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NONLINEAR DYNAMICS OF A DUALBACKPLATE 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 (ECS0097636), and financial support from a NSF CAREER award (CMS0348288) 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....... Singlebackplate condenser microphones .................. ................2 Dualbackplate 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 PULLIN INST ABILITIE S ................. ......... ...............126 ..... QuasiStatic Pullin due to an Applied DC Voltage ................. .......................126 Equilibrium Points and Local Stabilities ....._____ ..... ... .............._..128 Graphical Analysis ..................... ...............13 1 Critical QuasiStatic Pullin Voltage ............... ...... ....___ .......___..........3 QuasiStatic Pullin by a Sub critical Pitchfork Bifurcation .............. ................134 Potential Advantage of Geometric Nonlinearity .............. .....................136 Compact QuasiStatic Stable Operation Range ................. .......................138 QuasiStatic Pullin due to an Applied Acoustical Pressure ................. .................1 40 Equilibrium Points and Local Stabilities ................. .............................141 Critical QuasiStatic Pullin Pressure ................. ............. ......... .......14 Dynamic Pullin due to a Mechanical Shock Load .............. ....................14 Problem Formulation.......................... .........4 Equilibrium Points and Local Stabilities ................. .............................153 Phase Portrait and Basins of Attraction ................. ........... ............... ....155 Dynamic Pullin due to a Mechanical Shock Load ................ ............... .....158 Potential Advantage of Geometric Nonlinearity .............. .....................162 Dynamic Pullin due to an Acoustic Shock Load ........................... ...............164 Problem Formulation............... ..............16 Numerical Simulation Results ................. ...............167................ Simulated Dynamic Pullin Results ................. ...............169............... Effect of Damping on Dynamic Pullin ................... .......... ................ ...172 Effect of Geometric Nonlinearity on Dynamic Pullin .............. ................... 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 LeastSquares 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 21 Maj or previous work in capacitive silicon microphones. ........._._... ........_........3 1 31 Material properties and physical parameters of the 2000Pa microphone in metric units (material: polysilicon). ............. ...............67..... 32 Maj or specifications of the diaphragm mesh with converged displacement results. ............. ...............78..... 33 Comparison of nonlinear FEA and LEM results ......___ ... .....___ ................80 41 Given and extracted (via MTS solution) parameters for a linear case. ....................96 42 Given and extracted parameters (via MTS solution) for a weakly nonlinear case...97 43 Given and extracted (via MTS solution) parameters for a highly nonlinear case....98 44 Results of the maximum error and sum of residual squares for each test case......100 45 Given and extracted (via HB solution) parameters for the small THD case..........108 46 Given and extracted (via HB solution) parameters for the transition THD case. ..110 47 Given and extracted (via HB solution) parameters for the large THD case...........112 48 Results of the maximum error and sum of residual squares for each test case......1 12 49 Parameters used for the comparison of approximate and numerical solutions. .....121 51 Force parameters for a designed 2000Pa capacitive MEMS microphone .............128 52 Parameters for the numerical study of an Nwave excitation. ............. .................168 61 Re sults of sy stem parameters of the b ottom b ackplate excitati on ................... .......1 84 62 Results of system parameters of the top backplate excitation. ........._..... ..............187 63 Amplitudes and phase of the integrated averaged steadystate center displacement of the bottom backplate excitation. ................ ................. .... 189 64 Re sults of sy stem parameters of the b ottom b ackplate excitati on ................... .......1 90 65 Amplitudes and phase of the integrated averaged steadystate center displacement of the top backplate excitation. ................ ................ ......... 190 66 Results of system parameters of the top backplate excitation. .............. ..... ..........191 67 Theoretical mean values and uncertainties of system parameters for a given 95% confidence level............... ...............191. 68 Nominal values of system parameters of the microphone. ................ .................192 E1 Uncertainties caused by the fabrication process for a 95% confidence level. .......231 E2 Sensitivity coefficients used in the uncertainty analysis. ................... ...............23 E3 Theoretical mean values and uncertainties of system parameters caused by fabrication for a given 95% confidence level............... ...............236. LIST OF FIGURES Figure pg 11 Schematic of a dualbackplate capacitive MEMS microphone............... ...............3 12 Schematic of an electrical model of the microphone. ................ .......................3 21 A typical frequency response plot with a defined sensitivity and bandwidth. ...........8 22 Typical noise power spectral density plot for a microphone. ............. ...................9 23 Time histories and power spectra of a pure and two distorted sinusoidal waves.....10 24 A simplified model of an electrostatic microphone. ............. ......................15 25 Schematic of a capacitive microphone with an electret diaphragm. ........................ 19 26 A simplified qausistatic model of an electret microphone............... ...............2 27 Illustration of the critical bias charge of an electret microphone. ................... .........21 28 Schematic of a singlebackplate condenser microphone ................. ................ ...22 29 Illustration of mechanical and electrostatic forces for a singlebackplate condenser microphone............... ...............2 210 Illustration of quasistatic pullin of a singlebackplate condenser microphone. ....25 211 Simplified circuit of a singlebackplate condenser microphone with a pream plifier. ............. ...............26..... 212 Effect of cubic nonlinearity on the system frequency response ............... ...............38 213 Phase plane traj ectories around Eixed points of a dualbackplate capacitive MEMS microphone (sink points are indicated by blue crosses, and saddle points are indicated by blue circles)............... ...............40 214 Typical bifurcation diagrams for onedimensional autonomous systems: (a) Saddlenode bifurcation; (b) Pitchfork bifurcation; (c) Transcritical bifurcation....41 31 3D crosssection view of the microphone structure (not to scale). ........._._.............52 32 Topview photograph of the microphone ................. ...............52............... 33 Schematic of a clamped circular diaphragm under a transverse uniform pressure loading. .............. ...............53.... 34 Displacement components in the neutral plane of a circular diaphragm. ................56 35 Normalized mode shape for several pressure values (2000, 10000 and 100000 Pascal s) ................. ...............58................. 36 Repetitive pattern of holes in the top backplate. ................ ...........................64 37 Simplified lumped element model of a dualbackplate capacitive microphone. .....68 38 A nonlinear dynamic model of a dualbackplate capacitive microphone. ...............70 39 Free body diagram of the nonlinear dynamic model ................. .......................71 310 Nonlinear vs. linearized mechanical and electrical forces of a singlebackplate capacitive microphone............... ...............7 311 Nonlinear vs. linearized mechanical and electrical forces of a dualbackplate capacitive microphone............... ...............7 312 3D mesh of the diaphragm in CoventorWare 2003 ................. .......__. ........._.77 313 Transverse center deflections of the diaphragm under the uniform pressure. ..........78 314 Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thickness direction). ............. ...............79..... 315 Plot of simulated and modeled electrostatic forces for the top capacitor. ................81 316 Plot of simulated and modeled electrostatic forces for the bottom capacitor. .........82 41 Dynamic model for an electrical square wave excitation on the top backplate. ......86 42 Plot of electrostatic and approximate electrostatic forces. ................... ...............87 43 Comparison of simulated and MTSbased curve fitting center displacement results. ............. ...............96..... 44 Comparison of simulated and MTSbased curve fitting center displacement results. ............. ...............97..... 45 Comparison of simulated and MTSbased curve fitting center displacement results. ............. ...............99..... 46 Dynamic model for an electrical sinusoidal excitation on the bottom backplate...101 47 Simulated sinusoidal response of the diaphragm. ............. ......................0 48 Simulated power spectrum of the steady state displacement. ............. .................108 49 Simulated sinusoidal response of the diaphragm. ............. ......................0 410 Simulated power spectrum of the steady state displacement. ............. .................109 411 Simulated sinusoidal response of the diaphragm. ................ ........................11 1 412 Simulated power spectrum of the steady state displacement. ............. ..... ............11 1 413 Dynamic model for the sinusoidal acoustical pressure excitation. ................... .....113 414 Plot of net electrostatic and approximate net electrostatic forces. ................... ......1 14 415 Comparison of the steadystate nondimensional amplitudes of the approximate and numerical solutions for a linear case. ............. ...............122.... 416 Comparison of the steadystate nondimensional amplitudes of the approximate and numerical solutions for a weakly nonlinear case ................. ......._.._.. ......123 417 Comparison of the steadystate nondimensional amplitudes of the approximate and numerical solutions for a highly nonlinear case. ................... ...............12 51 Plot of the ND mechanical and net electrostatic forces ................. ................ ...13 1 52 Plot of the ND mechanical and net electrostatic forces ................. ................ ...133 53 A subcritical pitchfork bifurcation illustrating quasistatic pullin due to an applied DC voltage. ................. ...............135._._.. ..... 54 A subcritical pitchfork bifurcation illustrating quasistatic pullin due to an applied DC voltage (versus bias voltages). ............. ...............136.... 55 Plot of a nondimensional net electrostatic force and different nondimensional mechanical forces. .............. ...............137.... 56 Quasistatic stable operation range of the microphone in a 3D space.. ........._.......138 57 Quasistatic stable operation range of the microphone in a 3D space (versus DC voltages). ............. ...............139.... 58 Plot of the ND net restoring and electrostatic forces ................. ............. .......143 59 Plot of the ND net restoring and electrostatic forces ................. ............. .......145 510 Quasistatic pullin due to varying ND parameters ................. .......................146 511 Quasistatic pullin due to varying acoustic pressure and DC bias voltage. ..........147 512 Three commonly used nonlinear mechanical shock load models (impulse, half sine and triangle). ............. ...............152.... 513 Phase plane traj ectories around the equilibrium points ................. ................ ...155 514 Three basins of attraction for a DC bias of 25 V. ............. ......................156 515 Basins of attraction within the physical backplates for a DC bias of 25 V............157 516 Stable and unstable nondimensional center displacement responses with two initial nondimensional velocities. ............. ...............159.... 517 Phase plots of a stable response and a dynamic pullin due to a large initial velocity imposed by a mechanical shock load. ............. ...............159.... 