microphone through a better understanding of the nonlinear dynamics issues. Nonlinear Dynamics Issues This section presents an overview of the nonlinear dynamics of a dual-backplate capacitive MEMS microphone. Different types of nonlinearities are discussed and some physical mechanisms are provided. Further details concerning the nonlinear model will be discussed in Chapter 3. A simplified schematic of a typical dual-backplate capacitive MEMS microphone is shown in Figure 1-1. The backplates of the microphone are perforated to let the air pass through them and hence reduce the air-streaming resistance. Two gaps are formed between the diaphragm and each backplate respectively. A cavity under the bottom backplate is vented to the ambient pressure resulting in an AC measurement device. mmmmmmm Top backplate mmmmmmm Bottom backplate Backpate hle Cavity -Vent Figure 1-1. Schematic of a dual-backplate capacitive MEMS microphone. The three plates of the microphone are made conductive; therefore, two capacitors are formed between the diaphragm and each backplate. When an acoustic wave impinges on the microphone, the incident pressure deflects the middle diaphragm and thereby alters the capacitance of the two capacitors as shown in Figure 1-2. The differential capacitance change is detected through various types of interface circuitry [5] to determine the input sound pressure level. Top Backplate Diaphragm Bottom Backplate Figure 1-2. Schematic of an electrical model of the microphone. When the diaphragm with a fixed boundary undergoes a large deflection with respect to its thickness, geometric nonlinearity [6] occurs and the restoring spring force becomes nonlinear. Under large deflections, the diaphragm behaves like a mechanical Duffing's spring, in which a hardening nonlinearity is caused by the mid-plane stretching. This mechanical nonlinearity directly affects the upper limits of the linearity and dynamic range of the microphone. The details of geometric nonlinearity will be discussed in Chapter 3. The net electrostatic force on the diaphragm is a nonlinear function of the mechanical displacement of the diaphragm, the gaps and the applied voltages between the backplates and diaphragm respectively. The net electrostatic force has singularities at the backplates, which will affect the upper dynamic range of the microphone. Although larger applied voltages and smaller gaps are desirable, because they increase the sensitivity of the microphone, the nonlinear net electrostatic force increases dramatically with larger applied voltages and smaller gaps. When the increasing net electrostatic force overcomes the nonlinear mechanical force, pull-in occurs. In a phase portrait, the pull-in point is an unstable fixed point for the microphone, which means that the displacement of the diaphragm continuously increases until the diaphragm crashes into one of the backplates. Damping plays a very important role in determining the bandwidth and the dynamic response of the microphone. Damping in a dual-backplate capacitive MEMS microphone is dominated by viscous damping and a linearized version is usually used to approximate the actual damping. When the diaphragm vibrates, the gas flow between the diaphragm and backplates can be divided into the horizontal flow between the plates and the vertical flow through the backplate holes. Viscous damping caused by the horizontal gas flow is often called squeeze-film damping, and the viscous damping caused by the vertical gas flow is called holes resistance [7, 8]. The structural damping of the diaphragm mainly consists of the thermoelastic energy dissipation inside the diaphragm and the vibration energy dissipation in the compliant boundary of the diaphragm [9, 10]. Objective and Approach The objective of this research is to study the electromechanical nonlinear dynamics of a dual-backplate capacitive MEMS microphone. An investigation of the existing nonlinear dynamics issues is targeted to help improve the overall performance of the microphone. In addition, the knowledge gained from this study can be applied to other electrostatic devices, such as dynamic nano-indenters [l l, 12] and MEMS mass sensors [13, 14]. To achieve the above research goal, several approaches are employed in this dissertation. Specifically, the nonlinear dynamical system is modeled via lumped element modeling and a general form of the nonlinear governing equation is obtained. Approximate analytical solutions to the nonlinear governing equations are obtained with multiple scales and harmonic balance analyses. Pull-in instabilities are explored by both analytical and numerical approaches. Finally, the microphone is experimentally characterized and system parameters of the nonlinear dynamic model are identified from the measured data. Research Contributions The contributions of this dissertation are summarized as follows. * Development of a nonlinear dynamic model for a dual-backplate capacitive MEMS microphone and numerical solutions of the nonlinear governing equation. * Development of approximate analytical solutions of the nonlinear governing equations via multiple time scales (MTS) and harmonic balance (HB) methods. * Preliminary experimental characterization and application of the uncertainty analysis to the experimentally identified system parameters. Dissertation Organization The dissertation is organized into seven chapters. Chapter 1 introduces and describes the research work in this dissertation. The next chapter provides a background of the microphone, basics of nonlinear dynamics and a review of previous work done on both capacitive microphones and nonlinear dynamics of electrostatic MEMS devices. A nonlinear dynamical model for a dual-backplate capacitive MEMS microphone is developed in Chapter 3. Nonlinear finite element analyses are performed to verify some theoretical results. In Chapter 4, the multiple time scales and harmonic balance methods are applied to obtain approximate solutions of the nonlinear governing equations under the electrical square, electrical sinusoidal and sinusoidal acoustical excitations. Numerical tests are conducted to provide the validity ranges of approximate solutions. Chapter 5 focuses on the theoretical studies of pull-in instabilities, including both quasi- static and dynamic pull-ins. In Chapter 6, the approximate solutions obtained in Chapter 4 are applied to identify system parameters through a series of preliminary experiments. Preliminary uncertainty analysis is also conducted for the experimentally identified system parameters. Finally, conclusions and future work are provided in Chapter 7. CHAPTER 2 BACKGROUND This chapter provides background information about microphone basics and some introductory information about nonlinear dynamics. A review of the published work on both capacitive silicon microphones and nonlinear dynamics of electrostatic MEMS devices is also presented in this chapter. Microphone Basics A microphone is a transducer that converts unsteady pressure inputs into an electrical signal. So far, many transduction mechanisms have been developed for microphones; these include electrodynamic, piezoelectric, piezoresistive, capacitive, optical and contact (carbon) transduction mechanisms [2, 15]. Microphones are widely employed in a variety of applications such as sound field measurements [16, 17], hearing aids [18-20], telecommunications [16] and noise localization using acoustic arrays [21, 22]. Also, the use of microphones in ultrasonic and underwater applications have also been reported [23, 24]. Conventional versus Silicon Microphones Based on the manufacturing technique, microphones can be categorized into two major types: conventional and silicon micromachined microphones. Conventional microphones are usually fabricated from separate metal parts and polymer foils with most of the assembly process done by hand [16]. On the other hand, silicon microphones are fabricated from modern silicon micromachining technology. In comparison with conventional microphones, silicon microphones are easier to integrate with the sensing and supporting electronics, which offers the potential for higher performance by reducing parasitic elements in the sensing subsystem. Moreover, the batch fabrication of silicon microphones leads to lower costs since hundreds or thousands of devices are fabricated together on a single silicon wafer simultaneously [25]. Metrics of Performance The major performance metrics for a microphone are the sensitivity, bandwidth, dynamic range, and noise floor. The open-circuit sensitivity of a microphone is typically defined at some reference frequency (for example, 1 k defined as the ratio of the output voltage change (before the preamplifier) to the amplitude change of the sound pressure incident on the diaphragm [16]. ~ i -3db -3db i Sensitivity S i at 1.0 kHz f Bandwidth cuton -3db ~ 90 -180- Frequency (rad/s) Figure 2-1. A typical frequency response plot with a defined sensitivity and bandwidth. The bandwidth of a microphone is defined as the frequency range where a microphone ideally maintains a constant sensitivity [16]. In practice, as shown in Figure 2-1, the bandwidth is usually the frequency range from a -3dB low cut-on point to a high -3dB frequency point. 1/f noise \ Thermal noise ~II / fe (Comer frequency) Frequency Figure 2-2. Typical noise power spectral density plot for a microphone. Noise floor level is one of most important specifications of a microphone, since it determines the lowest measurable sound pressure level and affects the signal to noise ratio (SNR) of a microphone. For microphones, common noise sources could be environmental noise (such as power line, radio frequency interference, and environment vibration), noise in a microphone (such as thermomechanical noise, Johnson noise, shot noise and 1/f noise [26]) and noise in the interface electronics. Shown in Figure 2-2 is a typical noise power spectral density plot for a microphone. The corner frequency is where the power from 1/f noise equals the power from thermal noise. As seen from the plot, 1/f noise is dominant at low frequencies, while thermal and/or shot noise becomes important at high frequencies. In practice, a noise floor is typically specified by a linear method, an A-weighted approach, or a narrow-band method [27]. In a linear method, the noise is integrated over a specified frequency range (for example, 20 Hz to 20 k audio microphones) without any weighting. An A-weighted noise floor is obtained by integrating noise spectrum after amplitude-weighting, which simulates the perceived noise by the human ear [27]. A narrow-band noise floor is obtained by calculating the total noise within a very narrow frequency band, for example, a 1 Hz bin centered at 1 k microphone signals are often sampled and analyzed in the frequency domain. \ Time Frequency Figure 2-3. Time histories and power spectra of a pure and two distorted sinusoidal waves. The range between the upper and lower measurable sound pressure levels of a microphone is defined as the dynamic range [16]. The lower measurable pressure level is typically determined by its noise floor. The upper limit of the dynamic range is usually set by a specific sound pressure level, which results in 3% total harmonic distortion (THD) within the frequency range from 160 Hz to 1000 Hz for measurement microphones [28]. The THD is defined as follows THD = PH x 100%, (2.1) where PH is the sum of power within all harmonics, and P,,, is the sum of power within the fundamental and all harmonics [28-31]. Figure 2-3 provides an illustration of harmonic distortion for a sinusoidal wave in both time and frequency domains. As more harmonics are added to the original single- tone sinusoidal signal in the frequency domain, more distortion is observed in the time history plot. For capacitive microphones, the possible sources of harmonic distortion are the nonlinear elastic behavior of the diaphragm, the electrostatic nonlinearity and the preamplifier "clipping" [16, 27]. Transduction Mechanisms In this section, the basics of electromechanical transducers are introduced. This section also briefly reviews some major types of transduction mechanisms, including piezoelectric, piezoresistive, electrostatic and optical. A detailed discussion of capacitive microphones is provided in the next section. Introduction to Electromechanical Transducers An electromechanical transducer is a device that converts a mechanical input into an electrical output or vice versa [3]. Properties of electromechanical transducers include linear vs. nonlinear, reciprocal vs. non-reciprocal, conservative vs. non-conservative, and direct vs. indirect [3, 29]. An electromechanical transducer is linear if its output quantities are linear functions of the input quantities. Minor nonlinear harmonic distortion is generally allowed in linear electromechanical transducers [3]. The reciprocal property is used to describe the ability of an electromechanical transducer to convert signals in either direction between two different energy domains. The transduction coefficients are all reversible in reciprocal electromechanical transducers [3]. An electromechanical transducer is conservative if there is no energy dissipation during the transduction, otherwise it is non-conservative. An electromechanical transducer is indirect if there exists a transition energy domain between its input and output quantities, otherwise it is a direct transducer. Typically, transducers can be classified into two major categories: sensors and actuators. By definition, microphones fall into the category of a sensor. A wide range of transduction schemes are employed in electromechanical microphones, including piezoelectric, piezoresistive, optical and electrostatic. These types of microphones and their properties are briefly discussed in the following sections. Piezoelectric Microphones Some materials (for example, quartz) develop electric surface charges when compressed by a mechanical force, and this effect is referred to as the direct piezoelectric effect [32]. In addition, a mechanical strain is generated when an electric field is applied to these same materials, and this effect is referred to as the converse piezoelectric effect [32]. Those materials are often called piezoelectric materials. The relations between the electric quantities (electric displacement and field) and mechanical quantities (stress and strain) are generally described linearly by a set of piezoelectric coefficients [33]. A piezoelectric microphone typically consists of a thin diaphragm (cantilever, rectangular or circular plates) and a multilayer piezoelectric material [2]. When an incident sound wave impinges and deflects the diaphragm, the induced mechanical stress in the piezoelectric material results in an output voltage due to the piezoelectric effect. Piezoelectric microphones possess many advantages including inherently low power consumption [34]. Disadvantages of piezoelectric microphones include the relatively low sensitivity [35] and high noise level [34, 36]. For the electromechanical property, piezoelectric microphones are reciprocal, linear, conservative and direct transducers. Piezoresistive Microphones The piezoresistive property of a material is defined as the change in its resistivity due to a mechanical strain or stress. For silicon, the resistivity change is due to a change in the mobility (or number of charge carriers) [37]. The piezoresistive transduction scheme can be used to design silicon microphones. A piezoresistive microphone is constructed mainly by a diaphragm with two pairs of piezoresistors. When a sound wave impinges and deflects the diaphragm, the induced mechanical stresses in the two pairs of piezoresistors results in opposite strain changes, which leads to the opposite resistance changes. By implementing a fully active Wheatstone bridge, the resistance modulation in the two pairs of piezoresistors is further expressed by the output voltage change; therefore, the incoming sound pressure can be determined [2, 38]. Piezoresistive microphones have many advantages, such as the scaling, robustness, micromachining convenience, and the absence of a need for on-chip circuitry due to its low output impedance [38, 39]. However, piezoresistive microphones have some drawbacks, such as a high noise floor [21], high power consumption, temperature drift and thermal degradation of the piezoresistors due to Joule heating [38]. For the electromechanical property, piezoresistive microphones are linear, direct, non-reciprocal and non- conservative transducers. Optical Microphones The classification of optical transduction is generally based on the property of modulated light. Three common transduction schemes are: 1) intensity modulation, 2) phase modulation and 3) polarization modulation [40, 41]. An optical microphone transforms an acoustic signal into an electrical signal by modulating a reference light signal [40]. Unlike other types of microphones, an acoustic signal is first converted into an optical signal before it is converted to an electrical signal for optical microphones. The detection electronics can be remotely located away from the acoustic field, which makes optical microphones immune to harsh environments [42, 43], and less vulnerable to electromagnetic and radio frequency interference [36]. Disadvantages of optical microphones include the requirement of a stable reference optical source in a physical environment and the packaging since all the system components, such as light sources, optical sensor, and photo detectors, must be well aligned and positioned [42, 44]. Optical microphones are linear, non-reciprocal, non-conservative and indirect transducers. Electrostatic Microphones The electrostatic transduction mechanism has established more than two centuries for actuator applications [3]. To realize an electrostatic conversion between electrical and mechanical quantities, a capacitor with a deformable electrode is generally needed [29]. Since the electrostatic force in a capacitor is nonlinear by nature, it needs to be linearized to be suitable for the electrostatic conversion. The linearization process is typically enabled by one of two polarization schemes: a charge polarization or a voltage polarization [29]. The electrostatic transduction can be utilized to create microphones. There are two general types of electrostatic or capacitive microphones a condenser and an electret microphone. Condenser microphones are polarized with a constant voltage, while electret microphones are polarized with a constant permanent charge. Electrostatic microphones are linear, reciprocal, conservative and direct transducers. A capacitive microphone mainly consists of a diaphragm and a backplate, which are separated by a dielectric, which is usually an air gap. The diaphragm and backplate are made either of conducting materials or connected with electrodes to realize the capacitive detection mechanism [29, 45]. When the sound pressure deflects the diaphragm, the induced capacitance change between the diaphragm and backplate is detected via various types of interface circuitry [5]. Capacitive microphones have many advantages such as a relatively high sensitivity, a large bandwidth, an inherently low power consumption and a low noise floor [18, 46]. However, capacitive microphones have some potential issues such as electrostatic pull-in instability, output signal attenuation due to the parasitic capacitance, and deceased sensitivity at high frequencies due to the viscous damping of the perforated backplate [2]. Electrostatic Microphones This section provides details of electret microphones and two major types of condenser microphones. A summary of previous work on capacitive microphones is provided in the next section. F x Cm Movable diaphragm x 0--~ ~ \\ ,~ Fixed backplate Figure 2-4. A simplified model of an electrostatic microphone (adapted from Rossi [29]). First, we need to see how the electrostatic transduction is realized by the linearization process. Shown in Figure 2-4 is a simplified model of an electrostatic microphone in air. The backplate is assumed to be fixed and the diaphragm is movable and located at x =xo due to some loading (for example, acoustic pressure). The mechanical restoring force for the diaphragm is modeled by a spring. If a small displacement fluctuation x is assumed, the position of the diaphragm is given by x = xo x. (2.2) By using a parallel-plate assumption, the capacitance C is given by coA coA x= '! x 23 C o = X 1 = Co1 23 where so is the dielectric constant of the air, A is the area of the diaphragm and Co = (EoA)/xo is the mean capacitance. If a charge, Q, is applied to the movable diaphragm, and we assume Q! = Qo + Q! (2.4) where Qo is the mean charge and Q is the small charge fluctuation. Therefore, the voltage, V, across the diaphragm and backplate is VC 1- Vo + 1- (2.5) C o xo Co xo where Vo = eo/Co is the mean voltage. The stored electrical potential energy in the capacitor, U is given by [5] U = 2C2 2~2 1 .~ (2.6) Therefore, the electrostatic force acting on the diaphragm is dU, Q 2 Fe = (2.7) e dx' 2xoCo The mechanical restoring force is given by F =1 (2.8) where C, is the mechanical compliance of the spring. Therefore, the net force acting on the diaphragm is x Q2 F = (2.9) C, 2xoCo Eqs. (2.3), (2.5) and (2.9) are three nonlinear coupled equations. In order to linearize these equations to realize the electrostatic transduction, we must assume -1, (2.10) =1 (2.11) and 1 (2.12) Physically, this linearization process can be done by one of two polarization schemes: a charge polarization or a voltage polarization [29]. If a constant voltage Vo is applied across the diaphragm and backplate, a condenser microphone is created. An electret microphone is created if a constant charge Qo is stored permanently on the diaphragm or backplate. Once the polarization is applied, Eqs. (2.3), (2.5) and (2.9) can be linearized as C = = Co ,(.3 xo -X, x ,X xo xo xo: 2.3 V x (2.14) Co xo and Sx V F' o Q (2.15) where V' and F' are the fluctuating components of the voltage and force applied on the diaphragm, respectively. Eqs. (2.14) and (2.15) represent the linearized electrostatic coupling equations in the displacement and charge form. These two equations can be further rewritten in power variables [29] as follows 1, V V= I ov (2.16) and F = v (2.17) where I is the fluctuating component of the current passing through the diaphragm and v is the fluctuating component of the velocity of the diaphragm. In matrix form, the coupling equations are written as iIV 1 0o jax 100 Electret Microphones For an electret microphone, the permanent charge is usually fixed by a thin layer of charge-holding electret material on the backplate or diaphragm [47, 48]. The electret material is dominated by Teflon for conventional electret microphones and silicon dioxide/nitride for silicon electret microphones [47]. In electret microphones, integration of the electret layer on the diaphragm is a common choice since the backplate is usually perforated [48-50]. Backplate Air Gap Acoustic Pressure Figure 2-5. Schematic of a capacitive microphone with an electret diaphragm. A schematic of a capacitive microphone with an electret diaphragm is shown in Figure 2-5. An air gap separates a metalized electret diaphragm from a backplate. A metallization layer is used to charge the electret layer prior to the operation. At the interface between the diaphragm and electret, a charge layer with certain charge density (total charge divided by the cross section area) is formed to generate the electrical field inside the gap. One advantage of electret microphones is the absence of an external power supply and the potential for portable applications [46, 50]. However, integration of electret microphones with MEMS suffers from the poor quality of thin film electrets and charge loss due to humidity [46]. Diaprg Backplate Figure 2-6. A simplified qausi-static model of an electret microphone. Critical bias charge. To illustrate the idea of the critical bias charge for electret microphones, shown in Figure 2-6 is a simplified quasi-static model of an electret microphone. The mass and mechanical stiffness of the movable diaphragm are modeled by a point mass (m) and a spring (k) respectively. The nominal gap between the diaphragm and backplate is d, and x is a vertical displacement of the diaphragm. Since only the quasi-static process is considered here, the inertial force and damping force are neglected since they are time-dependent. In the following analysis, only the mechanical restoring and electrostatic forces are considered. The diaphragm and backplate are conductive, thus one capacitor is formed. For electret microphones, the electrostatic force, Fe, is given as [5] Fe (2.19) S2E,A As seen from Eq. (2.19), the electrostatic force is a quadratic function of the bias charge and independent of the vertical displacement x of the diaphragm. For a given constant bias charge, the electrostatic force remains as a constant force. By using a linear spring model, the mechanical restoring force, Fm, is given as Fm = kx, O <; x <; ,. (2.20) Electrostatic force Fe (Q > Qcr) Electrostatic force Fe (Q = Q ) Stable point Electrostatic force Fe (Q < Qcr) Mechanical force F o dm Displacement x d Figure 2-7. Illustration of the critical bias charge of an electret microphone. Shown in Figure 2-7 is a plot with both electrostatic and mechanical forces for different bias charge cases. When the bias constant charge Q is smaller than a critical value Qc;, the electrostatic force can be always balanced by the mechanical force and the diaphragm moves to a stable position as shown in the plot. When the constant bias charge Q reaches Qc;., the electrostatic force is greater than the mechanical force when x < d, and the diaphragm is forced to move to the rigid backplate position x = d,, which represents a non-functioning microphone. When the constant bias charge Q is greater than Qc, the electrostatic force is always greater than the mechanical force and the diaphragm always moves to the rigid backplate. Therefore, electret microphones must operate with a bias charge less that its critical value. To find the quasi-static critical bias charge, the electrostatic attraction force is set to be equal to the mechanical restoring force at x = d,, which results in or = kdo (2.21) Namely, the critical charge Qc;. is [51] Qc; = J2E,Aku. (2.22) Condenser Microphones The majority of condenser microphones can be categorized into single-backplate condenser microphones and dual-backplate condenser microphones based on the backplate configuration. In 1996, Bay et al. proposed a dual-diaphragm condenser microphone [19]; however, it was not fabricated. In 2002, Rombach et al. fabricated the first dual-backplate condenser microphone [52]. The details of two major types of condenser microphones are provided in the following. Single-backplate condenser microphones Diaphragm Air Gap Backplate Backplate hole Cvt Vent Figure 2-8. Schematic of a single-backplate condenser microphone. A typical single-backplate condenser microphone with a diaphragm and a backplate is shown in Figure 2-8. A backplate is perforated to reduce the air-streaming resistance, a vent is provided to equilibrate the pressure inside the cavity to the ambient atmospheric pressure. A capacitor is formed between the conductive backplate and diaphragm. Quasi-static pull-in. Based on a similar dynamic model as shown in Figure 2-6, for single-backplate condenser microphones, the electrostatic force Fe is given as [5] coAV2 Fe= (2.23) 2d2 where d = do x is the distance between the diaphragm and backplate. By using a linear spring model, the mechanical restoring force, Fm, is given by Fm = k(do -d). (2.24) To find the quasi-static pull-in voltage, the electrostatic attraction force is set to be equal to the mechanical restoring force at dPI = 2do/3 or xPI = do/3 which results in =k do (2.25) 2do 3 Namely, the critical pull-in voltage VPI in the constant bias voltage case is [5] VPI = .(2.26) It should be pointed out that the above analysis results are valid based on a simple parallel-plate assumption and a linear spring model. When the displacement of the diaphragm becomes nonlinear, geometric nonlinearity of the diaphragm needs to be considered to obtain more accurate results. As seen from Eq. (2.23), the electrostatic force is a nonlinear function of the applied voltage and the displacement of the diaphragm. Shown in Figure 2-9 is the plot with both electrostatic and mechanical forces when the constant bias voltage V is less than the critical pull-in voltage V, . Electrostatic force Fe (V < VPS Mechanical force Fm ~~~Unstable region Unstable fixed point Stable regions Stable fixed point a dol3 Displacement x Figure 2-9. Illustration of mechanical and electrostatic forces for a single-backplate condenser microphone (V < V, ). As seen from Figure 2-9, there are three regions and two equilibrium points. Below the stable fixed point, the electrostatic force is always greater than the mechanical force and attracts the diaphragm to the stable fixed point. Between stable and unstable fixed points, the mechanical force is always greater than the electrostatic force and pulls the diaphragm back to the stable fixed point. Above the unstable fixed point, the electrostatic force is always greater than the mechanical restoring force and the displacement of the diaphragm keeps increasing until the diaphragm crashes into the backplate. Therefore, the stable operation range of the microphone is from the rest position to the unstable fixed point when V < V, As the applied bias voltage becomes larger and reaches its critical pull-in value, the electrostatic force is larger than the mechanical restoring force except for one critical point (d, = 2do/3 or x, = do/3) as shown in Figure 2-10. When pull-in occurs, the two fixed points shown in Figure 2-9 move towards each other and coalesce at the pull-in position as shown in Figure 2-10. When the applied bias voltage is over its critical pull- in value, the electrostatic force is always larger than the mechanical restoring force and all regions in Figure 2-9 become unstable. SMechanical force F ElctosaicfoceF (=m l -Electrostatic force Fe (V= VPI) 0 ~Increased bias voltage V/ O dol3 Displacement x Figure 2-10. Illustration of quasi-static pull-in of a single-backplate condenser microphone (V > V ). Scaling. In practice, a condenser microphone needs to be connected to a preamplifier to make a measurement to avoid a signal attenuation or loss. The preamplifier serves as an impedance converter and it is commonly modeled as a source follower with an input capacitance C, and a gainHa The overall sensitivity far below resonance [2] of a single-backplate condenser microphone is compliance dominated and can be expressed as follows Sovevall = S,,S,HcHa, (2.27) where S,, is the mechanical sensitivity, S, is the electrical sensitivity, He is the capacitance signal attenuation due to the input capacitance of the preamplifier C, the parasitic capacitance C,, and the gain of the preamplifier Ha as shown in Figure 2-1 1. Microphone I Parasitic IPreamplifier ICapacitance C I I R I I 5Output Figure 2-11. Simplified circuit of a single-backplate condenser microphone with a preamplifier (adapted from Scheeper et al. [2]). The mechanical sensitivity S,, of the microphone is defined as the change of the diaphragm center deflection resulting from a unit change in the applied sound pressure level. For a circular diaphragm with a residual tensile stress a, the mechanical sensitivity, S,,, is given by [2] S = (2.28) "'8e,h~ where a and h are the radius and the thickness of the diaphragm respectively. It should be pointed out that Eq. (2.28) is obtained based on a pure linear mechanical membrane model, which does not account for the bending forces and moments. The electrical sensitivity S, is defined as the change in the voltage across the air gap resulting from a unit change of the air gap. If the electric field strength E in the air gap is assumed homogeneous, and the electrical sensitivity, S,, is given by S = E = ^, (2.29) where y, is the DC bias voltage. From the simplified circuit, as shown in Figure 2-11, the capacitive signal attenuation He is H= (2.30) where C 4A(2.31) and A = rea (2.32) The microphone capacitance is represented by C,,,,and A is the area of the diaphragm. For the designer, the gain of the preamplifier Ha is close to unity and signal attenuation is not desired. Ideally, the following can be obtained He = 1, (2.33) C,, >> C, + C,. (2.34) Ideally, the overall sensitivity is then changed into Smwoveal SmSe = Sope,,, (2.35) where Sope; is the open-circuit sensitivity. In practice, as the radius of the diaphragm decreases, the capacitance of the microphone C, also decreases. Therefore, Eq. (2.34) does not hold and the microphone will suffer from the output signal attenuation due to the parasitic capacitance. By using Eqs. (2.26) and (2.28), the critical pull-in voltage for a single-backplate condenser microphone can be rewritten as 1 8d 3 V, = o (2 36) P'Sm 27E0 If the applied bias voltage is always kept at a fixed fraction of the critical pull-in voltage, from Eqs.(2.28), (2.29), (2.31), (2.35) and (2.36), the following relations can be obtained C, (2.37) V, ~ d02 (2.38) and S ~ ~ ~V .IV (2.39) From Eq. (2.39), as the area of diaphragm becomes smaller, the open-circuit sensitivity decreases. Also to maintain Eq. (2.34) for the overall sensitivity, the gap needs to become smaller. As the gap becomes smaller, the critical pull-in voltage becomes smaller too; therefore, the open-circuit sensitivity eventually becomes smaller. Hence, from the sensitivity point of view, the gap and area of the diaphragm could be the inherent limitations with the miniaturization of single-backplate condenser microphones. The air-streaming resistance due to the holes of the backplate plays an important role in determining the frequency response or the bandwidth of condenser microphones. For a rigid circular backplate, the mechanical air-streaming resistance Rm is obtained as follows [53] 4pyra~4 1 1 3 1 1 (.0 R I --+-A A2(.0 S 3nd03 4 8 2 P 8 where pu is dynamic viscosity of the air, n is the total number of holes and Ap is the ratio of the total area of holes to the backplate area. Therefore the following scaling relation is obtained Rm (2.41) ndo3 As the gap becomes smaller, the air-streaming resistance increases quickly and decreases the bandwidth of the microphone. Therefore, from the bandwidth point of view, the gap could be one of inherent limitations with miniaturization of single- backplate condenser microphones. However, a very high number of holes (for example, hundreds or thousands) can be used to maintain a small air-streaming resistance for micromachined condenser microphones with narrow air gaps [20]. On the other hand, as the number of holes in the backplate increases, the backplate becomes more compliant and electrostatic pull-in will occur with a lower voltage. Dual-backplate condenser microphones A schematic of a typical dual-backplate condenser microphone is shown in Figure 1-1. It consists of two perforated backplates with a diaphragm in between, a cavity and a vent. Two capacitors are formed by the conductive backplates and diaphragm respectively. Due to the existence of symmetric backplates, a dual-backplate condenser microphone ideally can generate twice the signal of a single-backplate condenser microphone [17]. The configuration of symmetric backplates could lead to a higher sensitivity, a higher capacitance, a wider linear dynamic range [18] and a broader bandwidth. In addition, the applied bias voltage can be higher for a double backplate condenser microphone, since the electrostatic forces acting on both sides of the diaphragm can balance with each other. Furthermore, the dual-backplate condenser microphone has the potential to operate in a closed loop [54]. Details of a dynamic model for a dual-backplate condenser microphone are provided in Chapter 3. Pull-in issues associated with dual-backplate condenser microphone are studied in Chapter 5. Major Previous Work on Capacitive Silicon Microphones Major previous work and recent developments of capacitive silicon microphones [17, 49, 50, 52, 55-67] are listed in Table 2-1. It can be seen that there is a significant trend towards developing instrumentation grade silicon condenser microphones in terms of smaller size, larger sensitivity and signal-to-noise ratio, and broader bandwidth. Hohm and Gerhard-Multhaupt (1984) developed the first electret silicon microphone. Its backplate consisted of a 1 cm x 1 cm p-type silicon layer, a 2 pm SiO2 top electret layer and a 0.1 pm bottom aluminum electrode layer. A circular hole with a diameter of 1 mm was perforated in the center of the backplate. The diaphragm was an aluminum-coated Mylar foil with a thickness of 13 pum and a diameter of 8 mm. A Mylar foil was used as a spacer to form an air gap of 30 pum The reported open-circuit sensitivity was approximately 8.8 mV/lPa, but the measured sensitivity at the frequency of 1 kHz was approximately 3 mV /Pa due to the parasitic capacitance. Table 2-1. Maj or previous work in capacitive silicon microphones. Diapragm Air pper Sensitivity Noise Level Authors Area Gap Frequency (nzV/Pa) ( dBA ) (nzn?) (pn?) ( kHz) aHohm and Gerhard- =50.3 30 8.5 3.0 - Multhaupt [49] aSprenkels et al. [50] 6.0 20 >15 25a bBergqvist and Rudolf [55] 4.0 4.0 16 13a bBergqvist et al. [56] 4.0 2.0 20 1.6 40 bKuhnel and Hess [57] 0.64 2 20 1.8 43 bBourouina et al. [58] 1.0 7.5 10 2.4 38 bScheeper [59] 4.0 3.0 >14 7.80 30. bZou et al. [60] 1.0 2.6 9.0 14.2 - bSchafer et al. [61] 0.5 4.0 17 14 28 bTorkkeli et al. [62] 1.0 1.3 12 4.0 33.5 bRombach et al. [52, 63] 4.0 0.9 >20 13c 22.5 bScheeper et al. [17] = 12 20 20 22c 23 bHansen et al. [64] =0.01 1 100 7.3 63.6 bMartin et al. [65, 66] =0.17 2 230 0.28", d 42e bPedersen [67] =0.1 1.24 100 0.5 22e bLoeppert and Lee [68] =0.25 4 20--- aElectret microphone bCondenser microphone cOpen-circuit sensitivity dCharge amplifier el Hz bin Sprenkels et al. (1989) reported an electret silicon microphone. It had a metallized Mylar foil diaphragm with a thickness of 2.5 pn? and a 20 pn? air gap. A SiO2 layer with a thickness of 1.1 pn? was used as electret material and was biased by a permanent charge with an effective voltage of approximately -300 V. The open-circuit sensitivity was approximately 25 na / Pa at the frequency of 1 kH: . Bergqvist and Rudolf (1990) published the first silicon condenser microphones. A microphone with a 5 pn? thick p-type silicon diaphragm and a 4 pn? air gap demonstrated an open-circuit sensitivity of 13nzV/Pa at the frequency of 1 kHz. Microphones with a 8 pn? thick p-type silicon diaphragm showed a bandwidth ( f1 dB ) of 16 kH: Low parasitic capacitances (< 0.5 pF ) and a large number of holes in the backplate were reported on those microphones. Bergqvist et al. (1991) presented a condenser microphone with a 2 nm x 2 nan stress-free p-type silicon diaphragm and a 2 pn? air gap. The backplate was perforated with 640 acoustic holes per nant to reduce the air-streaming resistance; however, the backplate size was not reported. The microphone operated with a 5 V bias voltage. Its frequency response was flat within f3 dB from 2 to 20 kH: The measured microphone noise was dominated by the preamplifier noise and the equivalent noise level was 40 dBA The measured sensitivity was 1.6 naV /Pa at the frequency of 1 kH: The total harmonic distortion was less than 0.08% at a sound pressure level of 120dB . Kuhnel and Hess (1992) developed a condenser microphone with a specially designed backplate to reduce the air-streaming resistance. The microphone consisted of a 0.8 nam x 0.8 nan silicon nitride diaphragm with a thickness of 150 nm The backplate was structured with either anisotropically etched holes or plasma-etched grooves. Microphones with grooves in backplates and stress-free diaphragms demonstrated an open-circuit sensitivity of 10 naV Pa (1.8 naV Pa measured sensitivity due to the signal attenuation caused by the parasite capacitance) and the measured bandwidth was up to 2 0 kH: . Bourouina et al. (1992) developed a condenser microphone design without acoustic holes in the backplate. In order to lower the air-streaming resistance, a relatively thick air gap was used (5 or 7.5 pn? ). The diaphragm was made with heavily boron-doped silicon and had a residual tension stress of 70 2IPa. A Pyrex wafer was bonded to the diaphragm wafer by anodic bonding. A microphone with an air gap of 7.5 pn? and a bias voltage of 20 V showed a flat frequency response up to 10kH: and a sensitivity of 2.4nzV/Pa. The noise of the microphone mainly came from the preamplifier, and was measured to be 3 8 dBA . Scheeper (1993) presented a condenser microphone with a high density of acoustic holes in the backplate. The microphone possessed an open-circuit sensitivity of 7.8nzV/Pa, a capacitance of 8.6 pF and a flat frequency response from 0.1 kH: to 14 kH: (+2 dB ). Due to the existence of a narrow air gap, the microphone operated at a relatively low bias voltage to avoid pull-in. The pull-in voltage was not reported in the paper. Zou et al. (1997) developed a silicon condenser microphone with a corrugated diaphragm. The microphone had an air gap of 2.6 pn? and a 1 nam x 1 nan diaphragm with a residual tension stress of 70 M~Pa and a thickness of 1.2 pn? A corrugated diaphragm was fabricated by an anisotropic etching to improve the sensitivity by reducing the effect of high tension stress. A microphone with a corrugation depth of 8 pn? and a 10V bias voltage showed a measured sensitivity of 14.2nzV/Pa. The simulated bandwidth was reported to be 9kH:. The finite element method (FEM) and equivalent circuit method were used to predict the microphone performance. Schafer et al. (1998) from Knowles Acoustics, reported a silicon micromachined condenser microphone for the hearing aid application. The microphone had an air gap of 4 pn? and a circular silicon nitride diaphragm with a radius of 0.4 nan and a thickness of 0.75 pum. A highly compliant diaphragm was achieved by the support in the middle rather than the clamped support at the perimeter. A lumped element model was built to predict the sensitivity, bandwidth and noise floor of the microphone. The measured sensitivity of the microphone with a bias voltage of 12 V was 14 nzV/Pa at the frequency of 1 kHz . The measured A-weighted noise level was as low as 28 dBA and the measured resonant frequency was up to 17 kH: . Torkkeli et al. (2000) fabricated a capacitive microphone with a low-stress (2 M~Pa ) polysilicon diaphragm. The microphone had an air gap of 1.3 pn? and a Inan; x 1nan diaphragm with a thickness of 0.8 pum. The measured sensitivity of the microphone with a 2V bias voltage was 4nzV/Pa at the frequency of 1 kHz The measured A-weighted noise level was 33.5 dBA The measured capacitance was 11 pF and the bandwidth was 12 kH: . Rombach et al. (2002) fabricated the first dual-backplate silicon condenser microphone. Due to the symmetric arrangement of backplates, a thin air gap of 0.9 pn? was fabricated to generate a high electrical field and a high sensitivity with a low bias voltage. The microphone consisted of a 2 nam x 2 nan multilayer diaphragm with a thickness of only 0.5 pn? and an overall residual tensile stress of 45 M~Pa The total sensitivity with a bias voltage of 1.5V was measured to be 13nzV/Pa and the A- weighted equivalent noise level was measured to be 22.5 dBA The upper limit of its dynamic range was determined to be 118 dB and the total harmonic distortion at 80 dBA SPL was less than 0.26%. Scheeper et al. (2003) fabricated a condenser MEMS measurement microphone. The microphone had an air gap of 20 pn? and an octagonal silicon nitride diaphragm with a tensile stress of 340 2IPa and a thickness of 0.5 pn? The area of a circular diaphragm was approximately 11.95 am while the area of a square backplate was approximately 8 nm 2. The measured average open-circuit sensitivity was 22nzV/Pa with a 200V bias voltage and the measured noise level (including the preamplifier) was 23 dBA The measured frequency response was flat up to 20 kHz, and the resonance frequency was reported between 47 and 5 1kH: Other important specifications of this microphone such as the measured temperature coefficient, humidity coefficient for the sensitivity and 3% distortion limit were also reported. Hansen et al. (2004) reported a wide-bandwidth micromachined capacitive microphone based on radio frequency detection. The microphone consisted of a metallized rectangular silicon nitride membrane, which was suspended over a silicon substrate to form a small sealed volume. A sensitivity of 7.3nzV/lPa was measured with a gap of 1 pn; and a 70 pn? x 190 pn? membrane with a thickness of 0.4 pn? The measured bandwidth was flat within 0.5 dB over the range from 0.1 Hz to 100 kH: . However, the microphone suffered from its relatively high noise floor, which was measured to be 63.6 dBA . Martin et al. (2005) fabricated a dual-backplate capacitive MEMS microphone by using SUMMiT V process at Sandia National Laboratories. Due to the symmetric arrangement of backplates, a thin air gap of 2 pum was fabricated. The microphone consisted of a circular polysilicon diaphragm with a 230 pum radius and a thickness of 2.25 pum The measured sensitivity with a charge amplifier and a bias voltage of 9 V was 0.28 mV/IPa and the measured noise level was 42 d'BIJA at 1 kHz The measured dynamic range was over 118 dB and a linear response up to 160 dB was observed. The predicted resonance frequency was approximately 185.5 kHz and the measured resonance frequency was approximately 230 kHz [66]. Pedersen (2006) presented an aero-acoustic capacitive MEMS microphone. By utilizing the Knowles SiSonic MEMS microphone technology, the microphone consisted of a circular diaphragm with a 180 pum radius and a thickness of 2 pum An air gap of 1.24 pum was fabricated between the diaphragm and backplate. The measured sensitivity at 1 kHz was approximately 0.5 mV/Pa. The measured total harmonic distortion at 130 dB SPL was 1.5%. The predicted bandwidth was approximately 100 kHz and the measured noise level was 22 d'BIJA at 1 kHz . Loeppert and Lee in Knowles Electronics, LLC (2006) presented the first commercialized condenser MEMS microphone. Fabricated by Knowles's 10-mask, dual- poly process, the microphone consisted of a silicon nitride backplate with a thickness of 1.5 pum and a polysilicon circular diaphragm with an effective 280 pum radius and a thickness of 1 pum An air gap of 4 pum was fabricated between the diaphragm and backplate. The circular backplate of the microphone was perforated with circular holes with a radius of 4 pum and the porosity of the backplate was 22.8%. The measured capacitance of the microphone was 0.5 pF The measured frequency response was flat from 100 Hz to approximate 20 kHz However, the measured sensitivity and noise floor were not reported in the paper. Basics of Nonlinear Dynamics Nonlinear dynamics is the study of the time evolving behavior of a nonlinear system. These systems are often governed by nonlinear partial/ordinary differential and/or algebraic equations [69]. To date, nonlinear dynamics has been applied to many fields of engineering and science [69-77]. To help provide the reader with some familiarity to the field of nonlinear dynamics, some general terminologies and theories are introduced. In the sections that follow, several key concepts, such as nonlinearities, autonomy, fixed points, stability and bifurcation, are discussed. Nonlinearities Although there are many types of nonlinearities, the maj or nonlinearities in the field of engineering can be categorized into the following four types [6, 73]: * Material or constitutive nonlinearity, examples include the inelastic phenomena (plasticity, creep, hysteresis, etc.) in solid mechanics, and nonlinear resistor (for example, thermal effect on resistance) in electrical engineering. * Kinematic nonlinearity, examples include Coriolis acceleration and the convective acceleration term in the Navier-Stokes equations in fluid mechanics. * Nonlinear force or boundary, examples include the nonlinear electrostatic force, magnetic force and a beam attached to a nonlinear torsional spring at one of its pinned end. * Geometric nonlinearity, examples include a Hertzian contact model for two elastic bodies and a Duffing's spring. Steady State, Autonomous System and Fixed Point The steady state refers to the asymptotic behavior of the dynamical system as time goes to infinity [69]. The behavior prior to the steady state is often called transient [69]. An autonomous dynamical system is a system that does not contain time as an explicit independent variable [69]. Otherwise, a dynamical system is defined as a non- autonomous dynamical system if it explicitly depends on time [69]. An example of an autonomous dynamical system is the unforced Duffing's equation, mit+ b + k, x +k,x = 0, (2.42) where x is the displacement, m is the mass, b is the damping coefficient, and k, and k, are the linear and nonlinear spring stiffnesses, respectively. An example of a non-autonomous dynamical system is the damped Duffing's equation with an external harmonic excitation, mit+ b + k,x +k,x3 = A cos (st), (2.43) where A and 0Z are the forcing amplitude and frequency, respectively. 