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Nonlinear amplification techniques for inertial MEMS sensors Sharma, Mrigank 2013

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Nonlinear amplification techniques forinertial MEMS sensorsbyMrigank SharmaB.E., Anna University, 2006M.A.Sc., The University of British Columbia, 2008A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate Studies(Electrical and Computer Engineering)The University Of British Columbia(Vancouver)August 2013c? Mrigank Sharma 2013AbstractInertial sensors, specifically MEMS gyroscopes, suffer in performance withdown scaling. Non linear amplification techniques, such as parametric reso-nance, can be employed in many resonant structures to alleviate this degra-dation in performance, improve sensitivity and Signal to Noise Ratio (SNR).In this thesis the application of parametric resonance amplification anddamping to both modes of a vibratory gyroscope is carried out using spe-cialized combs. Gap-varying combs, which are usually used for the sensingmode are known for producing electrostatic spring modulations. They areused in this thesis to achieve parametric modulation in sense mode, for in-creasing spectral selectivity and to reduce the equivalent input noise angularrate (from 0.0046 deg/s/?Hz to 0.0026 deg/s/?Hz, for a parametric gainof 5). Additionally, novel shaped combs were used for performing parametricmodulation of the driven mode of a resonant gyroscope as well. Analyticalmodes for both types of parametric amplification are derived and experimen-tally verified. In order to study the effect of parametric modulation for largesignal operation, the dynamic pull-in process is analyzed and modeled in in-ertial MEMS sensors. The dynamic analytical model is derived and experi-mentally verified for parametric amplification. The dependence of dynamicpull-in voltage amplitudes on the values of externally-induced accelerations(e.g. Coriolis accelerations in the case of vibratory gyroscopes) is experimen-iiAbstracttally proven. The measurements indicate that the dynamic pull-in voltagesreduce from 100 V to 56 V for a designed and fabricated MEMS gyroscope(device A) and from 21.77 V to 17.3 V for a MEMS accelerometer (de-vice B), for an equivalent input acceleration signal of 0.319 ms?2, when thestructures are actuated at their resonance frequency. In order to further an-alyze the fundamental limitations of sensing at microscale, a separate noiseanalysis of MEMS resonant sensors is performed. The frequency-dependentdamping theory is used to suggest new optimization methods for the designof MEMS vibratory gyroscopes.iiiPrefaceAll chapters are based on the work conducted under the supervision of Dr.Edmond Cretu. This thesis was supported by a AUTO 21 grant for a projecttitled, ?Inertial Sensor cluster for adaptive path prediction?. This was ateam work, where Mr. Elie Hanna Sarraf and Dr. Edmond Cretu are partof many published manuscripts which were written by me. Chapter1 (inparticular needs and approach) has appeared in [49] [38] [37]. Parts of thework presented in Chapter 4 have appeared in an application note writtenby me for CMC Microsystems c? [121]. Chapter 5 is based on the publishedresearch article [38], which was co-authored by me and Dr. Edmond Cretu.My contributions included the model development for gyroscope, extrac-tion of damping and spring forces for the noise analysis, and the analysisof frequency-dependent noise for the gyroscope using MATLAB. Both de-signs of resonator and gyroscope were made by me using CoventorWare R?.The gyroscope was fabricated in SOIMUMPs R? fabrication technology ofMEMSCAP c?. The curve fitting for the Fig. 5.6 was done by Dr. EdmondCretu in Mathematica c?. Damping and spring forces for the resonator wereextracted by Ms. Akila Kannan using ConventorWare R?, as depicted inFig. 5.4. Chapter 6 is based on the published research articles [49] [122].My contribution included design, modeling and characterization of MEMSgyroscope, fabricated in SOIMUMPs R? technology. The concepts of para-ivPrefacemetric resonance, amplification and damping, were employed by me on thegyroscopes. I have designed and submitted for fabrication. Experimentalverification was carried by me using the Polytec PMA-500 c? (Planar MotionAnalyzer). Dr. Edmond Cretu laid the foundation for the research projectand explained the need for a nonlinear theory. Mr. Elie Hanna Sarraf helpedwith the set up of experiments and helped with the understanding of aspectsof the nonlinear control