518 Dynamic pullin due to a combination of DC bias voltage and a mechanical shock load. ........... ..... .._ ...............161... 519 Phase plane trajectories for an added geometric nonlinearity case. Sink points are indicated by blue crosses, and saddle points are indicated by blue circles. .....162 520 Expanded stable operation region of the microphone due to the added geometric nonlinearity ................. ...............163................ 521 A typical Nwave with an amplitude and a duration time ................. .................1 65 522 The Fourier transform of a typical Nwave ................. .............................166 523 Transient nondimensional center displacement response of diaphragm due to an Nwave with an amplitude of 125 dB SPL and a duration time of 2 ms................1 68 524 Transient nondimensional center velocity response of diaphragm due to an N wave with an amplitude of 125 dB SPL and a duration time of 2 ms. ........._.._......169 525 Dynamic pullin due to an Nwave with a ND amplitude of 1.29 and a ND duration time of 2.3. ............. ...............170.... 526 Threshold of dynamic pullin due to an Nwave ................. ........................170 527 Threshold of dynamic pullin due to an Nwave with a normalized pressure parameter ................. ...............172................ 528 Effect of damping on dynamic pullin threshold. ............. ......................173 529 Effect of geometric nonlinearity on dynamic pullin threshold. ................... .........174 61 Block diagram of the experiment setup ................. ...............178........... .. 62 Laser beam spot (red dot) impinges the diaphragm through the center hole of the top backplate. ............. ...............179.... 63 Simplified circuit to generate the high voltage signal ................. ......._.._.. ......181 64 Measured averaged center velocity response for an applied square wave with an amplitude of 5V ................. ...............182................ 65 Integrated center displacement response for an applied square wave with an amplitude of 5V ................. ...............183................ 66 Constructed phase plot for an applied square wave with an amplitude of 5V.......183 67 Comparison of integrated and curvefit center displacements for an applied square wave with an amplitude of 5V ................. ...............184........... .. 68 Measured averaged center velocity response for an applied square wave with an amplitude of 18V ................. ...............185............... 69 Integrated center displacement response for an applied square wave with an amplitude of 18V ................. ...............186............... 610 Constructed phase plot after 12 Cls for an applied square wave with an amplitude of 18V ................. ...............186................ 611 Comparison of integrated and curvefit center displacements for an applied square wave with an amplitude of 18V ................. ...............187............. 612 Measured averaged steadystate center velocity response (asterisk) for a sinusoidal excitation with an amplitude of 9V and a frequency of 1 14.4 k 613 Comparison of the integrated (red asterisk) and curvefitting (blue solid line) steadystate 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 DUALBACKPLATE 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 dualbackplate 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 quasistatic pullin voltage of the microphone is found to be approximately 41V both analytically and numerically. The phenomenon of qausistatic pullin 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 pullin. Further study shows that dynamic pullin could potentially take place below the critical quasistatic pullin voltage when the microphone is subj ect to a large mechanical shock load. The dynamic pullin due to an acoustical pulse, in the form of an Nwave, has been investigated by using numerical simulation. A dynamic pullin 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 pullin 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 leastsquares curvefitting 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. Pullin 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 pullin 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 dualbackplate 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 dualbackplate 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 dualbackplate capacitive MEMS microphone is shown in Figure 11. The backplates of the microphone are perforated to let the air pass through them and hence reduce the airstreaming 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 11. Schematic of a dualbackplate 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 12. 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 12. 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 midplane 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, pullin occurs. In a phase portrait, the pullin 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 dualbackplate 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 squeezefilm 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 dualbackplate 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 nanoindenters [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. Pullin 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 dualbackplate 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 dualbackplate 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 pullin instabilities, including both quasi static and dynamic pullins. 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 [1820], 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 opencircuit 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 21. 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 21, the bandwidth is usually the frequency range from a 3dB low cuton point to a high 3dB frequency point. 1/f noise \ Thermal noise ~II / fe (Comer frequency) Frequency Figure 22. 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 22 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 Aweighted approach, or a narrowband 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 Aweighted noise floor is obtained by integrating noise spectrum after amplitudeweighting, which simulates the perceived noise by the human ear [27]. A narrowband 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 23. 