8 ~k3 = 0 I ~ ~ ~ k U Y i I( cubic nonlinearity) I ~I Jk~ Forcing Frequency R Figure 2-12. Effect of cubic nonlinearity on the system frequency response. Shown in Figure 2-12 is the effect of cubic nonlinearity on the frequency response of the system defined in Eq. (2.43). As the cubic nonlinearity increases, more bending is observed in the response curve. Also, multiple steady-state solutions could be obtained in some frequency range, which indicates the system is nonlinear. To define the term Eixed point, consider the following general autonomous dynamical system as follows S= f (2), (2.44) where 2 represents a column vector of all state variables, and f is a column vector of all corresponding general functions. Physically, a Eixed point 2, (also known as steady state solution) corresponds to an equilibrium position of a system. Mathematically, it is the solution to the following equation f(2,)= 0. (2.45) Stability, Basin of Attraction and Phase Portrait For stability, a fixed point is locally stable if, given an initial condition sufficiently close to the Eixed point, the system eventually approaches this Eixed point. A Eixed point is globally stable if it is approached for all initial conditions [69, 75]. For a linear system, a locally stable fixed point is always globally stable. However, for a nonlinear system, starting from different initial conditions could result in different steady-state solutions. Local stability does not guarantee global stability, which leads to the concept of a basin of attraction. The domain of all initial conditions that eventually converges to a fixed point is called the basin of attraction (stability region) of the fixed point. More details about basins of attraction will be provided in Chapter 5. A phase portrait is a collection of all trajectories that represent the solutions (both transient and steady state solutions) of the governing equation in the state space [69, 77]. Phase portrait is one of most important tools in studying the behavior of nonlinear systems since usually there is no closed-form solution for most nonlinear systems. Outer Physiys cal border lines ~ Outer range r ange a,0.51 -15- 05 . Non-imeniona Cete Displceen Fiur 2-3 Phas pln rjcoisaon ie onsoada-aklt aaitv pont are iniae bybuice) I1 -, x, 2 where the two states x, and x2 are the non-dimensional center displacement and velocity of the diaphragm respectively, and other non-dimensional parameters are assumed to be fixed in the phase portrait. From the plot, we can see the coexistence of multiple fixed points: two locally stable sink points (attractors) outside the physical border lines, one locally stable sink point in the center and two unstable saddle nodes (repellors) between the center point and physical border lines respectively. Bifurcation When one or more parameters of the dynamical system are varied, the qualitative change of response behavior, such as a change in the number of stable solutions, is called a bifurcation [69]. The varying parameter is often called the bifurcation parameter. Diagrams, in which the variation of fixed point solutions and their stability are displayed in the space of state variables and bifurcation parameters, are often called bifurcation diagrams [69]. The critical location, at which a bifurcation occurs in the bifurcation diagrams, is called a bifurcation point [69]. SStable 8I Unstable Bifurcation Parameter a Figure 2-14. Typical bifurcation diagrams for one-dimensional autonomous systems: (a) Saddle-node bifurcation; (b) Pitchfork bifurcation; (c) Transcritical bifurcation (adapted from Nayfeh and Balachandran [69]). For simplicity, one-dimensional autonomous systems are used here to illustrate the basic idea of a bifurcation. Shown in Figure 2-14 are three typical static bifurcation diagrams. The saddle-node bifurcation diagram is generated by considering the following dynamical system x = a x2, (2.47) where a is the bifurcation parameter. In the saddle-node bifurcation, two solution branches start growing from the bifurcation point at a = 0 one is stable and another is unstable. The pitchfork bifurcation diagram is generated by considering the following dynamical system x = ax x3. (2.48) In the pitchfork bifurcation, initially, there is only one stable solution branch. As the bifurcation parameter increases, the stable solution branch evolves into three different solution branches: two stable solution branches outside and one unstable solution branch between them. The bifurcation point is located at a = 0 . Finally, the following dynamical system is considered to generate the transcritical bifurcation x= ax-x2. (2.49) In the transcritical bifurcation, stabilities of the original solution branches change into their counterparts when the bifurcation parameter is sweeping over its bifurcation point (in this example, it is located at a = 0 ). Previous Work on Nonlinear Dynamics of Electrostatic MEMS Devices Recently, many electrostatic 1VEMS devices, such as microphones [65], micropumps [78], microswitches [79], pressure sensors [80] and mass sensors [14], have been fabricated due to the advances of modern silicon micromachining technology. Many types of nonlinear forces and geometrical nonlinearities were encountered in those MEMS devices. To adequately study these nonlinear behaviors, one analysis approach is to assume a linear relationship between the physical quantities and neglect or linearize those nonlinearities. It turns out that the linear method produces erroneous results, for example, when the device is under the large deflection or large excitation. Therefore, one has to resort to a nonlinear analysis. In this section, previous work on nonlinear dynamics of electrostatic MEMS devices is summarized. In 1954, F. V. Hunt [3] studied pull-in instability for a single-sided electrostatic loudspeaker. By equating a linear mechanical restoring force to a nonlinear electrostatic force, he found that quasi-static pull-in (referred to it as "fall in" in his book) occurs at two thirds of the gap between the diaphragm and bottom electrode of the loudspeaker. Pull-in experiments were carried for the loudspeaker by monitoring the displacement of the diaphragm; pull-in occurred at a value of 78% of the gap. To explain the difference between the theoretical and measured critical pull-in locations, he suggested that the most likely reason was the exclusion of the curvature of diaphragm in the model. H. C. Nathanson and his coworkers [4] in 1967 studied pull-in of an electrostatic resonant gate transistor. To understand the electrostatic actuation in their device, a simplified mass-spring model based on the parallel-plate assumption was constructed and analyzed, and pull-in instability was predicted and explained by using the 1-D model. In 1968, Taylor [81] observed pull-in phenomenon when he increased the voltage between the two closed-spaced liquid drops. The two drops remained apart when the applied voltage was small, and coalesced when the voltage was beyond a threshold voltage. The threshold voltage is referred to as pull-in voltage. Puers and Lapadatu [51] studied electrostatic forces and their effects on capacitive mechanical MEMS sensors (accelerometer and pressure sensor). In a constant bias charge mode, the critical pull-in charge and mechanical force were obtained analytically based on a simple parallel-plate assumption and a linear spring model. The influence of damping on the critical pull-in charge was also discussed. In a constant bias voltage mode, the critical pull-in voltage and mechanical force were also obtained analytically. Moreover, possible solutions, such as the use of symmetrical structures, were suggested to avoid pull-in in a constant bias charge mode. Pedersen et al. [82] investigated the harmonic distortion in micromachined silicon condenser microphones. A quasi-static model, which includes a more realistic shape of the diaphragm deflection, was applied to study the harmonic distortion for frequencies far below the resonance of the diaphragm. Instead of an analytical series solution, an iterative numerical approach based on finite differences was implemented to find the quasi-static diaphragm deflection and open-circuit sensitivity of the microphone. The harmonic distortion generated by the sound pressure and nonlinear electrostatic force due to a bias DC voltage was studied numerically and experimentally for a silicon condenser microphone. Both numerical and experimental results showed that the total harmonic distortion of the microphone was linearly dependent on the applied sound pressure level and was a nonlinear function of the applied DC bias voltage. On the other hand, some researchers are looking for ways to take advantage of pull- in in their MEMS devices. Gupta and Senturia [80] presented a MEMS absolute pressure sensor utilizing pull-in. The device operated with two modes: non-contact and contact modes. During the contact mode, the device was dynamically actuated by an applied step voltage higher than the pull-in voltage, pull-in time from rest to contact was measured and found to be a nearly linear function of the absolute pressure inside the sensor. A simple 1-D lumped mass-spring-damper model was constructed to theoretically calculate the pull-in time for an electrostatically actuated fixed-fixed microbeam. The electrostatic force was modeled based on the parallel-plate assumption and the fringing field effect [83] was neglected. Also, simulated pull-in times were found to be in good agreement with the measured values. Zavracky et al. [79] reported a micromechanical switch with three electrodes: a source, a gate and a drain. When the applied voltage between the gate electrode and a microbeam, which connects to the source electrode and hangs over the gate and drain electrodes, was over the pull-in voltage, contact was made between the source and drain electrodes. Both numerical and analytical methods were used to investigate the pull-in of the microbeam; however, the analytical model based on the parallel plate assumption yielded poor results. A nonlinear spring constant of the microbeam was extracted from the numerical simulation results and applied back to the analytical model to obtain good results. Nemirovsky and Bochobza-Degani [84] presented a generalized model for the quasi-static pull-in parameters of electrostatic actuators with a single charge or voltage input. By setting the first and second derivatives of the total energy (the summation of mechanical and electrical energy) with respect to the displacement of the moving electrode plate to zero, two general algebraic equations were obtained to solve for the pull-in parameters for each type of input respectively. The obtained equations were applied to a wide range of case studies, including parallel-plate and tilted-plate electrostatic actuators with the fringing field capacitance, the parasitic capacitance, residual charges, constant external forces and nonlinear mechanical forces. Specifically, the analyses results showed that the addition of cubic mechanical nonlinearity in the form of a Duffing's hardening spring extends the stable range of electrostatic actuators quasi- statically. In addition, it was showed that the quasi-static pull-in parameters (such as voltage and displacement) can be affected by the constant external force or pressure. For example, the quasi-static pull-in voltage was reduced when the external force or pressure was in the same direction with the electrostatic force and vice versa. Zhang et. al. [13, 14, 85] investigated the nonlinear behavior of a parametric resonance-based MEMS mass sensor. Cubic nonlinearities were modeled for the mechanical restoring force of a fixed-fixed beam and the electrostatic force of a comb finger. The sensor was modeled by a lumped mass-spring-damper system with an external electrostatic force. A Duffing equation and a nonlinear Mathieu equation [69] were derived to model the behaviors of nonlinear harmonic and parametric resonances. A perturbation method (multiple scales method) [72] was used to explore the effect of damping and cubic nonlinearity on the parametric resonance. Two pitch-fork bifurcations and three different stability regions were finally obtained for the dynamic characteristics of the nonlinear Mathieu equation. Experimental frequency responses showed that the system is linear with a small AC excitation and nonlinear with a large AC excitation. Also some model parameters, such as the Q factor, cubic mechanical stiffness and linear electrostatic stiffness, were extracted from the experimental data. Nayfeh and his co-workers presented a series of nonlinear models for the electrically actuated annular plates [78], fixed-fixed rectangular beams [86, 87], simply supported rectangular plates [88], and clamped circular plates [89, 90]. The static deflections of those microstructures are determined by either an analytical reduced-order model (macromodel) or a numerical shooting method, which is widely used to determine the periodic solutions of autonomous and non-autonomous systems [69]. The shooting method is an iterative procedure and computationally costly, and becomes numerically unstable when pull-in is approached; therefore, it is not suitable for the prediction of pull- in. On the other hand, the reduced-order model [86-90] usually approximates the system dynamics with N coupled nonlinear ordinary-differential equations, which are obtained by the spatial discretization of the di stributed-parameter governing equation and associated boundary conditions via a Galerkin approach or finite-element method. The reduced-order model is robust up to the pull-in point, and it also has the capability to account for the in-plane residual stress, general material and geometric nonlinearities. Younis et al. [86] and Abdel-Rahman et. al. [91] applied the reduced-order model to simulate the dynamical behavior of a MEMS switch and predict its pull-in time. A saddle-node bifurcation of a microbeam was found due to the pull-in. Two deflection solution branches of the microbeam moved closer to each other as the DC voltage increased, and finally coalesced when pull-in voltage was reached. Based on the reduced-order model, they also calculated that the deflection at the pull-in is approximately 57% of the gap. Younis and Nayfeh [92] studied the nonlinear response of a resonant microbeam under an electrostatic actuation. A nonlinear model was first built to account for the mid- plane stretching and an electrostatic load with both DC and AC components. A perturbation method was applied directly to analyze the nonlinear forced response to a primary-resonance excitation of its first mode. The analysis results showed that the resonance frequency can be affected by the damping, mid-plane stretching, and electrostatic nonlinearity. This paper showed that the DC electrostatic load could result in both softening and hardening spring behaviors. In addition, the method of multiple scales was applied to investigate a three-to-one internal resonance between the first and second modes of the clamped-clamped beam. The analysis result showed that those two modes are nonlinearly uncoupled and therefore the internal resonance cannot be activated. Chowdhury et. al. [93] studied the nonlinear effects in a MEMS capacitive microphone. The microphone was modeled as a second-order single-degree-of-freedom system. The spring hardening effect due to the mid-plane stretching and the spring softening effect due to the nonlinear electrostatic force were presented. By using Taylor series expansion, the nonlinear electrostatic force was linearized and pull-in voltage was derived for a fully clamped square diaphragm with a built-in tension stress. Finally, nonlinear finite element analyses were carried out to verify the analytical results. Nadal-Guardia et al. [94] presented a compact 1-D lumped model of capacitive silicon sensors. By applying a perturbation method to the dynamic governing equations, transfer functions of the system were developed analytically. When comparing to the traditional equivalent analog circuit model, the obtained transfer functions can account for the effect of pull-in instability. Numerical simulations with the transfer functions were conducted, and the results were compared with the experimental data for a capacitive MEMS microphone. Both simulation and experimental results showed that pull-in instability improved the low-frequency response of the microphone for DC bias voltages close to its critical pull-in value. However, the developed transfer function could not predict the sensitivity change of the microphone as a function of the bias voltage. This limitation is mainly due to the application of constant model parameters, which was obtained for a certain DC voltage value, for a whole range of the bias voltage. Fargas-Marques and Shkel [95] studied both static and dynamic pull-in conditions for an electrostatic MEMS resonator based on the parallel-plate actuation. They used a 1- D lumped mass-spring-damper model and derived the condition for AC dynamic pull-in based on the kinetic and potential energy of the system. The experimental results for the pull-in voltages agreed well with their analytical results. The dynamic pull-in voltage was reported to be approximately 8% lower than the static pull-in voltage. Their energy analysis results also showed that the quality factor or damping of the system had an impact on the dynamic pull-in voltage. Elata and Bamberger [96] presented a purely theoretical study on the dynamic pull- in of electrostatic actuators when subjected to instantaneous application of DC voltages. Based on Hamilton's principle [97] and quasi-static equilibrium equations, dynamic pull- in for general undamped electrostatic actuators with multiple degrees of freedom and voltage sources was formulated. Specifically, the critical dynamic pull-in voltages were found approximately 8% lower than the corresponding quasi-static pull-in voltages for electrostatic actuators with parallel-plates, double parallel-plates, and clamed-clamped beams. For the electrostatic actuator with a block of comb drives with no initial overlap, the critical dynamic pull-in voltage was found approximately 16% lower than its quasi- static pull-in voltage. If damping exists in electrostatic actuators (which is always the case in practice), they concluded that the actual dynamic pull-in voltage was bounded between the quasi-static pull-in voltage and the dynamic pull-in voltage without damping. Although previous research work provided a good understanding of various aspects of nonlinear dynamics of electrostatic MEMS devices, none of these studies is helpful for a dual-backplate capacitive MEMS microphone. The rest of this dissertation is devoted to investigating the nonlinear dynamics issues associated with a dual-backplate capacitive MEMS microphone. CHAPTER 3 NONLINEAR DYNAMIC MODEL This chapter derives a nonlinear model for the dynamics of a dual-backplate MEMS microphone. First, general displacement solutions (small displacement by an analytical method and large displacement by an energy approach) of the middle diaphragm are provided based on the plate theory. Based on the general displacement solutions, lumped element modeling is used to extract the parameters of the diaphragm. Other lumped parameters of the microphone, including the damping coefficient and stiffness of the cavity, are also presented. After all the lumped parameters of the microphone are obtained, the general nonlinear governing equation is derived and the model nonlinearities are discussed. Finally, nonlinear finite element analyses (FEA) are carried out to verify the theoretical lumped stiffnesses of the diaphragm. Microphone Structure The dual-backplate capacitive MEMS microphone studied here has been fabricated using the SUMMiT V process at Sandia National Laboratories as well as facilities at the University of Florida for post-processing [65, 66]. A 3D cross-section view of the microphone is shown in Figure 3-1. It has a 2.25 Clm thick circular solid diaphragm with a 230 Clm radius and a 2 Clm gap between each circular perforated backplate. The 5 Clm radius holes in the backplates allow the incident acoustic pressure to act on the diaphragm. A cavity under the bottom backplate is formed via a deep reactive ion etch, and vented to the ambient pressure resulting in an AC measurement device. Pressure Back ,late Holes Top Backplate Diaphragm Bottom Backplate Cavity V, (t) V, (t) . Gaps ~Anchors Figure 3-1. 3D cross-section view of the microphone structure (not to scale). Figure 3-2 shows a microscope photograph of the microphone top with a field of view of approximately 1 mm x Imm The bond pads shown in the photograph enable electrical connections between the backplates and diaphragm. Bottom backplate Diaphragm Top backplate Bond pads 1 mm Figure 3-2. Top-view photograph of the microphone. Displacement Solutions of the Diaphragm The diaphragm is a key energy transduction component and plays a significant role in determining the performance of the microphone. Therefore, its mechanical behavior under the uniform transverse pressure loading is studied first. It is assumed that the circular polysilicon diaphragm is linearly elastic, isotropic and axisymmetric. Also, zero in-plane residual stress [98] and a clamped boundary condition are assumed in the analysis. Small Displacement Solution Shown in Figure 3-3, a clamped circular diaphragm is subject to a transverse uniform pressure loading p The radius and thickness of the diaphragm are denoted by a and h respectively. From the plate theory, if the transverse deflection of the diaphragm is much smaller than its thickness, the strain in the neutral plane of the diaphragm can be neglected and the solution in this case is called the small displacement solution [99]. r=0 r=a Figure 3-3. Schematic of a clamped circular diaphragm under a transverse uniform pressure loading. For the static small displacement solution, the governing differential equation for the transverse deflection w (r) in a polar coordinate system is given as follows [99] dl 1 d pr where r is the distance of any radial point along the radius and D is the flexural rigidity of the diaphragm defined as Eh3 D =. (3.2) 12 1- v' The parameters E and v are the Young's modulus and Poisson's ratio of polysilicon respectively. The boundary conditions are w r(a)= 0, (3.3) dr I= and w (0) < 00. (3.4) Solving Eq. (3.1) with Eqs. (3.3) and (3.4), the small displacement solution becomes w r)= (3.5) The center displacement of the diaphragm is then w = w(0) = pa (3.6) 64D Energy Method and Large Displacement Solution From the plate theory, if the transverse deflection of the diaphragm is comparable to its thickness, the strain in the neutral plane of the diaphragm cannot be neglected and the solution in this case is called the large displacement solution. For the large displacement solution, it can be assumed from the plate theory that the deflected surface in such a case is still symmetrical with respect to the origin of the circular diaphragm. In this section, an energy approach is used to obtain an approximate large displacement solution. A more rigorous theory of the nonlinear mechanics of transducer diaphragms including the effects of in-plane stress is given in the reference [100]. Procedure of energy method By applying an energy method [97, 99], trial functions with unknown coefficients for the transverse displacement and in-plane displacement need to be chosen first. Typical choices for the trial functions are polynomials and orthogonal functions such as sinusoidal or cosine functions. However, trial functions must satisfy the boundary condition, which limits the choices of trial functions. Once the trial displacement functions are assumed, the strain energy can be then calculated by using the strain-displacement relation (linear and nonlinear) and constitutive equation. Also, the work done by the external load can be calculated. The potential energy of the system is the summation of strain energy and work done by the external load. Finally, by applying the principle of minimum potential energy for the equilibrium condition, a set of linear algebraic equations is obtained and further solved to yield the unknown coefficients in the trial functions. Thereafter, the approximate solutions for the transverse displacement and in-plane displacement are determined. The next section presents the application of an energy method to solving the large displacement solution for a clamped circular diaphragm, and the detailed steps are provided in Appendix A. Large displacement solution The polar coordinate frame is introduced here to facilitate the analysis for a circular diaphragm. As mentioned previously, the radial displacement in the neutral plane of a diaphragm cannot be neglected when large transverse deflections occur. The displacement of a point in the neutral plane of a circular diaphragm is decomposed into two components: u(r) in the radial direction and w(r) perpendicular to the neutral plane as shown in Figure 3-4. The shape of the transverse deflection surface of a circular diaphragm is assumed to be similar with the one in a small displacement case, which is defined in Eq. (3.5) as follows w(r)=- w, 1- (3.7) where w, is the unknown center displacement that needs to be determined. The assumed transverse deflection satisfies the clamped boundary conditions defined in Eqs. (3.3) and (3.4). w(r) dB Figure 3-4. Displacement components in the neutral plane of a circular diaphragm. To satisfy the clamped boundary conditions, the radial displacement u(r) must vanish at the edge and the origin of the circular diaphragm; therefore, the following polynomial is used to approximate the complex expression of the radial displacement [99] u(r) = r(a -r)(C, + C r), (3.8) where C, and C, are the two unknown constants. Based on the von Karman plate theory [99, 101] and energy method (detailed derivation steps are provided in Appendix A), finally we can determine the three unknown coefficients for a polysilicon diaphragm (Poisson's ratio v = 0.22 is used in the derivation, note that the following results are only valid for this specific Poisson's ratio) as follows C, =1.2652 o- (3.9) a3 C2 =-1.8129 0-, (3.10) and wo = 2. (3.11) 64D w 1+0.4708 O h2 Eq. (3.11) can be rewritten as 2 4 wo 1+0.4708 w__ (3.12) h2 64D The final approximate expression for the large displacement solution can be determined by substituting the solution of wo from Eq. (3.12) into Eq. (3.7). As we can see from Eq. (3.11), the small factor 0.4708 w02 h2 TepfeSents a geometric nonlinearity (nonlinear spring hardening effect) due to the in-plane stretching when large displacements occur. The diaphragm can now be modeled as a nonlinear Duffing spring, where the two spring constants can be obtained via further lumped element modeling in the following section. As seen from Eq. (3.7), the assumed mode shape based on this energy approach is not affected by the applied pressure. Shown in Figure 3-5 is a plot of different normalized mode shapes for several pressure values. In the plot, three normalized mode shapes are generated based on the exact solution given in the reference [100]. As we can see from the plot, for larger pressure values (for example, 100000 Pa), the assumed mode shape used in the energy approach is not accurate. However, for our 2000Pa microphone design, the assumed mode shape used in the energy approach is in good agreement with the exact mode shape as shown in the above plot. ----***** Exact solution, p=2000 Pa 0.9 -*. ------ Exact solution, p=10000 Pa ----- Exact solution, p=100000 Pa 0.8~ Energy-based solution 0.7 - 0.6 ? \\ \ \ p 100000 Pa 0.4 \\ 0.3 s p =2000 Pa \ 0.2 - 0. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/a Figure 3-5. Normalized mode shape for several pressure values (2000, 10000 and 100000 Pascals). Lumped Element Modeling of the Microphone The microphone is a typical multi-domain (acoustical, mechanical and electrical) system and its dynamics are usually governed by a set of coupled nonlinear partial differential equations, which are difficult and complex to study. As we can see from Eq. (3.7), the displacement of the diaphragm is not uniform but distributed along the radial direction. The associated potential and kinetic energy are also distributed along the radial direction of the diaphragm, which indicates that the microphone is a distributed- parameter system. One alternative for approximating the coupled di stributed-parameter system dynamics is through the construction of a lumped-element model. In general, lumped element modeling is based on the assumption that the device length scale of interest is much smaller than the characteristic length scale (for example, wavelength) of the physical phenomena [29, 102, 103]. By using lumped element modeling, the spatial variations of the quantities of interest can be decoupled from the temporal variations. The coupled di stributed-parameter system can be then divided into many idealized lumped impedances (mass, stiffness and dissipation) [29]. The dynamic behavior of the system can be described only by ordinary differential equations with time being the only independent variable, which is comparatively easier (than coupled nonlinear partial differential equations) to analyze in practice. To carry out lumped element modeling, usually a lumped reference point needs to be defined first to lump a di stributed-parameter system into a single/multiple degree of freedom system. In the case of a circular diaphragm, the center point is chosen. In this section, the lumped parameters for the microphone are discussed. Since the diaphragm is a key part of the microphone and plays a crucial role in determining the overall performances of the microphone, the diaphragm is studied first by the lumped element modeling. Lumped Parameters of Diaphragm Before proceeding to extract lumped parameters, the reference point for the circular diaphragm is chosen to be at its center position r = 0. The general applied transverse pressure is assumed for simplicity to be a harmonic oscillation p(t) = poeaco', (3.13) where 0i is the angular frequency of the pressure oscillation. The dynamic small displacement solution is obtained by modifying Eq. (3.5) p(t)a4 2 "4 w(r,t) = 1-0 -1 esco'. (3.14) The velocity is v(r,tf)= -Iw(r, t)= je~w(r,t). (3.15) Therefore the displacement and velocity at the reference point are 64D and vo (t) = jew, (t). (3. 17) Lumped mass The total kinetic energy of the diaphragm is given by W, =v' (r, t) pdA = v? (r, t) p 2xir~dr, (3.18) where p' is the mass per unit area. Substituting Eqs. (3.15), (3.16) and (3.17) into Eq. (3.18) yields ira p' vo (t)M vo (t) W = 5 2 = ,,e 2 (.9 Therefore, the work-equivalent lumped mass is calculated as M~,,, = WK (3.20) J o t2 ]=~~~5 iah5 where p is the mass per unit volume and M~,,, is 1/5 of the mass of the actual diaphragm. Physically, the diaphragm with distributed deflections has been replaced by a rigid disk (piston) and a linear spring. In order to conserve the kinetic energy, a rigid disk with 1/5 of the mass of the actual diaphragm is used. Lumped linear stiffness and compliance The total potential energy of the diaphragm can be expressed as WPE = SFdx = py(t)2x~rdrMv(r, t). (3.21) From Eqs. (3.13), (3.14) and (3.16), we know 64D p(t) = 4 w(t), (3 .22) and Av (, t)= 1-r g,(t).(3.23) Substituting Eqs. (3.22) and (3.23) into Eq. (3.21) yields 64D )i~ r r~~ 64xD wt (t)(t=3a WPE 4 2x 1-n dr t)Av (t) (3.24) Therefore, the work-equivalent mechanical stiffness is calculated as k, =WP (3.25) ~E2 3a2 The mechanical compliance is 1 3a2 C, (3.26) ;ek, 64xiD Physically, the diaphragm with distributed deflections has been replaced by a rigid disk (piston) and a linear spring. By using the lumped stiffness or compliance, the conservation of potential energy in the physical diaphragm and an idealized linear spring is ensured. Lumped area To ensure the continuity of volumetric velocity in physical and lumped domains, the lumped area of the diaphragm needs to found. The volumetric velocity through the diaphragm is given by Q = v(r, t) dA = v (r, t) 2mrd. (3.27) From Eqs. (3.15) and (3.17), we have vr)= 1 ra~ vot (3.28) Substituting Eq. (3.28) into (3.27), we can get Q = ra vo (t) A Amvo (t). (3.29) Therefore the equivalent lumped area is calculated as ria2 A Ae = Q/ vo (t) = ,(3.30) 3 3 where Ae is 1/3 of the area A of the actual diaphragm. The distributed deflection of the diaphragm has been replaced by the translation of a rigid disk (piston) with an area that is 1/3 that of the actual diaphragm. Lumped cubic stiffness The lumped cubic stiffness of the diaphragm is developed in this section. Eq. (3.25) only shows the linear spring constant of the diaphragm for a small displacement case. For a large displacement case, a nonlinear Duffing spring model is considered for the diaphragm, which is F pAme =kwo + k3 03. (3.31) Comparing Eqs. (3.12) and (3.30), we have 64Dri k, = ,(3.32) 3a2 and 10.044Dri k =(3.33) 3 a2h2 Eq. (3.32) is same with Eq. (3.25) because both energy-based large displacement and small displacement solutions used a same deflection shape defined in Eq. (3.7). When the diaphragm vibrates, the equivalent structural damping of the diaphragm mainly consists of two parts: the thermoelastic energy dissipation in the diaphragm and the vibration energy dissipation in the compliant boundary of the diaphragm [9, 10]. Usually the equivalent structural damping is determined experimentally, it is neglected in the current model since it is small comparing with the dominant viscous damping. The experimental result for the dominant viscous damping will be discussed in Chapter 6. Lumped Damping Coefficient The two backplates of the microphone are designed to be perforated with a large number of holes. One purpose of these holes is to let the air pass through without deflecting the backplates and make them acoustically transparent. A repetitive pattern of holes in the top backplate is shown in Figure 3-6. The top backplate has a radius of 256 Clm and 557 holes with a radius of 5 Clm. A similar hole pattern exists for the bottom backplate, which has a radius of 213 Clm and 367 holes with a radius of 5 Clm. 512an Figure 3-6. Repetitive pattern of holes in the top backplate. When the diaphragm vibrates, the gas flow between the diaphragm and backplate can be divided into two parts, the horizontal gas flow between the parallel plates and the vertical gas flow through the backplate holes. Viscous damping caused by the horizontal gas flow is often called squeeze-film damping, and the viscous damping caused by the vertical gas flow is called holes resistance. The latter becomes important and cannot be neglected especially when the thickness of backplate is comparable to the gap thickness [7, 8]. Since the viscous damping plays a very important role in determining the bandwidth and noise floor of the microphone, its modeling becomes important for this research. For the squeeze-film damping in the gap, the mechanical lumped damping coefficients are given as follows [8, 53] be = B A,,) (3.34) and bbps~ 3Y d3nbdBAbp> (3.35) where pu is the dynamic viscosity of air, d, is the nominal gap between backplates and diaphragm, ntp and nbp are the numbers of holes in the top and bottom backplates respectively. The porosity of the top and bottom backplates is given by A,, and Abp respectively, and B( ) is a function defined as B(A = Iln --+-A A (3.36) 4 82 8 When the thickness of the backplate is comparable to the gap, the viscous damping due to the vertical gas flow through the backplate holes becomes important. By modeling the motion in the holes as a pressure-driven Poiseuille flow in a pipe, the mechanical lumped damping coefficients are given as follows [7, 8] h 8 psh n, b, (3.37) A,, and h 8 pr\ nbp hbbp (3.38) Ab p where h,, and h~ are the thicknesses of the top backplate and bottom backplate respectively. Therefore, the total viscous damping of the microphone is, b = s~lbp "+b h+hb h b (3.39) Lumped Stiffness of the Cavity The cavity of the microphone impedes the movement of diaphragm by storing potential energy and acts as a spring. Based on the lumped element assumption, the work-equivalent lumped acoustic compliance of the cavity is given as follows [104] Cca 2 (3.40) pco where Vc is the volume of the cavity, p is the density of air, and co is the isentropic speed of sound in air. It should be pointed out that Eq. (3.40) is valid for kde < 0.3, where k is the wave number [104] and de is the depth of the cavity. For the cylindrical cavity with a circular cross section, the volume is Vc = rac~2dc, (3.41) where and ac is the radius of the cavity. Therefore the mechanical lumped stiffness of the cavity is irac2 2 k = ,(3.42) CoC where (rac12 2 iS used to convert the acoustic stiffness into the mechanical stiffness. Since the lumped stiffness of the cavity is in parallel with the linear stiffness of the diaphragm, the first resonant frequency of the system is approximated by fnaur (3.k(;43) As seen from the Eq.(3.43), the existence of the cavity increases the bandwidth of the capacitive microphone. Based on the results of the lumped element modeling, the calculated nominal values of lumped parameters as well as the material properties and physical dimensions of the designed 2000Pa dual-backplate condenser microphone are summarized in Table 3-1 [65, 66]. Table 3-1. Material properties and physical parameters of the 2000Pa microphone in metric units (material: polysilicon). Parameter Nominal value Young's modulus (E) 1.60el l (Pa) Poisson's ratio (v) 0.22 Density (p) 2.23e3 (kg/m3) Thickness of the diaphragm (h ) 2.25e-6 (m) Radius of the diaphragm (a) 230e-6 (m) Linear spring constant of the diaphragm ( k, ) 202 (N/m) Cubic spring constant of the diaphragm (k3 ) 1.88el3 (N/m3) Lumped mass of the diaphragm (MIme ) 16.7e-11 (kg) Lumped area of the diaphragm ( Ame ) 5.54e-8 (m2) Gap (do) 2.00e-6 (m) Depth of the cavity ( de ) 650e-6 (m) Radius of the cavity (ac ) 187e-6 (m) Linear spring constant of the cavity ( ke ) 24.3 (N/m) Thickness of the top backplate ( h, ) 2.25e-6 (m) Thickness of the bottom backplate (h~ ) 2.50e-6 (m) Number of holes for top backplate ( n, ) 557 Number of holes for bottom backplate ( n ) 367 Ratio of the total holes area to the backplate area (A,, = A,,) 0.22 First resonant frequency of the microphone system ( fnature) 185 (k Total damping ratio 8.09e-2 *First resonant frequency of diaphragm 175 (k FEA results with fixed boundaries Further calculations show that the first natural frequency of the system is increased by approximately 5.9% due to the existence of cavity. If the cavity stiffness is neglected in Eq. (3.43), the first natural frequency of the diaphragm via LEM is approximately 175.3 k diaphragm is to solve the transverse bending wave equation for the circular diaphragm with a fixed boundary [105]. Further studies show that the theoretical first natural frequency of the diaphragm is approximately 173.3 k element assumption is approximately 1.1%. C'- C' trnsaen ndte oplineo h c ihaviie bewe h ipramadtpbto bakpats re negeced A implified lumped element model of da-cklth aaiie microphoneshw in Figure 3-7 is used to verify these assumptions. This model describes the microphone in the acoustic domain, where the mechanical lumped parameters derived in the above section are converted to their analogous acoustic parameters. Additionally, other features of the microphone, such as the vent resistance and the compliance of each backplate, are also included in this model [66]. Readers are referred to [66] for more details of constructing this lumped model. In the model, Iq represents the incident acoustic pressure. C;, Ca> b~a, and Cf are the acoustic compliances of the top backplate, diaphragm, bottom backplate and bottom cavity, respectively. C' and C'", are the acoustic compliance of the cavities between the diaphragm and top/bottom backplates, respectively. Ra and Rbap are the acoustic resistance of the top backplate and bottom backplate, respectively. Ma~ is the acoustic mass of the diaphragm and Ra is the acoustic vent resistance from the cavity to the incident acoustic pressure. For the cavities between the diaphragm and top/bottom backplates, each cavity volume is less than 0.5% of that of the bottom cavity. Therefore, the acoustic compliance of eah smal cavty (CanC ) is less than 0.5% of the compliance ( Ca) of the bottom cavity (the acoustic compliance of the cavity is proportional to its volume). The impedance of the cavities 1 ljjeC ) and 1 je~( a between the diaphragm and top/bottom backplates are much larger than the impedance of the bottom cavity and the diaphragm. Therefore, in the equivalent circuit, C'" and Cbac, can be treated as open circuit and are neglected in the lumped element model. As seen from Figure 3-7, for each backplate, its acoustic compliance is in parallel with its resistance (a RC loop is formed). Therefore, the impedance of each backplate will be approximately equal to its resistance [66] when f < = 1.3MHz, (3.44) 27taC, and f < = 3.3M~Hz, (3.45) where f is the frequency of the incident acoustic pressure. As seen from Figure 3-7, the pressure drop across each plate is proportional to the impedance of each plate (note that C~ and Cic are neglected). Based on the microphone design, the impedance of each backplate is much less than the impedance of the diaphragm [66] when Eqs. (3.44) and (3.45) are satisfied; therefore, the pressure drop across each backplate is negligible. Since the small backplate deflection is proportional to the pressure drop across it; therefore, the backplate deflection is negligible and the backplate can be treated as being acoustically transparent. Nonlinear Dynamic Model A general nonlinear dynamic model for the microphone is shown schematically in Figure 3-8. The top and bottom backplates are assumed to be rigid and have equal areas with the diaphragm. The diaphragm is modeled by a Duffing spring with two spring constants k, and k3 and a lumped mass Mme, with a lumped area Ae Top backplate x=0 m' m Pressure p Bottom backplate Figure 3-8. A nonlinear dynamic model of a dual-backplate capacitive microphone. Shown in Figure 3-9 is the free body diagram of the dynamic model, pA,,, is the lumped mechanical force caused by the incoming acoustic pressure, and M~,,,,f is the lumped inertia force due to the acceleration. For simplicity, during the derivation of the dynamic governing equation, the diaphragm is assumed to move downwards with a certain displacement x. 2aZ~j~a Tx For p Figure 3-9. Free body diagram of the nonlinear dynamic model. The nonlinear lumped spring force is given by F~nR= -k,x-kx (3.46) The lumped damping force is F = -bx(3.47) where x is the center velocity of the diaphragm. By using the equal-area parallel-plate assumption and neglecting the fringing field effect of the holes in the backplates, the total electrostatic co-energy, U, stored in the system is calculated as follows 1 1 EA, lEA, ,= 2 2d,+xzt f 2 d,-x""~b2() 3.8 where E is the dielectric constant of the air, and I ~(t) and IFL (t) are the instantaneous voltages applied to the top backplate and bottom backplate respectively. It follows that the net lumped electrostatic force is dUi EA,, I (t) L 2(t) Fe dx ~ 2m (d, + x)' (d, -) x)49 The lumped mechanical reaction force from the cavity is Fe = -kcx. (3.50) After obtaining all the lumped forces acting on the diaphragm, by applying Newton's second law, the general dynamic governing equation is ~Mp"e = C F7 = F,,, + Flanipig Fb + Fe + pAme (3.5 1) By substituting Eqs. (3.46), (3.47), (3.49) and (3.50) into Eq. (3.51), the governing equation is M~se =-k,+e~x-~x bi-A ,, (t) ITp (t)+p,,. (52 Mmp = k,+ k~x ,X hx-2 (dl + x)2 (do x)? A (.2 Rewriting Eq. (3.52) becomes M~sef~bi+(k +ke x kx= eA, I (t) I (t) +pA,p,. (3.53) I~~,,E~ (d +x x)k (d -c~ x) Discussion of Nonlinearities The above governing equation physically represents a general damped second- order system with a cubic mechanical nonlinearity and under both nonlinear electrostatic loading and uniform pressure loading. Since the cubic stiffness parameter k, is positive, it physically represents a spring hardening effect. The electrostatic forces between backplates and diaphragm in nature are nonlinear, even when the displacement of the diaphragm is not large. From the expressions of nonlinear lumped electrical forces shown in Eq. (3.53), the electrical nonlinearity is coupled with the mechanical nonlinearity, which indicates that the microphone system is an electromechanically coupled system. To facilitate the nonlinear analyses throughout this dissertation, it is necessary to quantify the mechanical and electrical nonlinearities in this section. The definitions are given by the following equations. For the mechanical nonlinearity, NL,, we have nonlinear mechanical force linearized mechanical force NL, = x 1 00% ~linearized mechanical force k,x +k,x' k,x x0%k,xl(.4 k,x k, where x is the center displacement under the interested pressure or electrical loading. Based on the above equation, we need to know the linear stiffness, cubic stiffness, and center displacement of the diaphragm to be able to calculate the mechanical nonlinearity. The expression for electrical nonlinearity, NLB, is nonlinear electrical force linearized electrical force NL = x100%. (3.55) e ~linearized electrical force To further quantify and gain physical insight into the definition of electrical nonlinearity, we consider a single-backplate capacitive microphone with an applied DC bias V, only. Following the definition in Eq. (3.49), the nonlinear electrostatic force is EA V2 Fe = "'" (3.56) e 2 (d,- )2 By using a Taylor's series expansion about x = 0, we can linearize the above nonlinear force as follows (3.57) Therefore, the electrical nonlinearity in this case is EA Yo E~mYo"O 1+2 2 (d -x)2 2d02 do NL o 2d02 12 dox do x)2 (dog + 2x) 1. (3.58) As we can see from the above equation, the electrical nonlinearity can be calculated based on the gap and center displacement of the diaphragm. The gap is an independent parameter; however, the center displacement of the diaphragm is dependent on the electrostatic loading (DC bias voltage in this case) and pressure loading. From Eqs. (3.54) and (3.58), clearly the electrical nonlinearity is coupled with the mechanical nonlinearity through the center displacement of the diaphragm. x 10-4 Nonlinear Mechanical Force -Linearized Mechanical Force SNonlinear Electrical Force @ 20V -Linearized Electrical Force @ 20V 0* 0 0.2 0.4 0.6 0.8 1 1 .2 Center displacement (m) x 10-6 Figure 3-10. Nonlinear vs. linearized mechanical and electrical forces of a single- backplate capacitive microphone. FeL = o2d 1+2 . An example is given here to help understand the above defined nonlinearities. Shown in Figure 3-10 is a plot of calculated nonlinear versus linearized mechanical and electrical forces by using the parameters in Table 3-1 for a given 20V DC voltage. As we can see from the plot, the electrical nonlinearity is dominant for the large center displacement of the diaphragm. For example, if the center displacement is 0.6 um the electrical nonlinearity at 20V is NL = 27.6% and the mechanical nonlinearity is NLM, = 3.4% . Similarly, if a dual-backplate capacitive microphone with an applied DC bias V, is considered, the electrical nonlinearity, followed by the definition in Eq. (3.55), is given as d6 NL, = -1. (3.59) d,: x d i+2x? From Eqs. (3.54) and (3.59), again the electrical nonlinearity is coupled with the mechanical nonlinearity through the center displacement of the diaphragm. An example is given here to help understand the electrostatic nonlinearity of a dual- backplate capacitive microphone. Shown in Figure 3-11 is a plot of calculated nonlinear versus linearized mechanical and electrical forces by using the parameters in Table 3-1 for a given 20V DC voltage. As we can see from the plot, the electrical nonlinearity becomes smaller and is on the same level with the mechanical nonlinearity, even when the center displacement of the diaphragm is large. For example, if the center displacement is 0.6 pum the electrical nonlinearity at 20V is NLe = 2.3% and the mechanical nonlinearity is NLM, = 3.4% Physicallly, the two opposite electrostatic forces in a dual-backplate capacitive microphone help to reduce the electric nonlinearity. x 10-4 Nonlinear Mechanical Force -Linearized Mechanical Force SNonlinear Electrical Force @ 20V -Linearized Electrical Force @ 20V 0 04 0.6 0.8 1 1.2 Center displacement (m) x 10-6 Figure 3-1 1. Nonlinear vs. linearized mechanical and electrical forces of a dual-backplate capacitive microphone. Since it is impossible to solve the governing nonlinear equation Eq. (3.53) in closed form, alternative approaches are used in this dissertation to study the nonlinear dynamic system. Nonlinear finite element analyses (FEA) are carried out in the next section to verify the theoretical lumped stiffnesses of the diaphragm. The approximate analytical solutions (specifically via multiple time scales and harmonic balance methods) and numerical simulations through the direct integration of nonlinear governing equation are provided in the next chapter. The instability analyses for pull-ins are carried out in Chapter 5. Nonlinear Finite Element Analyses In this section, nonlinear mechanical finite element analyses are carried out in CoventorWare 2003 [106] to extract the equivalent lumped stiffnesses (k,,k3) Of the diaphragm and the accuracy of theoretical lumped stiffnesses is verified. This section also provides discussions of the modeling error for the electrostatic force through the coupled electromechanical simulation. Stiffnesses of the Diaphragm Based on the solid model shown in Figure 3-1, a 3D mesh of the diaphragm is generated and shown in Figure 3-12. Based on this mesh, converged displacement results are achieved with the diaphragm under the applied uniform pressure. Some major specifications of the mesh are listed in Table 3-2. Mesh of diaphragm ' Zoom-in elements Figure 3-12. 3D mesh of the diaphragm in CoventorWare 2003. Table 3-2. Maj or specifications of the diaphragm mesh with converged displacement results. Parameter Value Volume element type Solid hexahedron Number of volume elements 82452 Number of nodes 43671 Average aspect ratio 1.6223 The material properties and physical dimensions of the diaphragm used in nonlinear FEA are taken partially from Table 3-1. The side surface of the diaphragm is fixed to be the boundary condition. To obtain the lumped stiffnesses of the diaphragm, different pressure loads with amplitudes varying from 10 to 4000 Pa are applied to the top surface of the diaphragm, and the nonlinear FEA are carried out to yield the center displacement of the diaphragm for each applied pressure respectively. x 10-6 1.2 ----- Ideal linear result -A-Nonlinear FEA result/ 1 ..9.P- Energy-based analytical result . ****EE-- Exact analytical result ** S0.8/ 8 0.4 0.2 500 1000 1500 2000 2500 3000 3500 4000 Applied pressure (Pa) Figure 3-13. Transverse center deflections of the diaphragm under the uniform pressure. Shown in Figure 3-13 is the plot of transverse center deflections of the diaphragm. The ideal linear, energy-based analytical as well as exact analytical deflection results (obtained from the reference [100]) are also plotted in Figure 3-13. As we can see from the plot, three sets of nonlinear deflection results agree very well with each other. The mechanical nonlinearity becomes important for the large applied pressure, for example, when the pressure value is above 2000Pa. gY D splacement U, -5 4E-01 -4 OE-01 -2 7E-01 -1 3E-01 2 7E-05 COV ENTOR Figure 3-14. Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thickness direction, unit: Clm). Shown in Figure 3-14 is a displacement contour plot of the diaphragm under a 2000Pa uniform pressure. As seen from the plot, the center deflection of the diaphragm is approximately 5.4e-7 m. Further calculation indicates that the mechanical nonlinearity ( NLM ) at 2000Pa is approximately 2.7% based on the definition in Eq. (3.54). The lumped linear and cubic stiffnesses can be extracted by curve-fitting the simulated nonlinear center displacements of the diaphragm with the formula in Eq. (3.3 1) and the final results are listed in Table 3-3. From Table 3-3, the differences for the linear stiffness k, and cubic stiffness k3 are approximately 1.3% and 5.0% respectively. Since the differences are small, the accuracy of the theoretical lumped stiffnesses of the diaphragm is verified by the nonlinear FEA results. In the following chapters, the theoretical lumped stiffnesses of the diaphragm will be used. Table 3-3. Comparison of nonlinear FEA and LEM results. Parameter Nonlinear FEA result LEM result Difference Linear stiffness k, (N/m) 199.7 202.2 1.3% Cubic stiffness k3 (N/m3) 1.979el3 1.880el3 5.0% Electrostatic Forces by CoSolveEM Simulations In the real microphone device, the area of three plates is not same. As shown in Figure 3-1, the top backplate has the largest area while the bottom backplate has the smallest area. In addition, the backplates are perforated with hundreds of holes. The capacitance between the backplate and diaphragm is reduced due to the backplate area loss; however, extra fringing fields generated by the holes could compensate the loss. The electrostatic forces in the previous sections are modeled based on the equal-area parallel-plate assumption; therefore, some errors exist between the modeled and real electrostatic forces. One possible way to examine the difference between the real and modeled electrostatic forces is through nonlinear finite element analyses, for example, the CoSolveEM (coupled electromechanical analysis) simulation in CoventorWare 2003. The simulation runs between the mechanical and electrostatic domains until a converged equilibrium point is found. By using the solid model shown in Figure 3-1, the CoSolveEM simulation takes into account the effects of unequal area, perforated holes, and the fringing field on the electrostatic force; therefore, the simulated electrostatic force can be treated as an accurate measure of the real electrostatic force. In the following, the 81 CoSolveEM simulations are conducted for top and bottom capacitors of the microphone respectively, and the simulation results are compared with the results based on lumped element modeling. During the simulations for the top capacitor, the side surfaces of the diaphragm and top backplate are assumed to be fixed and the applied DC voltage varies from 10 to 25V. When the applied DC voltage is greater than 25V, the simulation becomes diverged and quasi-static pull-in occurs. Similarly, for the bottom capacitor, the side surfaces of the diaphragm and bottom backplate are assumed to be fixed, and the simulated quasi-static pull-in voltage is approximately 33.5V. x 10-4 1.8 -9- Simulated electrostatic force 1.6 -EI- Modeled electrostatic force 1.4- P 1.- S0.8- w 0.6 0.4 10 15 20 25 Applied \loltage (V) Figure 3-15. Plot of simulated and modeled electrostatic forces for the top capacitor. Shown in Figure 3-15 is a plot of simulated electrostatic forces for the top capacitor when the applied DC voltage varies. The corresponding modeled electrostatic forces are also plotted in Figure 3-15 by using the parameters in Table 3-1. As seen from Figure 3- 15, the difference between the simulated and model electrostatic forces becomes larger when the applied voltage increases. Further calculations show that the difference at 10V is approximately 4% and 17% at 25V. x 10-4 -9 Simulated electrostatic force 3t E- Modeled electrostatic force 2.5 S1.5 0.5- 10 15 20 25 30 35 Applied voltage (V) Figure 3-16. Plot of simulated and modeled electrostatic forces for the bottom capacitor. For the bottom capacitor, similarly, shown in Figure 3-16 is the comparison plot of the simulated and modeled electrostatic forces. A similar trend of the difference between the simulated and model electrostatic forces is observed. And further calculations show that the difference at 10V is approximately 13% and 17% at 33.5V. Based on the results shown in Figure 3-15 and Figure 3-16, when the applied DC voltage increases, the error generated by the model with an equal-area parallel-plate assumption increases. Specifically, the error is up to 17% when the applied voltage is up |