theory. Dr. Rajashree Baskaran of Intel Inc. helpedwith the understanding of more advanced issues of nonlinear theory. Chap-ter 7 is based on two submitted articles([124], [125]), which is co-authoredby me, Mr. Elie Hanna Sarraf and Dr. Edmond Cretu. Design, modelingand characterization of gyroscope fabricated in SOI Tronics R? technologywere carried out by me. Modeling of shaped combs was assisted by Dr.Edmond Cretu. Mr. Siamak Moori provided assistance with characteriza-tion of gyroscope using Agilent 4294A impedance analyser. For the secondresonator implemented in SOIMUMPs R? technology, Mr. Elie Hanna Sarrafhelped with the phase control of the signals parametric resonance using PXIe1062Q DAQ controller, in order to experimentally validate the parametricresonance concepts. Chapter 8 is based on published article [123]. Com-plete analysis and experimental work is carried out by me. An accelerometerused in the chapter was designed and modeled by Mr.Elie Hanna Sarraf. Thecontribution of this thesis work is summed up as accepted and submit-ted research articles as following:[1] Mrigank Sharma, Elie Hanna Sarraf, Rajashree Baskaran and Ed-mond Cretu Parametric resonance: Amplification and Damping in MEMSgyroscopes. Sensors and Actuators A: Physical, pg.771-779, 2012.[2] Mrigank Sharma and Edmond Cretu, Frequency dependant noiseanalysis and damping in MEMS. J.Microsystem Technologies, 15(8), pp.1129-vPreface1139, 2009.[3] Mrigank Sharma, Elie Hanna Sarraf, Edmond Cretu, Novel dy-namic pull in MEMS gyroscopes, Proc.Eurosensors(2011),Volume 25, pp.5558, 2011.[4] Mrigank Sharma, Elie Hanna Sarraf, Edmond Cretu, Parametricamplification/damping in MEMS gyroscopes, Proc. IEEE MEMS (2011),pp. 617620[5] Mrigank Sharma, Akhila Kannan and Edmond Cretu, Noise analy-sis and noise based optimization for resonant MEMS structures. Proc.Symposiumon Design, Test, Integration and Packaging of MEMS/MOEMS - DTIP 2008,Nice : France (2008).[6] Mrigank Sharma,Application Note: System-Level Design and Mod-eling of a MEMS Gyroscope Using CoventorWare,https://www.cmc.ca/en/WhatWeOffer/Products/CMC-00025-75669.aspx[7] Mrigank Sharma, Elie Hanna Sarraf, Edmond Cretu, Shaped Combsand parametric amplification in inertial sensors, Proceedings of IEEE SEN-SORS 2013 (accepted), Baltimore, Maryland, USAOther contributions:[8] Manav, Greg Reynen,Mrigank Sharma, Edmond Cretu and Srikan-tha Phani, Ultrasensitive resonant MEMS Transducers with tunable couplingThe 17th International Conference on Solid-State Sensors, Actuators andMicrosystems June 16-20, 2013, Barcelona. SpainIn this article, I provided assistance with the experimental setup andcharacterization of the coupled resonators. I also aided in the layout designof the resonator.[9] Elie Hanna Sarraf, Ankit Kansal, Mrigank Sharma and EdmondCretu, FPGA-based Novel Adaptive Scheme using PN Sequences for Self-viPrefaceCalibration and Self-Testing of MEMS-based Inertial Sensors, J.ElectronTest (2012)28:599614In this article, the design, modeling and characterization of the gyroscopewas carried by me. I also provided assistance with the implementation ofthe electronic interface readout. Mr. Ankit Kansal worked closely with mewhile he was implementing the simulink models.[10] Elie Hanna Sarraf, Mrigank Sharma and Edmond Cretu, NovelBand-Pass Sliding Mode Control for Driving MEMS based Resonators, Sen-sors and Actuators, Physical A(186), January 2012, pp154-162[11] Elie Hanna Sarraf,Mrigank Sharma and Edmond Cretu, NovelSliding Model Control for MEMS-Based High Sensitive Resonators, 25thEUROSENSORS, Athens, Greece, September 2011In the above two research articles, gyroscope used for the research wasdesigned, modeled and characterized by me. Ch.4 covers the complete mod-eling of the sensor used for this research.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Coriolis effect . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Performance figures of merit . . . . . . . . . . . . . . . . . . 91.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Needs and approach . . . . . . . . . . . . . . . . . . . 141.4 Overview and thesis organization . . . . . . . . . . . . . . . 192 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Classification of gyroscopes . . . . . . . . . . . . . . . . . . . 212.1.1 Spinning mass gyroscope . . . . . . . . . . . . . . . . 22viiiTable of Contents2.1.2 Optical gyroscope . . . . . . . . . . . . . . . . . . . . 232.1.3 Fluidic gyroscopes . . . . . . . . . . . . . . . . . . . . 252.1.4 Non vibratory gyroscopes . . . . . . . . . . . . . . . . 