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 [2831]. Figure 23 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. nonreciprocal, conservative vs. nonconservative, 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 nonconservative. 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 onchip 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, nonreciprocal 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, nonreciprocal, nonconservative 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 pullin 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 24. 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 24 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 parallelplate 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 chargeholding 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 [4850]. Backplate Air Gap Acoustic Pressure Figure 25. Schematic of a capacitive microphone with an electret diaphragm. A schematic of a capacitive microphone with an electret diaphragm is shown in Figure 25. 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 26. A simplified qausistatic model of an electret microphone. Critical bias charge. To illustrate the idea of the critical bias charge for electret microphones, shown in Figure 26 is a simplified quasistatic 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 quasistatic process is considered here, the inertial force and damping force are neglected since they are timedependent. 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 27. Illustration of the critical bias charge of an electret microphone. Shown in Figure 27 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 nonfunctioning 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 quasistatic 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 singlebackplate condenser microphones and dualbackplate condenser microphones based on the backplate configuration. In 1996, Bay et al. proposed a dualdiaphragm condenser microphone [19]; however, it was not fabricated. In 2002, Rombach et al. fabricated the first dualbackplate condenser microphone [52]. The details of two major types of condenser microphones are provided in the following. Singlebackplate condenser microphones Diaphragm Air Gap Backplate Backplate hole Cvt Vent Figure 28. Schematic of a singlebackplate condenser microphone. A typical singlebackplate condenser microphone with a diaphragm and a backplate is shown in Figure 28. A backplate is perforated to reduce the airstreaming 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. Quasistatic pullin. Based on a similar dynamic model as shown in Figure 26, for singlebackplate 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 quasistatic pullin 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 pullin 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 parallelplate 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 29 is the plot with both electrostatic and mechanical forces when the constant bias voltage V is less than the critical pullin 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 29. Illustration of mechanical and electrostatic forces for a singlebackplate condenser microphone (V < V, ). As seen from Figure 29, 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 pullin 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 210. When pullin occurs, the two fixed points shown in Figure 29 move towards each other and coalesce at the pullin position as shown in Figure 210. 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 29 become unstable. SMechanical force F ElctosaicfoceF (=m l Electrostatic force Fe (V= VPI) 0 ~Increased bias voltage V/ O dol3 Displacement x Figure 210. Illustration of quasistatic pullin of a singlebackplate 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 singlebackplate 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 21 1. Microphone I Parasitic IPreamplifier ICapacitance C I I R I I 5Output Figure 211. Simplified circuit of a singlebackplate 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 211, 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 opencircuit 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 pullin voltage for a singlebackplate 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 pullin 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 opencircuit 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 pullin voltage becomes smaller too; therefore, the opencircuit 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 singlebackplate condenser microphones. The airstreaming 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 airstreaming 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 airstreaming 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 airstreaming 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 pullin will occur with a lower voltage. Dualbackplate condenser microphones A schematic of a typical dualbackplate condenser microphone is shown in Figure 11. 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 dualbackplate condenser microphone ideally can generate twice the signal of a singlebackplate 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 dualbackplate condenser microphone has the potential to operate in a closed loop [54]. Details of a dynamic model for a dualbackplate condenser microphone are provided in Chapter 3. Pullin issues associated with dualbackplate 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, 5567] are listed in Table 21. 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 signaltonoise ratio, and broader bandwidth. Hohm and GerhardMulthaupt (1984) developed the first electret silicon microphone. Its backplate consisted of a 1 cm x 1 cm ptype 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 aluminumcoated 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 opencircuit 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 21. 