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PAGE 1 NONLINEAR DYNAMICS OF A DUAL-BACKPLATE CAPACITIVE MEMS MICROPHONE By JIAN LIU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 Copyright 2007 by Jian Liu PAGE 3 To my parents and my beloved wife, Zhen PAGE 4 iv ACKNOWLEDGMENTS Financial support for this research was provided in part by a National Science Foundation grant (ECS-0097636), and financia l support from a NSF CAREER award (CMS-0348288) is also gratefully acknowledged. First, I would like to express my sincere gratitude to my advisor, Professor Mark Sheplak, for giving me the opportunity to work at the Interdiscipl inary Microsystems Group. His guidance over the years has been in valuable and his desire for excellence has had a very positive influence on me. I would like to thank my chair, Professor Brian P. Mann, for many valuable insightful talks and encouragement during this research. His guidance and help will always be cherishe d. I would also like to thank Professors Toshikazu Nishida, Louis N. Cattafesta, and Bhavani V. Sankar for their help and discussions on different aspects of this multid isciplinary research and for serving on my committee. My acknowledgements go to all my co lleagues at the In terdisciplinary Microsystems Group. Special thanks go to David Martin and Karthik Kadirvel for working together over the past five years a nd many valuable discussi ons on this research. In particular, I want to thank David Martin for helping me improve my English over the years. I would like to express my deep apprec iation to my parents for their endless support, understanding and guida nce throughout my life. Fina lly, I would like to thank PAGE 5 v my beloved wife, Zhen, for her love, patie nce and sacrifice, without which this dissertation would not have been done. I am forever grateful for her love. PAGE 6 vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...............................................................................................................x LIST OF FIGURES..........................................................................................................xii ABSTRACT....................................................................................................................xvi i CHAPTER 1 INTRODUCTION........................................................................................................1 Nonlinear Dynamics Issues..........................................................................................2 Objective and Approach...............................................................................................5 Research Contributions.................................................................................................5 Dissertation Organization.............................................................................................6 2 BACKGROUND..........................................................................................................7 Microphone Basics.......................................................................................................7 Conventional versus Silicon Microphones............................................................7 Metrics of Performance.........................................................................................8 Transduction Mechanisms..........................................................................................11 Introduction to Electromechanical Transducers..................................................11 Piezoelectric Microphones..................................................................................12 Piezoresistive Microphones.................................................................................13 Optical Microphones...........................................................................................13 Electrostatic Microphones...................................................................................14 Electrostatic Microphones..........................................................................................15 Electret Microphones...........................................................................................19 Condenser Microphones......................................................................................22 Single-backplate c ondenser microphones....................................................22 Dual-backplate condenser microphones.......................................................29 Major Previous Work on Capacitive Silicon Microphones........................................30 Basics of Nonlinear Dynamics...................................................................................37 Nonlinearities......................................................................................................37 Steady State, Autonomous System and Fixed Point...........................................37 Stability, Basin of Attraction and Phase Portrait.................................................39 PAGE 7 vii Bifurcation...........................................................................................................41 Previous Work on Nonlinear Dynamics of Electrostatic MEMS Devices.................42 3 NONLINEAR DYNAMIC MODEL..........................................................................51 Microphone Structure.................................................................................................51 Displacement Solutions of the Diaphragm.................................................................52 Small Displacem ent Solution..............................................................................53 Energy Method and Large Displacement Solution.............................................54 Procedure of energy method........................................................................55 Large displacement solution.........................................................................55 Lumped Element Modeling of the Microphone.........................................................58 Lumped Parameters of Diaphragm......................................................................59 Lumped mass................................................................................................60 Lumped linear stiffness and compliance......................................................61 Lumped area.................................................................................................62 Lumped cubic stiffness.................................................................................63 Lumped Damping Coefficient.............................................................................63 Lumped Stiffness of the Cavity...........................................................................66 Nonlinear Dynamic Model.........................................................................................70 Discussion of Nonlinearities.......................................................................................72 Nonlinear Finite Element Analyses............................................................................76 Stiffnesses of the Diaphragm...............................................................................77 Electrostatic Forces by CoSolveEM Simulations................................................80 Summary.....................................................................................................................83 4 APPROXIMATE SOLUTIONS OF NO NLINEAR GOVERNING EQUATIONS..84 Introduction.................................................................................................................84 Governing Equation for the Electr ical Square Wave Excitation................................86 Approximate MTS Solution for the Electrical Square Wave Excitation....................89 Approximate Solution by the MTS Method........................................................90 Discussion of the MTS Approximate Solution....................................................92 Approximate solution with zero initial conditions.......................................93 Approximate solutions in other applications................................................94 Validity Region of the Approximate MTS Solution............................................94 Results of linear case....................................................................................95 Results of weakly nonlinear case.................................................................96 Results of highly nonlinear case...................................................................98 Governing Equation for the Elec trical Sinusoidal Excitation...................................100 Approximate HB Solution for the Electrical Sinusoidal Excitation.........................103 Approximate Solution by a HB Method............................................................103 Validity Region of the Approximate HB Solution.............................................106 Results of small THD case.........................................................................106 Results of transition THD case...................................................................108 Results of large THD case..........................................................................110 Governing Equation for the Sinusoida l Acoustical Pressure Excitation..................113 PAGE 8 viii Approximate Solutions for the Sinusoi dal Acoustical Pressure Excitation..............116 HB Approximate Solution.................................................................................117 MTS Approximate Solution..............................................................................118 Validity Region of Approximate Solutions.......................................................120 Results of linear case..................................................................................121 Results of weakly nonlinear case...............................................................122 Results of highly nonlinear case.................................................................123 Summary...................................................................................................................125 5 PULL-IN INSTABILITIES......................................................................................126 Quasi-Static Pull-in due to an Applied DC Voltage.................................................126 Equilibrium Points and Local Stabilities...........................................................128 Graphical Analysis............................................................................................131 Critical Quasi-Static Pull-in Voltage.................................................................132 Quasi-Static Pull-in by a Subc ritical Pitchfork Bifurcation..............................134 Potential Advantage of Geometric Nonlinearity...............................................136 Compact Quasi-Static Stable Operation Range.................................................138 Quasi-Static Pull-in due to an Applied Acoustical Pressure.....................................140 Equilibrium Points and Local Stabilities...........................................................141 Critical Quasi-Static Pull-in Pressure................................................................144 Dynamic Pull-in due to a Mechanical Shock Load..................................................148 Problem Formulation.........................................................................................149 Equilibrium Points and Local Stabilities...........................................................153 Phase Portrait and Basins of Attraction.............................................................155 Dynamic Pull-in due to a Mechanical Shock Load...........................................158 Potential Advantage of Geometric Nonlinearity...............................................162 Dynamic Pull-in due to an Acoustic Shock Load.....................................................164 Problem Formulation.........................................................................................164 Numerical Simu lation Results...........................................................................167 Simulated Dynamic Pull-in Results...................................................................169 Effect of Damping on Dynamic Pull-in.............................................................172 Effect of Geometric Nonlin earity on Dynamic Pull-in.....................................173 Summary...................................................................................................................175 6 SYSTEM IDENTIFICATION BY PRELIMINARY EXPERIMENTS..................177 Experiment Setup and Procedures............................................................................177 Experiment Setup..............................................................................................177 Experiment Procedures......................................................................................179 Procedures for the electrical square wave excitation.................................179 Procedures for the electrical sinusoidal excitation.....................................180 Results of Electrical Square Wave Excitation..........................................................181 Results of Bottom Backplate Excitation............................................................182 Results of Top Backplate Excitation.................................................................185 Results of Electrical Sinusoidal Excitation...............................................................187 Results of Bottom Backplate Excitation............................................................188 PAGE 9 ix Results of Top Backplate Excitation.................................................................190 Discussion of Analysis Results.................................................................................191 Summary...................................................................................................................193 7 CONCLUSIONS AND FUTURE WORK...............................................................194 Conclusions...............................................................................................................194 Recommendations for Future Work..........................................................................196 APPENDIX A LARGE DISPLACEMENT ENER GY SOLUTION OF A CIRCULAR DIAPHRAGM..........................................................................................................199 B APPROXIMATE SOLUTION FOR A GENERAL NONLINEAR SECOND ORDER SYSTEM....................................................................................................204 Introduction...............................................................................................................204 Approximate Solution by the Multiple Time Scales Method...................................205 C COEFFICIENTS OF THE A PPROXIMATE HARMONIC BALANCE SOLUTION..............................................................................................................213 D APPROXIMATE SOLUTIONS FOR A SINUSOIDAL ACOUSTICAL PRESSURE EXCITATION.....................................................................................215 HB Approximate Solution........................................................................................217 MTS Approximate Solution.....................................................................................220 E UNCERTAINTY ANALYSIS.................................................................................227 Uncertainty Analysis Methods.................................................................................227 Uncertainty Sources..................................................................................................229 Uncertainty in the Experimental Data...............................................................229 Errors of Approximate Solutions an d Nonlinear Least-Squares Algorithms....230 Uncertainties Caused by the Fabrication Process..............................................231 Preliminary Uncertainty Analysis Results................................................................231 LIST OF REFERENCES.................................................................................................237 BIOGRAPHICAL SKETCH...........................................................................................248 PAGE 10 x LIST OF TABLES Table page 2-1 Major previous work in capacitive silicon microphones.........................................31 3-1 Material properties and physical pa rameters of the 2000Pa microphone in metric units (material: polysilicon).....................................................................................67 3-2 Major specifications of the diaphr agm mesh with conve rged displacement results.......................................................................................................................7 8 3-3 Comparison of nonlinear FEA and LEM results......................................................80 4-1 Given and extracted (via MTS solu tion) parameters for a linear case.....................96 4-2 Given and extracted parameters (via MTS solution) for a weakly nonlinear case...97 4-3 Given and extracted (via MTS solution) parameters for a highly nonlinear case....98 4-4 Results of the maximum error and sum of residual squares for each test case......100 4-5 Given and extracted (via HB solution) parameters for the small THD case..........108 4-6 Given and extracted (via HB solution) parameters for the transition THD case...110 4-7 Given and extracted (via HB solutio n) parameters for the large THD case...........112 4-8 Results of the maximum error and sum of residual squares for each test case......112 4-9 Parameters used for the comparison of approximate and numerical solutions......121 5-1 Force parameters for a designe d 2000Pa capacitive MEMS microphone..............128 5-2 Parameters for the numerical study of an N-wave excitation................................168 6-1 Results of system parameters of the bottom backplate excitation..........................184 6-2 Results of system parameters of the top backplate excitation................................187 6-3 Amplitudes and phase of the inte grated averaged steady-state center displacement of the bottom backplate excitation...................................................189 PAGE 11 xi 6-4 Results of system parameters of the bottom backplate excitation..........................190 6-5 Amplitudes and phase of the inte grated averaged steady-state center displacement of the top backplate excitation.........................................................190 6-6 Results of system parameters of the top backplate excitation................................191 6-7 Theoretical mean values and uncertainti es of system parameters for a given 95% confidence level......................................................................................................191 6-8 Nominal values of system parameters of the microphone.....................................192 E-1 Uncertainties caused by the fabricati on process for a 95% confidence level........231 E-2 Sensitivity coefficients used in the uncertainty analysis........................................235 E-3 Theoretical mean values and uncertain ties of system parameters caused by fabrication for a given 95% confidence level.........................................................236 PAGE 12 xii LIST OF FIGURES Figure page 1-1 Schematic of a dual-bac kplate capacitive MEMS microphone..................................3 1-2 Schematic of an electrical model of the microphone.................................................3 2-1 A typical frequency response plot with a defined sensitivity and bandwidth............8 2-2 Typical noise power spectral density plot for a microphone.....................................9 2-3 Time histories and power spectra of a pure and two distorte d sinusoidal waves.....10 2-4 A simplified model of an electrostatic microphone.................................................15 2-5 Schematic of a capacitive microphone with an electret diaphragm.........................19 2-6 A simplified qausi-static m odel of an electret microphone......................................20 2-7 Illustration of the critical bias charge of an electret microphone.............................21 2-8 Schematic of a single-bac kplate condenser microphone..........................................22 2-9 Illustration of mechanical and elect rostatic forces for a single-backplate condenser microphone..............................................................................................24 2-10 Illustration of quasi-static pull-in of a single-backplate condenser microphone.....25 2-11 Simplified circuit of a single-ba ckplate condenser microphone with a preamplifier..............................................................................................................26 2-12 Effect of cubic nonlinearity on the system frequency response...............................38 2-13 Phase plane trajectories around fixe d points of a dualbackplate capacitive MEMS microphone (sink points are indicate d by blue crosses, and saddle points are indicated by blue circles)....................................................................................40 2-14 Typical bifurcation diagrams for one-dimensional autonomous systems: (a) Saddle-node bifurcation; (b) Pitchfork bifu rcation; (c) Transcri tical bifurcation....41 3-1 3D cross-section view of the mi crophone structure (not to scale)...........................52 PAGE 13 xiii 3-2 Top-view photograph of the microphone.................................................................52 3-3 Schematic of a clamped circular diap hragm under a transverse uniform pressure loading......................................................................................................................53 3-4 Displacement components in the neutral plane of a circular diaphragm.................56 3-5 Normalized mode shape for seve ral pressure values (2000, 10000 and 100000 Pascals).....................................................................................................................58 3-6 Repetitive pattern of hol es in the top backplate.......................................................64 3-7 Simplified lumped element model of a dual-backplate capacitive microphone......68 3-8 A nonlinear dynamic model of a dual-backplate capacitive microphone................70 3-9 Free body diagram of the nonlinear dynamic model................................................71 3-10 Nonlinear vs. linearized mechanical and electrical forces of a single-backplate capacitive microphone..............................................................................................74 3-11 Nonlinear vs. linearized mechanical a nd electrical forces of a dual-backplate capacitive microphone..............................................................................................76 3-12 3D mesh of the diaphragm in CoventorWare 2003..................................................77 3-13 Transverse center deflections of the diaphragm under the uniform pressure...........78 3-14 Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thic kness direction)...........................................................................79 3-15 Plot of simulated and modeled elect rostatic forces for the top capacitor.................81 3-16 Plot of simulated and modeled electro static forces for the bottom capacitor..........82 4-1 Dynamic model for an electrical square wave excitation on the top backplate.......86 4-2 Plot of electrostatic and a pproximate electrostatic forces........................................87 4-3 Comparison of simulated and MTS-based curve fitting center displacement results.......................................................................................................................9 6 4-4 Comparison of simulated and MTS-based curve fitting center displacement results.......................................................................................................................9 7 4-5 Comparison of simulated and MTS-based curve fitting center displacement results.......................................................................................................................9 9 4-6 Dynamic model for an electrical sinuso idal excitation on the bottom backplate...101 PAGE 14 xiv 4-7 Simulated sinusoidal response of the diaphragm...................................................107 4-8 Simulated power spectrum of the steady state displacement.................................108 4-9 Simulated sinusoidal response of the diaphragm...................................................109 4-10 Simulated power spectrum of the steady state displacement.................................109 4-11 Simulated sinusoidal response of the diaphragm...................................................111 4-12 Simulated power spectrum of the steady state displacement.................................111 4-13 Dynamic model for the sinusoida l acoustical pressure excitation.........................113 4-14 Plot of net electrostatic and a pproximate net electrostatic forces..........................114 4-15 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions for a linear case...............................................................122 4-16 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions for a weakly nonlinear case.............................................123 4-17 Comparison of the steady-state non-dimensional amplitudes of the approximate and numerical solutions fo r a highly nonlinear case..............................................124 5-1 Plot of the ND mechanical and net electrostatic forces..........................................131 5-2 Plot of the ND mechanical and net electrostatic forces..........................................133 5-3 A subcritical pitchfork bi furcation illustrating quasistatic pull-in due to an applied DC voltage.................................................................................................135 5-4 A subcritical pitchfork bi furcation illustrating quasistatic pull-in due to an applied DC voltage (versus bias voltages).............................................................136 5-5 Plot of a non-dimensional net electros tatic force and different non-dimensional mechanical forces...................................................................................................137 5-6 Quasi-static stable operation ra nge of the microphone in a 3D space....................138 5-7 Quasi-static stable operation range of the microphone in a 3D space (versus DC voltages).................................................................................................................139 5-8 Plot of the ND net restori ng and electrostatic forces..............................................143 5-9 Plot of the ND net restori ng and electrostatic forces..............................................145 5-10 Quasi-static pull-in due to varying ND parameters................................................146 PAGE 15 xv 5-11 Quasi-static pull-in due to varying acoustic pressure and DC bias voltage...........147 5-12 Three commonly used nonlinear mechanic al shock load models (impulse, half sine and triangle)....................................................................................................152 5-13 Phase plane trajectories around the equilibrium points..........................................155 5-14 Three basins of attraction for a DC bias of 25 V...................................................156 5-15 Basins of attraction within the physic al backplates for a DC bias of 25 V............157 5-16 Stable and unstable non-dimensional center displacement re sponses with two initial non-dimensi onal velocities..........................................................................159 5-17 Phase plots of a stable response and a dynamic pull-in due to a large initial velocity imposed by a mechanical shock load.......................................................159 5-18 Dynamic pull-in due to a combination of DC bias voltage and a mechanical shock load...............................................................................................................161 5-19 Phase plane trajectories for an added geometric nonlinearity case. Sink points are indicated by blue crosses, and saddl e points are indicated by blue circles......162 5-20 Expanded stable operation region of th e microphone due to the added geometric nonlinearity.............................................................................................................163 5-21 A typical N-wave with an amplitude and a duration time......................................165 5-22 The Fourier transform of a typical N-wave............................................................166 5-23 Transient non-dimensional center displacement response of diaphragm due to an N-wave with an amplitude of 125 dB SPL and a duration time of 2 ms................168 5-24 Transient non-dimensiona l center velocity response of diaphragm due to an Nwave with an amplitude of 125 dB SPL and a duration time of 2 ms....................169 5-25 Dynamic pull-in due to an N-wave with a ND amplitude of 1.29 and a ND duration time of 2.3................................................................................................170 5-26 Threshold of dynamic pull-in due to an N-wave....................................................170 5-27 Threshold of dynamic pull-in due to an N-wave with a normalized pressure parameter................................................................................................................172 5-28 Effect of damping on dynamic pull-in threshold...................................................173 5-29 Effect of geometric nonlinear ity on dynamic pull-in threshold.............................174 6-1 Block diagram of the experiment setup..................................................................178 PAGE 16 xvi 6-2 Laser beam spot (red dot) impinges th e diaphragm through the center hole of the top backplate..........................................................................................................179 6-3 Simplified circuit to genera te the high voltage signal............................................181 6-4 Measured averaged center velocity res ponse for an applied square wave with an amplitude of 5V......................................................................................................182 6-5 Integrated center displa cement response for an applied square wave with an amplitude of 5V......................................................................................................183 6-6 Constructed phase plot for an applied square wave with an amplitude of 5V.......183 6-7 Comparison of integrated and curve -fit center displacements for an applied square wave with an amplitude of 5V....................................................................184 6-8 Measured averaged center velocity res ponse for an applied square wave with an amplitude of 18V....................................................................................................185 6-9 Integrated center displa cement response for an applied square wave with an amplitude of 18V....................................................................................................186 6-10 Constructed phase plot after 12 s for an applied square wave with an amplitude of 18V.....................................................................................................................186 6-11 Comparison of integrated and curve -fit center displacements for an applied square wave with an amplitude of 18V..................................................................187 6-12 Measured averaged steady-state cente r velocity response (asterisk) for a sinusoidal excitation with an amplitude of 9V and a frequency of 114.4 kHz......188 6-13 Comparison of the integrated (red aste risk) and curve-fitting (blue solid line) steady-state center displacement results for a sinusoidal excitation with an amplitude of 9V and a frequency of 114.4 kHz.....................................................189 PAGE 17 xvii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR DYNAMICS OF A DUAL-BACKPLATE CAPACITIVE MEMS MICROPHONE By Jian Liu May 2007 Chair: Brian P. Mann Cochair: Mark Sheplak Major Department: Mechanic al and Aerospace Engineering This work presents an investigation of the electromechanical nonlinear dynamics of a dual-backplate capac itive MEMS (microelectromechanical systems) microphone. A large displacement solution via an energy method has been utilized to provide linear and cubic lumped stiffnesses of the circular diaphragm of the microphone. A nonlinear dynamic model of the microphone is devel oped using lumped element modeling. Theoretical lumped stiffnesses of the diaphragm are verified by nonlinear finite element analyses and the errors for the linear and cubic stiffnesses are approximately 1.3% and 5.0% respectively. The critical quasi-static pull-in voltage of the microphone is found to be approximately 41V both analytically and numer ically. The phenomenon of qausi-static pull-in due to an applied DC voltage is illu strated by a subcritical pitchfork bifurcation. By using a phase portrait and basin of attrac tion, a mechanical shock load is related to PAGE 18 xviii dynamic pull-in. Further study shows that dyn amic pull-in could pot entially take place below the critical quasi-static pull-in voltage when the micr ophone is subject to a large mechanical shock load. The dynamic pull-in due to an acoustical pulse, in the form of an N-wave, has been investigated by using numerical simulation. A dynamic pull-in threshold curve has been obtai ned in terms of the duration time and amplitude of the Nwave for a given DC bias voltage. Studies of dynamic pull-in also show that several nonlinea rities (geometric, electrostatic and mechanical/acoustical shock) compete with each other. An increased electrostatic nonlinearity and/or an incr eased mechanical/acoustical shock load destabilize the system while an increased ge ometric nonlinearity helps to stabilize the microphone and expands the stable operational range. The multiple time scales and harmonic balance methods are applied to obtain approximate solutions of the nonlinear governin g equations under the electrical square, electrical sinusoidal and si nusoidal acoustical excitations. Based on the two approximate solutions for the electrical excitations and a nonlinear least-squares curve-fitting technique, system parameters are extracted from two types of experimental data. The preliminary uncertainty analysis, which includes only the uncertainties caused by fabrication, shows that the experimentally extracted linear natural frequency, damping ratio and nonlinear stiffness parameter fall w ithin their conservative theoretical ranges for a 95% confidence level. PAGE 19 1 CHAPTER 1 INTRODUCTION During the past three decades, the demand for reducing noise pollution, especially in communities surrounding airports, has increased. To abate the aircraft noise, the generation and propagation of noise source s must be characterized. Aeroacoustic measurement tools are required to both inve stigate mechanisms of noise generation and validate methods of noise reduction [1]. To enable aeroacoustic measurements, measurement microphones with the instrumentati on grade, cost effici ency, and small size must be developed. Currently, commer cial traditional microphones such as B&K condenser microphones are widely used in the field of aeroacoustic measurements; however, those microphones are co stly and not suitable for miniaturization. With the recent advancements in microelectromech anical systems (MEMS) technology, batch fabrication of microphones with smaller sizes and lower prices is now possible. A variety of transduction schemes, such as piezoelectric, piezoresistive, capacitive and optical, have been used in MEMS micr ophones [2]. Capacitive MEMS microphones have shown the potential to provide a dynami c range of 160 dB an d a bandwidth of 90 kHz, which are the key requirements to accommodate 1/8th-scale aeroacoustic testing [1]. However, the development of capacitiv e MEMS microphones comes with several issues. The electrostatic fo rce between the diaphragm and backplate of the microphone is inherently nonlinear. Pull-in instability [3, 4], a phenomenon in which the diaphragm collapses to the backplate or vice versa, o ccurs due to a large a pplied voltage and/or a large displacement. Since pull-in results in structural failure of the device when the PAGE 20 2 process is not reversible, the microphone needs to be carefu lly designed to operate in a stable domain. In addition, the mechanical restoring force of the diaphragm becomes nonlinear for large displacements. Both m echanical and electros tatic nonlinearities interact with each other, whic h adds more nonlinear distortion and lowers the fidelity of the microphone. Moreover, an increased elec trostatic nonlineari ty could potentially destabilize the microphone system while a larger mechani cal geometric nonlinearity could help to stabilize the microphone a nd expands the stable operation range. Therefore, a thorough understa nding of aforementioned non linear issues becomes vital for the success of the cap acitive MEMS microphone. This dissertation is part of a larger effo rt to develop a dual -backplate capacitive MEMS microphone for aeroacoustic applications. The focus of this dissertation is to investigate the electromechanical nonlin ear dynamics of the microphone through theoretical analysis, numerical simulation and preliminary ex perimental characterization. In summary, the study of this dissertation wi ll help improve the overall performances of the microphone through a better understan ding of the nonlinear dynamics issues. Nonlinear Dynamics Issues This section presents an overview of the nonlinear dynamics of a dual-backplate capacitive MEMS microphone. Different types of nonlinearities are discussed and some physical mechanisms are provided. Further details concerning the nonlinear model will be discussed in Chapter 3. A simplified schematic of a typical dua l-backplate capacitive MEMS microphone is shown in Figure 1-1. The backplates of the microphone are perfor ated to let the air pass through them and hence re duce the air-streaming resistan ce. Two gaps are formed PAGE 21 3 between the diaphragm and each backplate respectively. A cavity under the bottom backplate is vented to the ambient pressu re resulting in an AC measurement device. Top backplate Diaphragm Bottom backplate Backplate hole Cavity VentAir Gap Figure 1-1. Schematic of a dual-ba ckplate capacitive MEMS microphone. The three plates of the microphone are ma de conductive; therefore, two capacitors are formed between the diaphragm and each ba ckplate. When an acoustic wave impinges on the microphone, the incident pressure de flects the middle di aphragm and thereby alters the capacitance of th e two capacitors as shown in Figure 1-2. The differential capacitance change is detected through vari ous types of interface circuitry [5] to determine the input sound pressure level. Top BackplateBottom BackplateDiaphragm Figure 1-2. Schematic of an electrical model of the microphone. When the diaphragm with a fixed boundary undergoes a large deflection with respect to its thickness, geometric nonlinearity [6] occurs and the restoring spring force becomes nonlinear. Under large deflections the diaphragm behaves like a mechanical DuffingÂ’s spring, in which a hardening nonlineari ty is caused by the mid-plane stretching. PAGE 22 4 This mechanical nonlinearity directly affect s the upper limits of the linearity and dynamic range of the microphone. The details of ge ometric nonlinearity will be discussed in Chapter 3. The net electrostatic force on the dia phragm is a nonlinear function of the mechanical displacement of the diaphragm, th e gaps and the applied voltages between the backplates and diaphragm respectively. The net electrostatic force has singularities at the backplates, which will affect the upper dynamic range of the microphone. Although larger applied voltages and smaller gaps ar e desirable, because they increase the sensitivity of the microphone, th e nonlinear net electrostatic force increases dramatically with larger applied voltages a nd smaller gaps. When the incr easing net electrostatic force overcomes the nonlinear mechanical force, pull-in occurs. In a phase portrait, the pull-in point is an unstable fixed point for the micr ophone, which means that the displacement of the diaphragm continuously increases until the diaphragm crashes into one of the backplates. Damping plays a very important role in determining the bandwidth and the dynamic response of the microphone. Damping in a dual-backplate capacitive MEMS microphone is dominated by viscous damping and a linearized version is usually used to approximate the actual damping. When the di aphragm vibrates, the gas flow between the diaphragm and backplates can be divided into the horizontal flow between the plates and the vertical flow through the backplate holes. Viscous damping caused by the horizontal gas flow is often called squeeze-film damp ing, and the viscous damping caused by the vertical gas flow is called holes resistan ce [7, 8]. The structural damping of the PAGE 23 5 diaphragm mainly consists of the thermoelas tic energy dissipation inside the diaphragm and the vibration energy dissi pation in the compliant boundary of the diaphragm [9, 10]. Objective and Approach The objective of this resear ch is to study the electrom echanical nonlinear dynamics of a dual-backplate capacitive MEMS micr ophone. An investigat ion of the existing nonlinear dynamics issues is targeted to help improve the overall performance of the microphone. In addition, the knowledge gained from this study can be applied to other electrostatic devices, such as dynamic nano-indenters [ 11, 12] and MEMS mass sensors [13, 14]. To achieve the above research goal, several appr oaches are employed in this dissertation. Specifically, the nonlinear dynamical system is modeled via lumped element modeling and a general form of the nonlinear governing equation is obtained. Approximate analytical solutions to the non linear governing equations are obtained with multiple scales and harmonic balance analyses. Pull-in instabilities are explored by both analytical and numerical approaches. Finally, the microphone is experimentally characterized and system parameters of th e nonlinear dynamic model are identified from the measured data. Research Contributions The contributions of this disserta tion are summarized as follows. Development of a nonlinear dynamic model for a dual-backplate capacitive MEMS microphone and numerical solutions of the nonlinear governing equation. Development of approximate analytical solutions of the nonlinear governing equations via multiple time scales (MTS) and harmonic balance (HB) methods. Preliminary experimental characterizati on and application of the uncertainty analysis to the experimentally identified system parameters. PAGE 24 6 Dissertation Organization The dissertation is organized into seve n chapters. Chapter 1 introduces and describes the research work in this disserta tion. The next chapter provides a background of the microphone, basics of nonlinear dynamics and a review of previous work done on both capacitive microphones and nonlinear dynamics of electrostatic MEMS devices. A nonlinear dynamical model for a dual-b ackplate capacitive ME MS microphone is developed in Chapter 3. Non linear finite element analyses are performed to verify some theoretical results. In Chapter 4, the multip le time scales and harmonic balance methods are applied to obtain approximate solutions of the nonlinear governing equations under the electrical square, electrical sinusoida l and sinusoidal acoustical excitations. Numerical tests are conducted to provide the validity ranges of approximate solutions. Chapter 5 focuses on the theoretical studies of pull-in instabilities, including both quasistatic and dynamic pull-ins. In Chapter 6, the approximate solutions obtained in Chapter 4 are applied to identify system parameters through a series of prel iminary experiments. Preliminary uncertainty analysis is also conducted for the experimentally identified system parameters. Finally, conclusions a nd future work are provided in Chapter 7. PAGE 25 7 CHAPTER 2 BACKGROUND This chapter provides background informa tion about microphone basics and some introductory information about nonlinear dynamics A review of the published work on both capacitive silicon microphones and nonlin ear dynamics of electrostatic MEMS devices is also presented in this chapter. Microphone Basics A microphone is a transducer that conve rts unsteady pressure inputs into an electrical signal. So far, many transduction mechanisms have been developed for microphones; these include elec trodynamic, piezoelectric, pi ezoresistive, capacitive, optical and contact (carbon) tr ansduction mechanisms [2, 15]. Microphones are widely employed in a variety of applications such as sound field measurements [16, 17], hearing aids [18-20], telecommunications [16] and noi se localization using ac oustic arrays [21, 22]. Also, the use of microphones in ultrasoni c and underwater applic ations have also been reported [23, 24]. Conventional versus Silicon Microphones Based on the manufacturing technique, microphones can be categorized into two major types: conventional and silicon micromachined microphones. Conventional microphones are usually fabricated from separate metal parts and polymer foils with most of the assembly process done by hand [16]. On the other hand, silicon microphones are fabricated from modern silicon microm achining technology. In comparison with conventional microphones, silicon microphones ar e easier to integrate with the sensing PAGE 26 8 and supporting electronics, which offers the potential for higher performance by reducing parasitic elements in the sensing subsystem. Moreover, the batch fabrication of silicon microphones leads to lower cost s since hundreds or thousands of devices are fabricated together on a single silicon wafer simultaneously [25]. Metrics of Performance The major performance metrics for a mi crophone are the sensitivity, bandwidth, dynamic range, and noise floor. The open-circui t sensitivity of a mi crophone is typically defined at some reference frequency (for exam ple, 1 kHz as shown in Figure 2-1). It is defined as the ratio of the output voltage change (before the preamplifier) to the amplitude change of the sound pressu re incident on the diaphragm [16]. -180 90 -90 0 Frequency (rad/s)Phase (degree) Magnitude (dB) -3db Bandwidth f-3db -3db -3db fcuton Sensitivity at 1.0 kHz Figure 2-1. A typical frequency response plot with a define d sensitivity and bandwidth. The bandwidth of a microphone is defi ned as the frequency range where a microphone ideally maintains a constant sensitiv ity [16]. In practice, as shown in Figure PAGE 27 9 2-1, the bandwidth is usually the frequency ra nge from a -3dB low cut-on point to a high -3dB frequency point. c f Thermal noise1/f noiseFrequency(Corner frequency)Power Spectral Density Figure 2-2. Typical noise power spec tral density plot for a microphone. Noise floor level is one of most import ant specifications of a microphone, since it determines the lowest measurable sound pressu re level and affects the signal to noise ratio (SNR) of a microphone. For microphones, common noise sources could be environmental noise (such as power line, ra dio frequency interfer ence, and environment vibration), noise in a microphone (such as thermomechanical noise, Johnson noise, shot noise and 1/f noise [26]) and noi se in the interface electronics Shown in Figure 2-2 is a typical noise power spectral density plot for a microphone. The corner frequency is where the power from 1/f noise equals the pow er from thermal noise. As seen from the plot, 1/f noise is dominant at low frequencies, while thermal and/or shot noise becomes important at high frequencies. In practice, a noise floor is typically specified by a linear method, an A-weighted approach, or a narro w-band method [27]. In a linear method, the noise is integrated over a specified frequenc y range (for example, 20 Hz to 20 kHz for audio microphones) without any weighting. An A-weighted noise floor is obtained by integrating noise spectrum after amplitude-w eighting, which simulates the perceived PAGE 28 10 noise by the human ear [27]. A narrow-band noise floor is obtained by calculating the total noise within a very narrow frequency ba nd, for example, a 1 Hz bin centered at 1 kHz. For measurement microphones, a narrowband method is usually adapted since the microphone signals are often sampled a nd analyzed in the frequency domain. TimeFrequencyPower SpectrumAmplitude Figure 2-3. Time histories and power spectra of a pure and two distorted sinusoidal waves. The range between the upper and lower meas urable sound pressure levels of a microphone is defined as the dynamic range [16]. The lower measurable pressure level is typically determined by its noise floor. Th e upper limit of the dynamic range is usually set by a specific sound pressure level, whic h results in 3% total harmonic distortion (THD) within the frequency range from 160 Hz to 1000 Hz for measurement microphones [28]. The THD is defined as follows 100%,H TotalP THD P (2.1) PAGE 29 11 where H P is the sum of power within all harmonics, and totalP is the sum of power within the fundamental and all harmonics [28-31]. Figure 2-3 provides an illust ration of harmonic distortion for a sinusoidal wave in both time and frequency domains. As more ha rmonics are added to the original singletone sinusoidal signal in the frequency domai n, more distortion is observed in the time history plot. For capacitive microphones, th e possible sources of harmonic distortion are the nonlinear elastic behavior of the diaphr agm, the electrostatic nonlinearity and the preamplifier Â“clippingÂ” [16, 27]. Transduction Mechanisms In this section, the basics of electromech anical transducers ar e introduced. This section also briefly reviews some major t ypes of transduction mechanisms, including piezoelectric, piezoresistive, electrostatic and optical. A detailed discussion of capacitive microphones is provided in the next section. Introduction to Electromechanical Transducers An electromechanical transdu cer is a device that convert s a mechanical input into an electrical output or vice versa [3]. Propert ies of electromechanical transducers include linear vs. nonlinear, recipro cal vs. non-reciprocal, conser vative vs. non-conservative, and direct vs. indirect [3, 29]. An electromech anical transducer is linear if its output quantities are linear functions of the i nput quantities. Minor nonlinear harmonic distortion is generally allowed in linear electr omechanical transducers [3]. The reciprocal property is used to describe the ability of an electromechanical transducer to convert signals in either direction between two di fferent energy domains. The transduction coefficients are all reversible in reciproc al electromechanical transducers [3]. An PAGE 30 12 electromechanical transducer is conservative if there is no ener gy dissipation during the transduction, otherwise it is non-conservative. An electromechanical transducer is indirect if there exists a tr ansition energy domain between its input and output quantities, otherwise it is a direct transducer. Typically, transducers can be classified into two major categories: sensors and actuators. By definition, microphones fall into the category of a sensor A wide range of transduction schemes are employed in el ectromechanical microphones, including piezoelectric, piezoresistive, optical and electrostatic. These types of microphones and their properties are briefly disc ussed in the following sections. Piezoelectric Microphones Some materials (for example, quartz) develop electric surface charges when compressed by a mechanical force, and this effect is referred to as the direct piezoelectric effect [32]. In addition, a mech anical strain is generated when an electric field is applied to these same materials, and th is effect is referred to as the converse piezoelectric effect [32]. Those materials are ofte n called piezoelectric materials. The relations between the electric quantities (electric di splacement and field) and mechanical quantities (stress and strain) are generally described linearly by a set of piezoelectric coefficients [33]. A piezoelectric microphone typically consis ts of a thin dia phragm (cantilever, rectangular or circular plat es) and a multilayer piezoelectric material [2]. When an incident sound wave impinges and deflects the diaphragm, the induced mechanical stress in the piezoelectric material results in an out put voltage due to the piezoelectric effect. Piezoelectric microphones possess many advantages including inherently low power consumption [34]. Disadvantages of piezoel ectric microphones include the relatively low PAGE 31 13 sensitivity [35] and high noise level [34, 36]. For the electromechanical property, piezoelectric microphones are reci procal, linear, conservative and direct transducers. Piezoresistive Microphones The piezoresistive property of a material is defined as the change in its resistivity due to a mechanical strain or stress. For silicon, the resistivity change is due to a change in the mobility (or number of charge carriers) [37]. The piezoresistive transduction scheme can be used to design silicon microphones. A piezoresistive microphone is constructed mainly by a diaphragm with two pa irs of piezoresistors. When a sound wave impinges and deflects the diaphragm, the induce d mechanical stresses in the two pairs of piezoresistors results in opposite strain cha nges, which leads to the opposite resistance changes. By implementing a fully active Wheat stone bridge, the resistance modulation in the two pairs of piezoresist ors is further expressed by the output voltage change; therefore, the incoming sound pressure can be determined [2, 38]. Piezoresistive microphones have many advantages, such as the scaling, robustness, micromachining convenience, and the absence of a need fo r on-chip circuitry du e to its low output impedance [38, 39]. However, piezoresisti ve microphones have some drawbacks, such as a high noise floor [21], high power consumption, temp erature drift and thermal degradation of the piezoresist ors due to Joule heating [38] For the electromechanical property, piezoresistive microphones are linear, direct, non-reciprocal and nonconservative transducers. Optical Microphones The classification of optical transducti on is generally based on the property of modulated light. Three common transduction schemes are: 1) intensity modulation, 2) phase modulation and 3) polarization m odulation [40, 41]. An optical microphone PAGE 32 14 transforms an acoustic signal into an electrical signal by modulating a reference light signal [40]. Unlike other type s of microphones, an acoustic signal is fi rst converted into an optical signal before it is converted to an electrical signal for optical microphones. The detection electronics can be remotely located away from the acoustic field, which makes optical microphones immune to harsh en vironments [42, 43], and less vulnerable to electromagnetic and radio frequency inte rference [36]. Disadvantages of optical microphones include the requirement of a stab le reference optical source in a physical environment and the packaging since all the system components, such as light sources, optical sensor, and photo detectors, must be well aligned and positioned [42, 44]. Optical microphones are linear, non-reciprocal, non-cons ervative and indir ect transducers. Electrostatic Microphones The electrostatic transduction mechanism has established more than two centuries for actuator applications [3]. To realize an electrostatic conversio n between electrical and mechanical quantities, a capacitor with a deformable electrode is generally needed [29]. Since the electrostatic force in a capa citor is nonlinear by na ture, it needs to be linearized to be suitable for the electrosta tic conversion. The linearization process is typically enabled by on e of two polarization schemes: a charge polarization or a voltage polarization [29]. The electrostatic transduction can be util ized to create microphones. There are two general types of electrostatic or capaciti ve microphones Â– a condenser and an electret microphone. Condenser microphones are polar ized with a constant voltage, while electret microphones are polarized with a c onstant permanent charge. Electrostatic microphones are linear, reci procal, conservative a nd direct transducers. PAGE 33 15 A capacitive microphone mainly consists of a diaphragm and a backplate, which are separated by a dielectric, which is usually an air gap. The di aphragm and backplate are made either of conducting materials or connected with electrodes to realize the capacitive detection mechanis m [29, 45]. When the s ound pressure deflects the diaphragm, the induced capacitance change between the diaphragm and backplate is detected via various types of interface circ uitry [5]. Capacitive microphones have many advantages such as a relatively high sensit ivity, a large bandwidth, an inherently low power consumption and a lo w noise floor [18, 46]. Ho wever, capacitive microphones have some potential issues such as elec trostatic pull-in instability, output signal attenuation due to the parasitic capacitance, and deceased sensitivity at high frequencies due to the viscous damping of the perforated backplate [2]. Electrostatic Microphones This section provides details of elect ret microphones and two major types of condenser microphones. A summary of pr evious work on capac itive microphones is provided in the next section. Movable diaphragmC m 0 x Fixed backplate 0xx 'x x F Figure 2-4. A simplified model of an electros tatic microphone (adapted from Rossi [29]). PAGE 34 16 First, we need to see how the electrostatic transduction is realized by the linearization process. Shown in Figure 2-4 is a simplified model of an electrostatic microphone in air. The backplate is assumed to be fixed and the diaphragm is movable and located at 0 x x due to some loading (for exam ple, acoustic pressure). The mechanical restoring force for the diaphragm is modeled by a spring. If a small displacement fluctuation x is assumed, the position of the diaphragm is given by 0. x xx (2.2) By using a parallel-plate assumption, the capacitance C is given by 11 '' 00 0 000011, AA xx CC xxxxx (2.3) where 0 is the dielectric constant of the air, A is the area of the diaphragm and 000CAx is the mean capacitance. If a charge, Q is applied to the movable diaphragm, and we assume 0, QQQ (2.4) where 0Q is the mean charge and 'Q is the small charge fluctuation. Therefore, the voltage, V, across the diaphragm and backplate is ''' 0 000011, QQxQx VV CCxCx (2.5) where 000VQC is the mean voltage. The stored electrical potential energy in the capacitor, eU is given by [5] PAGE 35 17 22' 001. 