262.1.5 Vibratory gyroscopes . . . . . . . . . . . . . . . . . . 272.2 State of the art MEMS gyroscopes in automotive and con-sumer market . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Modeling of vibratory MEMS gyroscope . . . . . . . . . . . 453.1 Resonance characterization . . . . . . . . . . . . . . . . . . . 453.2 Dynamics of gyroscope . . . . . . . . . . . . . . . . . . . . . 483.2.1 Drive mode oscillations . . . . . . . . . . . . . . . . . 503.2.2 The Coriolis response . . . . . . . . . . . . . . . . . . 513.2.3 Mode matching and ?f . . . . . . . . . . . . . . . . . 543.3 Capacitive sensing methodology . . . . . . . . . . . . . . . . 563.4 System level MEMS design issues . . . . . . . . . . . . . . . 593.4.1 Quadrature error . . . . . . . . . . . . . . . . . . . . 593.4.2 Mechano-thermal (Brownian) noise . . . . . . . . . . 613.4.3 Electronic noise equivalent rate . . . . . . . . . . . . 624 Fabrication technology and modeling in ConventorWare R? 644.1 SOIMUMPS R? . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 SOI Tronics R? . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3 Design simulations . . . . . . . . . . . . . . . . . . . . . . . . 714.3.1 Dynamic response . . . . . . . . . . . . . . . . . . . . 724.3.2 Drive mode oscillations . . . . . . . . . . . . . . . . . 754.3.3 Sense mode oscillations . . . . . . . . . . . . . . . . . 77ixTable of Contents4.3.4 Mode analysis using FE analysis and final parametertuning . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Analytical modeling of damping . . . . . . . . . . . . . . . . 854.4.1 Slide film damping . . . . . . . . . . . . . . . . . . . 854.4.2 Squeeze film damping . . . . . . . . . . . . . . . . . . 865 Frequency dependent noise analysis and damping in MEMSinertial sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.1 Design criteria for frequency varying noise . . . . . . . . . . 915.2 Implementation of noise model on out of plane structure . . 925.3 Implementation of noise model on gyroscope (in-plane move-ment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.2 Step 2 FE analysis . . . . . . . . . . . . . . . . . . . . 1025.3.3 Macro-model extraction . . . . . . . . . . . . . . . . . 1055.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . 1096 Parametric resonance: amplification and damping in MEMSgyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1 Device structure: nonlinearity in MEMS coupled resonators . 1136.2 Parametric amplification in MEMS gyroscopes (secondary mode)1206.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 1246.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 1256.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1307 Novel sloped combs for parametric resonance . . . . . . . . 1327.1 Device structure and behavior characterization . . . . . . . . 136xTable of Contents7.1.1 Static measurements . . . . . . . . . . . . . . . . . . 1377.2 Nonlinearity measurements . . . . . . . . . . . . . . . . . . . 1397.3 Common mode analysis spring softening and parametric am-plification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.4 Parametric amplification in MEMS gyroscopes both in (pri-mary and secondary mode) . . . . . . . . . . . . . . . . . . . 1497.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1568 Concept of dynamic pull-in MEMS gyroscope . . . . . . . . 1588.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.2 Pull-in mechanism . . . . . . . . . . . . . . . . . . . . . . . . 1608.2.1 Static asymmetric pull-in phenomenon . . . . . . . . 1618.2.2 Static symmetric pull-in phenomenon . . . . . . . . . 1648.2.3 Resonant pull-in and symmetric dynamic pull-in . . . 1658.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 1698.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 1708.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1729 Conclusion and future work . . . . . . . . . . . . . . . . . . . 175Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184AppendicesA Experimental results of parametric amplification for SOITronics R? gyroscope . . . . . . . . . . . . . . . . . . . . . . . . 200B Characterization using optical planar motion analyzer . . 207xiTable of ContentsC Simulink models . . . . . . . . . . . . . . . . . . . . . . . . . . 212xiiList of Tables1.1 Scaling of sensing force for pressure sensor, accelerometer andgyroscope [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 MEMS gyroscope prototypes in literature . . . . . . . . . . . 352.2 Requirement for ESC application[79] . . . . . . . . . . . . . . 