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 cOpencircuit 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 opencircuit 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 ptype silicon diaphragm and a 4 pn? air gap demonstrated an opencircuit sensitivity of 13nzV/Pa at the frequency of 1 kHz. Microphones with a 8 pn? thick ptype 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 stressfree ptype silicon diaphragm and a 2 pn? air gap. The backplate was perforated with 640 acoustic holes per nant to reduce the airstreaming 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 airstreaming 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 plasmaetched grooves. Microphones with grooves in backplates and stressfree diaphragms demonstrated an opencircuit 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 airstreaming resistance, a relatively thick air gap was used (5 or 7.5 pn? ). The diaphragm was made with heavily borondoped 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 opencircuit 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 pullin. The pullin 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 Aweighted 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 lowstress (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 Aweighted 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 dualbackplate 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 opencircuit 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 widebandwidth 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 dualbackplate 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 aeroacoustic 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 10mask, 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 [6977]. 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 NavierStokes 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 nonautonomous 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 212. Effect of cubic nonlinearity on the system frequency response. Shown in Figure 212 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 steadystate 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 steadystate 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 closedform solution for most nonlinear systems. Outer Physiys cal border lines ~ Outer range r ange a,0.51 15 05 . Nonimeniona Cete Displceen Fiur 23 Phas pln rjcoisaon ie onsoadaaklt aaitv pont are iniae bybuice) I1 , x, 2 where the two states x, and x2 are the nondimensional 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 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 214. Typical bifurcation diagrams for onedimensional autonomous systems: (a) Saddlenode bifurcation; (b) Pitchfork bifurcation; (c) Transcritical bifurcation (adapted from Nayfeh and Balachandran [69]). For simplicity, onedimensional autonomous systems are used here to illustrate the basic idea of a bifurcation. Shown in Figure 214 are three typical static bifurcation diagrams. The saddlenode bifurcation diagram is generated by considering the following dynamical system x = a x2, (2.47) where a is the bifurcation parameter. In the saddlenode 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= axx2. (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 pullin instability for a singlesided electrostatic loudspeaker. By equating a linear mechanical restoring force to a nonlinear electrostatic force, he found that quasistatic pullin (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. Pullin experiments were carried for the loudspeaker by monitoring the displacement of the diaphragm; pullin occurred at a value of 78% of the gap. To explain the difference between the theoretical and measured critical pullin 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 pullin of an electrostatic resonant gate transistor. To understand the electrostatic actuation in their device, a simplified massspring model based on the parallelplate assumption was constructed and analyzed, and pullin instability was predicted and explained by using the 1D model. In 1968, Taylor [81] observed pullin phenomenon when he increased the voltage between the two closedspaced 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 pullin 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 pullin charge and mechanical force were obtained analytically based on a simple parallelplate assumption and a linear spring model. The influence of damping on the critical pullin charge was also discussed. In a constant bias voltage mode, the critical pullin voltage and mechanical force were also obtained analytically. Moreover, possible solutions, such as the use of symmetrical structures, were suggested to avoid pullin in a constant bias charge mode. Pedersen et al. [82] investigated the harmonic distortion in micromachined silicon condenser microphones. A quasistatic 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 quasistatic diaphragm deflection and opencircuit 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 pullin. The device operated with two modes: noncontact and contact modes. During the contact mode, the device was dynamically actuated by an applied step voltage higher than the pullin voltage, pullin time from rest to contact was measured and found to be a nearly linear function of the absolute pressure inside the sensor. A simple 1D lumped massspringdamper model was constructed to theoretically calculate the pullin time for an electrostatically actuated fixedfixed microbeam. The electrostatic force was modeled based on the parallelplate assumption and the fringing field effect [83] was neglected. Also, simulated pullin 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 pullin voltage, contact was made between the source and drain electrodes. Both numerical and analytical methods were used to investigate the pullin 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 BochobzaDegani [84] presented a generalized model for the quasistatic pullin 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 pullin parameters for each type of input respectively. The obtained equations were applied to a wide range of case studies, including parallelplate and tiltedplate 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 quasistatic pullin parameters (such as voltage and displacement) can be affected by the constant external force or pressure. For example, the quasistatic pullin 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 resonancebased