22eQQx U CCx (2.6) Therefore, the electrostatic fo rce acting on the diaphragm is 2 00. 2e edU Q F dxxC (2.7) The mechanical restoring force is given by ,m m x F C (2.8) where mC is the mechanical compliance of the spring. Therefore, the net force acting on the diaphragm is 2 00. 2mxQ F CxC (2.9) Eqs. (2.3), (2.5) and (2.9) are three non linear coupled equations. In order to linearize these equations to realize the electrostatic transduction, we must assume 01, x x (2.10) 01 Q Q (2.11) and 01 V V (2.12) Physically, this lineariza tion process can be done by one of two polarization schemes: a charge polarization or a voltage polarization [29]. If a constant voltage 0V is applied across the diaphragm and backplate, a condenser microphone is created. An PAGE 36 18 electret microphone is created if a constant charge 0Q is stored permanently on the diaphragm or backplate. Once the polarization is applied, Eqs. (2.3 ), (2.5) and (2.9) can be linearized as 1 '' 00 0 000011, AA x x CC x xxxx (2.13) '' 0 00, V Q Vx Cx (2.14) and '' 0 0,mV x FQ Cx (2.15) where 'V and 'F are the fluctuating components of the voltage and force applied on the diaphragm, respectively. Eqs. (2.14) and (2.15) represent the linea rized electrostatic c oupling equations in the displacement and charge form. These two equations can be furthe r rewritten in power variables [29] as follows ''' 0 001 v, V VI jCjx (2.16) and ''' 0 01 v,mV FI jCjx (2.17) where I is the fluctuating component of the current passing through the diaphragm and 'v is the fluctuating component of the velocity of the diaphragm. In matrix form, the coupling equations are written as PAGE 37 19 0 '' 00 '' 0 01 1 vmV jCjx VI V F jxjC (2.18) Electret Microphones For an electret microphone, the permanent ch arge is usually fixed by a thin layer of charge-holding electret materi al on the backplate or diaphr agm [47, 48]. The electret material is dominated by Teflon for c onventional electret microphones and silicon dioxide/nitride for silicon el ectret microphones [47]. In el ectret microphones, integration of the electret layer on the diaphragm is a common choice since the backplate is usually perforated [48-50]. Air GapBackplateDiaphragmElectretMetallization Acoustic Pressure Figure 2-5. Schematic of a capacitive mi crophone with an electret diaphragm. A schematic of a capacitive microphone w ith an electret diaphragm is shown in Figure 2-5. An air gap sepa rates a metalized electret diaphragm from a backplate. A metallization layer is used to charge the electret layer prior to the operation. At the interface between the diaphragm and electret, a charge layer with certain charge density (total charge divided by the cro ss section area) is formed to generate the electrical field inside the gap. One advant age of electret microphones is the absence of an external power supply and the potential for portable app lications [46, 50]. However, integration PAGE 38 20 of electret microphones with MEMS suffers from the poor quality of thin film electrets and charge loss due to humidity [46]. 0d 0 x F k, mABackplateDiaphragm Q x Figure 2-6. A simplified qausi-static model of an electret microphone. Critical bias charge. To illustrate the idea of the critical bias charge for electret microphones, shown in Figure 2-6 is a simplified quasi-static model of an electret microphone. The mass and mechanical stiffne ss of the movable diaphragm are modeled by a point mass (m) and a spring (k) respectively. The nom inal gap between the diaphragm and backplate is 0d and x is a vertical displacement of the diaphragm. Since only the quasi-static process is c onsidered here, the inertial force and damping force are neglected since they are time-dependent. In the following analysis, only the mechanical restoring and electrostatic forces are considered. The diaphragm and backplate are conductive, thus one capacitor is formed. For electret microphones, the electrostatic force, eF, is given as [5] 2 0. 2eQ F A (2.19) As seen from Eq. (2.19), the electrostatic force is a quadratic function of the bias charge and independent of the vertical displacement x of the diaphragm. For a given PAGE 39 21 constant bias charge, the electrostatic force re mains as a constant force. By using a linear spring model, the mechanical restoring force,mF, is given as 0, 0.mFkxxd (2.20) 0 Displacement xForce d0 Mechanical force Fm Electrostatic force Fe (Q < Qcr) Electrostatic force Fe (Q > Qcr) Electrostatic force Fe (Q = Qcr) Stable point Figure 2-7. Illustration of the critical bias charge of an electret microphone. Shown in Figure 2-7 is a plot with both electrostatic and mechanical forces for different bias charge cases. When the bias constant charge Q is smaller than a critical value crQ, the electrostatic force can be always balanced by the mechanical force and the diaphragm moves to a stable position as show n in the plot. When the constant bias charge Q reaches crQ, the electrostatic force is greater than the mechanical force when 0 x d and the diaphragm is forced to move to the rigid backplate position 0 x d which represents a non-functioning microphone. When the constant bias charge Q is greater than crQ, the electrostatic force is always great er than the mechanical force and the PAGE 40 22 diaphragm always moves to the rigid backpl ate. Therefore, el ectret microphones must operate with a bias charge le ss that its cri tical value. To find the quasi-static critical bias charge, the electrostatic attraction force is set to be equal to the mechanical restoring force at 0 x d which results in 2 0 0. 2crQ kd A (2.21) Namely, the critical charge crQ is [51] 002.crQAkd (2.22) Condenser Microphones The majority of condenser microphones can be categorized into single-backplate condenser microphones and dual-backplat e condenser microphones based on the backplate configuration. In 1996, Bay et al. proposed a dual-diaphragm condenser microphone [19]; however, it was no t fabricated. In 2002, Rombach et al. fabricated the first dual-backplate condenser microphone [52] The details of two major types of condenser microphones are provided in the following. Single-backplate condenser microphones Backplate Diaphragm Backplate hole Cavity VentAir Gap Figure 2-8. Schematic of a singl e-backplate condenser microphone. A typical single-backplate condenser micr ophone with a diaphragm and a backplate is shown in Figure 2-8. A bac kplate is perforated to reduce the air-streaming resistance, a PAGE 41 23 vent is provided to equilibrate the pressure inside the cavity to the ambient atmospheric pressure. A capacitor is formed between the conductive backplate and diaphragm. Quasi-static pull-in. Based on a similar dynamic model as shown in Figure 2-6, for single-backplate condenser micr ophones, the electrostatic force eF is given as [5] 2 0 2, 2eAV F d (2.23) where 0ddx is the distance between the diaphragm and backplate. By using a linear spring model, the mechanical restoring force,mF, is given by 0.mFkdd (2.24) To find the quasi-static pull-in voltage, the electrostatic attraction force is set to be equal to the mechanical restoring force at 023PIdd or 03PIxd which results in 2 00 0 2 02 3 2 2 3PIAVd kd d (2.25) Namely, the critical pull-in voltage P IV in the constant bias voltage case is [5] 3 0 08 27PIkd V A (2.26) It should be pointed out that the above an alysis results are valid based on a simple parallel-plate assumption and a linear spri ng model. When the displacement of the diaphragm becomes nonlinear, geometric nonl inearity of the diaphragm needs to be considered to obtain more accurate results. As seen from Eq. (2.23), the electrostat ic force is a nonlin ear function of the applied voltage and the displacem ent of the diaphragm. Show n in Figure 2-9 is the plot PAGE 42 24 with both electrostatic and mechanical forces when the constant bias voltage V is less than the critical pull-in voltage P IV. 0 Displacement xForce Electrostatic force Fe (V < VPI) Mechanical force Fm Unstable fixed point Stable fixed point Stable regions Unstable region d0/3 Figure 2-9. Illustration of mechanical and electrostatic forces for a single-backplate condenser microphone ( P IVV ). As seen from Figure 2-9, there are thr ee regions and two equilibrium points. Below the stable fixed point, the electrostatic force is always greater than the mechanical force and attracts the diaphragm to the stab le fixed point. Between stable and unstable fixed points, the mechanical force is always greater than the electrostatic force and pulls the diaphragm back to the stable fixed point. Above the unsta ble fixed point, the electrostatic force is always greater than the mechanical restoring force and the displacement of the diaphragm keeps increa sing until the diaphragm crashes into the backplate. Therefore, the stable operati on range of the microphone is from the rest position to the unstable fixed point when P IVV PAGE 43 25 As the applied bias voltage becomes larger and reaches its critical pull-in value, the electrostatic force is larger than the mechan ical restoring force except for one critical point ( 023PIdd or 03PIxd ) as shown in Figure 2-10. When pull-in occurs, the two fixed points shown in Figure 2-9 move to wards each other and coalesce at the pull-in position as shown in Figure 2-10. When the a pplied bias voltage is over its critical pullin value, the electrostatic force is always la rger than the mechanical restoring force and all regions in Figure 2-9 become unstable. 0 Displacement xForce Mechanical force Fm Electrostatic force Fe (V=VPI) Electrostatic force Fe (V>VPI) d0/3 Increased bias voltage V Figure 2-10. Illustration of quasi-static pull-in of a single-backplate condenser microphone ( P IVV ). Scaling. In practice, a condenser microphone needs to be connected to a preamplifier to make a measurement to avoid a signal attenuation or loss. The preamplifier serves as an impedance convert er and it is commonly modeled as a source follower with an input capacitanceiC and a gainaH. The overall sensitivity far below PAGE 44 26 resonance [2] of a single-backplate condens er microphone is compliance dominated and can be expressed as follows ,overallmecaSSSHH (2.27) where mS is the mechanical sensitivity, eS is the electrical sensitivity, cH is the capacitance signal attenuation due to th e input capacitance of the preamplifier iC, the parasitic capacitance pC, and the gain of the preamplifier aH as shown in Figure 2-11. + Output Preamplifier Parasitic Capacitance MicrophonemC p CiCbRbVaH Figure 2-11. Simplified circuit of a singl e-backplate condenser microphone with a preamplifier (adapted from Scheeper et al. [2]). The mechanical sensitivity mS of the microphone is defined as the change of the diaphragm center deflection resulting from a unit change in the applied sound pressure level. For a circular diaphrag m with a residual tensile stress0 the mechanical sensitivity, mS, is given by [2] 2 0, 8ma S h (2.28) where a and h are the radius and the thickness of th e diaphragm respectively. It should be pointed out that Eq. (2.28) is obtained based on a pure linear mechanical membrane model, which does not account for the bending forces and moments. PAGE 45 27 The electrical sensitivity eS is defined as the change in the voltage across the air gap resulting from a unit change of the ai r gap. If the electric field strength E in the air gap is assumed homogeneous, and the electrical sensitivity, eS, is given by 0,b eV SE d (2.29) where bV is the DC bias voltage. From the simplified circuit, as show n in Figure 2-11, the capacitive signal attenuation cH is ,m c mipC H CCC (2.30) where 0 0,mA C d (2.31) and 2.Aa (2.32) The microphone capacitance is represented by mC, and A is the area of the diaphragm. For the designer, the gain of the preamplifier aH is close to unity and signal attenuation is not desired. Ideal ly, the following can be obtained 1,cH (2.33) if .mipCCC (2.34) Ideally, the overall sensitivity is then changed into ,overallmeopenSSSS (2.35) where openS is the open-circuit sensitivity. PAGE 46 28 In practice, as the radius of the dia phragm decreases, th e capacitance of the microphone mC also decreases. Therefore, Eq. (2.34) does not hold and the microphone will suffer from the output signal attenuation due to the parasitic capacitance. By using Eqs. (2.26) and (2.28), the crit ical pull-in voltage for a single-backplate condenser microphone can be rewritten as 3 0 08 1 27 PI md V S (2.36) If the applied bias voltage is always kept at a fixed fract ion of the critical pull-in voltage, from Eqs.(2.28), (2.29), (2.31), (2.3 5) and (2.36), the follo wing relations can be obtained 0~,mA C d (2.37) 3 2 0~,PIVd (2.38) and 0000~~~.mbbb openSVAVAV S dhdd (2.39) From Eq. (2.39), as the area of diaphragm becomes smaller, the open-circuit sensitivity decreases. Also to maintain Eq. (2.34) for the overall sensitivity, the gap needs to become smaller. As the gap beco mes smaller, the critical pull-in voltage becomes smaller too; therefore, the open-circuit sens itivity eventually becomes smaller. Hence, from the sensitivity point of view, th e gap and area of the diaphragm could be the inherent limitations with th e miniaturization of single-ba ckplate condenser microphones. The air-streaming resistance due to the holes of the backplate plays an important role in determining the frequency response or the bandwidth of condenser microphones. PAGE 47 29 For a rigid circular backplate, the mechanical air-streaming resistance m R is obtained as follows [53] 4 2 3 0411311 34828mpp pa RlnAA ndA (0.40) where is dynamic viscosity of the air, n is the total number of holes and pA is the ratio of the total area of holes to the bac kplate area. Therefore the following scaling relation is obtained 3 01 ~,mR nd (2.41) As the gap becomes smaller, the air-str eaming resistance increases quickly and decreases the bandwidth of the microphone. Therefore, from the bandwidth point of view, the gap could be one of inherent limitations with miniaturization of singlebackplate condenser microphones. However, a very high number of holes (for example, hundreds or thousands) can be used to main tain a small air-streaming resistance for micromachined condenser microphones with narrow air gaps [20]. On the other hand, as the number of holes in the backplate increas es, the backplate becomes more compliant and electrostatic pull-in will occur with a lower voltage. Dual-backplate condenser microphones A schematic of a typical dual-backplate c ondenser microphone is shown in Figure 1-1. It consists of two perforated backplat es with a diaphragm in between, a cavity and a vent. Two capacitors are formed by th e conductive backplat es and diaphragm respectively. Due to the existence of symm etric backplates, a dual-backplate condenser microphone ideally can generate twice the signal of a single-backplate condenser PAGE 48 30 microphone [17]. The configur ation of symmetric backplat es could lead to a higher sensitivity, a higher capaci tance, a wider linear dynamic range [18] and a broader bandwidth. In addition, the applied bias voltage can be higher for a double backplate condenser microphone, since th e electrostatic forces ac ting on both sides of the diaphragm can balance with each other. Furthermore, the dual-backplate condenser microphone has the potential to ope rate in a closed loop [54]. Details of a dynamic model for a dua l-backplate condenser microphone are provided in Chapter 3. Pull-in issues associated with dual-backplate condenser microphone are studied in Chapter 5. Major Previous Work on Capa citive Silicon Microphones Major previous work and recent devel opments of capacitive silicon microphones [17, 49, 50, 52, 55-67] are listed in Table 2-1. It can be se en that there is a significant trend towards developing instrumentation gr ade silicon condenser microphones in terms of smaller size, larger sensitivity and si gnal-to-noise ratio, and broader bandwidth. Hohm and Gerhard-Multhaupt (1984) de veloped the first electret silicon microphone. Its backplate consisted of a 1cm 1cm p-type silicon layer, a 2m SiO2 top electret layer and a 0.1m bottom aluminum electrode layer. A circular hole with a diameter of 1 mm was perforat ed in the center of the backpl ate. The diaphragm was an aluminum-coated Mylar foil with a thickness of 13m and a diameter of 8 mm. A Mylar foil was used as a spacer to form an air gap of 30m The reported open-circuit sensitivity was approximately 8.8/ mVPa, but the measured sensitivity at the frequency of 1kHzwas approximately 3/ mVPa due to the parasitic capacitance. PAGE 49 31 Table 2-1. Major previous work in capacitive silicon microphones. Authors Diaphragm Area (2mm ) Air Gap (m ) Upper Frequency (kHz) Sensitivity (/mVPa) Noise Level (dBA) aHohm and GerhardMulthaupt [49] 50.3 30 8.5 3.0 -aSprenkels et al. [50] 6.0 20 >15 25a -bBergqvist and Rudolf [55] 4.0 4.0 16 13a -bBergqvist et al [56] 4.0 2.0 20 1.6 40 bKuhnel and Hess [57] 0.64 2 20 1.8 43 bBourouina et al [58] 1.0 7.5 10 2.4 38 bScheeper [59] 4.0 3.0 >14 7.8c 30. bZou et al [60] 1.0 2.6 9.0 14.2 -bSchafer et al [61] 0.5 4.0 17 14 28 bTorkkeli et al [62] 1.0 1.3 12 4.0 33.5 bRombach et al [52, 63] 4.0 0.9 >20 13c 22.5 bScheeper et al [17] 12 20 20 22c 23 bHansen et al [64] 0.01 1 100 7.3 63.6 bMartin et al [65, 66] 0.17 2 230 0.28c, d 42e bPedersen [67] 0.1 1.24 100 0.5 22e bLoeppert and Lee [68] 0.25 4 20 --aElectret microphone bCondenser microphone cOpen-circuit sensitivity dCharge amplifier e1 Hz bin Sprenkels et al. (1989) reported an electre t silicon microphone. It had a metallized PAGE 50 32 Mylar foil diaphragm with a thickness of 2.5 m and a 20 m air gap. A SiO2 layer with a thickness of 1.1 m was used as electret material and was biased by a permanent charge with an effective volta ge of approximately -300 V. The open-circuit sensitivity was approximately 25/ mVPa at the frequency of 1kHz. Bergqvist and Rudolf (1990) published the first silicon condenser microphones. A microphone with a 5 m thick p-type silicon diaphragm and a 4 m air gap demonstrated an open-circuit sensitivity of 13/mVPa at the frequency of 1.kHz Microphones with a 8m thick p-type silicon diaphragm showed a bandwidth ( 1 dB ) of 16kHz. Low parasitic capacitances (< 0.5 p F) and a large number of holes in the backplate were reporte d on those microphones. Bergqvist et al (1991) presented a conde nser microphone with a 2mm 2 mm stress-free p-type sili con diaphragm and a 2 m air gap. The backplate was perforated with 640 acoustic holes per 2mm to reduce the air-streaming resistance; however, the backplate size was not reported. The microphone operated with a 5 V bias voltage. Its frequency response was flat within 3 dB from 2 to 20kHz. The measured microphone noise was dominated by the pr eamplifier noise and the e quivalent noise level was 40dBA. The measured sensitivity was 1.6/mVPa at the frequency of 1kHz. The total harmonic distortion was less than 0.08 % at a sound pressure level of 120dB. Kuhnel and Hess (1992) developed a c ondenser microphone with a specially designed backplate to reduce the air-streami ng resistance. The mi crophone consisted of a 0.8mm0.8mm silicon nitride diaphrag m with a thickness of 150nm. The backplate was structured with either anisotropically etched holes or plasma-etched grooves. Microphones with grooves in backplates and stress-free diaphragms demonstrated an PAGE 51 33 open-circuit sensitivity of 10/ mVPa (1.8/ mVPa measured sensitivity due to the signal attenuation caused by the parasite capacitanc e) and the measured bandwidth was up to 20kHz. Bourouina et al (1992) developed a condenser microphone design without acoustic holes in the backplate. In or der to lower the air-streaming resistance, a relatively thick air gap was used (5 or 7.5m ). The diaphragm was made w ith heavily boron-doped silicon and had a residual tension stress of 70. M Pa A Pyrex wafer was bonded to the diaphragm wafer by anodic bonding. A microphone with an air gap of 7.5 m and a bias voltage of 20 V showed a flat freq uency response up to 10kHz and a sensitivity of 2.4/ mVPa. The noise of the microphone mainly came from the preamplifier, and was measured to be 38dBA. Scheeper (1993) presented a condenser mi crophone with a high density of acoustic holes in the backplate. The microphone possessed an open-circ uit sensitivity of 7.8/mVPa, a capacitance of 8.6 p F and a flat freque ncy response from 0.1kHz to 14kHz (2 dB). Due to the existence of a narrow air gap, the microphone operated at a relatively low bias voltage to avoid pull-in. The pull-in voltage was not reported in the paper. Zou et al (1997) developed a silicon condenser microphone with a corrugated diaphragm. The microphone had an air gap of 2.6 m and a 1mm1mm diaphragm with a residual tension stress of 70 M Pa and a thickness of 1.2 m A corrugated diaphragm was fabricated by an anisotropi c etching to improve the sensitivity by reducing the effect of high tension stress. A microphone with a corrugation depth of 8 m and a 10V bias voltage showed a m easured sensitivity of 14.2/ mVPa. The PAGE 52 34 simulated bandwidth was reported to be 9kHz. The finite element method (FEM) and equivalent circuit method were used to predict the micr ophone performance. Schafer et al (1998) from Knowles Acoustics, reported a silicon micromachined condenser microphone for the hearing aid applic ation. The microphone had an air gap of 4 m and a circular silicon nitride diaphragm with a radius of 0.4 mm and a thickness of 0.75.m A highly compliant diaphragm was achie ved by the support in the middle rather than the clamped support at the perimeter. A lumped element model was built to predict the sensitivity, bandwidth and noise floor of the microphone. The measured sensitivity of the microphone with a bias voltage of 12 V was 14 /mVPa at the frequency of 1kHz. The measured A-weighted noise level was as low as 28dBA and the measured resonant frequency was up to 17 kHz. Torkkeli et al. (2000) fabricated a capacitive microphone with a low-stress (2 M Pa) polysilicon diaphragm. The mi crophone had an air gap of 1.3m and a 1mm 1mm diaphragm with a thickness of 0.8.m The measured sensitiv ity of the microphone with a 2V bias voltage was 4/mVPa at the frequency of 1kHz. The measured A-weighted noise level was 33.5dBA. The measured capacitance was 11 p F and the bandwidth was 12kHz. Rombach et al. (2002) fabricated the first dua l-backplate silicon condenser microphone. Due to the symmetric arrangemen t of backplates, a thin air gap of 0.9m was fabricated to generate a high electrical field and a high sensitivity with a low bias voltage. The microphone consisted of a 2mm 2mm multilayer diaphragm with a thickness of only 0.5m and an overall residual tensile stress of 45 M Pa. The total sensitivity with a bias voltage of 1.5V was measured to be 13/mVPa and the A- PAGE 53 35 weighted equivalent noise le vel was measured to be 22.5dBA. The upper limit of its dynamic range was determined to be 118dB and the total harmonic distortion at 80 dBASPL was less than 0.26%. Scheeper et al. (2003) fabricated a condenser MEMS measurement microphone. The microphone had an air gap of 20m and an octagonal silicon nitride diaphragm with a tensile stress of 340 M Paand a thickness of 0.5m The area of a circular diaphragm was approximately 11.95 2mm, while the area of a square backplate was approximately 8 2mm. The measured average open-circuit sensitivity was 22/mVPa with a 200V bias voltage and the measured noise leve l (including the preamplifier) was 23dBA. The measured frequency response was flat up to 20,kHz and the resonance frequency was reported between 47 and 51kHz. Other important specificati ons of this microphone such as the measured temperature coefficient, humidity coefficient for the sensitivity and 3% distortion limit were also reported. Hansen et al. (2004) reported a wide-bandw idth micromachined capacitive microphone based on radio frequency detect ion. The microphone consisted of a metallized rectangular silicon nitride memb rane, which was suspended over a silicon substrate to form a small sealed volume. A sensitivity of 7.3/mVPa was measured with a gap of 1 m and a 70m 190m membrane with a thickness of 0.4m The measured bandwidth was flat within 0.5 dB over the range from 0.1 Hz to 100 kHz. However, the microphone suffered from its relatively high noise floor, which was measured to be 63.6 dBA. Martin et al. (2005) fabricated a dual-bac kplate capacitive MEMS microphone by using SUMMiT V process at Sandia Nationa l Laboratories. Due to the symmetric PAGE 54 36 arrangement of backplates a thin air gap of 2 m was fabricated. The microphone consisted of a circular polysilicon diaphragm with a 230 m radius and a thickness of 2.25 m The measured sensitivity with a charge amplifier and a bias voltage of 9 V was 0.28/mVPa and the measured noise level was 42 dBHz at 1 kHz. The measured dynamic range was over 118 dB and a linear response up to 160 dB was observed. The predicted resonanc e frequency was approximately 185.5 kHz and the measured resonance frequency was approximately 230 kHz [66]. Pedersen (2006) presented an aero-ac oustic capacitive MEMS microphone. By utilizing the Knowles SiSonic MEMS micr ophone technology, the microphone consisted of a circular diaphragm with a 180 m radius and a thickness of 2 m An air gap of 1.24 m was fabricated between the diaphragm a nd backplate. The measured sensitivity at 1 kHz was approximately 0.5 /mVPa. The measured total harmonic distortion at 130 dBSPL was 1.5%. The predicted bandwidth was approximately 100 kHz and the measured noise level was 22 dBHz at 1 kHz. Loeppert and Lee in Knowles Electr onics, LLC (2006) presented the first commercialized condenser MEMS microphone. Fabricated by KnowlesÂ’s 10-mask, dualpoly process, the microphone consisted of a sili con nitride backplate with a thickness of 1.5 m and a polysilicon circular diap hragm with an effective 280 m radius and a thickness of 1 m An air gap of 4 m was fabricated between the diaphragm and backplate. The circular backplate of the microphone was perforated with circular holes with a radius of 4 m and the porosity of the backplate was 22.8%. The measured capacitance of the microphone was 0.5 p F. The measured frequency response was flat PAGE 55 37 from 100 Hz to approximate 20kHz. However, the measured sensitivity and noise floor were not reported in the paper. Basics of Nonlinear Dynamics Nonlinear dynamics is the study of th e time evolving behavior of a nonlinear system. These systems are often governed by nonlinear partial/o rdinary differential and/or algebraic equations [ 69]. To date, nonlinear dynamics has been applied to many fields of engineering and sc ience [69-77]. To help prov ide the reader with some familiarity to the field of nonlinear dynamics, some general terminologies and theories are introduced. In the sections that follow, several key concepts, such as nonlinearities, autonomy, fixed points, stability and bifurcation, are discussed. Nonlinearities Although there are many types of nonlinea rities, the major nonlinearities in the field of engineering can be categorized into the following four types [6, 73]: Material or constitutive nonlinearity, examples include the inelastic phenomena (plasticity, creep, hysteresis, etc.) in solid mechanics, and nonlinear resistor (for example, thermal effect on resistance) in electrical engineering. Kinematic nonlinearity, examples include Coriolis acceleration and the convective acceleration term in the Navier-Stoke s equations in fluid mechanics. Nonlinear force or boundary, examples in clude the nonlinear electrostatic force, magnetic force and a beam attached to a nonlinear torsional spring at one of its pinned end. Geometric nonlinearity, examples include a Hertzian contact model for two elastic bodies and a DuffingÂ’s spring. Steady State, Autonomous System and Fixed Point The steady state refers to the asymptotic behavior of the dynamical system as time goes to infinity [69]. The behavior prior to the steady state is often called transient [69]. An autonomous dynamical system is a system that does not contain time as an explicit PAGE 56 38 independent variable [69]. Otherwise, a dynamical system is defined as a nonautonomous dynamical system if it explicitly depends on time [69]. An example of an autonomous dynamical system is the unforced DuffingÂ’s equation, 3 130,mxbxkxkx (2.42) where x is the displacement, m is the mass, b is the damping coefficient, and 1k and 3k are the linear and nonl inear spring stiffnesses, respectively. An example of a non-autonomous dynamical system is the damped DuffingÂ’s equation with an external harmonic excitation, 3 13cos,mxbxkxkxAt (2.43) where A and are the forcing amplitude and frequency, respectively. 1km 30cubic nonlinearity kDisplacement Amplitude (Steady State)Forcing Frequency 30 k Figure 2-12. Effect of cubic nonlinear ity on the system frequency response. PAGE 57 39 Shown in Figure 2-12 is the effect of cubic nonlinearity on the frequency response of the system defined in Eq. (2.43). As the cubic nonlinearity increases, more bending is observed in the response curve. Also, multip le steady-state solutions could be obtained in some frequency range, which i ndicates the system is nonlinear. To define the term fixed point, cons ider the following general autonomous dynamical system as follows x fx (2.44) where x represents a column vector of all state variables, and f is a column vector of all corresponding general functions. Physically, a fixed point e x (also known as steady state solution) corresponds to an equilibrium position of a system. Mathematically, it is the solution to the following equation 0.efx (2.45) Stability, Basin of Attraction and Phase Portrait For stability, a fixed point is locally stab le if, given an initia l condition sufficiently close to the fixed point, the sy stem eventually approaches th is fixed point. A fixed point is globally stable if it is approached for all initial conditions [69, 75]. For a linear system, a locally stable fixed point is always globally stable. However, for a nonlinear system, starting from different initial conditions coul d result in different steady-state solutions. Local stability does not guarant ee global stability, which leads to the concept of a basin of attraction. The domain of all initial conditions that eventually converges to a fixed point is called the basin of a ttraction (stability region) of the fixed point. More details about basins of attraction will be provided in Chapter 5. PAGE 58 40 A phase portrait is a collection of all trajec tories that represent the solutions (both transient and steady state solutions) of the gove rning equation in the state space [69, 77]. Phase portrait is one of most important tools in studying the behavior of nonlinear systems since usually there is no closed-f orm solution for most nonlinear systems. -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Non-Dimensional Center Dis p lacementNon-Dimensional Center Velocity Physical border lines Outer range Outer range Figure 2-13. Phase plane traj ectories around fixed points of a dual-backpl ate capacitive MEMS microphone (sink points are indica ted by blue crosses, and saddle points are indicated by blue circles). For the illustration purpose, shown in Figure 2-13 is an example phase portrait of a dual-backplate capacitiv e MEMS microphone under the electro static DC excitation only. The non-dimensional nonlinear governing equation in state-space form is 2 1 3 1 211 2 2 2 1, 2 1x x x xxx x x (2.46) PAGE 59 41 where the two states 1 x and 2 x are the non-dimensional center displacement and velocity of the diaphragm respectively, and other nondimensional parameters are assumed to be fixed in the phase portrait. From the plot, we can see the coexistence of multiple fixed points: two locally stable sink points (attractors) outside the physical bord er lines, one locally stable sink point in the center and two unstable saddle nodes (repellors) between the center point and physical border lines respectively. Bifurcation When one or more parameters of the dyna mical system are varied, the qualitative change of response behavior, such as a change in the number of stable solutions, is called a bifurcation [69]. The varying parameter is often called the bifurcation parameter. Diagrams, in which the variati on of fixed point solutions and their stability are displayed in the space of state variables and bifurcat ion parameters, are often called bifurcation diagrams [69]. The critical location, at wh ich a bifurcation occurs in the bifurcation diagrams, is called a bi furcation point [69]. Bifurcation Parameter Fixed Point Solution(a)(b)(c)StableUnstable Figure 2-14. Typical bifurcati on diagrams for one-dimensional autonomous systems: (a) Saddle-node bifurcation; (b) Pitchfor k bifurcation; (c) Transcritical bifurcation (adapted from Nayf eh and Balachandran [69]). For simplicity, one-dimensional autonomous systems are used here to illustrate the basic idea of a bifurcation. Shown in Figu re 2-14 are three typica l static bifurcation PAGE 60 42 diagrams. The saddle-node bifurcation di agram is generated by considering the following dynamical system 2, x x (2.47) where is the bifurcation parameter. In th e saddle-node bifurc ation, two solution branches start growing from the bifurcation point at 0 one is stable and another is unstable. The pitchfork bifurcation diagram is generated by considering the following dynamical system 3. x xx (2.48) In the pitchfork bifurcation, initially, there is only one stable solution branch. As the bifurcation parameter increases, the stable so lution branch evolves into three different solution branches: two stable solution branch es outside and one uns table solution branch between them. The bifurcation point is located at 0 Finally, the following dynamical system is c onsidered to generate the transcritical bifurcation 2. x xx (2.49) In the transcritical bifurcation, stabilities of the original solution branches change into their counterparts when the bi furcation parameter is sweepi ng over its bifurcation point (in this example, it is located at 0 ). Previous Work on Nonlinear Dynami cs of Electrostatic MEMS Devices Recently, many electrostatic MEMS de vices, such as microphones [65], micropumps [78], microswitches [79], pressure sensors [80] and mass sensors [14], have been fabricated due to the advances of modern silicon micromachining technology. PAGE 61 43 Many types of nonlinear forces and geometrical nonlinearities were en countered in those MEMS devices. To adequately study these nonlinear behaviors, one analysis approach is to assume a linear relationship between the physical quantities and neglect or linearize those nonlinearities. It turns out that the linear met hod produces erroneous results, for example, when the device is under the large de flection or large excita tion. Therefore, one has to resort to a nonlinear analysis. In this section, previous work on nonlinear dynamics of electrostatic ME MS devices is summarized. In 1954, F. V. Hunt [3] studied pull-in in stability for a single-sided electrostatic loudspeaker. By equating a linear mechanical restoring force to a nonlinear electrostatic force, he found that quasi-static pull-in (referred to it as Â“f all inÂ” in his book) occurs at two thirds of the gap between the diaphrag m and bottom electrode of the loudspeaker. Pull-in experiments were carried for the l oudspeaker by monitoring the displacement of the diaphragm; pull-in occurred at a value of 78% of the gap. To explain the difference between the theoretical and measured critical pull-in loca tions, he suggested that the most likely reason was the exclusion of the curvature of diaphragm in the model. H. C. Nathanson and his coworkers [4] in 1967 studied pull-in of an electrostatic resonant gate transistor. To understand the electrostatic actuation in their device, a simplified mass-spring model based on the para llel-plate assumption was constructed and analyzed, and pull-in instability was predic ted and explained by using the 1-D model. In 1968, Taylor [81] observed pull-in phenomenon when he increased the voltage between the two closed-spaced liquid drops. The two drops remained apart when the applied voltage was small, and coalesced when the voltage was beyond a threshold voltage. The threshold voltage is referred to as pull-in voltage. PAGE 62 44 Puers and Lapadatu [51] studied electrosta tic forces and their effects on capacitive mechanical MEMS sensors (acce lerometer and pressure sensor). In a constant bias charge mode, the critical pull-in charge and mechanical force were obtained analytically based on a simple parallel-plate assumption a nd a linear spring model. The influence of damping on the critical pull-in charge was al so discussed. In a constant bias voltage mode, the critical pull-in voltage and mechanic al force were also obtained analytically. Moreover, possible solutions, such as the use of symmetrical struct ures, were suggested to avoid pull-in in a cons tant bias charge mode. Pedersen et al. [82] investigated the ha rmonic distortion in micromachined silicon condenser microphones. A quasi-static model, which includes a more realistic shape of the diaphragm deflection, was applied to st udy the harmonic distortio n for frequencies far below the resonance of the diaphragm. Inst ead of an analytical series solution, an iterative numerical approach based on finite differences was implemented to find the quasi-static diaphragm deflec tion and open-circuit sensitiv ity of the microphone. The harmonic distortion generated by the sound pressu re and nonlinear elec trostatic force due to a bias DC voltage was studied numerically and experimentally for a silicon condenser microphone. Both numerical and experimental results showed that the total harmonic distortion of the microphone was linearly depe ndent on the applied sound pressure level and was a nonlinear function of the applied DC bias voltage. On the other hand, some researchers are l ooking for ways to take advantage of pullin in their MEMS devices. Gupta and Sentur ia [80] presented a MEMS absolute pressure sensor utilizing pull-in. Th e device operated with two m odes: non-contact and contact modes. During the contact mode, the device was dynamically actuated by an applied step PAGE 63 45 voltage higher than the pull-in voltage, pull-in time from rest to contact was measured and found to be a nearly linear function of th e absolute pressure inside the sensor. A simple 1-D lumped mass-spring-damper model was constructed to theoretically calculate the pull-in time for an electrostatically actua ted fixed-fixed microbeam The electrostatic force was modeled based on the parallel-pla te assumption and the fringing field effect [83] was neglected. Also, si mulated pull-in times were found to be in good agreement with the measured values. Zavracky et al. [79] reported a micromech anical switch with three electrodes: a source, a gate and a drain. When the appl ied voltage between the gate electrode and a microbeam, which connects to the source elec trode and hangs over the gate and drain electrodes, was over the pull-in voltage, cont act was made between the source and drain electrodes. Both numerical and analytical methods were used to investigate the pull-in of the microbeam; however, the analytical mode l based on the parall el plate assumption yielded poor results. A nonlinear spring consta nt of the microbeam was extracted from the numerical simulation results and applied back to the an alytical model to obtain good results. Nemirovsky and Bochobza-Degani [84] pr esented a generalized model for the quasi-static pull-in parameters of electrostatic actuators with a single charge or voltage input. By setting the first and second deri vatives of the total energy (the summation of mechanical and electrical energy) with re spect to the displacement of the moving electrode plate to zero, two ge neral algebraic equations were obtained to solve for the pull-in parameters for each type of input respectively. The obtained equations were applied to a wide range of case studies, including parallel-pla te and tilted-plate PAGE 64 46 electrostatic actuators with the fringing fi eld capacitance, the parasitic capacitance, residual charges, constant exte rnal forces and nonlinear mechanical forces. Specifically, the analyses results showed that the addition of cubic mechanical nonlinearity in the form of a DuffingÂ’s hardening spring extends the st able range of electrostatic actuators quasistatically. In addition, it was showed that the quasi-static pull-in parameters (such as voltage and displacement) can be affected by the constant exte rnal force or pressure. For example, the quasi-static pull-in voltage was re duced when the external force or pressure was in the same direction with the electrostatic force and vice versa. Zhang et. al. [13, 14, 85] investigated the nonlinear behavior of a parametric resonance-based MEMS mass sensor. Cubi c nonlinearities were modeled for the mechanical restoring force of a fixed-fixed beam and the electrostatic force of a comb finger. The sensor was modeled by a lump ed mass-spring-damper system with an external electrostatic force. A Duffing equation and a nonlinear Mathieu equation [69] were derived to model the behaviors of non linear harmonic and parametric resonances. A perturbation method (multiple scales method) [72] was used to explore the effect of damping and cubic nonlinearity on the para metric resonance. Two pitch-fork bifurcations and three different stability regions were finally obtained for the dynamic characteristics of the nonlinear Mathieu equa tion. Experimental frequency responses showed that the system is linear with a sm all AC excitation and nonlinear with a large AC excitation. Also some model parameters such as the Q factor, cubic mechanical stiffness and linear electrostatic stiffness, were extracted from the experimental data. Nayfeh and his co-workers presented a series of nonlinear models for the electrically actuated annular pl ates [78], fixed-fixed rectan gular beams [86, 87], simply PAGE 65 47 supported rectangular plates [88], and clampe d circular plates [ 89, 90]. The static deflections of those microstructures are determ ined by either an analytical reduced-order model (macromodel) or a numerical shooting me thod, which is widely used to determine the periodic solutions of autonomous a nd non-autonomous systems [69]. The shooting method is an iterative procedure and comput ationally costly, and becomes numerically unstable when pull-in is approach ed; therefore, it is not suitable for the prediction of pullin. On the other hand, the reduced-order model [86-90] us ually approximates the system dynamics with N coupled nonlinear ordinary-di fferential equations, which are obtained by the spatial discretization of the dist ributed-parameter governing equation and associated boundary conditions via a Galerkin approach or finite-element method. The reduced-order model is robust up to the pullin point, and it also has the capability to account for the in-plane residual stress, gene ral material and geom etric nonlinearities. Younis et al. [86] and Abdel-Rahman et. al. [91] applied the reduced-order model to simulate the dynamical behavior of a ME MS switch and predict its pull-in time. A saddle-node bifurcation of a microbeam was found due to the pull-in. Two deflection solution branches of the microbeam moved closer to each other as the DC voltage increased, and finally coalesced when pu ll-in voltage was reached. Based on the reduced-order model, they also calculate d that the deflection at the pull-in is approximately 57% of the gap. Younis and Nayfeh [92] studied the nonlin ear response of a resonant microbeam under an electrostatic actuation. A nonlinear model was first built to account for the midplane stretching and an electrostatic lo ad with both DC and AC components. A perturbation method was applied directly to analyze the nonlinear forced response to a PAGE 66 48 primary-resonance excitation of its first mode The analysis results showed that the resonance frequency can be affected by the damping, mid-plane stretching, and electrostatic nonlinearity. This paper showed that the DC electrostatic load could result in both softening and hardening spring behavi ors. In addition, the method of multiple scales was applied to investigate a three-toone internal resonance between the first and second modes of the clamped-clamped beam. Th e analysis result showed that those two modes are nonlinearly uncoupled and theref ore the internal resonance cannot be activated. Chowdhury et. al. [93] studied the non linear effects in a MEMS capacitive microphone. The microphone was modeled as a second-order single-d egree-of-freedom system. The spring hardening effect due to the mid-plane stretc hing and the spring softening effect due to the nonlinear electrosta tic force were presente d. By using Taylor series expansion, the nonlinear electrostatic force was linearized and pull-in voltage was derived for a fully clamped square diaphrag m with a built-in tension stress. Finally, nonlinear finite element analys es were carried out to veri fy the analytical results. Nadal-Guardia et al. [94] presented a compact 1-D lumped model of capacitive silicon sensors. By applying a perturbati on method to the dynamic governing equations, transfer functions of the system were deve loped analytically. When comparing to the traditional equivalent analog circuit model, the obtained transfer functions can account for the effect of pull-in instability. Numeri cal simulations with the transfer functions were conducted, and the results were compared with the experimental data for a capacitive MEMS microphone. Both simulation and experimental results showed that pull-in instability improved the low-freque ncy response of the microphone for DC bias PAGE 67 49 voltages close to its critical pull-in value. However, the devel oped transfer function could not predict the sensitivity change of the microphone as a function of the bias voltage. This limitation is mainly due to the application of constant model parameters, which was obtained for a certain DC voltage va lue, for a whole range of the bias voltage. Fargas-Marques and Shkel [95] studied bot h static and dynamic pull-in conditions for an electrostatic MEMS res onator based on the para llel-plate actuation. They used a 1D lumped mass-spring-damper model and de rived the condition for AC dynamic pull-in based on the kinetic and potential energy of the system. The experimental results for the pull-in voltages agreed well w ith their analytical results. The dynamic pull-in voltage was reported to be approximately 8% lower th an the static pull-in voltage. Their energy analysis results also showed that the quality factor or damping of the system had an impact on the dynamic pull-in voltage. Elata and Bamberger [96] presented a pur ely theoretical stud y on the dynamic pullin of electrostatic actuators wh en subjected to instantaneous application of DC voltages. Based on HamiltonÂ’s principle [97] and quasi -static equilibrium equations, dynamic pullin for general undamped electrostatic actuato rs with multiple degrees of freedom and voltage sources was formulated. Specifically, the critical dynamic pull-in voltages were found approximately 8% lower than the co rresponding quasi-static pull-in voltages for electrostatic actuators with parallel-plates, double parallel-plates, and clamed-clamped beams. For the electrostatic ac tuator with a block of comb drives with no initial overlap, the critical dynamic pull-in voltage was f ound approximately 16% lower than its quasistatic pull-in voltage. If damping exists in electrostatic actuators (which is always the PAGE 68 50 case in practice), they concluded that th e actual dynamic pull-in voltage was bounded between the quasi-static pull-in voltage and the dynamic pu ll-in voltage without damping. Although previous research work provide d a good understanding of various aspects of nonlinear dynamics of electrostatic MEMS de vices, none of these st udies is helpful for a dual-backplate capacitive MEMS microphone. Th e rest of this dissertation is devoted to investigating the nonlinear dynamics issues associated with a dua l-backplate capacitive MEMS microphone. PAGE 69 51 CHAPTER 3 NONLINEAR DYNAMIC MODEL This chapter derives a nonlinear model for the dynamics of a dual-backplate MEMS microphone. First, general displacement solutions (small displacement by an analytical method and large displacement by an energy approach) of the middle diaphragm are provided based on the plate th eory. Based on the general displacement solutions, lumped element modeling is used to extract the parameters of the diaphragm. Other lumped parameters of the microphone including the damping coefficient and stiffness of the cavity, are also presented. After all the lumped parameters of the microphone are obtained, the general nonlinea r governing equation is derived and the model nonlinearities are discusse d. Finally, nonlinear finite element analyses (FEA) are carried out to verify the theoretical lumped stiffnesses of the diaphragm. Microphone Structure The dual-backplate capacitive MEMS microphone studied here has been fabricated using the SUMMiT V process at Sandia National Laboratories as well as facilities at the University of Florida for post-processing [ 65, 66]. A 3D cross-section view of the microphone is shown in Figur e 3-1. It has a 2.25 m thick circular solid diaphragm with a 230 m radius and a 2 m gap between each circular perforated backplate. The 5 m radius holes in the backplates allow the incident acoustic pressure to act on the diaphragm. A cavity under the bottom backplat e is formed via a deep reactive ion etch, and vented to the ambient pressure resulting in an AC measurement device. PAGE 70 52 Bottom Backplate Top Backplate Diaphragm Cavity Backplate Holes Anchors Gaps Bottom Backplate Top Backplate Diaphragm Cavity Backplate Holes Anchors Gaps Pressure Bottom Backplate Top Backplate Diaphragm Cavity Backplate Holes Anchors Gaps Bottom Backplate Top Backplate Diaphragm Cavity Backplate Holes Anchors Gaps Pressure Pressure tpVtbpVt Figure 3-1. 