422.3 Sensitivity and full-scale-range for consumer gyroscopes[83] . 443.1 Mechanical gain of the resonator force to displacements [85] . 474.1 Dimensions of proof mass and suspensions . . . . . . . . . . . 734.2 Dimensions of area varying combs(Actuation) . . . . . . . . 764.3 Dimensions of gap varying combs(Sensing) . . . . . . . . . . . 784.4 Resonant modes and the stiffness for different modes obtainedfrom Saber architect AC plots . . . . . . . . . . . . . . . . . . 794.5 FEA Mech MM comparison for coarse and extra coarse meshes 845.1 Features of white noise analysis and frequency model withlumped model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.1 Parameters of two mass gyroscope . . . . . . . . . . . . . . . 180xiiiList of Figures1.1 Development stages of different MEMS devices . . . . . . . . 41.2 Resolution vs technology and applications of different gyro-scopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Coriolis Effect on the canon ball fired from north pole to equator 71.4 Toy gyroscope and the gyrocompass . . . . . . . . . . . . . . 101.5 Cost of MEMS development cycle . . . . . . . . . . . . . . . . 151.6 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Classification of gyroscopes . . . . . . . . . . . . . . . . . . . 212.2 Spinning gyroscope with spring . . . . . . . . . . . . . . . . . 222.3 Rate integrated gyroscope with feedback loop . . . . . . . . . 242.4 Ring laser gyroscope . . . . . . . . . . . . . . . . . . . . . . . 252.5 Fibre optical gyroscope . . . . . . . . . . . . . . . . . . . . . 262.6 Tuning fork gyroscope . . . . . . . . . . . . . . . . . . . . . . 282.7 Position vector relative to the inertial frame A and to rotatingreference frame B . . . . . . . . . . . . . . . . . . . . . . . . . 292.8 Two degrees-of-freedom mass spring damper system under-going rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.9 Hemispherical resonator gyroscope . . . . . . . . . . . . . . . 322.10 Ring gyroscope resonant modes . . . . . . . . . . . . . . . . . 33xivList of Figures2.11 Classification of non degenerate MEMS gyroscopes . . . . . . 342.12 Prototype of tuning fork gyroscope . . . . . . . . . . . . . . . 352.13 MEMS inertial navigation platform for cars . . . . . . . . . . 433.1 Bode plot for different quality factors . . . . . . . . . . . . . . 473.2 Step response for different quality factor (damping) values . . 483.3 System level vibratory MEMS gyroscope . . . . . . . . . . . . 493.4 Linearized actuation scheme . . . . . . . . . . . . . . . . . . . 523.5 Actuation scheme using gap varying combs . . . . . . . . . . 543.6 Impact of frequency mismatch . . . . . . . . . . . . . . . . . . 553.7 Schematic of the Sense mode readout of the gyroscope . . . . 573.8 Quadrature displacement . . . . . . . . . . . . . . . . . . . . 604.1 Fabricated inertial sensors during the period of this PhD. . . 654.2 Fabrication steps in SOIMUMPS R? . . . . . . . . . . . . . . . 684.3 SOIMUMPS R? gyroscope structure in Solid Modeler . . . . . 694.4 Fabrication steps in SOI Tronics R? . . . . . . . . . . . . . . . 704.5 Tronics Layout and Solid modeler image . . . . . . . . . . . . 714.6 Flow chart for system level modeling in CoventorWare R? . . . 724.7 Four crab legs and beams mechanically connected for ACanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.8 Crab leg beam dimensions and FEA resonant modes . . . . . 744.9 Dynamic response and impact of beam width on the reso-nance behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 754.10 Architect connections for transient analysis for the drive mode 774.11 Transient analysis results for drive mode oscillations . . . . . 784.12 Complete architect model with sensing scheme . . . . . . . . 80xvList of Figures4.13 AC analysis response of the final structure with angular ve-locity along z direction . . . . . . . . . . . . . . . . . . . . . . 814.14 Transient response of the final structure with angular velocityalong z direction . . . . . . . . . . . . . . . . . . . . . . . . . 824.15 6 resonant modes obtained from FE analysis of MEMech . . . 834.16 a)Impact of variation of length (lb2) on resonant modes b)Frequency tuning using DC bias . . . . . . . . . . . . . . . . 844.17 Squeeze film damping variation with frequency of the de-signed gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . 885.1 Macro model of resonator with frequency varying noise . . . . 925.2 FE analysis result for out of plane resonator for damping study 945.3 FE analysis result of damping coefficient for out of plane res-onator for damping study . . . . . . . . . . . . . . . . . . . . 