MEMS mass sensor. Cubic nonlinearities were modeled for the mechanical restoring force of a fixedfixed beam and the electrostatic force of a comb finger. The sensor was modeled by a lumped massspringdamper 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 pitchfork 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 coworkers presented a series of nonlinear models for the electrically actuated annular plates [78], fixedfixed 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 reducedorder model (macromodel) or a numerical shooting method, which is widely used to determine the periodic solutions of autonomous and nonautonomous systems [69]. The shooting method is an iterative procedure and computationally costly, and becomes numerically unstable when pullin is approached; therefore, it is not suitable for the prediction of pull in. On the other hand, the reducedorder model [8690] usually approximates the system dynamics with N coupled nonlinear ordinarydifferential equations, which are obtained by the spatial discretization of the di stributedparameter governing equation and associated boundary conditions via a Galerkin approach or finiteelement method. The reducedorder model is robust up to the pullin point, and it also has the capability to account for the inplane residual stress, general material and geometric nonlinearities. Younis et al. [86] and AbdelRahman et. al. [91] applied the reducedorder model to simulate the dynamical behavior of a MEMS switch and predict its pullin time. A saddlenode bifurcation of a microbeam was found due to the pullin. Two deflection solution branches of the microbeam moved closer to each other as the DC voltage increased, and finally coalesced when pullin voltage was reached. Based on the reducedorder model, they also calculated that the deflection at the pullin 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 primaryresonance excitation of its first mode. The analysis results showed that the resonance frequency can be affected by the damping, midplane 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 threetoone internal resonance between the first and second modes of the clampedclamped 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 secondorder singledegreeoffreedom system. The spring hardening effect due to the midplane 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 pullin voltage was derived for a fully clamped square diaphragm with a builtin tension stress. Finally, nonlinear finite element analyses were carried out to verify the analytical results. NadalGuardia et al. [94] presented a compact 1D 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 pullin 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 pullin instability improved the lowfrequency response of the microphone for DC bias voltages close to its critical pullin 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. FargasMarques and Shkel [95] studied both static and dynamic pullin conditions for an electrostatic MEMS resonator based on the parallelplate actuation. They used a 1 D lumped massspringdamper model and derived the condition for AC dynamic pullin based on the kinetic and potential energy of the system. The experimental results for the pullin voltages agreed well with their analytical results. The dynamic pullin voltage was reported to be approximately 8% lower than the static pullin voltage. Their energy analysis results also showed that the quality factor or damping of the system had an impact on the dynamic pullin 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 quasistatic equilibrium equations, dynamic pull in for general undamped electrostatic actuators with multiple degrees of freedom and voltage sources was formulated. Specifically, the critical dynamic pullin voltages were found approximately 8% lower than the corresponding quasistatic pullin voltages for electrostatic actuators with parallelplates, double parallelplates, and clamedclamped beams. For the electrostatic actuator with a block of comb drives with no initial overlap, the critical dynamic pullin voltage was found approximately 16% lower than its quasi static pullin voltage. If damping exists in electrostatic actuators (which is always the case in practice), they concluded that the actual dynamic pullin voltage was bounded between the quasistatic pullin voltage and the dynamic pullin 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 dualbackplate capacitive MEMS microphone. The rest of this dissertation is devoted to investigating the nonlinear dynamics issues associated with a dualbackplate capacitive MEMS microphone. CHAPTER 3 NONLINEAR DYNAMIC MODEL This chapter derives a nonlinear model for the dynamics of a dualbackplate 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 dualbackplate 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 postprocessing [65, 66]. A 3D crosssection view of the microphone is shown in Figure 31. 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 31. 3D crosssection view of the microphone structure (not to scale). Figure 32 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 32. Topview 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 inplane residual stress [98] and a clamped boundary condition are assumed in the analysis. Small Displacement Solution Shown in Figure 33, 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 33. 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 inplane 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 inplane 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 straindisplacement 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 inplane 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 34. 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 34. 