3D cross-secti on view of the microphone st ructure (not to scale). Figure 3-2 shows a microscope photograph of the microphone top with a field of view of approximately 11mmmm The bond pads shown in the photograph enable electrical connections between the backplates and diaphragm. Figure 3-2. Top-view phot ograph of the microphone. Displacement Solutions of the Diaphragm The diaphragm is a key ener gy transduction component an d plays a significant role in determining the performance of the micr ophone. Therefore, its mechanical behavior Bond pads 1 m m 1 mm Top backplate Diaphragm Bottom backplate PAGE 71 53 under the uniform transverse pressure loading is studied first. It is assumed that the circular polysilicon diaphragm is linearly elas tic, isotropic and axisymmetric. Also, zero in-plane residual stress [98] and a cl amped boundary condition are assumed in the analysis. Small Displacement Solution Shown in Figure 3-3, a clam ped circular diaphragm is subject to a transverse uniform pressure loading p The radius and thickness of the diaphragm are denoted by a and h respectively. From the plate theory if the transverse deflection of the diaphragm is much smaller than its thickness the strain in the neutral plane of the diaphragm can be neglected and the solution in this case is called the small displacement solution [99]. 0 r ra p h Figure 3-3. Schematic of a clamped circul ar diaphragm under a transverse uniform pressure loading. For the static small displacement solution, the governing differential equation for the transverse deflection wr in a polar coordinate system is given as follows [99] 1 2dddwpr r drrdrdrD (3.1) where r is the distance of any radi al point along the radius and D is the flexural rigidity of the diaphragm defined as PAGE 72 54 3 2. 121 Eh D (3.2) The parameters E and are the YoungÂ’s modulus and PoissonÂ’s ratio of polysilicon respectively. The boundary conditions are 0,radw wa dr (3.3) and 0.w (3.4) Solving Eq. (3.1) with Eqs. (3.3) and (3.4), the small displacement solution becomes 2 2 41. 64 par wr Da (3.5) The center displacement of the diaphragm is then 4 00. 64 p a ww D (3.6) Energy Method and Large Displacement Solution From the plate theory, if the transverse deflection of the diaphragm is comparable to its thickness, the strain in the neutral plane of the diaphragm cannot be neglected and the solution in this case is called the larg e displacement solution. For the large displacement solution, it can be assumed from the plate theory that the deflected surface in such a case is still symmetrical with respect to the origin of the ci rcular diaphragm. In this section, an energy approach is used to obtain an approximate large displacement solution. A more rigorous theory of the nonl inear mechanics of transducer diaphragms including the effects of in-plane stre ss is given in the reference [100]. PAGE 73 55 Procedure of energy method By applying an energy method [97, 99], trial functions with unknown coefficients for the transverse displacement and in-plane displacement need to be chosen first. Typical choices for the trial functions are polynomials and orthogona l functions such as sinusoidal or cosine functions. However, trial functions must satisfy the boundary condition, which limits the choices of trial functions. Once the trial displacement functions are assumed, the strain energy can be then calculated by using the strain-displacemen t relation (linear and nonlinear) and constitutive equation. Also, the work done by the external load can be calculated. The potential energy of the system is the summ ation of strain energy and work done by the external load. Finally, by applying the principle of mini mum potential energy for the equilibrium condition, a set of linear algebraic equations is obtained and further solved to yield the unknown coefficients in the tria l functions. Thereafter, the approximate solutions for the transverse displacement and in-plane displ acement are determined. The next section presents the application of an energy met hod to solving the large displacement solution for a clamped circular diaphragm, and the detailed steps are provided in Appendix A. Large displacement solution The polar coordinate frame is introduced here to facilitate the analysis for a circular diaphragm. As mentioned previously, the radi al displacement in th e neutral plane of a diaphragm cannot be neglected when larg e transverse deflections occur. The displacement of a point in the neutral plane of a circular diaphragm is decomposed into two components: ur in the radial direction and wr perpendicular to the neutral PAGE 74 56 plane as shown in Figure 3-4. The shape of th e transverse deflection surface of a circular diaphragm is assumed to be similar with the one in a small displacement case, which is defined in Eq. (3.5) as follows 2 2 0 21, r wrw a (3.7) where 0w is the unknown center displacement that needs to be determined. The assumed transverse deflection satisfie s the clamped boundary conditions defined in Eqs. (3.3) and (3.4). d rdrO ur wr a Figure 3-4. Displacement components in the neutral plane of a circular diaphragm. To satisfy the clamped boundary condi tions, the radial displacement ur must vanish at the edge and the origin of the circular diaphragm; therefore, the following polynomial is used to approximate the comple x expression of the radial displacement [99] 12,urrarCCr (3.8) where 1C and 2C are the two unknown constants. PAGE 75 57 Based on the von Krmn plate theory [99, 101] and energy method (detailed derivation steps are provided in Appendix A), finally we can determine the three unknown coefficients for a polysili con diaphragm (PoissonÂ’s ratio 0.22 is used in the derivation, note that the following results are only valid for this specific PoissonÂ’s ratio) as follows 2 0 1 31.2652, w C a (3.9) 2 0 2 31.8129, w C a (3.10) and 4 0 2 0 21 64 10.4708 pa w w D h (3.11) Eq. (3.11) can be rewritten as 2 4 0 0 210.4708. 64 w p a w hD (3.12) The final approximate expression for the large displacement solution can be determined by substituting the solution of 0w from Eq. (3.12) into Eq. (3.7). As we can see from Eq. (3.11), the small factor 22 00.4708 wh represents a geometri c nonlinearity (nonlinear spring hardening effect) due to the in-plane stretching when large displacements occur. The diaphragm can now be modeled as a non linear Duffing spring, where the two spring constants can be obtained via further lumped element modeling in the following section. As seen from Eq. (3.7), the assumed mode shape based on this energy approach is not affected by the applied pressure. Show n in Figure 3-5 is a plot of different normalized mode shapes for several pressure va lues. In the plot, three normalized mode PAGE 76 58 shapes are generated based on the exact soluti on given in the referenc e [100]. As we can see from the plot, for larger pressure values (for example, 100000 Pa), the assumed mode shape used in the energy approach is not accurate. However, for our 2000Pa microphone design, the assumed mode shap e used in the energy approa ch is in good agreement with the exact mode shape as shown in the above plot. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w(r)/w0r/a Exact solution, p=2000 Pa Exact solution, p=10000 Pa Exact solution, p=100000 Pa Energy-based solution p = 2000 Pa p = 100000 Pa Figure 3-5. Normalized mode shape for se veral pressure values (2000, 10000 and 100000 Pascals). Lumped Element Modeling of the Microphone The microphone is a typical multi-domain ( acoustical, mechanical and electrical) system and its dynamics are usually govern ed by a set of coupl ed nonlinear partial differential equations, which are difficult and co mplex to study. As we can see from Eq. (3.7), the displacement of the diaphragm is not uniform but distri buted along the radial direction. The associated potential and kine tic energy are also dist ributed along the radial PAGE 77 59 direction of the diaphragm, which indica tes that the microphone is a distributedparameter system. One alternative for approximating the c oupled distributed-parameter system dynamics is through the construction of a lump ed-element model. In general, lumped element modeling is based on the assumption that the device length scale of interest is much smaller than the characteristic lengt h scale (for example, wavelength) of the physical phenomena [29, 102, 103]. By usin g lumped element modeling, the spatial variations of the quantities of interest can be decoupled from the temporal variations. The coupled distributed-parameter system can be then divided into many idealized lumped impedances (mass, stiffness and dissi pation) [29]. The dynamic behavior of the system can be described only by ordinary diffe rential equations with time being the only independent variable, which is comparatively easier (tha n coupled nonlin ear partial differential equations) to analyze in practice. To carry out lumped element modeling, us ually a lumped reference point needs to be defined first to lump a distributed-parameter system into a single/multiple degree of freedom system. In the case of a circular diap hragm, the center point is chosen. In this section, the lumped parameters for the micr ophone are discussed. Since the diaphragm is a key part of the microphone and plays a cr ucial role in determining the overall performances of the microphone, the diaphrag m is studied first by the lumped element modeling. Lumped Parameters of Diaphragm Before proceeding to extract lumped parameters, the reference point for the circular diaphragm is chosen to be at its center position 0 r The general applied transverse pressure is assumed for simplicity to be a harmonic oscillation PAGE 78 60 0(),jt p tpe (3.13) where is the angular frequency of the pressure oscillation. The dynamic small displacement solution is obtained by modifying Eq. (3.5) 22 22 4 4 0,11. 6464jtpta pa rr wrte DaDa (3.14) The velocity is v,,,. rtwrtjwrt t (3.15) Therefore the displacement and velo city at the reference point are 4 0 00,, 64jtpa wtwte D (3.16) and 00v. tjwt (3.17) Lumped mass The total kinetic energy of the diaphragm is given by 2'2' 011 v,v,2, 22a KEWrtdArtrdr (3.18) where' is the mass per unit area. Substituting Eqs. (3.15), (3.16) and (3.17) into Eq. (3.18) yields 22 2' 00vv 522KEmett a WM (3.19) Therefore, the work-equivalent lumped mass is calculated as 2 2'2 0v 2 55meKEt aah MW (3.20) PAGE 79 61 where is the mass per unit volume and me M is 1/5 of the mass of the actual diaphragm. Physically, the diaphragm with distributed deflections has been replaced by a rigid disk (piston) and a linear spring. In order to conser ve the kinetic energy, a rigid disk with 1/5 of the mass of the actual diaphragm is used. Lumped linear stiffness and compliance The total potential energy of the diaphragm can be expressed as 2,.PEWFdxptrdrdwrt (3.21) From Eqs. (3.13), (3.14) and (3.16), we know 0 464 D p twt a (3.22) and 2 2 0,1. r dwrtdwt a (3.23) Substituting Eqs. (3.22) and (3.23) into Eq. (3.21) yields 02 2 2 0 00 4 2 006464 21. 32wta PEwt DrD Wrdrwtdwt aaa (3.24) Therefore, the work-equivalent mechan ical stiffness is calculated as 2 0 1 264 2 3PEwt D kW a (3.25) The mechanical compliance is 2 113 64mea C kD (3.26) PAGE 80 62 Physically, the diaphragm with distributed deflections has been replaced by a rigid disk (piston) and a linear spring. By using the lumped stiffness or compliance, the conservation of potential energy in the physic al diaphragm and an idealized linear spring is ensured. Lumped area To ensure the continuity of volumetric velocity in physical and lumped domains, the lumped area of the diaphragm needs to found. The volumetric velocity through the diaphragm is given by 0v,v,2.a AQrtdArtrdr (3.27) From Eqs. (3.15) and (3.17), we have 2 2 0,1v. r vrtt a (3.28) Substituting Eq. (3.28) into (3.27), we can get 2 00vv. 3mea QtAt (3.29) Therefore the equivalent lumped area is calculated as 2 0/v, 33meaA AQt (3.30) where meA is 1/3 of the area A of the actual diaphragm. The distributed deflection of the diaphragm has been replaced by the translation of a rigid disk (piston) with an area that is 13 that of the actual diaphragm. PAGE 81 63 Lumped cubic stiffness The lumped cubic stiffness of the diaphragm is developed in this section. Eq. (3.25) only shows the linear spring consta nt of the diaphragm fo r a small displacement case. For a large displacement case, a nonlin ear Duffing spring model is considered for the diaphragm, which is 3 1030. meFpAkwkw (3.31) Comparing Eqs. (3.12) and (3.30), we have 1 264 3 D k a (3.32) and 3 2210.044 D k ah (3.33) Eq. (3.32) is same with Eq. (3.25) becau se both energy-based large displacement and small displacement solutions used a same deflection shape defined in Eq. (3.7). When the diaphragm vibrates, the equivalent structural damping of the diaphragm mainly consists of two parts: the thermoelastic en ergy dissipation in the diaphragm and the vibration energy dissipation in the compliant boundary of th e diaphragm [9, 10]. Usually the equivalent structural damping is determin ed experimentally, it is neglected in the current model since it is small comparing with the dominant viscous damping. The experimental result for the dominant viscous damping will be discussed in Chapter 6. Lumped Damping Coefficient The two backplates of the microphone are de signed to be perfor ated with a large number of holes. One purpose of these hol es is to let the air pass through without deflecting the backplates and ma ke them acoustically transparent. A repetitive pattern of PAGE 82 64 holes in the top backplate is shown in Figure 3-6. The top backplat e has a radius of 256 m and 557 holes with a radius of 5 m. A similar hole pattern exists for the bottom backplate, which has a radius of 213 m and 367 holes with a radius of 5 m. Figure 3-6. Repetitive pattern of holes in the top backplate. When the diaphragm vibrates, the gas fl ow between the diaphragm and backplate can be divided into two parts, the horizontal gas flow between the parallel plates and the vertical gas flow through the backplate holes Viscous damping caused by the horizontal gas flow is often called squeeze-film damp ing, and the viscous damping caused by the vertical gas flow is called holes resistance. The latter becomes important and cannot be neglected especially when the thickness of b ackplate is comparable to the gap thickness [7, 8]. Since the viscous damping plays a very important role in determining the bandwidth and noise floor of the microphone, its modeling becomes important for this PAGE 83 65 research. For the squeeze-film damping in the gap, the mechanical lumped damping coefficients are given as follows [8, 53] 4 3 04 3s tptp tpa bBA nd (3.34) and 4 3 04 3s bpbp bpa bBA nd (3.35) where is the dynamic viscosity of air, 0d is the nominal gap between backplates and diaphragm, tpn and bpn are the numbers of holes in the top and bottom backplates respectively. The porosity of the t op and bottom backplates is given by tpA and bpA respectively, and B is a function defined as 211311 4828 Bln (3.36) When the thickness of the backplate is comp arable to the gap, the viscous damping due to the vertical gas flow through the back plate holes becomes important. By modeling the motion in the holes as a pressure-driven Poiseuille flow in a pipe, the mechanical lumped damping coefficients are given as follows [7, 8] 28 ,tptp h tp tphn b A (3.37) and 28 ,bpbp h bp bphn b A (3.38) PAGE 84 66 where tph and bph are the thicknesses of the top backplate and bottom backplate respectively. Therefore, the total viscous damping of the microphone is, s shh tpbptpbpbbbbb (3.39) Lumped Stiffness of the Cavity The cavity of the microphone impedes th e movement of diaphragm by storing potential energy and acts as a spring. Based on the lumped element assumption, the work-equivalent lumped acoustic compliance of the cavity is give n as follows [104] 2 0, a c cV C c (3.40) where cV is the volume of the cavity, is the density of air, and 0c is the isentropic speed of sound in air. It should be poin ted out that Eq. (3.40) is valid for 0.3ckd where k is the wave number [104] and cd is the depth of the cavity. For the cylindrical cavity with a circular cross section, the volume is 2,cccVad (3.41) where and ca is the radius of the cavity. Therefor e the mechanical lumped stiffness of the cavity is 2 2,c c a ca k C (3.42) where 2 2 ca is used to convert the acoustic sti ffness into the mechanical stiffness. Since the lumped stiffness of the cavity is in parallel with the linear stiffness of the diaphragm, the first resonant freque ncy of the system is approximated by PAGE 85 67 11 2c nature mekk f M (3.43) As seen from the Eq.(3.43), the existence of the cavity increases the bandwidth of the capacitive microphone. Based on the results of the lumped element modeling, the calculated nominal values of lumped parameters as well as the material properties and physical dimensions of the designed 2000P a dual-backplate condenser microphone are summarized in Table 3-1 [65, 66]. Table 3-1. Material proper ties and physical parameters of the 2000Pa microphone in metric units (material: polysilicon). Parameter Nominal value Young's modulus ( E ) 1.60e11 (Pa) Poisson's ratio ( ) 0.22 Density ( ) 2.23e3 (kg/m3) Thickness of the diaphragm (h) 2.25e-6 (m) Radius of the diaphragm (a) 230e-6 (m) Linear spring constant of the diaphragm (1k ) 202 (N/m) Cubic spring constant of the diaphragm (3k ) 1.88e13 (N/m3) Lumped mass of the diaphragm (me M ) 16.7e-11 (kg) Lumped area of the diaphragm (meA ) 5.54e-8 (m2) Gap (0d ) 2.00e-6 (m) Depth of the cavity (cd ) 650e-6 (m) Radius of the cavity (ca ) 187e-6 (m) Linear spring constant of the cavity (ck ) 24.3 (N/m) Thickness of the top backplate (tph ) 2.25e-6 (m) Thickness of the bottom backplate (bph ) 2.50e-6 (m) Number of holes for top backplate (tpn ) 557 Number of holes for bottom backplate (bpn ) 367 Ratio of the total holes ar ea to the backplate area (tptp A A ) 0.22 First resonant frequency of the microphone system (nature f ) 185 (kHz) Total lumped damping coefficient (b) 3.15e-5 (Ns/m) Total damping ratio 8.09e-2 *First resonant frequenc y of diaphragm 175 (kHz) *First resonant frequency of bottom backplate 203 (kHz) *First resonant frequency of top backplate 130 (kHz) *FEA results with fixed boundaries PAGE 86 68 Further calculations show that the first na tural frequency of the system is increased by approximately 5.9% due to the existence of cavity. If the cavity stiffness is neglected in Eq. (3.43), the first natural frequency of the diaphragm via LEM is approximately 175.3 kHz. One way to check the accuracy of th e modeled first natural frequency of the diaphragm is to solve the transverse bendi ng wave equation for the circular diaphragm with a fixed boundary [105]. Fu rther studies show that the theoreti cal first natural frequency of the diaphragm is approximately 173.3 kHz. The error due to the lumped element assumption is approximately 1.1%. a tpCa tp R a v R a d M a dCa cCa bp R a bpC a tcCa bcCinPTop BPDiaphragmBottom BP Figure 3-7. Simplified lumped element mode l of a dual-backplat e capacitive microphone (adapted from Martin et al. [66]). In the current lumped element model, the backplates are assumed to be acoustically transparent and the compliance of the cavi ties between the diaphragm and top/bottom backplates are neglected. A simplified lu mped element model of the microphone shown in Figure 3-7 is used to verify these assu mptions. This model describes the microphone in the acoustic domain, where the mechanical lumped parameters derived in the above section are converted to their analogous acousti c parameters. Additiona lly, other features of the microphone, such as the vent resistan ce and the compliance of each backplate, are PAGE 87 69 also included in this model [66]. Readers are referred to [66] for more details of constructing this lumped model. In the model, inP represents the incide nt acoustic pressure. a tpC, a dC a bpC and a cC are the acoustic compliances of the top b ackplate, diaphragm, bottom backplate and bottom cavity, respectively. a tcC and a bcC are the acoustic compliance of the cavities between the diaphragm and top/bot tom backplates, respectively. a tp R and a bp R are the acoustic resistance of the top backplate and bottom backplate, respectively. a d M is the acoustic mass of the diaphragm and a v R is the acoustic vent resistance from the cavity to the incident acoustic pressure. For the cavities between the diaphragm and top/bottom backpl ates, each cavity volume is less than 0.5% of that of the bottom cavity. Therefore, the acoustic compliance of each small cavity (a tcC and a bcC ) is less than 0.5% of the compliance (a cC ) of the bottom cavity (the acoustic compliance of the ca vity is proportional to its volume). The impedance of the cavities 1 and 1aa tcbcjCjC between the diaphragm and top/bottom backplates are much larger than the impedance of the bottom cavity and the diaphragm. Therefore, in the equivalent circuit, a tcC and a bcC can be treated as open circuit and are neglected in the lumped element model. As seen from Figure 3-7, for each backplat e, its acoustic compliance is in parallel with its resistance (a RC loop is formed). Therefore, the impedance of each backplate will be approximately equal to its resistance [66] when 1 1.3, 2aa tptp f MHz RC (3.44) PAGE 88 70 and 1 3.3, 2aa bpbp f MHz RC (3.45) where f is the frequency of the incident acoustic pressure. As seen from Figure 3-7, the pressure drop across each plate is proportional to the impedance of each plate (note that a tcC and a bcC are neglected). Based on the microphone design, the impedance of each backplate is much less than the impedance of the diaphragm [66] when Eqs. (3.44) and (3.45) are satisfied; therefore, the pressure drop across each backplate is neglig ible. Since the small backpl ate deflection is proportional to the pressure drop across it; therefore, th e backplate deflection is negligible and the backplate can be treated as be ing acoustically transparent. Nonlinear Dynamic Model A general nonlinear dynamic model for the microphone is shown schematically in Figure 3-8. The top and bottom backplates ar e assumed to be rigid and have equal areas with the diaphragm. The diaphragm is m odeled by a Duffing spring with two spring constants 1k and 3k and a lumped mass me M with a lumped areameA 13 kkb MA meme Pressure p k c tpVtbpVt0 x 0d0dDiaphragmTop backplateBottom back p late Figure 3-8. A nonlinear dynamic model of a dual-backplate capacitive microphone. PAGE 89 71 Shown in Figure 3-9 is the free body diagram of the dynamic model, me p A is the lumped mechanical force caused by the incoming acoustic pressure, and me M x is the lumped inertia force due to the acceleration. For simplicity, during the derivation of the dynamic governing equation, the diaphragm is assumed to move downwards with a certain displacement x D iaphragm me M xx F orpdampingFme p A s pringFe F c F Figure 3-9. Free body diagram of the nonlinear dynamic model. The nonlinear lumped spring force is given by 3 13.springFkxkx (3.46) The lumped damping force is dampingFbx (3.47) where x is the center velocity of the diaphragm. By using the equal-area parallel-plate assumption and neglecting the fringing field effect of the holes in the backplates the total electrost atic co-energy, U, stored in the system is calculated as follows 222 00111 222meme iitpbp iAA UCVVtVt dxdx (3.48) PAGE 90 72 where is the dielectric constant of the air, and tpVt and bpVt are the instantaneous voltages applied to the top backplate and botto m backplate respectively. It follows that the net lumped electrostatic force is 22 22 00. 2tpbp me eVtVt A dU F dx dxdx (3.49) The lumped mechanical reacti on force from the cavity is .ccFkx (3.50) After obtaining all the lumped forces acting on the diaphragm, by applying NewtonÂ’s second law, the general dynamic governing equation is .meispringdampingecme M xFFFFFpA (3.51) By substituting Eqs. (3.46), (3.47), (3.49) and (3.50) into Eq. (3.51), the governing equation is 22 3 13 22 00. 2tpbp me mec meVtVt A M xkkxkxbxpA dxdx (3.52) Rewriting Eq. (3.52) becomes 22 3 13 22 00. 2tpbp me mec meVtVt A M xbxkkxkxpA dxdx (3.53) Discussion of Nonlinearities The above governing equation physically represents a general damped secondorder system with a cubic mechanical nonlinea rity and under both non linear electrostatic loading and uniform pressu re loading. Since the cubic stiffness parameter 3k is positive, it physically represents a spring hardening e ffect. The electrosta tic forces between backplates and diaphragm in nature are non linear, even when the displacement of the PAGE 91 73 diaphragm is not large. From the expressi ons of nonlinear lumped electrical forces shown in Eq. (3.53), the electrical nonlinea rity is coupled w ith the mechanical nonlinearity, which indicates that the micr ophone system is an electromechanically coupled system. To facilitate the nonlinear analyses throughout this dissertation, it is necessary to quantify the mechanical and electrical nonlinearit ies in this section. The definitions are given by the following equations. For the mechanical nonlinearity, mNL we have 100% mnonlinearmechanicalforcelinearizedmechanicalforce NL linearizedmechanicalforce 3 2 1313 11 100%,kxkxkxk x kxk (3.54) where x is the center displacement under the interested pressure or electrical loading. Based on the above equation, we need to know the linear stiffness, cubic stiffness, and center displacement of the diaphr agm to be able to calculate the mechanical nonlinearity. The expression for electrical nonlinearity, eNL is 100%. enonlinearelectricalforcelinearizedelectricalforce NL linearizedelectricalforce (3.55) To further quantify and gain physical in sight into the defi nition of electrical nonlinearity, we consider a single-backplat e capacitive microphone with an applied DC bias 0V only. Following the definition in Eq. (3 .49), the nonlinear electrostatic force is 2 0 2 0. 2me eAV F dx (3.56) By using a TaylorÂ’s series expansion about 0x we can linearize the above nonlinear force as follows PAGE 92 74 2 0 2 0012. 2me eLAV x F dd (3.57) Therefore, the electrical non linearity in this case is 22 00 2 2 3 00 0 0 2 2 0 00 2 00 12 22 1. 2 12 2meme e meAVAV x dd dx d NL AV x dxdx dd (3.58) As we can see from the above equation, th e electrical nonlinearity can be calculated based on the gap and center displacement of th e diaphragm. The gap is an independent parameter; however, the center displacemen t of the diaphragm is dependent on the electrostatic loading (DC bias voltage in this case) and pressure loading. From Eqs. (3.54) and (3.58), clearly the electrical nonl inearity is coupled with the mechanical nonlinearity through th e center displacement of the diaphragm. 0 0.2 0.4 0.6 0.8 1 1.2 x 10-6 0 1 2 x 10-4 Center displacement (m)Forces (N) Nonlinear Mechanical Force Linearized Mechanical Force Nonlinear Electrical Force @ 20V Linearized Electrical Force @ 20V Figure 3-10. Nonlinear vs. linearized mechan ical and electrical forces of a singlebackplate capacitive microphone. PAGE 93 75 An example is given here to help unders tand the above defined nonlinearities. Shown in Figure 3-10 is a plot of calculate d nonlinear versus linearized mechanical and electrical forces by using the parameters in Table 3-1 for a given 20V DC voltage. As we can see from the plot, the electrical nonlin earity is dominant for the large center displacement of the diaphragm. For exam ple, if the center displacement is 0.6 m the electrical nonlineari ty at 20V is 27.6%eNL and the mechanical nonlinearity is 3.4%mNL Similarly, if a dual-back plate capacitive microphone w ith an applied DC bias 0V is considered, the electrical nonlinearity, followed by the definition in Eq. (3.55), is given as 6 0 2 2222 001. 2ed NL dxdx (3.59) From Eqs. (3.54) and (3.59), again the elec trical nonlinearity is coupled with the mechanical nonlinearity through the center displacement of the diaphragm. An example is given here to help understa nd the electrostatic non linearity of a dualbackplate capacitive microphone. Shown in Figu re 3-11 is a plot of calculated nonlinear versus linearized mechanical and electrical fo rces by using the parameters in Table 3-1 for a given 20V DC voltage. As we can s ee from the plot, the electrical nonlinearity becomes smaller and is on the same level w ith the mechanical nonlinearity, even when the center displacement of th e diaphragm is large. For example, if the center displacement is 0.6 m the electrical nonlinearity at 20V is 2.3%eNL and the mechanical nonlinearity is 3.4%mNL Physically, the two oppos ite electrostatic forces in a dual-backplate capacitive microphone he lp to reduce the el ectric nonlinearity. PAGE 94 76 0 0.2 0.4 0.6 0.8 1 1.2 x 10-6 0 1 2 x 10-4 Center displacement (m)Forces (N) Nonlinear Mechanical Force Linearized Mechanical Force Nonlinear Electrical Force @ 20V Linearized Electrical Force @ 20V Figure 3-11. Nonlinear vs. linearized mechanical and electrical forces of a dual-backplate capacitive microphone. Since it is impossible to solv e the governing nonlinear equa tion Eq. (3.53) in closed form, alternative approaches are used in th is dissertation to st udy the nonlinear dynamic system. Nonlinear finite element analyses (F EA) are carried out in the next section to verify the theoretical lumped stiffnesses of the diaphragm. The approximate analytical solutions (specifically via multiple time scales and harmonic balance methods) and numerical simulations through the direct in tegration of nonlinear governing equation are provided in the next chapter. The instabil ity analyses for pull-in s are carried out in Chapter 5. Nonlinear Finite Element Analyses In this section, nonlinear mechanical fin ite element analyses are carried out in CoventorWare 2003 [106] to extract th e equivalent lumped stiffnesses (1k ,3k ) of the PAGE 95 77 diaphragm and the accuracy of theoretical lump ed stiffnesses is verified. This section also provides discussions of the modeling error for the elect rostatic force through the coupled electromechanical simulation. Stiffnesses of the Diaphragm Based on the solid model shown in Figure 3-1, a 3D mesh of the diaphragm is generated and shown in Figure 3-12. Based on this mesh, converged displacement results are achieved with the diaphragm under the applied uniform pressure. Some major specifications of the mesh are listed in Table 3-2. Mesh of diaphragm Zoom-in elements Figure 3-12. 3D mesh of the di aphragm in CoventorWare 2003. PAGE 96 78 Table 3-2. Major specifications of the di aphragm mesh with c onverged displacement results. Parameter Value Volume element type Solid hexahedron Number of volume elements 82452 Number of nodes 43671 Average aspect ratio 1.6223 The material properties and physical di mensions of the diaphragm used in nonlinear FEA are taken partially from Table 31. The side surface of the diaphragm is fixed to be the boundary condition. To obtain the lumped stiffnesses of the diaphragm, different pressure loads with amplitudes varying from 10 to 4000 Pa are applied to the top surface of the diaphragm, and the nonlinea r FEA are carried out to yield the center displacement of the diaphragm for each applied pressure respectively. 0 500 1000 1500 2000 2500 3000 3500 4000 0 0.2 0.4 0.6 0.8 1 1.2 x 106 Applied pressure (Pa)Center displacement (m) Ideal linear result Nonlinear FEA result Energy-based analytical result Exact analytical result Figure 3-13. Transverse center deflections of the diaphragm under the uniform pressure. PAGE 97 79 Shown in Figure 3-13 is the plot of transver se center deflections of the diaphragm. The ideal linear, energy-based analytical as well as exact analytical deflection results (obtained from the reference [100] ) are also plotted in Figure 3-13. As we can see from the plot, three sets of nonlinear deflection re sults agree very well w ith each other. The mechanical nonlinearity becomes important fo r the large applied pressure, for example, when the pressure value is above 2000Pa. Figure 3-14. Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thickness direction, unit: m ). Shown in Figure 3-14 is a displacement contour plot of the diaphragm under a 2000Pa uniform pressure. As seen from the plot, the center deflec tion of the diaphragm is approximately 5.4e-7 m. Further calculati on indicates that the mechanical nonlinearity (mNL ) at 2000Pa is approximately 2.7% ba sed on the definition in Eq. (3.54). The lumped linear and cubic stiffnesses can be extracted by curve-fitting the simulated nonlinear center displacements of the diaphragm with the formula in Eq. (3.31) PAGE 98 80 and the final results are listed in Table 3-3. From Table 3-3, the differences for the linear stiffness 1k and cubic stiffness 3k are approximately 1.3% and 5.0% respectively. Since the differences are small, the accuracy of the theoretical lumped stiffnesses of the diaphragm is verified by the nonlinear FEA results. In the following chapters, the theoretical lumped stiffnesses of the diaphragm will be used. Table 3-3. Comparison of non linear FEA and LEM results. Parameter Nonlinear FEA result LEM result Difference Linear stiffness 1k (N/m) 199.7 202.2 1.3% Cubic stiffness 3k (N/m3) 1.979e13 1.880e13 5.0% Electrostatic Forces by CoSolveEM Simulations In the real microphone device, the area of th ree plates is not same. As shown in Figure 3-1, the top backplate has the larges t area while the bottom backplate has the smallest area. In addition, the backplates are perforated with hundreds of holes. The capacitance between the backplate and diaphr agm is reduced due to the backplate area loss; however, extra fringing fields generate d by the holes could compensate the loss. The electrostatic forces in the previous sections are modeled based on the equal-area parallel-plate assumption; therefore, some errors exist between the modeled and real electrostatic forces. One possible way to examine the difference between the real and modeled electrostatic forces is through nonlinear finite element analyses, for example, the CoSolveEM (coupled electromechanical anal ysis) simulation in CoventorWare 2003. The simulation runs between the mechanical and electrostatic domains until a converged equilibrium point is found. By using the solid model shown in Figure 3-1, the CoSolveEM simulation takes into account the effects of unequal area, perforated holes, and the fringing field on the electrostatic force; therefore, the simulated electrostatic force can be treated as an accurate measure of the r eal electrostatic force. In the following, the PAGE 99 81 CoSolveEM simulations are conducted for t op and bottom capacitors of the microphone respectively, and the simulation results are compared with the re sults based on lumped element modeling. During the simulations for the top capacitor, the side surfaces of the diaphragm and top backplate are assumed to be fixed and th e applied DC voltage varies from 10 to 25V. When the applied DC voltage is greater th an 25V, the simulation becomes diverged and quasi-static pull-in occurs. Similarly, for the bottom capacitor, the side surfaces of the diaphragm and bottom backplate are assumed to be fixed, and the simulated quasi-static pull-in voltage is approximately 33.5V. 10 15 20 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10-4 Applied voltage (V)Electrostatic force (N) Simulated electrostatic force Modeled electrostatic force Figure 3-15. Plot of simulate d and modeled electrostatic fo rces for the top capacitor. Shown in Figure 3-15 is a plot of simulate d electrostatic forces for the top capacitor when the applied DC voltage varies. The corresponding modeled electrostatic forces are PAGE 100 82 also plotted in Figure 3-15 by using the paramete rs in Table 3-1. As seen from Figure 315, the difference between the simulated and model electrostatic forces becomes larger when the applied voltage increases. Further ca lculations show that the difference at 10V is approximately 4% and 17% at 25V. 10 15 20 25 30 35 0 0.5 1 1.5 2 2.5 3 3.5 x 10-4 Applied voltage (V)Electrostatic force (N) Simulated electrostatic force Modeled electrostatic force Figure 3-16. Plot of simulate d and modeled electrostatic fo rces for the bottom capacitor. For the bottom capacitor, similarly, shown in Figure 3-16 is the comparison plot of the simulated and modeled electrostatic forces A similar trend of the difference between the simulated and model electrostatic forces is observed. And further calculations show that the difference at 10V is appr oximately 13% and 17% at 33.5V. Based on the results shown in Figure 3-15 and Figure 3-16, when the applied DC voltage increases, the error ge nerated by the model with an equal-area parallel-plate assumption increases. Specifically, the error is up to 17% when the applied voltage is up PAGE 101 83 to 25V for the top capacitor, and the error is up to 17% when the applied voltage is up to 33.5V for the bottom capacitor. Summary In this chapter, a nonlinear model fo r a dual-backplate MEMS microphone is developed. Specifically, lumped element modeling is applied to the microphone and lumped parameters of the microphone are extracted including the cubic mechanical stiffness of the diaphragm. Based on th e lumped parameters, the nonlinear governing equation is obtained and non linearities in the governing equation are discussed. Theoretical lumped stiffnesses of the diaphr agm are verified by nonlinear finite element analyses and the differences for the linear stiffness 1k and cubic stiffness 3k of the diaphragm are approximately 1. 3% and 5.0% respectively. PAGE 102 84 CHAPTER 4 APPROXIMATE SOLUTIONS OF NONLINEAR GOVERNING EQUATIONS A primary goal for this research is the development of a system characterization approach for the microphone w ith electrical exci tation. This chap ter provides the mathematical background and derivation of th e formulae used in the characterization experiments to extract real system parameters. Specifically, the multiple time scales (MTS) and harmonic balance (HB) methods are applied to two nonlinear governing equations to obtain their appr oximate solutions. Further di scussion and validity check for each approximate solution are also provided. In addition, this chapter presents approximate analytical solutions for th e microphone under the sinusoidal acoustical pressure excitation. Major fi ndings and contributions are su mmarized at the end of this chapter. Introduction To facilitate the analysis in this chapter, the general nonlinear governing equation Eq. (3.53) in Chapter 3 is further rewritten by using the system parameters (combination of lumped parameters) as follows 22 23 00 22 002, 2tpbpVtVt x xxxpt dxdx (4.1) where system parameters are defined as 01,cmekkM (4.2) 0, 2meb M (4.3) PAGE 103 85 3,mek M (4.4) and ,me meA M (4.5) where 0 is the first system resonant frequency, is the damping ratio, is the nonlinear stiffness parameter and is the ratio of the lumped area over lumped mass. Also the gap, 0d is treated here as a system pa rameter although it is a structural dimension of the microphone. As we can see from Eq. (4.1), along with the defined system parameters, the dynamic system response can be determined fo r given electrical a nd pressure inputs. Although system parameters can be calculated theoretically from LEM, it is well-known that some of the modeling assumptions are idealizations. For example, the actual device possesses a finite amount of compliance in the diaphragm boundary conditions, and variations in physical dimensions and materi al properties, which are usually caused by the fabrication process, can of ten substantially alter the mode l parameters of the device. Therefore, some modeling errors exist be tween the theoretical and actual system parameters, and characterization experiments are required to determ ine the actual system parameters to evaluate the accuracy of the th eoretical model. In the following sections, the mathematical background and derivation of the formulae used in the characterization experiments are presented. Details of the experiments as well as the analysis results are discussed in Chapter 6. PAGE 104 86Governing Equation for the El ectrical Square Wave Excitation During the characterization tests for the el ectrical square wave excitation, no acoustical pressure exists on the diaphr agm and only a uni-polar square wave Vt is applied directly to bottom (or top) backpl ate with diaphragm and the other backplate electrically grounded. The expression for the applied uni-polar squa re wave is given by 0 000 0 00 2 0 1 2 T VnTtnT Vt T nTtnT (4.6) where 0V and 0T are the voltage amplitude and period of the square wave respectively, and 0, 1, 2, ...n 13 kkb M A meme 0 x k cDiaphragm0d 0V0 FBottom backplateTop backplate Figure 4-1. Dynamic model for an electrical sq uare wave excitation on the top backplate. Since the backplates are assumed to be symmetric around the center diaphragm, it turns out that the two governi ng equations are similar when e ither of backplates is under the electrical square wave excitation. For simplicity, the following analysis only studies PAGE 105 87 the case of an excitation volta ge being applied to the top backplate. The dynamic model for this study is shown in Figure 4-1. During the time 0 002 T nTtnT an electrostatic force acts on the diaphragm to force it to vibrate, and the corresponding governing e quation can be reduced to 2 23 00 00 2 02. 2 V xxxx dx (4.7) A TaylorÂ’s series expansion for the a bove nonlinear electrostatic force up to 3rd order results in 23 22 0000 2 2 0000 01234. 22 VV xxx dddd dx (4.8) 0 1 2 3 4 5 6 x 10-7 3 4 5 6 7 8 9 10 11 12 x 10-5 Center displacement (m)Force (N) Electrostatic force Approximate electrostatic force V0 = 30V V0 = 25V Figure 4-2. Plot of elec trostatic and approximate electrostatic forces for 0V25V and 0V30V respectively. PAGE 106 88 In order to verify th e accuracy of the 3rd order TaylorÂ’s seri es expansion in Eq. (4.8), by using the parameters in Table 3-1, the electrostati c and approximate electrostatic forces based on the 3rd order TaylorÂ’s series expansion are plotted in Figure 4-2 for two different voltages 0V25V and 0V30V respectively. Those two voltages are chosen since they are used in the numer ical validity tests in the next section. As we can see from the plot, the two electrostatic forces ar e similar for each voltage case. Further calculations show that the appr oximation error caused by the 3rd order TaylorÂ’s series expansion is up to 3.0% for 0V25V and 3.1% for 0V30V when the center displacement is up to 0.6m. Mathematically, the approximation error becomes larger if the center displacement is over 0.6m. Physically, the curvatur e of the diaphragm needs to be considered if the center displacem ent becomes large (for example, x0.6m ). The parallel plate assumption used to model th e electrostatic force (l eft hand side of Eq. (4.8)) might not be able to hold for the large displacement case. Therefore, the 3rd order TaylorÂ’s series approximation in Eq. (4.8 ) carries a maximum error of approximately 3.1% when the center displacement is 0x0.6m and the applied voltage is 0V30V. To proceed, substitution of Eq (4.8) into (4.7) yields 2222 223 00000000 00 3452 000032 2. 22 VVVV xxxxx dddd (4.9) Physically, Eq. (4.9) represents a damped s econd order system with both quadratic and cubic nonlinearities and a non-zero extern al step loading. During the time PAGE 107 89 0 001 2 T nTtnT no electrostatic force acts on th e diaphragm, the diaphragm is under the free vibration and the corresponding governing equation is 23 0020. xxxx (4.10) The above equation is the so -called damped DuffingÂ’s e quation, in which only the cubic mechanical nonlinearity exists. Eq. (4. 10) is a special case of Eq. (4.9) when the applied voltage 0V is zero. The solution of Eq. (4.9) is important since it can be easily tailored and applied to other applications. In the following section, the MTS method is utilized to find its approx imate analytical solution. Approximate MTS Solution for the Electrical Square Wave Excitation From Chapter 3, we know that the non linearities in the designed microphone are small. In the field of nonlinear dynamics the commonly used methods to find the approximate solution of a weak ly nonlinear system are the method of harmonic balance, the method of averaging and perturbation me thods. The method of multiple time scales is one of widely used perturbation me thods with high accuracy and efficiency. The basic idea of the MTS method is to c onsider the expansion of the time response to be a function of multiple independent time variables, or scales, instead of a single time variable [72]. The different scales will allo w us to Â“observeÂ” the system behavior on the different scales and also capture the characteristics of differ ent dynamics (such as the fast dynamics of the linear response and slow dynami cs of the nonlinear response). Typically, the approximate solution is assumed as some integer order expansion of a small positive dimensionless parameter, which is artificially introduced to order terms in the governing equation. This parameter eventually is subs tituted back into the solution, and the final results are often obtained independent of th is book-keeping paramete r [69]. In this PAGE 108 90 section, the MTS approach is adopted to find the approximate analytical solution to Eq. (4.9). Some discussions of the appr oximate solution are also presented. Approximate Solution by the MTS Method To proceed with the approximate solution, we need to non-dimensionalize the governing equation in Eq. (4.9), which finally leads to '''23 12345, yyyyy (4.11) where 0, x y d (4.12) ', dy y d (4.13) 2 '' 2, dy y d (4.14) 0, t (4.15) 01,cmekkM (4.16) 1 0,meb M (4.17) 2 00 2 23 001,me meAV M d (4.18) 2 00 3 32 003 2me meAV Md (4.19) 22 3000 4 232 0002 ,me memekdAV MMd (4.20) and PAGE 109 91 2 00 5 32 00. 2me meAV Md (4.21) In Eq. (4.11), y and 'y are the non-dimensional center disp lacement and velocity of the diaphragm respectively, 0 is the linear natural frequency of the system, is the dimensionless time, and 1 through 5 are non-dimensional coeffi cients. By using the MTS method, the approximate solution of Eq (4.11) is assumed as a second order expansion in terms of a small positive parameter 2 001210122012,,,,,,, yyyy (4.22) where the multiple independent time scales are defined as 2 012,, and (4.23) The coefficients in Eq. (4.11) are further ordered to show up in the 2O by doing the following substitution 2222 1123344, and (4.24) Eqs. (4.22), (4.23) and (4.24) are then substituted into Eq. (4.11), by collecting and equating the coefficients (detailed derivati on steps are provided in Appendix B), finally the approximate solution for y is obtained as follows 12 121 22 53524540 2 0020 2 221233 cos, 28 R yRee (4.25) where 0 R and 0 are constants determined by the initial conditions. From Eqs. (4.15), (4.23) and (4.24) we have the following expressions 00,t (4.26) 1210,t (4.27) PAGE 110 92 22 3524523545 20 22 222323 22t (4.28) and 44 11. (4.29) Combining Eqs. (4.12), (4.24), (4.25), (4.26) (4.27), (4.28) and (4.29), finally the MTS solution for x t is given by 10 101 22 52354540 2 000200 5 2 12 2 2233 cos. 8 2t tR xtdRdete (4.30) Discussion of the MTS Approximate Solution From Eq. (4.30), during the up-stroke of the square wave, the transient center displacement response is an exponentially decaying cosine function oscillating around some DC offset. The decaying speed is uniqu ely determined by the quotient of system damping coefficient over lumped mass 10 meb M. The oscillation frequency is affected by all nonlinearities (3 and 4 ) and external step loading (5 ). Also further analysis shows that the phase an gle is affected by all nonlinea rities and external loading. If an initial displacement 0 is imposed and the diaphragm st arts from rest, the resulting transient displacement x t becomes 10 102 1 2 52354540 2 0020 5 2 2 120 2 2233 cos1, 8 2t txtdete d (4.31) where PAGE 111 93 5 000 2.d (4.32) Approximate solution with zero initial conditions If the initial displacement is zero (00 ), the resulting transient displacement becomes 10 101 22 235454 2 20 5 2 120 2 2233 cos1, 8 2t t ss ssssx xtxxete d (4.33) where s s x represents the steady state displacement and is defined as follows 5 0 2.ss x d (4.34) Eq. (4.33) will be used in Chapter 6 for the system parameter extraction from the experimental data. The identification process is outlined as follows. For a square wave excitation with a given amplitude of 0V, a nonlinear least-squares curve-fitting technique is applied and the real values of 0 0,d1 2 3 4 and 5 are obtained by curvefitting Eq. (4.33) with the experimental up-stroke center displacement response of the diaphragm. After obtaining the values of 0 and 0d from the experimental data, other system parameters and are extracted as follows 1, 2 (4.35) 2 450 2 04 ,d (4.36) and 23 500 2 002 .d V (4.37) PAGE 112 94Approximate solutions in other applications If there are no extern al step loading and quadratic non linearity in the system, which means 30, (4.38) and 50. (4.39) Then Eq. (4.31) is reduced into 10 101 2 40 2 020 2 1203 cos1, 8t txtete d (4.40) which is the general expression derived in th e reference [107]. The oscillation frequency in this case is given by 20 As we can see from Eq. (4.18), as the square wave voltage amplitude 0V increases, the oscillation frequency decreases due to the electrostatic spring-softening effect. In terms of applications, Eq. (4.31) can be applied to study the large-angle free-fall response of a parametrically ex cited pendulum [107]. Also Eq (4.31) can be altered and applied to find the approximate solution to Eq. (4.10) for the free vibration of the diaphragm during the down-stroke of the app lied square wave. The resulting transient displacement response becomes 002 2 0 00 2 03 cos1. 16ttxtete (4.41) Validity Region of the Approximate MTS Solution Since Eq. (4.33) basically represents an approximate solution to Eq. (4.7) via the MTS method, it is important to find out wher e the solution fails, in other words, its PAGE 113 95 validity region. In this section, numerical simulations via the ODE45 command in MATLAB are used to determine the validity region of Eq. (4.33). The ODE45 command is based on an explicit, variable-step, Runge-Kutta (4,5) integration formula, which utilizes a 4th order predictor and a 5th order corrector [108, 109]. First, we conduct three test simulations with three different applied voltages (0V) and given theoretical system parameters (0 , and 0d). The three different voltages are chosen such that the system responses fall into the linear, w eakly nonlinear and highl y nonlinear regions respectively. Then, based on the simulate d system responses, the approximate MTS solution is applied to extract system parameters for each te st case. Finally, extracted system parameters are compared with their given values a nd errors are calculated for each test case respectively. From the errors obtained, we can obtain the validity region for the approximate MTS solution. Results of linear case For the linear test case, the applied voltage is chosen to be 05VV Shown in Figure 4-3 is the comparison of simulated and MTS-based nonlinear least squares curve fitting center displacement results. As seen from the plot, the simulated and MTS solution results agree very well and th e maximum mechanical and electrical nonlinearities (see Eqs. (3.54) and (3.58) fo r their definitions) are calculated to be 0%mNL and 0%eNL Table 4-1 summarizes the resu lts for the linear test case and the given input system parameters are gene rated based on the theoretical LEM (also see Table 3-1). From the table, the maximum error of 0.89% occurs for the nonlinear stiffness parameter, and other errors are smaller. PAGE 114 96 0 1 2 3 4 5 6 7 x 10-5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10-8 Time (s)Center displacement (m) Simulated result MTS solution result Figure 4-3. Comparison of simulated and MT S-based curve fitting center displacement results (05VV ). Table 4-1. Given and extracted (via MTS solution) parameters for a linear case. Parameter Given Extracted Error Damping ratio 8.091e-2 8.050e-2 0.51% Natural frequency 002f (kHz) 185.5 185.4 0.054% Nonlinear parameter (3//Nmkg) 1.128e23 1.138e23 0.89% Ratio (2/mkg) 332.2 332.0 0.060% Gap 0d (m) 2.000e-6 1.990e-6 0.50% Results of weakly nonlinear case For the weakly nonlinear test case, th e applied voltage is chosen to be 025VV Shown in Figure 4-4 is the comparison of simulated and MTSbased nonlinear least squares curve fitting center displacement re sults. As we can see from the plot, the simulated and MTS solution results still agree very well. Part of transient center PAGE 115 97 displacement response is within the nonlinear range (for ex ample, the maximum transient displacement is larger than 0.3m ). 0 1 2 3 4 5 6 7 x 10-5 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10-7 Time (s)Center displacement (m) Simulated result MTS solution result Figure 4-4. Comparison of simulated and MT S-based curve fitting center displacement results (025VV ). Table 4-2. Given and extracted parameters (via MTS solution) for a weakly nonlinear case. Parameter Given Extracted Error Damping ratio 8.091e-2 8.030e-2 0.75% Natural frequency 002f (kHz) 185.5 184.1 0.22% Nonlinear parameter (3//Nmkg) 1.128e23 1.139e23 0.98% Ratio (2/mkg) 332.2 329.9 0.69% Gap 0d (m) 2.000e-6 1.980e-6 1.0% In this case, the maximum mechanical and electrical nonlinearities are calculated to be 1.3%mNL and 9.9%eNL Table 4-2 summarizes the results for the weakly nonlinear test case and the maxi mum error of 1.0% occurs for the gap, and other errors are smaller. PAGE 116 98Results of highly nonlinear case It turns out from the simulation that we cannot achieve higher mechanical nonlinearities ( 10%) from our physical microphone sy stem (2000Pa design) due to the electrostatic pull-in. To be able to generate higher mech anical nonlineari ties and further test the validity of the MTS solution in a highly nonlinear regi on, we numerically increase the nonlinear stiffness parameter, for example, 333 kk or *3 (where the asterisk sign represents new parameters). With an applied voltage 030VV and the new nonlinear stiffness parameter the plot of simulated an d MTS-based nonlinear least squares curve fitting center disp lacements is shown in Figure 45. Also part of transient center displacement response is within the nonlinear range (for example, the maximum transient displacement is larger than 0.5m ). The maximum mechanical and electrical nonlinearities are calculated to be 10%mNL and 28.6%eNL Table 4-3. Given and extracted (via MTS solution) parameters for a highly nonlinear case. Parameter Given Extracted Error Damping ratio 8.091e-2 7.700e-2 4.8% Natural frequency 002f (kHz) 185.5 182.7 1.5% Nonlinear parameter (3//Nmkg) 3.384e23 3.690e23 9.0% Ratio (2/mkg) 332.2 338.9 2.0% Gap 0d (m) 2.000e-6 1.970e-6 1.5% Table 4-3 summarizes the results for this case and the maximum error of 9.0% occurs for the nonlinear stiffness parameter. It should be pointed out that in this case the 3rd TaylorÂ’s series approximation for the electr ostatic force is accurate with an error of approximately 3.1% (Figure 4-2). PAGE 117 99 0 1 2 3 4 5 6 7 x 10-5 0 1 2 3 4 5 6 7 x 10-7 Time (s)Center displacement (m) Simulated result MTS solution result Figure 4-5. Comparison of simulated and MT S-based curve fitting center displacement results (030VV ,* 333 kk). Table 4-4 summarizes the nonlinear curve-fi tting results of the maximum error and sum of residual squares (an indicator of the accuracy of the curve-fitting), which is defined as follows [108] 2 ,, 1,npts risimulatediMTS iSxtxt (4.42) where rS is the sum of residual squares, npts is the total number of simulated (MTSbased) points, ,isimulated x t is the simulated displacement value for thi point, and ,iMTS x t is the MTS-based displacement value for thipoint. As we can see from Table 4-4, the maximum error and rS increases as the nonlinearity (mechanical or electrical) becomes larg er, which means the MTS PAGE 118 100 approximate solution is getting less accura te. The maximum error for the weakly nonlinear case is 1.0% and will be incr easing as the nonlinearity (mechanical or electrical) exceeds 10%. Therefore, we can conclude the MTS approximate solution holds with an error of less than 1.0% for a weakly nonlinear system with the electrostatic nonlinearity up to 10%. Table 4-4. Results of the maximum error and sum of residual squares for each test case. Test case Maximum error Sum of residual squares (rS) Linear case 0.89% 2.0e-17 Weakly nonlinear case 1.0% 6.5e-15 Highly nonlinear case 9.0% 8.0e-14 Governing Equation for the Elec trical Sinusoidal Excitation During the characterization tests for the elec trical sinusoidal exc itation, no acoustic pressure is incident on the diaphr agm and a designed electrical signal Vt is applied directly to either top or bottom backplate with diaphr agm and the other backplate electrically grounded. Physically, the designed signal has 60 sinus oidal cycles in three quarters of its period and is equal to zero fo r the remainder of the peri od. During the sinusoidal excitation in each period, the system goes to th e steady state. The transient response dies out completely when no voltage is applied during the last portion of each period. The mathematical expression for the designed excitation signal is 0 0000 0 003 cos, 4 ,0, 1, 2, ... 3 0, 1 4T VtnTnTtnT Vtn T nTtnT (4.43) PAGE 119 101 where 0V is the voltage amplitude and 0T is the period of the designed signal. Since there are 60 sinusoidal cycl es in three quarters of 0T, the angular sinusoidal excitation frequency is given by 0 0180 22. 3 460 T T (4.44) Since the backplates are assumed to be symmetric around the center diaphragm, it turns out that the two governi ng equations are similar when e ither of backplates is under the electrical sinusoidal excitation. For simp licity, the following anal ysis only studies the case of an excitation voltage being applied to the bottom b ackplate. The dynamic model for an electrical sinusoidal excitation on th e bottom backplate is shown in Figure 4-6. 0 x 0 13 kkb M A meme k c0dDiaphragmTop backplateBottom backplate 0cos Vt Figure 4-6. Dynamic model for an electri cal sinusoidal excitation on the bottom backplate. During the time 0 003 1, 4 T nTtnT no electrostatic force acts on the diaphragm, the governing equation for the free vibration of diaphragm is 23 0020. xxxx (4.45) If the diaphragm starts from re st and an initial displacement, 0 is imposed, the resulting transient displacement response is given by Eq. (4.41). PAGE 120 102 During the time 0 003 4 T nTtnT an electrostatic force acts on the diaphragm and forces it to vibrate. Based on th e dynamic model in Figure 4-6, the governing equation is derived as 22 0 23 0 00 2 0cos 2. 2 Vt xxxx dx (4.46) The square of the sinusoidal i nput contains both constant and harmonic terms as follows 22 22 00 0coscos2. 22 VV Vtt (4.47) By introducing the non-dimensional center di splacement, Eq. (4.46) can be further rewritten as 02 3 123 2cos2 1 FFt yyyy y (4.48) where 0, x y d (4.49) 102, (4.50) 2 20, (4.51) 2 30, d (4.52) 2 00 0 3 0, 4 V F d (4.53) and 2 00 2 3 0. 4 V F d (4.54) PAGE 121 103 Further rearrangement of Eq. (4.48) results in 2 3 123021cos2. yyyyyFFt (4.55) Eq. (4.55) represents a general damp ed second order nonlinear system under harmonic excitation. It should be pointed out that no approximation (for example, the 3rd order TaylorÂ’s series) is made for the elec trostatic force in Eq. (4.46) or (4.48). Therefore, the TaylorÂ’s series approximati on error for the electrostatic force is not applicable in the validity tests for the following harmonic balance solution. Approximate HB Solution for the Electrical Sinusoidal Excitation In this section, an approximate analytical solution to Eq. (4.55) has been investigated by the harmonic balance method. The basic idea of the harmonic balance method is to consider a periodic solution of the time response in a form of a truncated Fourier series, after substituting the periodic solution into the governing ODE and equating the coefficient of each of the lowest N+1 harmonics to zero, a system of N+1 algebraic equations will be obt ained and solved to finally yield the approximate solution [72]. In agreement with the input excitations of Eq. (4.55), the approximate steady-state solution of Eq. (4.55) is assumed to be a tr uncated Fourier series of the following form 02cos2, yAAt (4.56) where 0A and 2A are the amplitudes of the cons tant and second harmonic terms respectively, and is the phase of the second harmonic term. Approximate Solution by a HB Method To facilitate the analysis, it is convenie nt to introduce a complex form for the harmonic terms in Eq. (4.55) PAGE 122 104 2 322 2 12301. 2ititF yyyyyFee (4.57) Eq. (4.56) is then substituted into Eq. (4.57) from which the constant term and the real and imaginary parts of the second ha rmonic term are finally found to be 11112213310Const: B BBCF (4.58) 211222233221 Real: 2 B BBCF (4.59) and 3113223333Imag: 0, BBBC (4.60) the expressions of coefficients ij B and iC (1,2,3 i and 1,2,3 j ) are summarized in Appendix C. It follows that 02FF from Eqs. (4.53) and (4.54); therefore, Eqs. (4.58), (4.59) and (4.60) can be further condensed into the following matrix form B UC (4.61) where 11121314 21222324 31323334, BBBB BBBBB BBBB (4.62) 1 2 3 0, U F (4.63) 1 2 3, C CC C (4.64) PAGE 123 105 and the expressions for the coefficients 4i B (1,2,3 i ) are provided in Appendix C. Since both matrix B and vector C only contain the spectrum properties (0A 2A and ) of the steady-state response of the diaphragm, and vector U only contains all the unknown system parameters, Eq. (4.61) can be used to identify syst em parameters from the characterization experiments. The identification process is outlined as fo llows. For a given sinusoidal excitation frequency a nonlinear least-squares curve-fitting technique is applied to extract 0A 2A and from the time history of the steadystate displacement response. Matrix B and vector C are then calculated respectively from the equations in the Appendix C. Since there are four unknowns (1 2 3 and 0F ) in vector U and the rank of matrix B is three, Eq. (4.61) is underdetermined. Therefore, a second test is performed at a different excitation frequency to provide more equations to make the system overdetermined (two redundant equations) and the four unknowns are solved via the linear least-squares method. Once 1 2 3 and 0F (or 2F ) are estimated, system parameters 0 and are extracted by solving Eqs. (4 .50), (4.51), (4.52), and (4.53) simultaneously as follows, 02, (4.65) 1 2, 2 (4.66) 3 2 0, d (4.67) and PAGE 124 106 3 00 2 004 dF V (4.68) Validity Region of the Approximate HB Solution Eq. (4.56) basically represents an approximate solution to Eq. (4.55) via the HB method, it has two terms: a constant and a second harmonic. It is important to find out if Eq. (4.56) is still a good approximation when other harmonics exist, in other words, its validity region needs to be determined in te rms of the total harmonic distortion (THD). The THD is defined in Eq. (2.1) in Chapter 2. As seen from Eqs. (4.46) and (4.47), a frequency doubling is obtained due to the electrostatic nonlin earity. It should be pointed out that the THD in the approximate harmonic balance solution (Eq. (4.56)) is 0% since only one harmonic, due to the frequenc y doubling caused by the electrostatic nonlinearity, is considered. In this section, numerical simulation is used to determine the validity region of the HB approximate solution. Firs t, three test simulations ar e conducted for three different applied voltages (0V ) and given input system parameters (0 , and 0d ), the three different voltages are chosen such that the system responses have different THD levels (<0.1%, 1.0% and 10.6%) respectivel y. Then, based on the simulated system responses, the HB approximate solution is appl ied to extract system parameters for each test case. Finally, extracted system paramete rs are compared with their given values and errors are calculated for each test case respec tively. From the errors obtained, we can conclude the validity region for the HB approximate solution. Results of small THD case For the small THD case, the appl ied voltage is chosen to be 05 VV. The simulated sinusoidal displacement response of the diaphragm for the excitation with a PAGE 125 107 frequency of 92.7kHz (half of the systemÂ’s natural fre quency) is shown in Figure 4-7. From the plot, we can see that the steady stat e displacement amplitude is small and within the linear range. 0 0.2 0.4 0.6 0.8 1 1.2 x 10-4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10-8 Time (s)Center displacement (m) Figure 4-7. Simulated sinusoida l response of the diaphragm (05 VV 292.7 kHz ). Shown in Figure 4-8 is the power spectrum of the steady state displacement of the diaphragm, it has a DC component and the ca lculated THD is approximately less than 0.1%. Finally, the extracted system para meters via the approximate HB solution are listed in Table 4-5. Note that the second fre quency test has been conducted to be able to extract system parameters but the results are neglected for the conciseness of this section. As we can see from Table 4-5, the maximum error of 1.5% occurs for the damping ratio, and other errors are smaller. PAGE 126 108 0 200 400 600 800 1000 10-10 10-8 10-6 10-4 10-2 Frequency (kHz)Power (m2)THD < 0.1% Figure 4-8. Simulated power spectru m of the steady state displacement (05 VV 292.7 kHz ). Table 4-5. Given and extracted (via HB solution) parameters for the small THD case. Parameter Given Extracted Error Damping ratio 8.091e-2 8.212e-2 1.5% Natural frequency 002 f (kHz) 185.5 186.4 0.49% Nonlinear parameter (3// Nmkg ) 1.128e23 1.133e23 0.44% Ratio (2/ mkg ) 332.2 328.9 0.99% Results of transition THD case For the transition THD case, the applied voltage is chosen to be 020 VV. The simulated sinusoidal displacement response of the diaphragm for the excitation with a frequency of 92.7kHz is shown in Figure 4-9. From the plot, we can see that the steady state displacement amplitude is within the quasi-linear range and the maximum mechanical nonlinearity is approximately 1.5%. PAGE 127 109 0 0.2 0.4 0.6 0.8 1 1.2 x 10-4 -3 -2 -1 0 1 2 3 4 x 10-7 Time (s)Center displacement (m) Figure 4-9. Simulated sinusoida l response of the diaphragm (020 VV 292.7kHz ). 0 200 400 600 800 1000 10-8 10-6 10-4 10-2 100 Frequency (kHz)Power (m2)THD = 1% Figure 4-10. Simulated power spectru m of the steady state displacement (020 VV 292.7kHz ). PAGE 128 110 Shown in Figure 4-10 is the power spectrum of the steady state di splacement of the diaphragm, it has a DC component and the calculated THD is approximately 1.0%. Finally, the extracted system parameters via the approximate HB solution are listed in Table 4-6. As we can see fr om Table 4-6, the maximum e rror of 4.8% occurs for the damping ratio, and other errors are smaller. Table 4-6. Given and extracted (via HB solu tion) parameters for the transition THD case. Parameter Given Extracted Error Damping ratio 8.091e-2 7.701e-2 4.8% Natural frequency 002 f (kHz) 185.5 190.1 2.5% Nonlinear parameter (3// Nmkg ) 1.128e23 1.116e23 1.1% Ratio (2/ mkg ) 332.2 345.5 4.0% Results of large THD case For the large THD case, it turns out from the simulation that we cannot achieve higher harmonic distortions (10% THD) from our physical microphone system (2000Pa design) due to the electrostatic pull-in. To be able to gene rate higher harmonic distortions and further test th e validity of the HB solution, we numerically increase the nonlinear stiffness parameter, for example, ** 3340 kk or **40 (where the double asterisk sign represents new parameters). With the new nonlinear stiffness parameter ** the plot of the simulated sinusoidal displacement response of the diaphragm is s hown in Figure 4-11 for the excitation with a frequency of 92.7kHz and a voltage of 45.V As seen from Figur e 4-11, the steady state displacement amplitude is within the nonlin ear range (for example, the steady state displacement amplitude is larger than 0.5m ). PAGE 129 111 0 0.2 0.4 0.6 0.8 1 1.2 x 10-4 -6 -4 -2 0 2 4 6 8 10 x 10-7 Time (s)Center displacement (m) Figure 4-11. Simulated sinusoida l response of the diaphragm (045 VV,** 3340 kk, 292.7kHz ). 0 100 200 300 400 500 600 700 800 900 1000 10-10 10-8 10-6 10-4 10-2 100 Frequency (kHz)Power (m2)THD = 10.6% Figure 4-12. Simulated power spectrum of the steady state displacement (045 VV, ** 3340 kk, 292.7kHz ). PAGE 130 112 Shown in Figure 4-12 is the power spectrum of the steady state di splacement of the diaphragm, it has a DC component and the calculated THD is approximately 10.6%. Finally, the extracted system parameters via the approximate HB solution are listed in Table 4-7. Table 4-7. Given and extracted (via HB so lution) parameters for the large THD case. Parameter Given Extracted Error Damping ratio 8.091e-2 7.603e-2 6.0% Natural frequency 002 f (kHz) 185.5 203.2 9.5% Nonlinear parameter ** (3// Nmkg ) 4.512e24 4.285e23 5.0% Ratio (2/ mkg ) 332.2 355.4 7.0% As we can see from Table 4-7, the maximu m error of 9.5% occurs for the natural frequency. Table 4-8 summar izes the maximum error and sum of residual squares for each test case, the sum of residual squares rS for Eq. (4.61) is defined as follows [108] 2 64 11.^2.rijji ijSsumBUCBUC (4.69) Table 4-8. Results of the maximum error and sum of residual squares for each test case. Test case Maximum error Sum of residual squares (rS ) Small THD case 1.5% 4.9e18 Transition THD case 4.8% 2.9e21 Large THD case 9.5% 4.2e22 As we can see from the ta ble, the maximum error and rS increase as the THD becomes large, which means the HB approxima te solution is getting less accurate. The maximum error for the 10.6%THD case is 9.5% and will be increasing as the THD becomes larger. Therefore, we can conclude the HB approximate solution holds with an error of less than 9.6% for a nonlinear system with a THD up to 10.6%. PAGE 131 113 Governing Equation for the Sinusoida l Acoustical Pressure Excitation Previous studies focus on the respons es of the microphone under electrical excitations. Since microphones serve as pressure sensors and sense ac oustical signals, it is necessary to investigate the microphone resp onse to acoustic excitation. This section presents the study of the st eady state response of the microphone when excited by a typical sinusoidal acoustical pr essure signal. The study of the transient response of the microphone when subject to shock (mechanical or acoustical) loads, in the form of dynamic pull-in, is presented in Chapter 5. MA meme Pressure p t Diaphragm bVbV0d0d 13 kkk c 0 x Fb x T op b ac k p l ateBottom back p late Figure 4-13. Dynamic model for the sinus oidal acoustical pressure excitation. Figure 4-13 shows the dynamic model for the sinusoidal acoustical pressure excitation. In the model, pt represents a sinusoidal acous tical pressure signal. The expression for the applied sinusoida l pressure wave is given by 0cos,p p tpt (4.70) where 0 p and p are the amplitude and driving frequency of the sinusoidal PAGE 132 114 acoustical wave, respectively. In this case, the general governing equation in Eq. (3.53) of Chapter 3 can be reduced to 22 3 130 22 00cos, 2mebb mecmepAVV M xbxkkxkxApt dxdx (4.71) where bV is the applied DC bias voltage. 0 1 2 3 4 5 6 x 10-7 0 1 2 3 4 5 6 7 8 x 10-6 Center displacement (m)Force (N) Electrostatic net force with Vb = 9V Approximate electrostatic net force with Vb = 9V Figure 4-14. Plot of net electrostatic and approximate net electrostatic forces for bV9V Unfortunately, due to the complexity of Eq (4.71), no closed-form solution exists. To find an approximate solution to Eq. (4.71), first we need to approximate the nonlinear net electrostatic force. Further TaylorÂ’s series e xpansion for the nonlinear net electrostatic force up to the 3rd order results in 2222 3 000 22 35 00 0024 2mebbmebmebAVVAVAV x x dd dxdx (4.72) PAGE 133 115 In order to verify the accuracy of the 3rd order TaylorÂ’s series expansion used in the above equation, by using the parameters in Table 3-1, the net electrostatic and approximate net electrostatic forces are pl otted in Figure 4-14 for an applied voltage 0V9V. This voltage is chosen sinc e it is used in the numerical validity test in the next section. As we can see from the plot, the two net el ectrostatic forces ar e close to each other when the center displacement is up to 0.6m. Further calculations show that the maximum approximation error caused by the 3rd order TaylorÂ’s series expansion is 2.3%. The approximation error becomes larger if the center displacement is over 0.6m. Physically, the curvature of the diaphragm needs to be considered if the center displacement becomes large. The parallel plate assumption used to m odel the net electrostatic force (left hand side of Eq. (4.72)) might not be able to hol d for the large displacement case. Therefore, the 3rd order TaylorÂ’s series approximation in Eq. (4.72) carries a maximum error of approximately 2.3% when the center displacement is 0x0.6m and the applied voltage is 0V9V. To proceed, substitution of Eq. (4 .72) into Eq. (4.71) yields 22 3 00 130 35 0022 cos.mebmeb mecmepAVAV M xbxkkxkxApt dd (4.73) Physically, the above equation represents a damped second order system with only cubic mechanical and electrical nonlinearities and an external excitation. Mathematically, Eq. (4.73) represents a forced da mped DuffingÂ’s equation [72]. PAGE 134 116Approximate Solutions for the Sinusoid al Acoustical Pressure Excitation In this section, the steady-state approxima te solutions to Eq. (4.73) are obtained by both harmonic balance and multiple time scal es methods. First, letÂ’s define two equivalent spring constants as follows, 2 0 1 3 02meb EAV k d (4.74) and 2 0 3 5 02 ,meb EAV k d (4.75) where 1Ek is a linear electrical spring constant and 3Ek is a cubic electrical spring constant. Prior to finding the approximate solutions, we need to non-dimensionalize the governing equation in Eq. (4.73), which finally leads to '''32cos,yyyyP (4.76) where 0, x y d (4.77) ', dy y d (4.78) 2 '' 2, dy y d (4.79) 0, t (4.80) 011,cEmekkkM (4.81) 0, 2meb M (4.82) PAGE 135 117 2 33 0 2 0,E mekk d M (4.83) 0 2 00,me mepA P M d (4.84) and 0,p (4.85) where y and 'y are the non-dimensional center di splacement and velocity of the diaphragm respectively, 0 is the modified natural frequency of the system, is the dimensionless time, is the modified damping ratio, is the modified nonlinear stiffness parameter, P is the external forcing parameter and is the ratio of the pressure driving frequency over the systemÂ’s natural fr equency. As seen from Eqs. (4.81) and (4.83), physically the linear and cubic stiffnesses of the diaphragm are reduced by the applied DC voltage or the electrostatic softening nonlinearity. HB Approximate Solution The harmonic balance approximate solution to Eq. (4.76) in the steady state is expressed in a Fourier se ries as follows [72] 0cos,n nyan (4.86) where na is the amplitude of the nth harmonic and is the phase angle. After substituting Eq. (4.86) into Eq. (4.76), by collecting and equating the coefficients of the harmonics, the approximate solution for y can be obtained (detailed derivations are provide d in Appendix D). In agreem ent with the input excitation PAGE 136 118 of Eq. (4.76), for a first-order approxim ation, the harmonic balance solution for y in the steady state is given by 1cos,ya (4.87) where 1a and are determined from the following equations respectively, 2 2 2624222 11193 1120, 162 aaaP (4.88) and 1 22 12 tan. 10.75 a (4.89) From Eqs. (4.77), (4.80), (4.85), (4.87) and (4.89), finall y we can obtain the following approximate harmonic balance soluti on for Eq. (4.73) in the steady state as follows 1 010 22 12 costan, 10.75pxtydadt a (4.90) where 1a can be determined by solving Eq. (4.88) for given parameters. MTS Approximate Solution For the multiple time scales solution, it is convenient to introduce the following transformations to rewrite the gove rning equation in Eq. (4.76) (4.91) (4.92) and 2, Pf (4.93) where is a small perturbation parameter, and f are new introduced variables. PAGE 137 119 Substituting Eqs. (4.92), and (4.93) into Eq. (4.76) yields '''322cos.yyyyf (4.94) If we are seeking a first order approximation to the above equation, which takes the form 001101,,, yyy (4.95) where 0y and 1y are unknown functions that need to be determined, 0 and 1 are the fast time scale (representing the fast dynami cs) and slow time scale (representing the slow dynamics) respectively, 0 (4.96) and 1. (4.97) After substituting Eq. (4.95) into Eq. (4.94), by collecting and equating the coefficients (detailed derivations are pr ovided in Appendix D), the first-order approximate solution for y in the steady state is obtained as follows 1 1 12 cossin,a ya P (4.98) where 1a is the root of the following equation, 642222 11193 0. 644 aaaf (4.99) The non-dimensional parameters in Eq. (4.99) are defined as 1 (4.100) (4.101) PAGE 138 120 and 2P f (4.102) From Eqs. (4.77), (4.80), (4.85) and (4. 98), finally the multiple time scales solution for x t in the steady state is 1 1 0102 cossin,pa xtydadt P (4.103) where 1a can be determined by solving Eq. (4.99) for given parameters. Validity Region of Approximate Solutions In this section, the steady-state approxima te solutions obtained in Eqs. (4.90) and (4.103) are compared to the steady-state num erical solution to the full-blown nonlinear equation in Eq. (4.71). The numerical solu tion to the full-blow n nonlinear equation in Eq. (4.71) is obtained by doing the direct integration via the ODE45 command in MATLAB. To check the validity of the appr oximate solutions, three different pressure amplitudes are chosen such that the steady-st ate system responses fall into the linear, weakly nonlinear and highly non linear regions, respectively. For each case, three different solutions (HB, MTS and numerical) are compared to obtain the errors of the approximate solutions. For this study, a bias DC voltage of 9V is chosen since it will be used for the future experiments and the parameters from Table 3-1 are used to calculate non-dimensional parameters as shown in Table 4-9. Three di fferent pressure amplitudes are chosen to be 100Pa, 350Pa and 700Pa, respectively. Th e corresponding non-dimensional forcing parameters are also listed in Table 4-9. PAGE 139 121 Table 4-9. Parameters used for the compar ison of approximate and numerical solutions. Parameter Value Acoustical pressure amplitudes 0 p 100.0, 350.0 and 700.0 (Pa) Bias DC voltage (bV ) 9.000 (V) Modified natural frequency ( 02 ) 181.4 (kHz) Modified damping ratio ( ) 8.270e-2 Modified nonlinear parameter ( ) 2.556e-1 External forcing parameters ( P ) 1.279e-2, 4.477e-2 and 8.954e-2 As seen from Eqs. (4.71), (4.88) and (4.99), the non-dimensional amplitude 1a represents the most difference between th e approximate (HB and MTS) and numerical solutions. Therefore, only the non-dime nsional displacement amplitudes of the approximate and numerical solutions are stud ied in the following when the frequency of the incident acoustical pressure is swept. Results of linear case For the linear test case, the pressure amplitude is chosen to be 0100 p Pa. Shown in Figure 4-15 is the comparison of the non -dimensional amplitudes of the approximate and numerical solutions for the frequency sweep of the incident acoustical pressure. From the plot, the maximum non-dimensional amplitude is approximately 0.08, where the corresponding mechanical and electrical non linearities (see Eqs. (3.54) and (3.59) for their definitions) are calculated to be 0%mNL and 0%eNL As seen from Figure 415, the difference between the non-dimensional amplitudes of the approximate and numerical solutions in the steady state is small around 1. Further studies show that the maximum amplitude error between the ha rmonic balance and numerical solutions is approximately 1.2% when 0.12. For the multiple time scales solution, the maximum amplitude error is approximately 2.1% when 0.951.05. Away from 1, the non-dimensional amplitudes of the ha rmonic balance and numerical solutions PAGE 140 122 are in good agreement with each other, while the non-dimensional amplitude difference between the multiple time scales and numerical solutions becomes larger. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Frequncy ratio ( = p/0)Non-dimensional amplitude (a1) HB solution MTS solution Numerical solution p0 = 100 Pa Vb = 9 V Figure 4-15. Comparison of the steady-st ate non-dimensional amplitudes of the approximate and numerical solutions for a linear case. Results of weakly nonlinear case For the weakly nonlinear test case, the pressure amplitude is chosen to be 0350 p Pa. Shown in Figure 4-16 is the compar ison of the non-dimensional amplitudes of the approximate and numerical solutions in the steady state. From the plot, the maximum non-dimensional amplitude is approximately 0.27, where the corresponding mechanical and electrical nonlinear ities are calcula ted to be 2.7%mNL and 1.5%eNL. As seen from Figure 4-16, the difference between the non-dimensional amplitudes of the approximate and numerical so lutions in the steady state is small around 1. Further studies show that the maximum amplitude error between the harmonic PAGE 141 123 balance and numerical solutions is approximately 3.0% when 0.12. For the multiple time scales solution, the maximum am plitude error is approximately 4.5% when 0.951.05. Away from 1 the non-dimensional amplitudes of the harmonic balance and numerical solutions are still in good agreement with each other, while the non-dimensional amplitude difference between the multiple time scales and numerical solutions becomes larger. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 0.15 0.2 0.25 0.3 Frequncy ratio ( = p/0)Non-dimensional amplitude (a1) HB solution MTS solution Numerical solution p0 = 350 Pa Vb = 9 V Figure 4-16. Comparison of the steady-st ate non-dimensional amplitudes of the approximate and numerical solutions for a weakly nonlinear case. Results of highly nonlinear case For the highly nonlinear test case, the pr essure amplitude is chosen to be 0700 p Pa. Shown in Figure 4-17 is the compar ison of the non-dimensional amplitudes of the approximate and numerical solutions in the steady state. From the plot, the maximum non-dimensional amplitude is approximately 0.55, where the corresponding PAGE 142 124 mechanical and electrical nonlinear ities are calculated to be 10.6%mNL and 24.8%eNL. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequncy ratio ( = p/0)Non-dimensional amplitude (a1) HB solution MTS solution Numerical solution p0 = 700 Pa Vb = 9 V Figure 4-17. Comparison of the steady-st ate non-dimensional amplitudes of the approximate and numerical solutions for a highly nonlinear case. As seen from Figure 4-17, the differen ce between the non-dimensional amplitudes of the approximate and numerical solutions in the steady state becomes larger around 1. Further studies show that the maximu m amplitude error between the harmonic balance and numerical solutions is approximately 4.5% when 0.12. For the multiple time scales solution, the maximum amplitude error is approximately 7.5% when 0.951.05. Away from 1 the harmonic balance and numerical solutions are still in good agreement with each other, while the large difference exists between the multiple time scales and numerical solutions. Also it can be observed that the resonance PAGE 143 125 occurs after 1 and all three response curves are be nding to the right, which indicates the existence of a DuffingÂ’s ha rdening spring in the system. Based on the results from the above th ree test cases, among the two obtained approximate solutions, the HB approximate so lution is more accurate to predict the microphone response under the sinusoidal acous tical pressure excitation. Away from 1, the multiple time scales solution will generate large errors and therefore it is not recommended for use. In conc lusion, the HB approximate solution holds with an error of less than 4.5% for the microphone under a bias voltage of 9V and a sinusoidal acoustical pressure excitation with an amplitude of up to 750Pa. Summary In this chapter, nonlinear an alyses are presented for th e response behavior due to the electrical square and sinusoidal excitations The approximate analytical solutions to those two governing equations are obtained vi a the MTS and HB methods respectively. Discussions and validity regi ons of the derived MTS and HB approximate solutions are also provided. The obtained approximate solutions will be applied in Chapter 6 to extract system parameters experimentally. This ch apter also provides HB and MTS approximate solutions for the microphone under the sinuso idal acoustical pres sure excitation. Comparison of the obtained two approximate so lutions to the numerical solution in the steady state is presented. The major contri butions made in this chapter are 1) a MTS approximate solution to a general damped seco nd-order system with quadratic and cubic nonlinearities and non-zero step loading as defined in Eq. (4.11); and 2) a HB approximate solution to a ge neral inhomogeneous nonlinear damped second order system under the harmonic excitation as defined in Eq. (4.55). PAGE 144 126 CHAPTER 5 PULL-IN INSTABILITIES As noted in the previous chapters, the microphone must be carefully designed to avoid unstable parameter domains and possible structural failure. This chapter focuses on the theoretical and numerical analyses of pull-in instabilities of the microphone. The next section examines the microphone quasi-sta tic pull-in inst ability, due to an applied DC voltage, through a standard nonlinear analysis process. Then, quasi-static pull-in due to a combination of an applied DC voltage and static pressure is pr ovided. In addition, the effect of a mechanical shock load on dynamic pull-in instability is also discussed. By using a phase portrait and the ba sins of attraction, a mechani cal shock load is related to dynamic pull-in. Finally, dynamic pull-in due to an acoustical pulse, in the form of an Nwave, is investigated via numerical simulation. Quasi-Static Pull-in due to an Applied DC Voltage During the quasi-static pull-in analysis in this section, it is assumed that no acoustical pressure exists on the diaphragm ( 0pt ) and only DC voltages are applied to the top and bottom backplates Since all the time derivatives go to zero in the quasistatic process, based on the ge neral nonlinear governing equati on in Eq. (3.53) of Chapter 3, finally the corresponding governing equa tion for the static response is given by 22 3 13 22 00. 2mebbAVV kxkx dxdx (5.1) Notice that the stiffness of the cavity is not included in the mechanical restoring force since the process is quasi-static. With the rearrangement of th e net electrostatic PAGE 145 127 force, Eq. (5.1) can be further changed into 32 130 2 22 02.mebx kxkxAdV dx (5.2) It should be pointed out that no approximation (for example, the 3rd order TaylorÂ’s series) is made for the net electrostatic fo rce throughout this chapter. To proceed, we need to non-dimensionalize the above sta tic equation to simplify the analysis 3 2 20, 1ssy fyyy y (5.3) where 2 21sy y is the non-dimensional net electrostatic force, 3syy is the nondimensional mechanical force, f y is a nonlinear function of y and represents a net non-dimensional force applied on the diaphragm. Other non-dimensional parameters in Eq. (5.3) are defined as 0, x y d (5.4) 23 3030 110,skdkd kkd (5.5) and 2 22 0 3 10102 2 ,meb meb sAV AVd kdkd (5.6) where y is the ND center displacement of the diaphragm, s is a ND mechanical force parameter, and s is a ND electrostatic force parameter with a square dependence on the DC voltage. Physically, y is a ratio of the center displ acement of the diaphragm over the PAGE 146 128 nominal gap, which serves as a reference displacement. The parameter s represents a ratio of the nonlinear component of the spring force, at the reference displacement, over its linear component The parameter s represents a ratio of the nonlinear electrostatic force over the linear component of the spring force at the reference displacement. To further investigate the stability of Eq. (5.3), first we need to find equilibrium points of the governing equation and study th eir local stabilities. Equilibrium Points and Local Stabilities The equilibrium points of the system can be obtained by findi ng the roots of Eq. (5.3). Physically, at equilibrium points, th e nonlinear mechanical spring force equals to the nonlinear electrostatic for ce. Therefore, the equilibr ium points satisfy the following nonlinear equation 3 2 20. 1e sese ey yy y (5.7) The above nonlinear equation is a sevent h order nonlinear algebraic equation. which does not have an analy tical solution. However, it is possible to obtain an analytical solution for this particular problem. As seen from Eq. (5.7), 0ey is always one of seven roots. Therefor e, Eq. (5.7) can be changed in to a sixth order equation with only even terms, which can be solved analytically. Table 5-1. Force parameters for a designed 2000Pa capacitive MEMS microphone. Parameter Value DC bias voltage (bV ) 20 (V) ND mechanical force parameter ( s ) 0.37 ND electrostatic force parameter ( s ) 0.24 In order to facilitate the analysis, some values for the non-dimensional parameters are required. By using the physical parame ters in Table 3-1 for the designed 2000Pa PAGE 147 129 MEMS microphone, non-dimensional parameters ar e calculated and listed in Table 5-1 for a DC voltage of 20V (it should be pointed out that this DC voltage value is chosen just for illustration purposes). From Table 5-1, we can see that the mech anical and electrostatic nonlinearities of the system are on the same order. As we will discuss later, a larger mechanical geometric nonlinearity 3 s kor stabilizes the system while a larger electrostatic nonlinearity bsVor destabilizes the system. Those two t ypes of nonlinearities compete with each other and some interesting phenomena occur when one of them becomes dominant. For example, when the value of bsVor reaches a certain extrem e point, quasi-static pullin occurs since the geometric nonlinearity becomes insignificant comparing with the competing electrostatic nonlinearity. After substituting the values of s and s into Eq. (5.7), five real solutions are finally obtained. Therefore, the fi ve corresponding equi librium points are 10,ey (5.8) 2,30.74,ey (5.9) and 4,51.18.ey (5.10) The negative sign in above equations mean s the diaphragm moves towards the top backplate while the positive sign means the diaphragm moves towards the bottom backplate. The above equilibrium points ar e symmetric around the rest position due to the symmetric structure assumpti on utilized in the modeling. PAGE 148 130 To study the local stability of each equili brium point, we need to linearize the net nonlinear force around each equilibrium point. To realize a linearization, mathematically, the Jacobian matrix needs to be introduced. By defi nition, the Jacobian is a matrix containing all of the first partial de rivatives with respect to each state variable. For this study, the calculated Jacobian matrix only has one element as follows 8642 3 2319933331 1ssssssyyyy f J y y (5.11) The local stability is determined from the eigenvalues of the Jacobian matrix evaluated at each equilibrium point. For exampl e, if all the eigenvalues of an equilibrium point have negative real parts, the equilibrium point is a locally stable sink point. If some, but not all, of the eigenvalues of an equilibrium point have positive real parts, while other eigenvalues have negative real parts, the equilibrium point is a locally unstable saddle point [69]. Evaluating the Jacobian matrix in Eq. (5 .11) at each equili brium point yields 10.757, J (5.12) 2,36.86, J (5.13) and 4,524.4. J (5.14) The eigenvalues of the above Jacobian matr ices can be easily calculated since each matrix has only one element. Therefore, based on the criteria of local stability, the first equilibrium point is a locally stable sink poi nt. The second and third equilibrium points are locally unstable saddle points, the fourth and fifth equilibrium points are locally stable sink points. PAGE 149 131Graphical Analysis One alternative to analyze th e local stabilities of equili brium points is by examining the two ND nonlinear forces graphically. Figur e 5-1 shows a plot of the ND mechanical and net electrostatic forces for an applie d DC voltage of 20V. The border lines are located at 1 y, which correspond to two physical backplates. Mathematically, 1 y are two singular points for Eq. (5.3), where the ND net electrostatic force goes to infinity. The range between two border lines is the physical operation range of the diaphragm. Figure 5-1. Plot of the ND mechanical and net electrostatic forces for 0.37s and 0.24s From Figure 5-1, the five possible equilibrium points are: 1) a locally stable point attractor (node 1) at the rest position of the diaphragm (0 y ); 2) two unstable saddle nodes (nodes 2 and 3), situated at 0.74 y which repel motions either toward the desirable stable center position or towards quasi-static pull-in at the border lines; and 3) PAGE 150 132 two locally stable point at tractors (nodes 4 and 5 in the figure), located at 1.18 y which, although physically do not exist, attrac t relatively larger amplitude displacements. Further analysis shows that there are seve ral stable and unstable quasi-static regions in Figure 5-1. In the region between th e unstable saddle nodes 2 and 3, the ND mechanical force is always greater than the ND net electrostatic force; therefore, the system is always quasi-statically stable. The region between unstable saddle nodes 2 and 3 is called the 1-D basin of at traction of the center equilibrium point, in which all initial displacements quasi-statically converge into the center equilibrium point. In the regions between the physical border lines and nodes 2 and 3 respectively, the ND mechanical force is always less than th e ND net electrostatic force; therefore, the system is unstable and the diaphragm moves qu asi-statically to the backplates. In the regions outside the border lines, the system is stable only mathematically. In the following analysis, these non-physical regions will be discarded. Critical Quasi-Static Pull-in Voltage From the previous analysis, the quasi-st atically stable oper ation range of the microphone is defined by the 1-D basin of at traction of the center equilibrium point. Further analysis shows that the stable ope rational range depends on the applied DC voltage. For example, the operation range is 0.740.74y or 1.48m1.48mx when the microphone is biased by a 20V DC voltage. The idea of a 1-D basin of attraction can be further applied to find the critical pullin voltage. Shown in Figure 5-2 is a plot of the ND mech anical force versus ND net electrostatic forces for diffe rent DC voltages (20V, 30V a nd 41V). As the DC voltage increases, the ND net electrostatic force grow s, and the 1-D basin of attraction of the PAGE 151 133 center equilibrium point shrinks quickly. At a critical pull-in voltage, the 1-D basin of attraction of the center equilibrium point is totally lost, which means no stable region exists and the diaphragm is attracted by th e ND net electrostatic force to either of backplates. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Non-dimensional center displacement yNon-dimensional force ND mechanical force with s = 0.37 ND electrostatic force with s = 0.24 ND electrostatic force with s = 0.55 ND electrostatic force with s = 1.0 s = 0.24s = 1.0 Figure 5-2. Plot of the ND mechanical and net electrostatic forces. For these calculations, 0.37s and s has values of 0.24, 0.55 and 1.0. The largest eigenvalue of the Jacobian ma trix in Eq. (5.11) goes to zero at 0 y because the system is neutrally stable when the critical pullin voltage is reached. From Eq. (5.11), we have 010.s yJ (5.15) Substituting Eq. (5.6) into Eq. (5.15), finall y the critical quasi-static pull-in voltage b SPIV is PAGE 152 134 3 1041 2b SPI mekd VV A (5.16) As we can see from Eq. (5.16), the crit ical quasi-static pull-in voltage is only determined by the air gap, the equivalent linear stiffness and lumped area of the diaphragm respectively. Furthe r calculations show that the critical quasi-static pull-in voltage in Eq. (5.16) is approximately 30% higher than that of a corresponding singlebackplate condenser microphone [3]. For the quasi-static pull-in of the single-backplate and dual-backplate condenser microphones, another distinction is that it occurs at 2/3 of the gap for a single-backplate microphone while at the rest position for a dual-backplate microphone [3]. As seen from Figure 2-10, for the quasi -static pull-in of a single-backplate microphone, when the applied DC voltage re aches the critical pull-in voltage, the equilibrium center displacement of the diaphragm is 1/3 of the gap. At this equilibrium point, any small perturbation of the center displacement leads to pull-in since the electrostatic force is larger than the mechani cal restoring force and can not be balanced. For a dual-backplate microphone, as shown in Figure 5-2, the equilibrium center displacement of the diaphragm is located at its rest position when a critical pull-in voltage is applied. Similarly, any small displ acement perturbation around the equilibrium point leads to pull-in since the electrostatic force is larger than the mechanical restoring force and can not be balanced. Quasi-Static Pull-in by a Subcritical Pitchfork Bifurcation Bifurcation diagrams provide a succinct representation of the changes in the equilibrium solutions as a single control parameter is varied in a quasi-static manner. Here, we investigate the quantitative change of equilibrium solutions for Eq. (5.3) as the PAGE 153 135 applied DC voltage is varied. To this end, we fix the non-dimensional mechanical force parameter s and vary the non-dimensional electrostatic force parameter s by changing the applied DC bias value from 1 to 60V. Show n in Figure 5-3 is the plot of a subcritical pitchfork bifurcation illust rating quasi-static pull-in. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Non-dimensional electrostatic force parameter sFixed point ye Stable Unstable Critical quasi-static pull-in s = 1 Figure 5-3. A subcritical pitchf ork bifurcation illustrating quas i-static pull-in due to an applied DC voltage. Instead of using the ND electrostatic force parameter s in the bifurcation plot, the DC bias voltage bV can be used as a bifurcation para meter and the regenerated plot is shown in Figure 5-4. Within the physical border lines (1y ), initially there are three different solution branches: two unsta ble solution branches (for example, 2,30.74ey in Eq. (5.9) when 20bVV) outside and one stable soluti on branch between them. As the bifurcation parameter bV increases, three solution br anches converge and finally PAGE 154 136 evolve into one unstable solution branch. Th e bifurcation point, at which a bifurcation occurs, is located at 41b SPIVV Therefore, the critical quasi-static pull-in point is a subcritical pitchf ork bifurcation. 5 10 15 20 25 30 35 41 45 50 55 60 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 DC bias voltage Vb (V)Fixed point ye Stable Unstable Critical quasi-static pull-in (Vb)SPI = 41V Figure 5-4. A subcritical pitchf ork bifurcation illustrating quas i-static pull-in due to an applied DC voltage (ver sus bias voltages). Potential Advantage of Geometric Nonlinearity As shown in Figure 5-5, increasing geometric nonlinearity (3 or s k ) allows designers to relocate saddle node s at desired positions. This effectively expands the desired 1-D basin of attraction of the stable equilibrium point located at 0 y, and hence increases the operational range of the mi crophone for a given DC voltage below its critical value. PAGE 155 137 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Non-dimensional center displacement yNon-dimensional force ND electrostatic force with s = 0.24 ND mechanical force with s = 0.30 ND mechanical force with s = 0.90 ND mechanical force with s = 1.8 s = 1.8s = 0.30 Figure 5-5. Plot of a nondimensional net electrosta tic force and different nondimensional mechanical forces. For these calculations, 0.24s and s has values of 0.30, 0.90 and 1.8. To find a scaling relation for the geometri c nonlinearity, we need to revisit the expressions of 1k 3k and s From Eqs. (3.32), (3.33) and Eq. (5.5), we have 22 300 2 1,skdd kh (5.17) where h is the thickness of the diaphragm and 0d is the gap. From Eq. (5.17), the quasi-static operat ional range of the microphone for a given DC voltage below its critical value can be increased by enlarging the gap and/or decreasing the thickness of the diaphragm. Mo reover, from Eq. (5.16), the critical quasistatic pull-in voltage increas es as the gap beco mes larger, which physically makes sense and is desired to furthe r stabilize the microphone. PAGE 156 138 From the microphone performance perspective, the sensitivity can be increased by decreasing the thickne ss of the diaphragm while keepi ng other parameters fixed. However, the dynamic range and linearity of the microphone are affected unfavorably by the addition of geometric nonlinea rity as shown in Figure 5-5. Compact Quasi-Static Stable Operation Range To better understand the relations between the quasi-static st able operation range and two nonlinearities, a general compact 3D plot is created by using the solutions of Eq. (5.7) for different values of s and s Shown in Figure 5-6 is a stable range of the diaphragm versus s and s in a 3D space. Figure 5-6. Quasi-static stable operati on range of the microphone in a 3D space. In Figure 5-6, the upper boundary surface consists of all positive unstable equilibrium points for given values of s and s (for example, node 2 in Figure 5-1 when 0.37s and 0.24s), and the lower boundary surface consists of all negative PAGE 157 139 unstable equilibrium points (for exampl e, node 3 in Figure 5-1 when 0.37s and 0.24s). Hence, the volume defined by the physical limits of three axes as well as the upper and lower boundary surfaces represents a quasi-static stable ope ration range of the microphone in a 3D space. Figure 5-7. Quasi-static stable operation ra nge of the microphone in a 3D space (versus DC voltages). Instead of using the ND electrostatic force parameter s in the 3D plot, the DC bias voltage bV can be used and the regenerated plot is shown in Figure 5-7. From the plot, if the value of s is fixed, as the DC bias voltage increases, the co rresponding stable operation range becomes smaller. When a cr itical pull-in voltage is reached, the operation range shrinks to zero. However, fo r a fixed value of DC bias voltage, as the parameter s increases, the stable operation range beco mes larger. Again, the addition of geometric nonlinearity helps stabilize the syst em while an increased DC bias voltage or PAGE 158 140 electrostatic nonlinearity destab ilizes the system. The stable operation range plots shown in Figure 5-6 and Figure 5-7 also provide a possible tool to choose design parameters, such as the DC bias voltage, physical dimens ions of the diaphragm and gap distance, for a desired operation range of the microphone. Ho wever, it should be pointed out that the dynamic range and linearity of the microphone are affected unfavorably by the addition of mechanical geometric nonlinearity. Quasi-Static Pull-in due to an Applied Acoustical Pressure In this section, quasi-static pull-in due to an applied static acoustic pressure is studied. The model for the analysis is show n in Figure 4-13. To study the quasi-static pull-in due to an applied acous tical pressure, the applied DC voltage is kept lower than the critical quasi-static pull-in voltage derive d in the previous section, and the applied pressure difference on the diaphragm is assumed to be time-independent 0 p tp Since all the time derivatives go to zero in the quasi-static process, based on the general nonlinear governing equation in Eq (3.53) of Chapter 3, finally the corresponding governing equation for the static response is given by 22 3 130 22 00. 