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 inplane 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 35 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~ Energybased 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 35. Normalized mode shape for several pressure values (2000, 10000 and 100000 Pascals). Lumped Element Modeling of the Microphone The microphone is a typical multidomain (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 stributedparameter system dynamics is through the construction of a lumpedelement 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 stributedparameter 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 stributedparameter 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) = 10 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 workequivalent 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)= 1r 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 1n dr t)Av (t) (3.24) Therefore, the workequivalent 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 energybased 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 36. 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 36. 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 squeezefilm 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 squeezefilm 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 pressuredriven 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 workequivalent 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 dualbackplate condenser microphone are summarized in Table 31 [65, 66]. Table 31. 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 ) 2.25e6 (m) Radius of the diaphragm (a) 230e6 (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 ) 16.7e11 (kg) Lumped area of the diaphragm ( Ame ) 5.54e8 (m2) Gap (do) 2.00e6 (m) Depth of the cavity ( de ) 650e6 (m) Radius of the cavity (ac ) 187e6 (m) Linear spring constant of the cavity ( ke ) 24.3 (N/m) Thickness of the top backplate ( h, ) 2.25e6 (m) Thickness of the bottom backplate (h~ ) 2.50e6 (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 8.09e2 *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 dacklth aaiie microphoneshw in Figure 37 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 37, 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 37, 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 38. 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 38. A nonlinear dynamic model of a dualbackplate capacitive microphone. Shown in Figure 39 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 39. Free body diagram of the nonlinear dynamic model. The nonlinear lumped spring force is given by F~nR= k,xkx (3.46) The lumped damping force is F = bx(3.47) where x is the center velocity of the diaphragm. By using the equalarea parallelplate assumption and neglecting the fringing field effect of the holes in the backplates, the total electrostatic coenergy, 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 biA ,, (t) ITp (t)+p,,. (52 Mmp = k,+ k~x ,X hx2 (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 singlebackplate 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 104 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 106 Figure 310. 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 310 is a plot of calculated nonlinear versus linearized mechanical and electrical forces by using the parameters in Table 31 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 dualbackplate 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 311 is a plot of calculated nonlinear versus linearized mechanical and electrical forces by using the parameters in Table 31 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 dualbackplate capacitive microphone help to reduce the electric nonlinearity. x 104 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 106 Figure 31 1. Nonlinear vs. linearized mechanical and electrical forces of a dualbackplate 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 pullins 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 31, a 3D mesh of the diaphragm is generated and shown in Figure 312. 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 32. Mesh of diaphragm ' Zoomin elements Figure 312. 3D mesh of the diaphragm in CoventorWare 2003. Table 32. 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 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 nonlinear FEA are carried out to yield the center displacement of the diaphragm for each applied pressure respectively. x 106 1.2  Ideal linear result ANonlinear FEA result/ 1 ..9.P Energybased 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 313. Transverse center deflections of the diaphragm under the uniform pressure. Shown in Figure 313 is the plot of transverse center deflections of the diaphragm. The ideal linear, energybased analytical as well as exact analytical deflection results (obtained from the reference [100]) are also plotted in Figure 313. 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 4E01 4 OE01 2 7E01 1 3E01 2 7E05 COV ENTOR Figure 314. Displacement contour of the diaphragm under the 2000Pa uniform pressure (not to scale in the thickness direction, unit: Clm). Shown in Figure 314 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 5.4e7 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 curvefitting the simulated nonlinear center displacements of the diaphragm with the formula in Eq. (3.3 1) and the final results are listed in Table 33. From Table 33, 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 33. 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 31, 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 equalarea parallelplate 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 31, 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 quasistatic pullin occurs. Similarly, for the bottom capacitor, the side surfaces of the diaphragm and bottom backplate are assumed to be fixed, and the simulated quasistatic pullin voltage is approximately 33.5V. x 104 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 315. Plot of simulated and modeled electrostatic forces for the top capacitor. Shown in Figure 315 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 315 by using the parameters in Table 31. 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 104 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 316. Plot of simulated and modeled electrostatic forces for the bottom capacitor. For the bottom capacitor, similarly, shown in Figure 316 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 315 and Figure 316, when the applied DC voltage increases, the error generated by the model with an equalarea parallelplate assumption increases. Specifically, the error is up to 17% when the applied voltage is up 