2mebb meAVV kxkxpA dxdx (5.18) With the rearrangement of the net electrostatic force, the governing equation in Eq. (5.1) can be further changed into 32 1300 2 22 02.memebx kxkxpAAdV dx (5.19) The above static equation is non-dime nsionalized by using the non-dimensional displacement y 0yxd PAGE 159 141 3 2 20, 1sssy fyyyP y (5.20) where f y is a nonlinear function of y and represents a net non-dimensional force applied on the diaphragm. The parameter s P represents the nondimensional force induced by the acoustical pre ssure and is defined as 0 10.me s p A P kd (5.21) The other non-dimensional parame ters in Eq. (5.20) are define d in Eqs. (5.5) and (5.6), respectively. To further invest igate the stability of Eq. (5.20) similarly we need to find equilibrium points of the governing equa tion and study their lo cal stabilities. Equilibrium Points and Local Stabilities The equilibrium points of the system can be obtained by finding the roots of Eq. (5.20). Physically, at equilibrium points, the net restoring force (the difference between the nonlinear mechanical spring force and the fo rce caused by the pres sure) equals the net electrostatic force. Therefore, the equilibrium points ey satisfy the following nonlinear equation 3 2 20. 1e seses ey yyP y (5.22) The above nonlinear equati on can be rewritten as 222 32221110seeeeseseyyyyPyy (5.23) or 75432122210.sesesesesesesyyPyyPyyP (5.24) PAGE 160 142 Eq. (5.24) basically is a general seventh order eq uation and there is no analytical solution. To proceed with the analysis, a numerical approach is employed to find the roots of the above equation. In order to carry ou t the numerical analysis, some values for the non-dimensional parameters are required. To be consistent with we did in the previous section and also for illustration purposes, a DC voltage of 20V is chosen here. Therefore, the corresponding non-dimensional parameters s and s are given in Table 5-1 for the designed 2000Pa MEMS microphone. Also for illustration purpo ses, a pressure difference of 1000Pa is chosen and the corresponding non-dimensional parameter s P is calculated to be 0.14.sP After substituting the values of s s and s P into Eq. (5.24), five real solutions and two complex conjugate roots are finally obtained via the roots command (computes the roots of a polynomial) in MATLAB. The two comple x conjugate roots are not physical and neglected in the following an alysis. The final five corresponding equilibrium points are given as 10.18,ey (5.25) 20.71,ey (5.26) 30.76,ey (5.27) 41.19,ey (5.28) and 51.18.ey (5.29) PAGE 161 143 The negative sign in above equations mean s the diaphragm moves towards the top backplate while the positive sign means the diaphragm moves towards the bottom backplate. The above equilibrium points ar e non-symmetric around the rest position due to the pressure loading. Figure 5-8. Plot of the ND net restoring and electr ostatic forces with 0.37s 0.14sP and 0.24s The local stabilities of equilibrium poi nts are studied by examining the two nondimensional (ND) nonlinear forces graphically Figure 5-8 shows a plot of the ND net restoring and electrostatic fo rces for an applied pressure difference of 1000Pa and an applied DC voltage of 20V. Again, the border lines are located at 1 y, which correspond to two physical backplates. From Figure 5-8, the five possible equilibrium points are: 1) a locally stable point attractor (node 1) at 0.18 y; 2) two unstable saddle nodes (nodes 2 and 3), situated at PAGE 162 144 0.71 y and 0.76 y; and 3) two locally stable po int attractors (nodes 4 and 5), located at 1.19 y and 1.18 y. In the region between unstable saddle nodes 2 and 3, the ND net restoring force is always greater th an the ND net electrostat ic force; therefore, the system is always quasi-statically stab le. The region between unstable saddle nodes 2 and 3 is called the 1-D basin of attractio n of the equilibrium point located at 0.18 y. In the regions outside the border lines, the system is stable only mathematically. In the following analysis, these non-physical regions will be discarded. Critical Quasi-Static Pull-in Pressure From the previous analysis, the quasi-st atically stable oper ation range of the microphone is defined by the 1D basin of attraction between nodes 2 and 3. Further analysis shows that the stable operational ra nge depends on the applied pressure for a given DC voltage. The idea of a 1-D basin of attraction can be further applied to find the critical pull-in pressure when the applied DC voltage is fixed. Shown in Figure 5-9 is a plot of different ND net rest oring forces for different pressure values: 1000Pa, 2000Pa and 2464Pa 0.14,0.27 and 0.34sP The parameter 0.37s is used to calculate the ND net restoring forces. Also, a ND net electros tatic force is plotted in Figure 5-9 for an applied DC voltage of 20V 0.24s. In Figure 5-9, as the value of s P increases, the ND net restoring force decreases, and the stable region between nodes 1 and 2 sh rinks accordingly. At a critical pull-in pressure point, nodes 1 and 2 merge together a nd the stable region between is totally lost, which means the diaphragm will be attracted by the net electrostatic force to the bottom backplate at 1 y. After numerical iterations, for an applied DC voltage of 20V, the PAGE 163 145 critical value of s P is found to be 0.34sP which corresponds to a critical pull-in pressure of 2464Pa. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Non-dimensional center displacement yNon-dimensional force ND net electrostatic force with s = 0.24 ND net restoring force with Ps = 0.14 ND net restoring force with Ps = 0.27 ND net restoring force with Ps = 0.34 Ps = 0.14 Ps = 0.34 Figure 5-9. Plot of the ND net restoring and electrostatic forces. For these calculations, 0.37s 0.24s and s P has values of 0.14, 0.27 and 0.34. Since the net electrostatic force is symmetric around the center point at 0 y further analysis shows that quasi-static pull-in occurs at the top backplate (at 1 y ) when a critical negative pressure value of -2464Pa is reached. In summary, with 20bVV, the microphone system is quasi-statically stable when 0.34,sSPIPP (5.30) or 002464,SPI p pPa (5.31) where SPIP or 0SPIp is the critical quasi-static pull-in parameter at 20bVV. PAGE 164 146 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 sPsStable region (I) Qausi-static pull-in (II) PSPI(s) 6e-4 Goes to + as s decreses to zero Figure 5-10. Quasi-static pull-in due to varying ND parameters s and s P The above pull-in analysis has been ca rried out for a sing le value of the ND electrostatic force parameter s For other values of s similar numerical analyses can be carried out. The result for quasi-sta tic pull-in due to varying ND parameters s and s P is shown in Figure 5-10. As the ND parameter s increases, the critical value of the ND parameter s P decreases. When the ND parameter s is less than its critical value of 1.0s and the ND parameter s P is less than its corresponding critical value of SPIP the system stays stable; otherwise, quasi-static pul l-in occurs. Therefore, a stable region (I) and an unstable region (II) are formed in the s sP plane. Physically, the ND parameters s and s P both destabilize the system. It should be pointed out that s in the plot of Figure 5-10 only has the value ranging from 6e-4 (1bVV ) to 1.0. As the s PAGE 165 147 further decreases to zero (0bV ), the required pressure goes to infinity. Physically, when 0bV, no pull-in occurs. 1 5 10 15 20 25 30 35 40 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 DC bias voltage Vb (V)Pressure p0 (Pa)Stable region (I) Qausi-static pull-in (II) Goes to + as Vb decreses to zero (p0)SPI (Vb) Figure 5-11. Quasi-static pullin due to varying acoustic pr essure and DC bias voltage. Instead of using the non-dimensional parameters s and s P in Figure 5-10, the physical DC bias voltage bV and acoustic pressure 0 p can be used. Based on Table 5-1, the regenerated plot is shown in Figure 5-11. As the DC bias voltage increases, the critical pull-in pressure value decreases, whic h means less pressure is required to cause quasi-static pull-in. When the DC bias voltage is less than its critical value of 41b SPIVV and the acoustic pressure amplitude is less than its corresponding critical value of 0SPIp, the system stays stable; otherwise, quasi-static pull-in occurs. As the DC bias voltage is over 41b SPIVV no acoustic pressure is required to cause quasistatic pull-in. Figure 5-11 also shows that quasi-static pull-in can potentially take place PAGE 166 148 below 41b SPIVV due to the addition of large acoustic pressure. Again, as the applied DC bias voltage further decreases to zero (0bV ), the required pressure goes to infinity. Physically, when 0bV, no pull-in occurs. Dynamic Pull-in due to a Mechanical Shock Load Although previous quasi-sta tic pull-in analyses provi de a good understanding of a stable operation range of a dual-backplate capacitive microphone, it is based on static analysis and does not account for dynamic effects, such as shock (mechanical or acoustical) loads or an AC voltage excitation w ith a frequency close to the first resonant frequency of the microphone. Since dynamic pull-in can dangerously take place below the quasi-static pull-in voltage due to the complex interac tion of nonlinearities, the study of dynamic effects on the pull-in limit is b ecoming important for ma ny electrostatically actuated MEMS devices. Previous studies [13, 90-92, 95] addressed some aspects of dynamic pull-in instability due to an AC volta ge excitation. Mechanical or acoustical shocks could potentially occur during the fabrication, employment and operation of MEMS devices including microphones [110-112] In addition, for the success of commercialization of MEMS devices, evaluation of reliability needs to be conducted in various test environments (pressure, mechan ical shock, temperature, humidity, etc.) [112]. Therefore, it is necessa ry to investigate the effect of shock loads on the dynamic pull-in instability of the microphone. Dynamic pull-in due to an acoustical shock load, specifically in the form of an Nwave, is presented in the next section. In this section, dynamic pull-in due to a mechanical shock load is studied for the microphone. First, the dynamic pull-in problem is formulated. The basins of attracti on for the microphone are then obtained using PAGE 167 149 numerical simulation. Finally, the basins of attraction are utilized to map the state of the system immediately after the mechan ical shock load to dynamic pull-in. Problem Formulation A mechanical shock load, such as shock loads generated by drop tests on a hard surface, is usually specifie d by the acceleration (for example, 1000g or 10000g) [112]. For modeling and analysis purposes in this section, the acceleration generated by the mechanical shock load is converted into an equivalent pressure m p t, which is defined as ,g mmat pt A (5.32) where gat is the acceleration of the mechanical shock load, m is the mass of the diaphragm and A is the area of the diaphragm. In the following analysis, only the equivalent pressure m p t is used. The corresponding acceleration can be obtained by .gmatAptm During the analysis of dynamic pull-in due to a mechanical shock load, it is assumed that no AC voltage ex citations exist between the diaphragm and backplates. Based on the general nonlinear governing equati on in Eq. (3.53) of Chapter 3, we can derive the governing equation of dynamic pull-in due to a mechanical shock load as follows 22 3 13 22 00. 2mebb mecmmeAVV M xbxkkxkxptA dxdx (5.33) It is convenient to study the non-dimensiona l form of Eq. (5.33) by introducing the non-dimensional center displacement and time PAGE 168 150 0, x y d (5.34) and 0. t (5.35) Eq. (5.33) can be rewritten as '''3 2 2 2 002, 1ddmy yyyyp d y (5.36) where 'y and ''y are the first and second derivativ es of the non-dimensional center displacement with respect to the non-dimensional time respectively. The nondimensional force parameters are 2 30 1,d ckd kk (5.37) and 2 3 102 .meb d cAV kkd (5.38) The other parameters in Eq. (5.36) have alr eady been defined in Eqs (4.3)-(4.5). Notice that the stiffness of the cavity ck is included in the non-dimensional force parameters d and d since the dynamic process is co nsidered in this section. As seen from Eq. (5.36), the mechanical s hock load serves as an external excitation for the nonlinear system. Since there is no clos ed-form solution to Eq. (5.36), we need to find some assumptions to simplify the study. First, it is assumed that a mechanical shock load has a very short duration time of t starting at 0t Eq. (5.36) can be then simplified into PAGE 169 151 0 '''3 2 2 0, for 0 2, 0, for 1ddP y yyyy y (5.39) where the non-dimensional force generated by the pressure is 2 00,mPp d (5.40) and the non-dimensional duration time is 00. t (5.41) By assuming 001 t based on the concept of convolution integral [113, 114], the original problem in Eq. (5.36) can be further simplified as a following equation subject to an initial non-dimensional velocity 0v imposed by the mechanical shock load, '''3 2 220. 1ddy yyyy y (5.42) The total momentum 0 M generated by the mechanical shock load is 0 0,t memmep M AptdtAA (5.43) where 0 t pmAptdt is defined as a characteristic parameter of the mechanical shock load. Physically it represents the area under any arbitrary me chanical shock signal and its unit is Pas From Eq. (5.39), the initi al non-dimensional velocity 0v is obtained as follows 00 0 000000v.t mem mep mememeAptdt AA M M dMdMd (5.44) An actual mechanical shock load, such as shock loads generated by drop tests on a hard surface, has an irregular shape [112] For modeling and analysis purposes, the PAGE 170 152 actual mechanical shock load is approximated by a simpler shaped signal, characterized by an amplitude of 0 p and a very short duration time of t mptmpt mpt t t t 000 p A p A p A ttt0p0p0p Figure 5-12. Three commonly used nonlinear m echanical shock load models (impulse, half sine and triangle). Shown in Figure 5-12 are three commonly used nonlin ear shock load models (impulse, half sine and triangle signals) [1 15]. For an impulse shock load with an amplitude of 0 p and a duration time of t the characteristic parameter becomes 0.pApt For a half sinusoidal shock load, the characteristic parameter becomes 02 .p p t A Finally the characteristic parameter for a mechanical triangular shock load is 0. 2p p t A After the initial non-dimensional velocity is obtained, we proceed to study Eq. (5.42). Eq. (5.42) can be further rewritten in the state-space form as follows 12 '3 1 2211 2 2 1, 2 1ddyy y yyyy y (5.45) PAGE 171 153 where 1yy and 2yy are the non-dimensional center displacement and velocity respectively. To investigate the dynamics of the above nonlinear system, we first need to find equilibrium points of the governing e quation and study their local stabilities. Equilibrium Points and Local Stabilities The equilibrium points of Eq. (5.45) can be obtained by setting 10 y and 20 y Namely, the equilibrium points * 1 1 20 y y y y satisfy the following nonlinear equation **3 1 11 2 *2 10. 1dy yy y (5.46) Eq. (5.46) has a same function form of Eq. (5.7). Similarly, to solve Eq. (5.46), some values of the non-dimensional parame ters are required. By using the physical parameters in Table 3-1 for the designed 2000Pa MEMS microphone and an applied DC voltage of 25V (this DC bias va lue is chosen here just for the illustration purpose), finally five real solutions to Eq. (5.46) are obtained as follows 1 10, y (5.47) 1 2,30.68, y (5.48) and 1 4,51.22. y (5.49) To study the local stability of each equilibri um point, we need to find the Jacobian matrix of Eq. (5.45), which is finally calculated to be PAGE 172 154 8642 1111 3 2 101 319933331 2 1ddddddyyyy J y (5.50) Evaluating the Jacobian matrix at each equi librium point defined from Eqs. (5.47) to (5.49) yields 101 0.6610.162 J (5.51) 2,301 3.590.162 J (5.52) and 4,501 19.50.162 J (5.53) The eigenvalues of the above matrices are 10.0810.081 0.0810.081Ji Eig i (5.54) 2,31.81 1.98JEig (5.55) and 4,50.0814.41 0.0814.41Ji Eig i (5.56) Based on the eigenvalues of each matrix, the fi rst equilibrium point is a locally stable sink point. The second and third equilibrium points are locally unstable saddle points, and the fourth and fifth equilibrium points are locally stable sink points. PAGE 173 155Phase Portrait and Basins of Attraction Based on the local stabilities of all equilibrium points, we can generate a phase plot of Eq. (5.45) for a DC bias voltage of 25V. The phase plane trajectories for different initial conditions are generated by using the streamslice command in MATLAB. -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Non-Dimensional Center Displacement (y)Non-Dimensional Center Velocity (y') Physical border lines Outer range Outer range Figure 5-13. Phase plane tr ajectories around the equilibrium points for a DC bias 25bVV (sink points are indicated by blue crosses, and saddle points are indicated by blue circles). The phase portrait in Figure 5-13 consists of three basins of attraction, illustrating that nonlinearities cause a sens itivity to initial conditions and co-existence of multiple solutions. The five possible equi librium solutions to Eq. (5.42) or (5.45) are: 1) a stable equilibrium point at the rest position (0 y ) of the diaphragm, that attracts the motions starting from the initial conditions within the physical boundaries; 2) two saddle nodes, situated at 0.68 y, that limit the amplitude of initial conditions by re pelling motions PAGE 174 156 either toward the desirable rest position or towards pull-in at th e physical border lines (top and bottom backplates); and 3) two non-phys ical stable equilibrium points, located outside the physical border lines (1.22 y ), that attract the motions from the relatively larger amplitude initial conditions. The th ree corresponding basins of attraction are shown in Figure 5-14. Figure 5-14. Three basins of attraction for a DC bias 25bVV. To generate the basins of attraction, numer ical simulations are carried out. First, we start by simulating Eq. (5.45) for a grid of initial conditions ( non-dimensional velocity and displacement), then we record both the in itial condition and the final states. After the overnight simulation is finished, the initial c onditions with final states close to the rest position (0 y) are plotted together to form a ba sin of attraction. Those initial conditions with final states close to the other two stable attractors (1.22 y) are plotted PAGE 175 157 to form other two basins of attraction respectivel y. Shown in Figure 5-14 is the plot of all three basins of attraction fo r a DC bias voltage of 25V. The blue shaded region corresponds to the basin of attr action for the attractor at 0 y The red shaded region corresponds to the basin of attractio n for the attractor located at 1.22y Finally, for the attractor located at 1.22y its basin of attraction is shaded by a magenta color. Figure 5-15. Basins of attr action within the physical b ackplates for a DC bias 25bVV As mentioned earlier, the motion of the diaphragm in a physical microphone device is constrained by backplates. Physically, the two special lines at 1y in Figure 5-14 represent boundaries imposed by backplates. Mathematically, they cause singularities to the governing equation in Eq. (5.33). As s een from Figure 5-14, mathematically any PAGE 176 158 initial condition starting outside the basin of attraction of the stable attractor at 0 y will be attracted to either 1.22y or 1.22 y Now the physical limits imposed by backpl ates are considered. Shown in Figure 515 is a plot of basins of attraction between physical backplates. Physically, due to the constraints imposed by backplates, any initi al condition starting out side the basin of attraction of the stable attractor at 0 y will finally stop at either of backplates and dynamic pull-in occurs. Therefore, the basin of attraction of the rest position represents the physical stable operation domain of the microphone. Dynamic Pull-in due to a Mechanical Shock Load With the phase portrait in Figure 5-13, we can relate the initial non-dimensional velocity imposed by a mechanical shock load to dynamic pull-in. If the diaphragm is at rest and the initial non-dimensional velocity imposed by a mechanical shock load is outside the physical stable ba sin of attraction in Figure 515, the diaphragm finally goes to one of backplates and dynamic pull-in occu rs. Otherwise, the diaphragm finally goes back to its rest position. For example, wh en the non-dimensional displacement is zero and the applied DC voltage is 25V, the stable range of the non-dimensional velocity from the phase portrait is 00.53v0.53. (5.57) In order to verify the above range of the initial non-dime nsional velocity is stable, simulations are carried out in MATLAB. Shown in Figure 5-16 are simulated nondimensional center displacement responses w ith two initial non-dime nsional velocities 0v0.53 and 0.54. As seen from the plot, the system goes unstable when the initial velocity is 0.54 while the system stays st able when the initial velocity is 0.53. PAGE 177 159 0 10 20 30 40 50 60 -0.5 0 0.5 1 Non-dimensional time ()Non-dimensional center displacement (y) Initial non-dimensional velocity v0 = 0.53 Initial non-dimensional velocity v0 = 0.54 Figure 5-16. Stable and unstable non-dimen sional center displacement responses with two initial non-dimensional velocities (0v0.53 and 0.54, 25bVV ). -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Non-dimensional center displacement (y)Non-dimensional center velocity (y') Initial non-dimensional velocity v0 = 0.53 Initial non-dimensional velocity v0 = 0.54 Towards dynamic pull-in Rest position Figure 5-17. Phase plots of a stable response (0v0.53, 25bVV ) and a dynamic pull-in due to a large initial velocity imposed by a mechanical shock load (0v0.54, 25bVV ). PAGE 178 160 Figure 5-17 shows the corres ponding phase plots of simu lated stable and unstable responses. As seen from the figure, dyna mic pull-in occurs when an initial nondimensional velocity is 0v0.54 and an applied DC voltage is 25bVV. Please note that due to the singularity at the dynamic pull-in point 1 y the simulation stops a little bit earlier before the dynamic pull-in. By using the general expression of 0v in Eq. (5.44), the mechanical shock load can be related to dynamic pull-in as follows 000.530.53.me p meA A Md (5.58) Substituting the values of the lumped area, lumped mass, and natural frequency of the diaphragm into the above equation, we have 3333.72103.7210 or 3.7210,pppcAAA (5.59) where pc A is defined as a critical value of p A for dynamic pull-in. Eq. (5.59) shows a relation between dynamic pull-in and a characteristic shock load parameter pA when an applied DC voltage is 25V. When the absolute value of pA of a mechanical shock load is great er than its critical value 33.7210pcAPas dynamic pull-in occurs at 25V; otherwise, the system stays stable and spirals into its rest position. The dynamic pull-in conditions in this case are 325 3.7210,bp DPIVVifA (5.60) where b D PIV is the critical dynamic pull-in voltage. PAGE 179 161 Specifically, if the impulse, half sine and triangle signal models are used as mechanical shock loads, the products of th e amplitude and durati on time of each model are given as follows 3 0Impulse signal model: 3.7210, pt (5.61) 3 0Half sine signal model: 5.8410, pt (5.62) and 3 0Triangle signal model: 7.4310. pt (5.63) The above dynamic pull-in analysis has been carried out for a single DC voltage of 25 V. For other applied DC voltages, simila r analyses can be carried out. The final results for dynamic pull-in due to a combin ation of DC bias voltage and mechanical shock load are shown in Figure 5-18. 1 5 10 15 20 25 30 35 40 43 45 50 0 1 2 3 4 5 6 7 8 9 x 10-3 DC bias voltage Vb (V)Shock load parameter Apc (Pa s) Dynamic pull-in (II) Stable region (I) Figure 5-18. Dynamic pull-in due to a combina tion of DC bias voltage and a mechanical shock load. PAGE 180 162 As the DC bias voltage increases in Figur e 5-18, the critical characteristic shock load parameter decreases, which means less shock loading is required to cause dynamic pull-in. In addition, a stable region (I) and an unstable region (II) are formed in the pcbAV plane. When the DC bias voltage is less than a cr itical value of 43bVV (this pull-in voltage value can be theoretically calculated by replacing 1k in Eq. (5.16) with 1 ckk) and a characteristic shock load paramete r is less than its corresponding critical value, the system stays stable; otherwise, dyna mic pull-in occurs. As the DC bias voltage is over 43V, no shock loading is required to cause dynamic pull-in. Figure 5-18 also verifies that dynamic pull-in can potentially take place below a critical quasi-static pull-in voltage due to a large mechanical shock load. Potential Advantage of Geometric Nonlinearity -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Non-dimensional center displacement (y)Non-dimensional center velocity (y') Figure 5-19. Phase plane trajectories for an added geometric nonlinearity case (* 3310 kk, 25bVV). Sink points are indicated by blue crosses, and saddle points are indicated by blue circles. PAGE 181 163 As we discussed in the prev ious section, an increased electrostatic nonlinearity and/or an increased mechanical shock load destabilize the system. As a competing factor, however, a larger geometric nonlineari ty helps to stabilize the system. This section focuses on a study of the potentia l advantage of geometric nonlinearity. Studies show that increasing the geom etric nonlinearity, in the form of a mechanical hardening spring, provides the abi lity to design the position of saddle points at desired locations. For example, shown in Figure 5-19 is a plot of phase plane trajectories for an added ge ometric nonlinearity with 3310 kk (or *10dd ) and an applied DC voltage of 25bVV. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Non-dimensional center displacement (y)Non-dimensional center velocity (y') Expanded stable region k3k310 k310 k3 Figure 5-20. Expanded stable operation re gion of the microphone due to the added geometric nonlinearity (3310 kk,25bVV ). In Figure 5-19, two saddle points are repositioned to 0.82 y which effectively expands the basin of attraction for th e stable center equilibrium point (0 y). Hence the dynamic stable operation range of the microphone for a given DC voltage is expanded as PAGE 182 164 shown in Figure 5-20. Also, an increased geometric nonlinearity enables the microphone to be stabilized for larger amplitude mechanical shock loads and DC bias voltages. However, the dynamic range and linearity of the microphone are affected unfavorably by the addition of geometric nonlinea rity as pointed out earlier. Dynamic Pull-in due to an Acoustic Shock Load In the previous section, the dynamic pullin due to a mechanical shock load was studied. In this section, th e transient response of the microphone is investigated for a sharp acoustic pulse Â– such as the acoustic im pulse of a car door slam. Dynamic pull-in could occur if the transient di splacement of the diaphragm is over certain threshold. In practice, the acoustical signal generated by a real car door slam is complex and strongly depends on the cavity shape (i.e., inside of the car), boundary conditions (i.e., doors, windows, seats), and the excitati on (structural characteristics of the door that is slammed) [116, 117]. To simplify and further provide physical insight into th e study, an N-wave [118] is used in this section to model th e actual acoustic pulse generated by a car door slam. The details of an N-wave will be pr ovided during the analysis. Based on the quasistatic pull-in study in the previous section, the study provided in this section will further differentiate between DC and AC pull-in responses. Problem Formulation During the analysis of dynamic pull-in due to an acoustic pulse, it is assumed that no AC voltage excitation exists between the diaphragm and backplates. This assumption is applied to alleviate the complex inte raction between the dynamic acoustical and electrical signals. Furthermore, this allows us to focus our efforts on the dynamic effect of an acoustic signal. Base d on the general nonlin ear governing equation in Eq. (3.53) of PAGE 183 165 Chapter 3, the governing equati on of dynamic pull-in due to an acoustical pulse is as follows 22 3 13 22 00, 2 mebb mecameAVV M xbxkkxkxptA dxdx (5.64) where a p t is the pressure generated by the acoustic pulse. By using the ND center displacement and time, as defined in Eqs. (5.34) and (5.35), Eq. (5.64) can be rewritten as '''3 2 2 2 002, 1dday yyyyp d y (5.65) where 'y and ''y are the first and second derivatives of the ND center displacement with respect to the ND time respectively, and the other pa rameters have already been defined in Eqs (4.3)-(4.5) and Eqs. (5.37)-(5.38). a p t0T00 p t0 p Figure 5-21. A typical N-wave with an amplitude of 0 p and a duration time of 0T For modeling and analysis purposes, the actu al acoustic pulse is approximated by a simpler N-wave, characterized by an amplitude of 0 p and a duration time of 0T as shown PAGE 184 166 in Figure 5-21 [118]. For a typical N-wave starting at 0t its mathematical expression is given by [118] 00 0 00, for 0 12, for 0. 0, for at t p tptT T tT (5.66) The Fourier transform of Eq. (5.36) by fu rther mathematical ma nipulation is given by 0 000 001 sin 11 2 cos. 11 2 22aT ppTT TT (5.67) The corresponding Fourier transf orm defined in Eq. (5.67) is plotted in Figure 5-22. 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 T0|pa()| / (p0 T0) Figure 5-22. The Fourier transform of a typical N-wave. PAGE 185 167 As we can see from the plot, the magnitude at each frequency is proportional to the product of the amplitude and duration time of the N-wave. For a smaller duration time, a larger pressure amplitude is required to ma intain the magnitude at each frequency. Additionally, the magnitude sp ectrum is continuous and the frequency content is determined by the duration time. Further anal ysis shows that the maximum magnitude of the Fourier transform occurs at max04.16. T In the time domain, Eq. (5.36) can be further rewritten into '''3 0 2 2 0 00, for 0 212, for 0, 1 0, for dddy yyyyP y (5.68) where the ND pressure amplitude is 000 22 000010meme d mecpAppA P dMdkkd (5.69) and the ND duration time is 000. T (5.70) As seen in Eq. (5.68), the acoustical exci tation term has two discontinuities at 0 and 0 Also, Eq. (5.68) has both mechanical and electrical nonlineari ties. Since it is impossible to obtain an analytical solution to Eq. (5.68), numerical si mulation is utilized in this section to study the transient re sponse of the microphone due to an N-wave excitation. Numerical Simulation Results In order to facilitate the numerical study, some values for the ND parameters in Eq. (5.68) are required. For the purpos e of illustration, a DC voltage of 20V is chosen. Also, PAGE 186 168 an N-wave possessing a 2 ms duration time and a 125 dB SPL 0 log20-6 dBSPLpe amplitude is chosen for this study based on the references [116, 117]. By using the physical parameters in Table 3-1 for the designed 2000Pa MEMS microphone, parameters are cal culated and listed in Table 5-2. Table 5-2. Parameters for the numerical study of an N-wave excitation. Parameter Value DC bias voltage (bV ) 20 (V) Damping ratio ( ) 8.1e-2 ND mechanical force parameter (d ) 0.33 ND electrostatic force parameter (d ) 0.22 ND pressure amplitude (dP ) 4.4e-3 ND duration time (0 ) 2.3 0 10 20 30 40 50 60 70 80 -3 -2 -1 0 1 2 3 x 103 Non-dimensional center displacement yNon-dimensional time Figure 5-23. Transient non-dimensional center displacement response of diaphragm due to an N-wave with an amplitude of 125 dB SPL and a duration time of 2 ms. Based on the parameters in Table 52, numerical simulations via the ODE45 command in MATLAB are carried out for E q. (5.68). Shown in Figure 5-23 is the PAGE 187 169 transient non-dimensional center displacement response. The corresponding transient non-dimensional center velocity response is shown in Figure 5-24. As seen from both figures, the transient response decays to zero and the microphone system is stable. 0 10 20 30 40 50 60 70 80 -3 -2 -1 0 1 2 3 x 10-3 Non-dimensional center velocity y'Non-dimensional time Figure 5-24. Transient non-dime nsional center velocity response of diaphragm due to an N-wave with an amplitude of 125 dB SPL and a duration time of 2 ms. Simulated Dynamic Pull-in Results From the previous simulation, the microphone system is stable when subject to a DC voltage of 20V and an N-wave posse ssing a 2 ms duration time and a 125 dB SPL amplitude. In order to simulate the dynamic pull-in, either the amplitude or the duration time of the N-wave needs to be increased in the time domain. If the DC voltage and the duration time of the N-wave are fixed, furt her simulation shows that the ND amplitude threshold for dynamic pull-in is approxima tely 1.29, which physically corresponds to approximately 174 dB SPL. PAGE 188 170 0 2 4 6 8 10 12 1 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Non-dimensional center displacement yNon-dimensional time Duration time: 0 = 2.3 Amplitude peak: Pd = 1.29 Figure 5-25. Dynamic pull-in due to an Nwave with a ND amplitude of 1.29 and a ND duration time of 2.3. 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Non-dimensional duration time 0Non-dimensional pressure peak PdUnstable region Stable region Border line Figure 5-26. Threshold of dynamic pull-in due to an N-wave. PAGE 189 171 Shown in Figure 5-25 is the simula ted transient non-dimensional center displacement response for this dynamic pull-in which occurs for the top backplate. Since the duration time and amplitude are two characteristics of an N-wave in the time domain, it is important to simulate the dynami c pull-in threshold in terms of these two parameters. By using the damping ratio, th e ND electrostatic and mechanical force parameters in Table 5-2, a plot of the dyna mic pull-in threshold is shown in Figure 5-26 in terms of the ND duration time and pre ssure peak. Shown in Figure 5-27 is the regenerated plot of the dyna mic pull-in threshold in te rms of the normalized dynamic pressure peak 0 p with respect to the correspondi ng quasi-static pull-in pressure 0 SPIp As seen from those two plots, above the threshold curve, dynamic pull-in occurs; otherwise, the microphone system stay s stable. Also, when the ND duration time is 027 a smaller pressure amplitude is re quired to cause dynamic pull-in as the duration time increases. This means the tota l input energy (proportional to the square of the magnitude of the Fourier transform functi on as shown in Eq. (5.67) or Figure 5-22) around the system resonance 0 is sufficient to cause dynamic pull-in. However, after 07.0 since there is less energy around the system resonance 0 as the duration time keeps increasing, a larger pressure amplitude is required to increase the total amount of input energy to cause dynamic pull-in. Also, as shown in Figure 5-27, when dynami c pull-in occurs, th e critical dynamic pressure peak 0 p is always greater than the corres ponding quasi-static pull-in pressure 0SPIp. The difference between those two pr essure values becomes minimal when PAGE 190 17207.0 One of reasons is probably due to the complex interaction of different nonlinearities as shown in Eq. (5.68). 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 Non-dimensional duration time 0Non-dimensional pressure p0/(p0)SPI Border lineUnstable region Stable Figure 5-27. Threshold of dynamic pull-in due to an N-wave with a normalized pressure parameter. Effect of Damping on Dynamic Pull-in Since damping plays an important role in determining the dynamic response of the system, it is necessary to st udy the effect of damping on th e dynamic pull-in limit. By keeping other parameters in Table 5-2 fixe d and simulating Eq. (5.68) with different damping ratios 0,0.08 and 0.80, a plot of different dynamic pull-in threshold curves is shown in Figure 5-28. As seen from the plot, as the damping ratio increases, a larger pressure amplitude is required to cause dynamic pull-in. As the damping increases, the amplitude of the displacement response becomes smaller under the same N-wave excitation conditions. PAGE 191 173 Therefore, a larger N-wave excitation amp litude is required to drive the displacement amplitude of the diaphragm over its pull-in threshold. Although larger damping helps stabilize the microphone system from the pull-in point of vi ew, it should be pointed out that the bandwidth of the microphone suffers fr om larger damping. Therefore, this tradeoff needs to be considered when designing a microphone with better performance. 2 3 4 5 6 7 8 9 10 0 1 2 4 6 8 10 12 14 Non-dimensional duration time 0Non-dimensional pressure p0/(p0)SPI Pull-in threshold with = 0 Pull-in threshold with = 0.08 Pull-in threshold with = 0.80 = 0 = 0.8 Figure 5-28. Effect of damping on dynamic pull-in threshold. Effect of Geometric Nonlinearity on Dynamic Pull-in Previous studies show that geometric non linearity helps stab ilize the microphone system. In this section, the effect of geom etric nonlinearity on th e dynamic pull-in limit is investigated to check if it also stabilizes the microphone system due to an N-wave excitation. By keeping other parameters in Table 5-2 fixed and simulating Eq. (5.68) PAGE 192 174 with different geometric nonlinearities 0.33,1.65 and 3.30ddd, a plot of different dynamic pull-in threshol d curves is shown in Figure 5-29. 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 Non-dimensional duration time 0Non-dimensional pressure p0/(p0)SPI Pull-in threshold with d = 0.33 Pull-in threshold with d = 1.65 Pull-in threshold with d = 3.30 d = 0.33d = 3.3 Figure 5-29. Effect of geometric nonli nearity on dynamic pull-in threshold. As seen from the plot in Figure 5-29, as the geometric nonlinearity increases, a larger pressure amplitude is required to cause dynamic pull-in. As the geometric nonlinearity increases, the large deflection amplitude of the diaphragm becomes smaller under the same N-wave excitation conditions. Therefore, a larger N-wave excitation amplitude is required to drive the displacem ent amplitude of the diaphragm over its pullin threshold. From the above analysis, larger geometric nonlinearity is preferre d to stabilize the microphone system. However, as pointed out earlier, the dynamic ra nge and linearity of the microphone are affected unfavorably by the addition of geometric nonlinearity. PAGE 193 175 Therefore, this trade-off needs to be cons idered when designing a microphone with better performance. Summary In this chapter, pull-in instabilities of a dual-backplat e capacitive MEMS microphone have been studied. First, quasi-static pull-in instability due to an applied DC voltage has been studied by bo th analytical and graphica l methods. The quasi-static operation range of the microphone is obtained for a given DC voltage by a 1-D basin of attraction. The critical quasi-s tatic pull-in voltage is found to be 41V analytically and the phenomenon of quasi-static pull-in is illustra ted by a subcritical pitchfork bifurcation. Also, quasi-static pull-in due to a combinati on of an applied DC voltage and a static pressure has been studied numerically. A qua si-static pull-in threshold curve has been obtained in terms of the applied DC vo ltage and static pr essure amplitude. In addition, investigation of dynamic pull-in due to a mechanical shock load is presented. By using a phase portrait and the ba sins of attraction, a mechanical shock load is related to dynamic pull-in. A relation betw een the critical DC voltage and critical characteristic parameter of a shock load has been established when dynamic pull-in occurs. Further study shows that dynamic pu ll-in can potentially take place below the critical quasi-static pull-in voltage du e to a large mechanical shock load. The investigation of dynamic pull-in due to an acoustical pulse, in the form of an N-wave, is provided in this chapter. By numerical simulations, a dynamic pull-in threshold curve has been obtai ned in terms of the duration time and amplitude of the Nwave for a given DC bias voltage. The eff ects of the damping and geometric nonlinearity on the dynamic pull-in limit are also studied. PAGE 194 176 Studies in this chapter also show that se veral nonlinearities (geometric, electrostatic and mechanical/acoustical shock) compete with each other. An in creased electrostatic nonlinearity and/or an increased mechanical/a coustical shock load destabilize the system while a larger geometric nonlinearity helps to stabilize the microphone and expands its stable operational range. PAGE 195 177 CHAPTER 6 SYSTEM IDENTIFICATION BY PRELIMINARY EXPERIMENTS This chapter focuses on the nonlinear iden tification of system parameters through a series of preliminary experiments. The work presented in this chapter is organized as follows. First, the experimental setup a nd procedures are desc ribed. Based on two approximate solutions of the electrical excita tions discussed previously in Chapter 4, and a nonlinear least-squares curve-fitting techniqu e, system parameters are extracted from the center displacement data. Conservative uncertainty ranges of experimentally extracted system parameters are obtained by the uncertainty analysis. Experimentally extracted system parameters are then compared with their theoretical values and discussions of analysis results are presented. Experiment Setup and Procedures The characterization experiment of the dual-backplate capacitive MEMS microphone was conducted using a laser Doppler vibrometer (Polytec MSV 300) in the Interdisciplinary Microsystems Laboratory at the University of Florida. The goal of the experiments is to determine the system parameters of the nonlinear dynamic model for a fabricated device. Experiment Setup Generally, a laser Doppler vibrometer operates based on th e detection of the Doppler shift of cohere nt laser light, which is reflected from a small area of the test object. The component of velocity which lies along the axis of the laser beam is determined by the Doppler frequency shift in the reflected laser beam [119]. The block PAGE 196 178 diagram of the experiment setup is shown in Figure 6-1. The microphone under test is positioned on the stage of the microscope (O lympus BX60), with a 100x objective lens [120]. The output of an external function ge nerator (Agilent 33220A) is used to provide electrical signals to excite the microphone and is also connect ed to the Â“RefÂ” channel of the laser vibrometer. The Â“SyncÂ” signal coming out of the f unction generator is connected to the laser vibrometer to trigge r the data acquisition process. The laser vibrometer generates the input laser beam a nd receives the resulti ng interference optical signal from the microscope. The resulting optic al signal is converted into an electrical signal by a photodetector inside the vibrometer and subsequen tly decoded to generate the velocity output [119]. A data acquisition PC then acquires data from both reference and velocity channels in the laser vibrometer. Figure 6-1. Block diagram of the experiment setup. To be able to measure the center veloci ty response of the diaphragm during the experiments, a laser beam is adjusted an d shone through the cent er hole of the top backplate as shown in the side view of Figure 6-1. Figure 6-2 shows an optical photograph (zoom-in top view) of a laser beam spot, which was positioned inside the center hole of the top bac kplate. From the experimental pi cture, the diameter of the laser PAGE 197 179 spot is estimated to be approximately 4 m. A sampling rate of 2.56 MHz was used to record the velocity for 3.2 ms; 100 ensemble averages were used to minimize the noise in the measured velocity data. Figure 6-2. Laser beam spot (red dot) impinge s the diaphragm through the center hole of the top backplate. Experiment Procedures To be able to extract the system paramete rs from the experimental data, two types of experiments were conducted. In the fi rst set of experiments, the microphone was excited with an electrical square wave to study its transient respons e using the previously presented MTS approximate solution. For the second series of experiments, the microphone was excited with an electrical sinusoidal wave to study its steady state response using the previously presented HB approximate so lution. In the following sections, the general procedur es to conduct those two types of experiments are described respectively. Procedures for the electrical square wave excitation For the electrical square wave excitati on, some major steps to conduct one test using the top backplate for excitation are as follows: PAGE 198 180 1. Adjust the laser beam from the vibrometer in the focus and position the laser spot inside the center hole of the top backplate so that the center velocity response of the diaphragm can be measured 2. Electrically ground the diaphragm and bottom backplate 3. Initially, apply 1 kHz uni-polar square wave with a voltage amplitude of 05 V V and 50% duty cycle directly to th e top backplate of the microphone 4. Measure the dynamic transient response of th e center velocity of the diaphragm via the laser vibrometer system shown in Figur e 6-1 and save the time-history data for the further analysis 5. Change the applied uni-polar s quare wave voltage amplitude 0V to 15 and 30 V and repeat step 4 for the two a pplied voltages respectively. Similarly, for the tests using the botto m backplate for excitation, the middle diaphragm and top backplate are electrically grounded. A 1 kHz uni-polar square wave with a voltage amplitude of 05 V V and 50% duty cycle is directly applied to the bottom backplate, and the above steps 3, 4 and 5 are repeated. Procedures for the electrical sinusoidal excitation For the electrical sinusoidal excitation, some major step s to conduct one test using the top backplate fo r excitation are listed in the following, 1. Adjust the laser beam from the vibrometer in the focus and position the laser spot inside the center hole of th e top backplate so that the cen ter velocity response of the diaphragm can be measured 2. Electrically ground the diaphragm and bottom backplate 3. Initially, apply the designe d sinusoidal signal with a voltage amplitude of 05 V V and a frequency around one half of the linea r resonant frequency of the diaphragm directly to the top b ackplate of the microphone 4. Measure the dynamic steady state response of the center velocity of the diaphragm via the laser vibrometer system and save the time-history data for the further analysis 5. Keep the voltage amplitude fixed (05 V V ) and change the excitation frequency close to first frequency point and take the measurement again PAGE 199 181 6. Change the applied sinusoidal wave voltage amplitude 0V to 20 and 25 V and repeat steps 4 and 5 for the tw o applied voltages respectively. Similarly, for the tests using the botto m backplate for excitation, the middle diaphragm and top backplate are electrica lly grounded. A sinus oidal signal with a voltage amplitude of 05 V V and a frequency of 92.7 kHz is directly applied to the bottom backplate, and the above steps 3, 4, 5 and 6 are repeated. Figure 6-3. Simplified circuit to ge nerate the high voltage signal. To ensure the repeatability, for each ty pe of experiment, one test using the top/bottom backplate for excitation is repeated 5 times in different time slots. As shown in Figure 6-3, the high-voltage (20 V ) signal required by the ex periments is generated by using an Agilent function generator (33220A) and an operational amplifier (AD823AN, Analog Devices, Inc.). Results of Electrical Square Wave Excitation In this section, the experimental results fo r the electrical square wave excitation are presented. System parameters are extracted from the transient experimental data via a MTS approximate solution, which is discussed in Eqs. (4.33)-(4.37) of Section 4.3 (Approximate MTS Solution for the Electrical Square Wave Excitation). For the electrical square wave excitation, the dynami c response of the center velocity of the diaphragm is recorded for 100 ensembles. The trapezoidal rule is a pplied to numerically integrate the measured average center veloc ity to yield the center displacement [108]. PAGE 200 182 Because the time step is small (0.39 s), the numerical integration error is small and can be neglected (the details ar e provided in the uncertainty analysis in Appendix E). Results of Bottom Backplate Excitation Shown in Figure 6-4 is the measured ensemb le-averaged center velocity (up-stroke) response of the diaphragm excited by a uni-pol ar 1 kHz square wave with an amplitude of 5V. As seen from the plot, the cent er velocity decays quickly to zero. The corresponding integrated center displacement re sponse is shown in Figure 6-5. The drift in the steady state of the integr ated center displacement res ponse is due to the experiment setup because the laser vibrometer does not m easure the static velocity. It should be pointed out that the system parameters are ex tracted only from the transient response data of the integrated center displacement. Ther efore, the accuracy of the extracted system parameters is not affected by the undesired drift. 0 20 40 60 80 100 120 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time ( s)Center Velocity (mm/s) Figure 6-4. Measured averaged center velocity response for an applie d square wave with an amplitude of 5V. PAGE 201 183 0 20 40 60 80 100 120 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10-3 Time ( s) C enter Displacement ( m ) Figure 6-5. Integrated center di splacement response for an applied square wave with an amplitude of 5V. -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10-3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Center Displacement ( m)Center Velocity (mm/s) Figure 6-6. Constructed phase plot for an applied square wave with an amplitude of 5V. PAGE 202 184 A phase plot is constructed in Figure 6-6 based on the measured velocity and integrated displacement responses. The system spirals into a fixed point. By using Eq. (4.34) in Chapter 4, a nonlinear least-squares curve-fitting procedure is carried out in MATLAB to obtain system parameters 0 , and 0d from the transient response data of the integrated center displacement. Figure 6-7 shows a comparison plot of the integrated and curve-fit center displacement re sults. The system parameters identified from the curve fit to the analytical approxi mate solution are summar ized in Table 6-1. 12 14 16 18 20 22 24 26 -3.5 -3 -2.5 -2 -1.5 -1 x 10-3 Time ( s)Center Displacement ( m) Figure 6-7. Comparison of integrated and curv e-fit center displacemen ts for an applied square wave with an amplitude of 5V. Table 6-1. Results of system paramete rs of the bottom b ackplate excitation. System parameter Curve-fitting result 0d (m) 2.052e-6 02 (Hz) 188.0e3 6.970e-2 (N/m3/ kg) 1.224e23 (m2/ kg) Â†112.6 Â†value for 5V only. PAGE 203 185Results of Top Backplate Excitation Shown in Figure 6-8 is a measured ense mble-averaged center velocity response (up-stroke) of the diaphragm excited by a uni-p olar 1 kHz square wave with an amplitude of 18 V. As seen from the plot, the cen ter velocity decays quickly to zero. The corresponding integrated center displacement response is shown in Figure 6-9. Similarly, based on the measured velocity and integrated displacement responses, the constructed phase plot after 12 s is generated in Figure 6-10, showing the system spirals into a fixed point. 0 20 40 60 80 100 120 -25 -20 -15 -10 -5 0 5 10 15 20 25 Time ( s)Center Velocity (mm/s) Figure 6-8. Measured averaged center velocity response for an applie d square wave with an amplitude of 18V. Similarly, by using Eq. (4.34) in Chapter 4, a set of system parameters can be extracted via a nonlinear least-squares curve-fi tting technique from the transient response data of the integrated center displacement. Shown in Figure 6-11 is the comparison plot of the integrated and curve-fit center displ acement results. The nonlinear least-squares PAGE 204 186 curve-fit results are in good agreement with the integrat ed results. The system parameters identified from the curve fit to the analytical approximate solution are summarized in Table 6-2. 0 20 40 60 80 100 120 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time ( s)Center Displacement ( m) Figure 6-9. Integrated center di splacement response for an applied square wave with an amplitude of 18V. 0.05 0.06 0.07 0.08 0.09 -25 -20 -15 -10 -5 0 5 10 15 Center Displacement ( m)Center Velocity (mm/s) Figure 6-10. Constructe d phase plot after 12 s for an applied square wave with an amplitude of 18V. PAGE 205 187 15 20 25 30 35 40 45 50 0.045 0.05 0.055 0.06 0.065 0.07 0.075 Time ( s)Center Displacement ( m) Figure 6-11. Comparison of integrated and curve-fit center displacements for an applied square wave with an amplitude of 18V. Table 6-2. Results of system parameters of the top backplate excitation. System parameter Curve-fitting result 0d (m) 1.950e-6 02 (Hz) 196.0e3 7.791e-2 (N/m3/ kg) 1.088e23 (m2/ kg) Â‡231.9 Â‡value for 18V only. Results of Electrical Sinusoidal Excitation In this section, the experimental results for the electrical si nusoidal excitation are presented. System parameters are extracted fr om the steady-state experimental data via a HB approximate solution, which is discu ssed in Eqs. (4.61)-(4.64) of Section 4.5 (Approximate HB Solution for the Electrical Sinusoidal Excitation). For the electrical sinusoidal excitation, a designed signal define d by Eqs. (4.43) and (4.44) in Chapter 4 is PAGE 206 188 applied directly to either bottom or t op backplates of the microphone. The dynamic response of the center velocity of the diaphragm is recorded for 100 ensembles. As mentioned previously, two tests need to be conducted to extract system parameters based on a HB approximate method. For the first te st, the sinusoidal exci tation frequency is chosen to be around one half of the linear resonant frequenc y of the diaphragm, and the excitation frequency of the second test is chosen to be close to the frequency of the first test so that each system parameter is not cha nged. Also, the trapezoidal rule is applied to numerically integrate the measured averag e center velocity to yield the center displacement [108]. Results of Bottom Backplate Excitation For the first test, the amplitude of si nusoidal excitation si gnal is 9V and its frequency is chosen to be 114.4 kHz. Shown in Figure 6-12 is the measured time-history of the ensemble-averaged center velocity in the steady state. 100 110 120 130 140 150 160 -20 -15 -10 -5 0 5 10 15 20 Time ( s)Center velocity (mm/s) Figure 6-12. Measured averaged steady-state center velocity response (asterisk) for a sinusoidal excitation with an amplitude of 9V and a frequency of 114.4 kHz. PAGE 207 189 100 110 120 130 140 150 160 -0.01 -0.005 0 0.005 0.01 0.015 Time ( s)Center displacement (m) Figure 6-13. Comparison of the integrated (red asterisk) and curve-fitting (blue solid line) steady-state center displacement results for a sinusoidal excitation with an amplitude of 9V and a frequency of 114.4 kHz. Table 6-3. Amplitudes and phase of the in tegrated averaged steady-state center displacement of the bottom backplate excitation. Parameter First test Second test Difference /2 (kHz) 114.4 114.8 0.3% 0A ( m) 1.457e-3 1.440e-3 1.1% 2A ( m) 1.321e-2 1.323e-2 0.2% (rad) -1.453 -1.539 5.9% By applying the trapezoidal rule, the corresponding time response of the integrated averaged center displacement in the steady state is shown in Figure 6-13. By using Eq. (4.57) in Chapter 4, a nonlinea r least-squares curve-fitting technique is implemented in MATLAB to extract the amplitudes and phase (0A 2A and ) from the steady-state center displacement response. The curve-fit center displacement response is also plotted in Figure 6-13, and the curve-fit parameters ar e summarized in Table 6-3. Similarly, for the second test, the frequency of the si nusoidal excitation is 114.8 kHz while its PAGE 208 190 amplitude is still 9 V. The final curve-fit resu lts of the second test are listed in Table 6-3. With the parameters in Table 6-3, system parameters 0 and can be extracted by using Eq. (4.65) in Chapter 4 for the botto m backplate excitation. The final set of system parameters are listed in the following table. Table 6-4. Results of system paramete rs of the bottom b ackplate excitation. System parameter Curve-fitting result 02 (kHz) 230.3 5.482e-2 (N/m3/ kg) 1.470e23 (m2/ kg) Â†67.48 Â†value for 9V only. Results of Top Backplate Excitation During the experiments using the top backpl ate for excitation, the amplitude of the sinusoidal excitation signal is 8.3V and the frequencies of the two tests are 114.0 kHz and 114.4 kHz respectively. After obtaining the m easured ensemble-averaged steady-state center velocity responses, a nonlin ear least-squares curve-fitti ng is carried out to extract the amplitudes and phase (0A 2A and ) for each test respectively. Finally, the curvefit results are summarized in Table 6-5. Si milarly, with the parameters in Table 6-5, system parameters 0 and can be extracted by using Eq. (4.65) in Chapter 4. The final set of system parameters are listed in Table 6-6. Table 6-5. Amplitudes and phase of the in tegrated averaged steady-state center displacement of the t op backplate excitation. Parameter First test Second test Difference /2 (kHz) 114.0 114.4 0.4% 0A ( m) 6.08e-4 6.08e-4 0.2% 2A ( m) 4.198e-3 4.037e-3 0.4% (rad) -2.223 -2.263 1.8% PAGE 209 191 Table 6-6. Results of system parameters of the top backplate excitation. System parameter Curve-fitting result 02 (kHz) 218.6 5.533e-2 (N/m3/ kg) 1.470e23 (m2/ kg) Â‡30.08 Â‡value for 8.3V only. Discussion of Analysis Results In the previous section, no uncertainties are assumed with the measured velocity and the physical dimensions/material prope rties of the microphone. However, there always exist some uncertain ty sources in the experime nts and physical microphone devices, for example, the variations of phys ical dimensions and material properties, which can be caused by the fabrication process. Therefore, uncertainty analysis of the experimentally extracted identified system parameters needs to be conducted. The detailed uncertainty analysis is included in Appendix E and the final uncertainty analysis results are summarized in Table 6-7. Table 6-7. Theoretical mean values and uncer tainties of system parameters for a given 95% confidence level. System parameter Mean Uncertainty Percentage 02 (Hz) 193.5e3 10.20e3 5.3% 7.272e-2 1.337e-2 18% (N/m3/ kg) 1.219e23 0.1409e23 12% (m2/ kg) 329.2 2.901 0.9% From Table 3-1 in Chapter 3, the nomina l values of system parameters of the microphone are calculated and listed in Table 6-8. As we can see from Table 6-7 and Table 6-8, the nominal values of system parameters fall within their theoretical uncertainty ranges for a given 95% confidence level respectively. Also from Table 6-1, Table 6-2 and Table 6-7, the experimentally extracted linear natural frequency, damping PAGE 210 192 ratio and nonlinear stiffness parameter ba sed on the MTS approximate solution fall within their conservative theoretical ranges for a 95% confidence level respectively. Table 6-8. Nominal values of syst em parameters of the microphone. System parameter Nominal value 02 (Hz) 185.5e3 8.091e-2 (N/m3/ kg) 1.127e23 (m2/ kg) 332.2 From Table 6-4, Table 6-6 and Table 6-7, the experimentally extracted system parameters, obtained from the HB approximate solution, do not fall within their conservative theoretical ranges. Possible reas ons are the relatively large bias error caused by the HB approximate solution/algorithm (see Chapter 4 for details, primarily due to the fact that higher order harmonics were not included in the HB solution) and the conservative nature of the uncertainty analysis (see Appendix E for details). The differences between system parame ters identified through square and sinusoidal wave excitation e xperiments respectively (Table 6-1, Table 6-2, Table 6-4, Table 6-6) are mainly due to the different dual-backplate capacitive microphones tested during each type of experiment. For either el ectrical square or sinus oidal wave excitation experiments, discrepancies between the experi mentally identified system parameters of both bottom and top backplate excitations resp ectively are mainly due to the different bottom and top capacitors, such as une qual air gaps and electrode areas. It is shown that is a constant for a given set of parameters from the lumped element modeling based on a simple equalarea parallel-plate assumption. However, analysis results shown in Tabl e 6-1, Table 6-2, Tabl e 6-4, Table 6-6 indicate that it is a function of the applied voltage, which mean s that the equivalent area for calculating electrostatic forces depends on the applied voltage (assuming the lumped mass is fixed). PAGE 211 193 The changing experimental behavior of could be due to several reasons: 1) in a physical device, the top backplate area is larger than the diaphragm area, and the diaphragm area is larger than the bottom backplate area; therefore the equal-area assumption is not accurate in practice [79, 93, 121, 122]; 2) the backplates are perforated with holes, which decreases the overlapping area and generates extra fringing field effect when calculating the electrostatic forces; a nd 3) the overlapping area and fringing field effect could change due to the different be nding shapes of the plates when different voltages are applied between the ba ckplate and diaphragm. Since plays an important role in determining the elec trostatic force and the sensit ivity of the microphone for a given bias voltage, further investigati on needs to be conducted in the future. Summary In this chapter, the nonlin ear identification of system parameters of the microphone through a series of preliminary experiments is presented. The measured center velocity and integrated center displacement results for both electrical square and sinusoidal wave excitation experiments are provided. Base d on the approximate MTS and HB solutions presented in Chapter 4, and a nonlinear le ast-squares curve-fitt ing technique, system parameters are extracted from two types of experimental data respectively. The uncertainty analysis in Appendi x E shows that the experimental ly extracted linear natural frequency, damping ratio and nonlinear stiffness parameter based on the MTS approximate solution fall within their c onservative theoretical ranges for a 95% confidence level. PAGE 212 194 CHAPTER 7 CONCLUSIONS AND FUTURE WORK This chapter summarizes the important re sults presented in this dissertation. Concluding remarks as well as recommenda tions for future work are discussed. Conclusions An investigation of the electromechanic al nonlinear dynamics of a dual-backplate capacitive MEMS microphone has been presen ted through the theoretical analysis, numerical simulation and preliminary ex perimental characterization. A large displacement solution via an energy method has been utilized to provide linear and cubic lumped stiffnesses of the circular dia phragm of the microphone. A nonlinear dynamic model of the microphone has been develope d based on lumped element modeling. Theoretical lumped stiffnesses of the diaphr agm are verified by nonlinear finite element analyses (FEA) and the errors for the linear and cubic stiffnesses are approximately 1.3% and 5.0% respectively. The coupled nonlinear electromechanical simulations show that the difference between the simulated and modele d electrostatic forces is 17% at 25V for the top capacitor and 17% at 33.5V for the bottom capacitor. Quasi-static pull-in due to an applied DC voltage has been investigated by both analytical and graphical approaches. The cr itical quasi-static pull-in voltage is found to be approximately 41V. The quasi-static ope ration range of the microphone is obtained for a given DC voltage by a 1-D basin of at traction. The phenomenon of qausi-static pull-in is illustrated by a subc ritical pitchfork bifurcation. A potential advantage of the mechanical geometric nonlinearity is explore d, and the quasi-static stable operation range PAGE 213 195 of the microphone can be increased by enlargin g the gap and/or decreasing the thickness of the diaphragm. Further studies show that the stable operational range of the microphone could benefit from the addition of geometric nonlinearity into the system. However, the linearity and dynamic range of the microphone are affected unfavorably by the addition of geometric nonlinearity. Investigation of dynamic pull-in due to a mechanical shock load has also been presented. A non-dimensional governing equation has been formulated for the dynamic pull-in study. By using a phase portrait and ba sin of attraction, a mechanical shock load is related to dynamic pull-in. A relation betw een the critical DC voltage and critical characteristic parameter of a mechanical s hock load has been established when dynamic pull-in occurs. Further study shows that dynami c pull-in can potentially take place below the critical quasi-static pull-in voltage due to a large mechanical shock load. The investigation of dynamic pull-in due to an acous tical pulse, in the fo rm of an N-wave, is provided. By using numerical simulations, a dynamic pull-in thresh old curve has been obtained in terms of the duration time and amp litude of the N-wave for a given DC bias voltage. The effects of the damping and geometric nonlinearity on the dynamic pull-in limit are also studied. Studies of dynamic pull-in s how that several nonlinearities (geometric, electrostatic and mechanical/acoustical shock) compete with each other. An in creased electrostatic nonlinearity and/or an increased mechanical/a coustical shock load destabilize the system while an increased geometric nonlinearity he lps to stabilize the microphone and expands its stable operational range. PAGE 214 196 Nonlinear approximate analytical soluti ons to the governing equations for the microphone under both electrical square and sinuso idal excitations are obtained to extract system parameters from experimental data. Specifically, for the first time, an approximate analytical solution to a general damped second-order system with quadratic and cubic nonlinearities and non-zero step load ing is obtained via a multiple time scales (MTS) method. An approximate analytic al solution to a general inhomogeneous nonlinear damped second order system under the harmonic excitation is obtained via a harmonic balance (HB) method. Different forms of the derived MTS approximate solution are provided and can be applied to many other applicat ions, such as the study of a large-angle motion of a simple pendulum [72] Discussions and validity regions of the derived MTS and HB approximate solutions are provided. Als o, two steady-state approximate solutions for the microphone un der the sinusoidal acoustical pressure excitation are provided vi a HB and MTS methods. A series of preliminary experiments are conducted and system parameters are extracted from two types of experimental data respectively. The pr eliminary uncertainty analysis, which includes only the uncertain ties caused by fabrication, shows that the experimentally extracted linear natural freque ncy, damping ratio and nonlinear stiffness parameter fall within their conservative theoretical ranges for a 95% confidence level. Recommendations for Future Work In the first-generation dual-backplat e capacitive MEMS microphone, the top backplate has the largest area while the botto m backplate has the smallest area. The thickness of the top backplate is same with the diaphragm and smaller than the thickness of the bottom backplate. Further analysis shows that the top backplate has the highest compliance, followed by the middle diaphragm and bottom backplate, respectively. PAGE 215 197 Since the stable operational range of the microphone will be reduced by the compliant backplates; therefore, in the future design, two backplates n eed to be further stiffened. Theoretically, there are many ways to achieve this goal, such as increasing the thickness of the backplate or using so me material with a higher Y oungÂ’s modulus than that of polysilicon. Practically, however, due to the physical constr aints imposed by the SUMMiT V process at Sandia Na tional Laboratories, the thic kness, residual stress and the material property of each layer are fixe d [66, 98]. One possible way to stiffen the backplates of the microphone is through the depo sition of extra layers with desired tensile stresses on each backplat e during the post-processing. Some additional future contributions could be: Experimental results presented are prelim inary, and relatively large errors could possibly exist for the extracted system pa rameters. To obtain more accurate and reliable results, characterization experiment s need to be repeated for the applied electrical excitations with a wide range of voltage amplitudes on more released microphone devices. Also, the presented uncerta inty analysis in th is dissertation is conservative since it only considers th e uncertainties caused by the fabrication process. More accurate uncertainty analyses might be conducted. The stiffnesses of the backplates might be experimentally characterized via a laser vibrometer. The top backplate can be acces sed directly by a laser beam. To be able to characterize the bottom backplate, the microphone chip might need to be flipped such that the laser beam can impi nge the bottom backplate through the back cavity. Experiments for the microphone under the sinusoidal pressure wave excitation might be carried out in a plane wave t ube (PWT) with an expanded bandwidth over the linear natural frequency of the micr ophone. The obtained experiment results can be compared with the theoretical analys is results presented in this dissertation. Pull-in experiments for the microphone unde r the acoustic pressure and electrical voltage excitations might be carried out. The obtained experimental results can be compared with the theoretical analysis results presented in this dissertation. Since dynamic pull-in can dangerously ta ke place below the quasi-static pull-in voltage due to the complex interacti on of nonlinearities, the study of dynamic effects on the pull-in limit becomes very important for many electrostatically actuated MEMS devices. The dynamic pull-in analyses for the AC voltage PAGE 216 198 excitation with a frequency close to the first resonant frequency of the microphone could be carried out. This study could incl ude an approximate analytical solution for a parametrically excited system via a multiple time scales method, and bifurcation and instability analyses. PAGE 217 199 APPENDIX A LARGE DISPLACEMENT ENERGY SOLUTI ON OF A CIRCULAR DIAPHRAGM The detailed steps to derive the large displacement solution for a clamped circular diaphragm based on the energy method [97, 99] are presented in this appendix. As mentioned in Chapter 3, the pola r coordinate is introduced to facilitate the analysis for a circular diaphragm. When large displaceme nts occur, the radial displacement in the neutral plane of the diaphragm cannot be negl ected. The displacement of a point in the neutral plane of a circular diaphragm is decomposed into two components: ur in the radial direction and wr perpendicular to the neutral plane as shown in Figure 3-4. The shape of the transverse deflection surf ace of a circular diaphragm is assumed to be similar with the small displacement case, which is defined in Eq. (3.7) as follows 2 2 0 21, r wrw a (A.1) where 0w is the unknown center displacement and n eeds to be determined. The assumed transverse deflection satisfie s the clamped boundary conditions 0.radwr wa dr (A.2) To satisfy the clamped boundary c onditions, the radial displacement ur must vanish at the edge and the origin of the circular diaphragm; therefore, the following polynomial is used to approximate the comp lex expression of the radial displacement [99] PAGE 218 200 12, urrarCCr (A.3) where1C and 2C are the two unknown constants. Based on the von Krmn plate theory, the strain-dis placement relations [99, 101] in the radial and tangential directions are given by 21 2rdudw drdr (A.4) and u r (A.5) By applying HookeÂ’s law for the 2D case [ 123], the following relations can be obtained 11 ,rrrNN EEh (A.6) and 11 ,rrNN EEh (A.7) wherer and are the stresses in the radial and tangential directions respectively, and rN and N are the corresponding forces per unit length in the radial and tangential directions. Solving rN and N from Eqs. (A.6) and (A.7) simultaneously yields 2, 1rrEh N (A.8) and 2. 1rEh N (A.9) Therefore the strain energy due to the transverse bending wr is calculated as 2 2 2 22 2 0 2222 001232 23a bw DwwwwD Vrdrd rrrrrra (A.10) PAGE 219 201 The strain energy due to the in-plane stretching ur is 22 0000. 2222aa rrrr sN N Vhrdrdrdrd (A.11) The total strain energy is the sum of two st rain energy components due to the transverse bending and in-plane stretching re spectively, and it is given by .bsVVV (A.12) Also the external work done by the uniform transverse pressure is calculated as 2 2 0 00. 3a p wrrdrdpaw (A.13) The total potential energy is obtained as .bsVVV (A.14) According to the principle of the mi nimum potential energy [97] when an equilibrium is reache d, the three unknowns: 0w (the center displacement), 1C and 2C can be determined by solving the following equations simultaneously 1 2 00 0. 0 C C d dw (A.15) Specifically, for a polysilicon diaphragm, the PoissonÂ’s ratio is 0.22 Substituting Eqs. (A.1), (A.3), (A.4), (A.5), (A.8), and (A.9) into Eq. (A.11), and evaluating the result with 0.22 yield 4252 12010.85330.99040.293sVEhaCEhaCEhawC PAGE 220 202 4 6222 0 202 20.38520.14341.006. w EhaCEhwaCEh a (A.16) By substituting Eqs. (A.10), (A.16) and (A.13) into Eq. (A.14), finally, we can solve Eqs. (A.15) simultaneously as follows 2 0 1 31.2652, w C a (A.17) 2 0 2 31.8129, w C a (A.18) and 4 0 2 0 21 64 10.4708 pa w w D h (A.19) Substituting Eqs (A.17) and (A.18) back into Eqs. (A.16) and (A.12), we can obtain the results for the strain energy in terms of 0w 4 0 222.511.sw VD ah (A.20) and 22 00 2232 10.2354. 3bsww VVVD ah (A.21) The final approximate expression for the larg e transverse displacement solution can be obtained by substituting the solution of 0w from Eq. (A.19) into Eq. (A.1). Rewritting Eq. (A.19) becomes 2 4 0 0 210.4708. 64 w p a w hD (A.22) PAGE 221 203 As we can see from Eq. (A.22), a cubi c equation for the center displacement 0w is obtained. The small factor 22 00.4708 wh represents a geometri c nonlinearity (nonlinear spring hardening effect) due to the in-plane stretching when large displacements occur. As seen from Eq. (A.1), the assumed mode shape based on this energy approach is not affected by the applied pressure. Show n in Figure 3-5 is a plot of different normalized mode shapes for several pressure va lues. In the plot, three normalized mode shapes are generated based on the exact soluti on given in the referen ce [100]. As we can see from Figure 3-5, for larger pressure values (for example, 100000 Pa), the assumed mode shape used in the energy approach is not accurate. However, for our 2000Pa microphone design, the assumed mode shape in the energy approach is in good agreement with the exact mode shape as shown in Figure 3-5. PAGE 222 204 APPENDIX B APPROXIMATE SOLUTION FOR A GENERAL NONLINEAR SECOND ORDER SYSTEM This appendix provides the mathematical background and detailed derivation of the formula used in the electrical square wave excitation experiments to extract system parameters for a dual-backplate capacitive MEMS microphone. Introduction From Chapter 4, for the characterizati on experiments with the Polytec laser vibrometer, no acoustical pressu re exists on the diaphragm and only a uni-polar square wave Vt is applied directly to bottom (or top) backplate with diaphragm and the other backplate electrically grounded. During the up -stroke of the square wave, and if the voltage is applied to the top backplate, the approximate governing equation for the forced vibration of the diaphragm is as follows, 23 12345, xxxxx (B.1) where 1,meb M (B.2) 2 100 2 3 0,cme memekkAV MMd (B.3) 2 00 3 4 03 2me meAV M d (B.4) 2 300 4 5 02 ,me memekAV MMd (B.5) PAGE 223 205 and 2 00 5 2 0. 2me meAV M d (B.6) Physically, Eq. (B.1) represents a more general damped second order system with both quadratic and cubic nonlinearities (3 and 4 ) and a non-zero external step loading (5 ). During the down-stroke of the square wave, the governing equation for the free vibration of diaphragm is 3 130.mecMxbxkkxkx (B.7) The above equation can be rewritten as ''3 1240, xxxx (B.8) where 1,meb M (B.9) 0' 1 22 0,c V mekk M (B.10) and 0' 3 44 0.V mek M (B.11) Eq. (B.7) or (B.8) is a special case of the general equation Eq. (B.1). Since there is no closed-form solution to Eq. (B.1), in the next section, its approximate solution is obtained by a multiple time scales method. The approximate solution to Eq. (B.7) or (B.8) can be then obtained by altering the general approximate so lution to Eq. (B.1). Approximate Solution by the Mu ltiple Time Scales Method To proceed with the approximate solution, we need to non-dimensionalize the governing equation in Eq. (B.1) which finally leads to PAGE 224 206 '''23 12345, yyyyy (B.12) where 0, x y d (B.13) ', dy y d (B.14) 2 '' 2, dy y d (B.15) 0, t (B.16) 01,cmekkM (B.17) 1 0,meb M (B.18) 2 00 2 23 001,me meAV M d (B.19) 2 00 3 32 003 2me meAV Md (B.20) 22 3000 4 232 0002 ,me memekdAV MMd (B.21) and 2 00 5 32 00. 2me meAV Md (B.22) The approximate solution of E q. (B.12) is assumed as a seco nd order expansion in terms of a small positive parameter (a book keeping parameter), 2 001210122012,,,,,,, yyyy (B.23) where the multiple independent time scales are defined as PAGE 225 207 2 012,, and (B.24) Therefore, the time derivatives with respect to t become the following expansion terms of the partial derivatives with respect to the corresponding time scales 2 012, d DDD d (B.25) and 2 2222 001021 222, d DDDDDD d (B.26) where 012 012, and ddd DDD ddd (B.27) The coefficients in Eq. (B.12) are further ordered to show up in the 2O by doing the following substitution 2222 1123344, and (B.28) Then Eq. (B.12) is changed into ''2'22223 1345. yyyyy (B.29) After the substitution of Eqs. (B.23), (B.24), (B.25), (B.26) and (B.27) into Eq. (B.29), the individual terms in the above equation are ''22 00010012 yDyDDyDy 2223 020100110222,DDyDyDDyDyO (B.30) 2'23 1100,yDyO (B.31) 222223 012,yyyyO (B.32) 22223 330,yyO (B.33) PAGE 226 208 and 23233 440.yyO (B.34) The following expressions are obtained after separating Eq. (B.29) into orders of 022 0005: ,ODyy (B.35) 122 011010: 2,ODyyDDy (B.36) and 222223 022011020101003040: 22.ODyyDDyDDyDyDyyy (B.37) The general solution to Eq. (B.35) is 005 01212 2,,,iiyAeAe (B.38) where 5 2 is a particular solution, and 12, A and 12, A are omplex conjugates. The following expression is obtained by substituting Eq. (B.38) into (B.36) 0022 0111121122,,.iiDyyieDAeDA (B.39) Elimination of the secular terms (the te rm that makes the solution divergent and unbounded) in Eq. (B.39) requires that 112, DA and 112, DA are zero, which means that A and A are only functions of 2 Therefore, the solution for Eq. (B.39) can be written as 0011212,,,iiyBeBe (B.40) where 12, B and 12, B are complex conjugates. Substituting Eqs. (B.38) and (B.40) into the right hand side of Eq. (B.37) yields 0001111222,iiDDyiDBeiDBe (B.41) PAGE 227 209 0002022222,iiDDyiDAeiDAe (B.42) 2 100,Dy (B.43) 0010011,iiDyiAeiAe (B.44) 002 55 3033 2222...,iiyAeAeOHT (B.45) and 0022 3 55 4044 22 2233...,iiyAAAeAAAeOHT (B.46) where ...OHT are other harmonic terms that are ne glected in the following analysis. Elimination of the secular terms in Eq. (B.37) requires 02 2 55 12134 2 22: 22230,ieiDBiDAiAAAA (B.47) and 02 2 55 12134 2 22: 22230.ieiDBiDAiAAAA (B.48) Eqs. (B.47) and (B.48) are not independent because they are complex conjugates. Hence, if one of them is satisfied, the other is automatically satisfied. Only Eq. (B.47) is considered in the following analysis. Since A is only a function of the time scale 2 it follows from Eq. (B.47) that 1B Therefore 1 x is unbounded as 1 unless 10.DB (B.49) Using Eq. (B.49), Eq. (B.47) can be rewritten as 2 2 55 21344 2 2222330. iDAiAAAA (B.50) PAGE 228 210 The polar forms for the 2A and 2A can be written as 2221 2iARe (B.51) and 2221 2iARe (B.52) where R is the amplitude of 0 x and is the phase angle. Substituting Eq. (B.51) into (B.50) and separating each term into the real and imaginary components results in 2 22222sincoscossin, dRddRd iDARRi dddd (B.53) 11111 sincos, 22iARRi (B.54) 22 5555 3434 22 22221 2323cos 2 AR 2 55 34 2 221 23sin, 2 R i (B.55) and 2 33 44433 3cossin. 88AARRi (B.56) Therefore, the real and imaginary component s in the left hand side of Eq. (B.50) are 2 3 55 344 2 22213 Real: 23cos 28d RRR d 1 21 sin0, 2 dR R d (B.57) and PAGE 229 211 2 3 55 344 2 22213 Imag: 23sin 28d RRR d 1 21 cos0. 2 dR R d (B.58) Solving Eqs. (B.57) and (B.58) simultaneously results in 1 21 0, 2dR R d (B.59) and 2 2 35245 4 2 2223 3 0. 28d R d (B.60) The solutions for R and are 121 2 20, RRe (B.61) and 1222 3524540 220 2 21233 28 R e (B.62) where 0 R and 0 are constants determined by the in itial conditions. Combining Eqs. (B.23), (B.38), (B.51), (B.61) and (B.62), th e approximate solution for the transient motion y is 0055 0012220 22,,cosiiyyAeAeR 12 121 22 53524540 2 0020 2 221233 cos. 28 R Ree (B.63) From Eqs. (B.16), (B.24) and (B.28) we have the following expressions 00, t (B.64) 1210, t (B.65) PAGE 230 212 22 3524523545 20 22 222323 22 t (B.66) and 44 11. (B.67) Using Eqs. (B.13), (B.28), (B.63) to (B.67), finally the general multiple time scales solution for x t is 10 101 22 52354540 2 000200 5 2 12 2 2233 cos. 8 2t tR xtdRdete (B.68) PAGE 231 213 APPENDIX C COEFFICIENTS OF THE APPROXIM ATE HARMONIC BALANCE SOLUTION The expressions for the coefficients in Eqs. (4.62) and (4.64) ar e provided in this appendix as follows 110, B (C.1) 2232 120200203 2, 2 B AAAAAA (C.2) 4245323224 130202000202023153 256, 284 B AAAAAAAAAAAA (C.3) 141, B (C.4) 23 212020221 2sin, 4 BAAAAAA (C.5) 23 22220220133 2cos, 228 BAAAAAA (C.6) 35323423 23222020202020351553 34cos, 816422 BAAAAAAAAAAAA (C.7) 241 2 B (C.8) 23 312022201 2cos, 4 BAAAAAA (C.9) 23 32220220133 2sin, 228 BAAAAAA (C.10) 35323423 33222020202020351553 34sin, 816422 BAAAAAAAAAAAA (C.11) PAGE 232 214 340, B (C.12) 2222 120244, CAAA (C.13) 223 22022203 224cos, 2 CAAAAAA (C.14) and 223 32022203 224sin. 2 CAAAAAA (C.15) PAGE 233 215 APPENDIX D APPROXIMATE SOLUTIONS FOR A SINUSOIDAL ACOUSTICAL PRESSURE EXCITATION This appendix provides the mathema tical background and derivation of the approximate solutions for the microphone un der the sinusoidal acoustical pressure excitation. From Chapter 4, during the sinus oidal acoustical pressure excitation, no AC excitations exist between the diaphragm and backplates. DC voltages (bV ) are applied to bias the bottom and top backplates with respect to the diaphragm. The governing equation is as follows, 22 3 130 22 00cos, 2mebb mecmepAVV M xbxkkxkxApt dxdx (D.1) where bV is the applied DC bias voltage, 0 p and p are the amplitude and driving frequency of the acoustical plane wave respectively. A TaylorÂ’s series expansion for the nonlin ear net electrostatic force up to the 3rd order results in 2222 3 000 22 35 00 0024 2mebbmebmebAVVAVAV x x dd dxdx (D.2) For the accuracy of the above 3rd order TaylorÂ’s series expans ion, please see Figure 4-14 in Chapter 4 for details. Substituti on of Eq. (D.2) into (D.1) yields 3 11330cos,mecEEmep M xbxkkkxkkxApt (D.3) where the two equivalent electrical spring constant s are defined as follows, PAGE 234 216 2 0 1 3 02 ,meb EAV k d (D.4) and 2 0 3 5 02 .meb EAV k d (D.5) Physically, Eq. (D.3) represents a damp ed second order system with only cubic mechanical and electrical nonlinearities and an external excitation. Mathematically, Eq. (D.3) represents a forced damped Duffi ngÂ’s equation, and there is no closed-form solution to it. In the following section, two approximate methods (harmonic balance method and multiple time scales method) are us ed to obtain its approximate solutions in the steady state. Prior to finding the approximate solutions in the steady state, we need to nondimensionalize the governing equation in Eq. (D.3), which finally leads to '''32cos, yyyyP (D.6) where 0, x y d (D.7) ', dy y d (D.8) 2 '' 2, dy y d (D.9) 0, t (D.10) 011,cEmekkkM (D.11) 0, 2meb M (D.12) PAGE 235 217 2 33 0 2 0,E mekk d M (D.13) 0 2 00,me mepA P M d (D.14) and 0,p (D.15) where y and 'y are the non-dimensional center di splacement and velocity of the diaphragm respectively, 0 is the modified natural frequency of the system, is the dimensionless time, is the modified damping ratio, is the modified nonlinear stiffness parameter, P is the external forcing parameter and is the ratio of the pressure driving frequency over the sy stemÂ’s natural frequency. HB Approximate Solution The harmonic balance approximate soluti on to Eq. (D.6) in the steady state is expressed in a Fourier se ries as follows [72] 0cos,n nyan (D.16) where na is the amplitude of the nth harmonic and is the phase angle. Therefore, the time derivatives are 0sin nn nyna (D.17) and ''22 0cos,nn nyna (D.18) where PAGE 236 218 .nn (D.19) If only the first three terms (0,1 and 2n ) in Eq. (D.16) are considered, by substituting Eqs. (D.16), (D.17) and (D.18) into Eq. (D.6), we have 22 1122112201122cos4cos2sin2sincoscos aaaaaaa 12 32222 001102210201cos21cos2 3cos3cos33 22 aaaaaaaaa 12 3322 11111222111cos21cos2 31 coscos33cos3cos 4422 aaaaaa 33 222201210121231 coscos33cos3coscos. 44 aaaaaaaaP (D.20) Also we have the following identities 1coscoscossinsin (D.21) and 1sinsincoscossin. (D.22) For the first order approximation, the low order terms (constant, cos and sin ) are kept and higher order terms ( cos2 sin2, etc.) are neglected. The low order terms can be collected as follo ws by using Eqs. (D.20), (D.21) and (D.22) 222 001233 Constant:10, 22 aaaa (D.23) 3222 110022133 cos:133cos2sin, 42 aaaaaaaP (D.24) PAGE 237 219 and 3222 110022133 sin:133sin2cos0. 42 aaaaaaa (D.25) For Eq. (D.23), one solution of 0a is 00. a (D.26) Since we are seeking a first or der approximation, the second or der term in Eq. (D.16) is set to zero, which means 20. a (D.27) Substituting Eqs.(D.26) and (D.27) into Eqs. (D.24) and (D.25) respectively, we have 32 1113 cos: 1cos2sin 4 aaaP (D.28) and 32 1113 sin: 1sin2cos0. 4 aaa (D.29) By squaring Eqs. (D.28) and (D.29) re spectively, and adding the two resulting equations together, we have 2 2 322 1113 12. 4 aaaP (D.30) Namely, 2 2 2624222 11193 1120, 162 aaaP (D.31) where 1a can be solved for a given set of para meters. From Eq. (D.29), we can solve as follows PAGE 238 220 1 22 12 tan. 10.75 a (D.32) Therefore, the steady-state approxi mate solution for Eq. (D.6) is 1cos. ya (D.33) Finally, the steady-state approxima te solution for Eq. (D.6) is 1 010 22 12 costan, 10.75pxtydadt a (D.34) where 1a can be determined by solving Eq. (D.31) for given parameters. MTS Approximate Solution To find the multiple time scales solution, first we need to change the original governing equation defined in Eq. (D.6) by introducing the following transformations [72] (D.35) (D.36) 2, Pf (D.37) and 1. (D.38) where is a small perturbation parameter. f and are new introduced variables. Substituting Eqs. (D.36), (D.37) a nd (D.38) into Eq. (D.6) yields '''322cos. yyyyf (D.39) Since 1 (for example, Table 4-9), we can ha ve the following power expression for y with a first-order approximation, PAGE 239 221 001101,,, yyy (D.40) where 0y and 1y are two unknown functions that need to be determined, 0 and 1 are the fast time scale (represen ting the fast dynamics) and slow time scale (representing the slow dynamics) respectively, 0 (D.41) and 1. (D.42) Therefore, the time derivatives with respect to t become the following expansion terms of the partial derivatives with respect to the corresponding time scales 0 1 01 01,, d d yy yyDyDy dd (D.43) and 2 ''222 0011 2,2, yyDyDDyDy (D.44) where 0 0, D (D.45) and 1 1. D (D.46) Substituting Eqs. (D.40), (D.43) and (D.44) into Eq. (D.39) yields 222 00110101010122 DDDDyyyyDDyy 322233 00110101332cos. yyyyyyf (D.47) PAGE 240 222 By collecting low order terms and neglecting higher order terms 2O from Eq. (D.47), we have 2 000Constant: 0, Dyy (D.48) and 23 01110000001: 222cos. DyyDDyDyyf (D.49) A general solution to Eq. (D.48) is 0011101,cos, ya (D.50) where 11a is the amplitude and 1 is the phase angle. It is convenient to rewrite Eq. (D.50) in a complex form as follows 000011111,,iiyAeAe (D.51) where 111111 2iAae (D.52) and 111111 2iAae (D.53) It follows that 000011iiDyiAeiAe (D.54) and 00'' 10011.iiDDyiAeiAe (D.55) Substituting Eqs. (D.54) and (D.55) into Eq. (D.49) and neglecting higher order terms 2O finally leads to PAGE 241 223 00002'' 011111122iiiiDyyiAeAeiAeAe 00 00 122 1111332. 2ii ii iee AAeAAefe (D.56) Elimination of the secular terms (which makes the solution divergent and unbounded) in the above equation requires 0 1'2 1111: 230 i ieiAAAAfe (D.57) and 0 1'2 1111: 230.i ieiAAAAfe (D.58) Eqs. (D.57) and (D.58) are not independent because they are complex conjugates. Hence, if one of them is satisfied, the other is automatically satisfied. Only Eq. (D.57) is considered in the following analys is. From Eq. (D.52), we have 11''' 11111 111 22iid AAaeaie d (D.59) Substituting Eqs. (D.52), (D.53) and (D .59) into Eq. (D.57) results in 1111 0 1''3 11111113 : 20. 2228iiii i ieiaeaieaeaefe (D.60) Eliminating 1ie on both sides of Eq. (D.60) leads to 1 0''3 11113 : 0. 8i ieiaaiaafe (D.61) Separating Eq. (D.61) into real and imaginary parts yields 111Imag: sin0. aaf (D.62) and '3 1113 Real: cos0. 8 aaf (D.63) PAGE 242 224 Finally, we obtain the follo wing equations by rearranging Eqs. (D.62) and (D.63) respectively 111sin, aaf (D.64) and '3 1113 cos. 8 aaf (D.65) The above equations represent an nonlinear non-autonomous (time-dependent) system. In order to make them easy to solve, we introduce the following transformation 111. r (D.66) Eqs. (D.64) and (D.65) are then changed into the following equations 11sin, aafr (D.67) and '3 1113 cos. 8 araafr (D.68) The above equations represent an autonomous sy stem and are easier to solve. Since we only consider the steady state solution around the fixed point (or equilibrium point), it follows 110sin, aafr (D.69) and '2 1 13 0cos. 8 f rar a (D.70) Eliminating all r terms in Eqs. (D.69) and (D.70) leads to 2 2 222 1113 8 aaaf (D.71) PAGE 243 225 After rearrangement, we finally obtain the following equation for the amplitude of the first order term in Eq. (D.40) 642222 11193 0, 644 aaaf (D.72) where 1 (D.73) (D.74) and 2 P f (D.75) After we solve 1a from Eq. (D.72) for given parameters, 1 is solved by using Eqs. (D.66), (D.69), (D.73) and (D.74) as follows 11 11 111122 1 sin1sin. aa r PP (D.76) Therefore, for a given set of parameters, 1a and 1 can be solved from Eqs. (D.72) and (D.76) respectively. Since we only c onsider a first-order approximation solution for y from Eqs. (D.40) and (D.50), the approxi mate solution in the steady state for Eq. (D.39) is 1 1 001110112 ,coscossin. a yyaa P (D.77) Finally, the steady-state approximate solution for x t is PAGE 244 226 1 1 0102 cossin, a xtydad P (D.78) where 1a can be determined by solving Eq. (D.72) for given parameters. PAGE 245 227 APPENDIX E UNCERTAINTY ANALYSIS This appendix presents an uncertainty analysis of the experimentally identified system parameters. In Chapter 6, no uncer tainties are assumed with the measured velocity and the physical dimensions/mater ial properties of the microphone. However, there always exist some uncertainty sources in the experiments and physical microphone devices, for example, the variations of phys ical dimensions and material properties, which can be caused by the fabrication process. To carry out uncertainty analysis for the identified system parameters in this appendix, first, different uncertainty analysis approaches are briefly discusse d. Then, uncertainties are discussed from the following possible sources: 1) experimental velocity and displacement results; 2) approximate solutions and nonlinear least-squares algorithms; and 3) the fabrication process. Finally, a linear analytical method is chosen and app lied to propagate the uncertainties caused only by fabrication into the corresponding uncer tainty ranges of experimentally extracted system parameters. Uncertainty Analysis Methods In practice, three differ ent approaches are employe d to conduct uncertainty analysis: an analytical method, a pertur bation method and the M onte Carlo method [124130]. For relatively simple equations, an an alytical method is commonly used to conduct the quantitative uncertainty analysis for the statistical error propaga tion [126, 129, 130]. If the input variables of the equation or sy stem are statistically independent and only the total small uncertainty for each input variab le is considered, a linear version of the PAGE 246 228 analytical method can be obtained by using a 1st order Taylor series expansion [126]. Since it is easy to implement and not computa tionally costly, a linear analytical method is widely used in many engineering fields. Since the data reduction equations for the system parameters of the microphone are obtained via LEM, a linear analytical method will be used for the uncertainty analysis. For more complex equations, an analytical approach is much more difficult to apply. Therefore, in these cases, its use is not practical or possi ble and some numerical methods are introduced to perform the uncer tainty analysis. Two major numerical methods are briefly discussed in the following. In a perturbation approach, in stead of finding sensitivity coefficients (derivatives) analytically, sensitivity coefficients ar e computed numerically [124]. A small perturbation (usually total uncertainty) for each input independent variable is assumed in a perturbation method and nonlinear effects c ould be included into the propagation. Some major steps for a perturba tion method are desc ribed as follows. First, the change for the desired output function is calculated wh en each input variable is perturbed by its uncertainty estimate for a give n confidence interval, with ot her input variables fixed at their respective mean values. The total uncertainty for the desired output function is computed as a root-sum-square of the resp ective changes due to all the perturbed input variables [131]. The Monte Carlo method is the most co mmonly applied numerical technique for uncertainty analysis [125, 127]. Due to the inherent approximation in the linear analytical approach, uncertainty analys es conducted directly by the Monte Carlo simulations yield more accurate results. However, the Monte Carlo simulations are PAGE 247 229 computationally costly, especi ally in the cases where hundr eds or thousands of input variables are needed. To carry out the M onte Carlo simulations, some major steps are summarized as follows. First, for each input variable, one value of its uncertainty is generated randomly from the computer for a given confidence interval. Then a Â“measuredÂ” value of this variable is obtaine d by adding the genera ted uncertainty value to its mean value. After obtaining all Â“measur edÂ” values of all inde pendent variables, the desired output result is calcu lated using the data reductio n equation [127]. The above process simulates running the experiment once and needs to be repeated N times to achieve converged statistics, where N usually is a relatively large integer number (for example, 10000). After running the simulations the mean, variance and other statistical moments of the distribution of the simulated output result are computed respectively. Finally, after obtaining converg ed statistics, the uncertain ty for the simulated output result is estimated. In practice, the Monte Carlo simulations can be performed in many ways. For example, running simulation in MA TLAB is a typical choice in the field of engineering. Uncertainty Sources There exist many uncertainty sources in the experiments and physical microphone devices, some contributes sign ificantly to the uncertainties of system parameters and some contributions are small and can be neglected. Uncertainty contributors are discussed in the following section. Uncertainty in the Experimental Data The measurement velocity resolution of the laser vibrometer is 1.5 m/s [119]. It can be calculated that the minimum detectable displacement is approximately 2.4pm at 100 kHz and 0.24pm at 1 MHz. The random uncer tainty of the measured center velocity PAGE 248 230 currently cannot be obt ained since the laser vibrometer only gave out the final mean value of each measured velocity although 100 en semble averages were used. If all 100 ensemble data points for each velocity can be obtained, we can calculate the standard deviation and mean value of each velocity, and finally the random uncertainty can be calculated for a given c onfidence interval [127]. The trapezoidal rule was used to numerica lly integrate the meas ured center velocity to yield the center displacemen t. Since 8192 sample points were used within a 3.2 ms record time, the sampling time step (t ) is approximately 0.39 s. The integration truncation error [108] for the tr apezoidal rule typically is 3Ot which is approximately on the order of 2010 and can be neglected. Also, the random uncertainty of the integrated center displa cement currently is not available due to the reason for the measured velocity. As we can see from the above analysis, the bias errors for both center velocity and displacement are small and can be neglected in the un certainty propagation. Errors of Approximate Solutions a nd Nonlinear Least-Squares Algorithms To obtain the errors caused by approximate solutions and nonlinear least-squares algorithms, three different applied voltages a nd given theoretical system parameters (0 , and ) are used to run simulations with the full-blown nonlinear equation in MATLAB. The three different voltages are chosen such that the system responses fall into the linear, weakly nonlinear and highly nonlinear regions respec tively. Then, based on the simulated system responses, the approxi mate solutions and nonlinear least-squares algorithms are applied to extract system para meters for each test case respectively. Finally, extracted system parameters are compar ed with their given values and errors are calculated for each test case resp ectively. The detailed results are summarized in Chapter PAGE 249 231 4 under the sections of validity regions of MTS and HB approximate solutions. In conclusion, the maximum error caused by the MTS approximate solution and nonlinear least-squares curve-fitting algorithm is less than 1% for the electrical square wave excitation case. The maximum error caus ed by the HB approximate solution and nonlinear least-squares curve-fitting algorithm is less than 4.4% for the electrical sinusoidal wave excitation case. Uncertainties Caused by the Fabrication Process From the reproducibility data of the fabri cation process [98], fo r a 95% confidence interval, the uncertainties of some structural parameters are listed in Table E-1. Also from the reference [132], the uncertainty ra nge of the YoungÂ’s modulus of polysilicon is obtained and listed in Table E-1. Table E-1. Uncertainties caused by the fabric ation process for a 95% confidence level. Parameter Mean Uncertainty Thickness of the diaphragm (m) 2.27e-6 0.2e-7 Thickness of the top backplate (m) 2.27e-6 0.12e-7 Thickness of the bottom backplate (m) 2.51e-6 0.6e-8 Gap between the diaphragm and top backplate (m) 2.00e-6 0.5e-6 Gap between the diaphragm and bottom backplate (m) 2.20e-6 0.5e-6 YoungÂ’s modulus of polysilicon (Pa) 173e9 20.0 e9 Preliminary Uncertainty Analysis Results From the previous discussion of the uncer tainty sources, the random uncertainties in the experimental data are currently not avai lable. And only the bias errors caused by the approximate solutions and nonlinear le ast-squares curve-fi tting algorithms are available. Therefore, in this section, onl y the uncertainties caused by the fabrication process are considered and propagated into th e uncertainty ranges of system parameters. It should be pointed ou t that the estimated uncertainty rang es of system parameters in this PAGE 250 232 section are conservative since other two un certainty sources are not included in the uncertainty analysis. From the reproducibility data of the fabri cation process [98], fo r a 95% confidence interval, we have 2.270.02 hhhm (E.1) 2.270.012 tptptphhhm (E.2) 2.510.006 bpbpbphhhm (E.3) 2.00.5 tptptpdddm (E.4) and 2.20.5 bpbpbpdddm (E.5) where h, tph and bph are the thicknesses of the diaphragm, top and bottom backplates respectively. tpd and bpd are the gaps between the diaphragm and top and bottom backplates respectively. Also the bar sign above each variable denotes the mean value and the sign denotes the correspond ing variation. Same sym bols apply for notations in the following analysis. For the YoungÂ’s modul us of polysilicon, from the reference [132], we have 17320 GPa. EEE (E.6) Since the uncertainty range for the radius of the diaphragm is not provided in the reproducibility data of the fabr ication process [98], in the following analysis, we assume that the radius of the diaphragm is 230. am From Chapter 3, we have PAGE 251 233 3 11 2 226416 ,, 3 91 DEh kkEh a a (E.7) and 33 22 2210.0440.837 ,, 1 DEh kkEh ah a (E.8) where 1k and 3k are the lumped stiffnesses of the diaphragm respectively, and they are functions of E and h respectively if other parameters are specified. Also we can have 2, 5memeah MMh (E.9) and 44 332288 44 ,,,, 33tptpbpbp tpbptpbptpbp tptpbpbptpbphnhn aa bbddhhBABA ndndAA (E.10) where me M is the lumped mass of the diaphragm and b is the total lumped damping. From Chapter 4, we have 1 1 0, ,,c c memekEhk kk Eh MMh (E.11) 00,,, ,,,,,, 22,tpbptpbp tpbptpbp memebddhh b ddhhEh M MhEh (E.12) 3 3, ,,memekEh k Eh M Mh (E.13) and ,meme memeAA h M Mh (E.14) where 0 is the first linear natural frequency, is the damping ratio, is the nonlinear stiffness parameter and is the ratio of the lumped area over lumped mass. PAGE 252 234 To propagate the uncertainties of the physical quantities defined from Eqs. (E.1) to (E.6) into the uncertainties of 0 , and respectively, a linear analytical method [126, 127, 133] is used here. First, we need to calculate the mean values for each system parameter. By using Eqs. (E.1) to (E.10), the mean values for each system parameter are calculated as follows 1 0, 12163 rad/s, c mekEhk e Mh (E.15) 0,,, 7.272-2, 2, tpbptpbp mebddhh e MhEh (E.16) 3 3, 1.21923 N/m/kg, mekEh e Mh (E.17) and 2329.2 m/kg. me meA Mh (E.18) The uncertainties defined in Eqs. (E.1) to (E.6) are assumed to be uncorrelated and small; therefore, a 1st order Taylor series approximati on can be used to propagate these uncertainties into the uncertainties of system parameters as follows, 22 00 064.093 rad/s, Ehe Eh (E.19) 2222 22tpbptpbp tpbptpbpddhhEh ddhhEh 1.337-2, e (E.20) 22 31.40922 N/m/kg, Ehe Eh (E.21) PAGE 253 235 and 2 22.901 m/kg, hh hh (E.22) where the partial derivatives are sensitivity coefficients, which relate the changes of the input quantities into the changes of the out put quantities in a ro ot-sum-square fashion [129, 130]. Table E-2 lists all the sensitivity coefficients used in the uncertainty analysis and the final uncertainty analysis results are summarized in Table E-3. Table E-2. Sensitivity coefficients used in the uncertainty analysis. Sensitivity coefficient Value 0 E 3.172e-6 11 sPa 0 h 4.573e11 11 sm tpd -1.772e4 1m bpd -1.837e4 1m tph 1.209e4 1m bph 7.964e3 1m E -1.897e-13 1m h -5.938e4 1m E 7.047e11 11 kgm h 0 41 Nmkg h -1.450e8 1 mkg PAGE 254 236 Table E-3. Theoretical mean values and un certainties of system parameters caused by fabrication for a given 95% confidence level. System parameter Mean Uncertainty Percentage 02 (Hz) 193.5e3 10.20e3 5.3% 7.272e-2 1.337e-2 18% (N/m3/ kg) 1.219e23 0.1409e23 12% (m2/ kg) 329.2 2.901 0.9% PAGE 255 237 LIST OF REFERENCES 1. Mueller, T.J., Aeroacoustic Measurements 2002, New York: Springer. p. 1-23, 158-179. 2. Scheeper, P.R., Van der Donk, A.G. H., Olthuis, W. and Bergved, P., A Review of Silicon Microphones. Sensors and Actuators A, 1994. 44: p. 1-11. 3. Hunt, F.V., Electroacoustics: the Analysis of Transduction, and its Historical Background 1954, New York: Acoustical Society of America. p. 3-9, 103-105, 181-187, 194-199. 4. 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PAGE 266 248 BIOGRAPHICAL SKETCH Jian Liu received a B.E. in aerospace e ngineering and a M.S. in mechanical engineering from Nanjing Univer sity of Aeronautics and Astrona utics, Nanjing, China, in 1998 and 2001 respectively. He also received a M.S. in aerospace engineering from the University of Florida, Gainesville, FL, in 2003. He is currently a doctoral student in the Department of Mechanical and Aerospace Engin eering at the University of Florida. His dissertation focuses on the nonlinear dynamics of a dual-backplate capacitive MEMS microphone. |