955.4 Normalized damping and stiffness forces extracted from FEAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Normalized damping(red) and stiffness(blue) forces vs Squeezenumber (?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.6 a Equivalent input acceleration noise, b equivalent displace-ment noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.7 a Equivalent input acceleration noise, b equivalent displace-ment noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.8 Gyroscope structure in analysis with area and gap varyingcombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.9 Slide film damping-actuation (spring force-green negligible,damping force-red increases at high frequency) . . . . . . . . 103xviList of Figures5.10 Squeeze film damping- spring and damping forces comparisonbetween FE analysis(straight lines) and analytical modeling(circles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.11 Squeeze film damping- damping coefficient comparison be-tween FE analysis(dashed lines) and analytical modeling (cir-cles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.12 Comparison of displacement noise, red-white noise displace-ment and blue-frequency varying noise displacement . . . . . 1075.13 Comparison of mechano thermal equivalent rate based on con-stant damping (red) and frequency dependent damping (blue)with electronic equivalent rate noise (green) . . . . . . . . . . 1086.1 Gyroscope fabricated in SOIMUMPs R? (25 ?m) technology.Red/Blue markers-Actuation/Sensing combs . . . . . . . . . . 1146.2 Simulation results of the Duffing Oscillator with normalizedforce(F=01,2,3,4.) Positive cubic nonlinearity(? > 0) . . . . 1166.3 Simulation results of the Duffing Oscillator with normalizedforce(F=01,2,3,4.) Negative cubic nonlinearity(? < 0) . . . . 1176.4 Dynamic characterization of Mathieu equation with stableand unstable points . . . . . . . . . . . . . . . . . . . . . . . . 1186.5 Experimental results obtained for Duffing oscillator(1,2,3) andparametric amplification on the secondary mode of the MEMSgyroscope. Parametric resonance differs from Duffing?s due topresence of instability region . . . . . . . . . . . . . . . . . . 1216.6 Stability points VS the parametric resonance frequency. Pointsindicate the experimental results and line is the theoretical fit 122xviiList of Figures6.7 Hysteresis phenomenon observed in the secondary mode ofMEMS gyroscope. Frequency sweep up is blue line and sweepdown in green line. Distinct jumps occur as shown with blackarrow marks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.8 Device concept and working principle . . . . . . . . . . . . . . 1256.9 Sense comb drives with gap-varying fingers . . . . . . . . . . 1266.10 Experimental (points) and theoretical (line) parametric gainvs. pump voltage amplitude . . . . . . . . . . . . . . . . . . . 1276.11 Influence of the parametric amplification on the equivalentbandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.12 Equivalent input angular rate noise (blue) and equivalent out-put displacement due to the mechano-thermal noise (red) vs.parametric gain . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.13 Phase dependence of the normalized gain (experimental andsimulated) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.14 SNR vs. normalized gain (experimental and theoretical fit). . 1317.1 Shaped comb drive with dimension definitions and surfaceelectric potential (Simulation) . . . . . . . . . . . . . . . . . . 1337.2 Non inter-digitated combs with dimension definitions and sur-face electric potential . . . . . . . . . . . . . . . . . . . . . . 1357.3 0y displacement obtained with a 10 V DC, for sloped shapedfingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.4 0y displacement obtained with a 10 V DC, for non interdigi-tated fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.5 Surface electric field obtained with a 10 V DC actuation volt-age, for slope shaped fingers . . . . . . . . . . . . . . . . . . . 139xviiiList of Figures7.6 Surface electric field obtained with a 10 V DC actuation volt-age, for non inter digitated fingers . . . . . . . . . . . . . . . 1407.7 Image of resonator with sloped shaped and area varying combs1417.8 FE analysis results for resonant modes . . . . . . . . . . . . . 1427.9 A) DC bias characterization for shaped combs and B)Schematicfor half actuation and DC bias tuning scheme . . . . . . . . . 1437.10 Static displacement VS DC bias on one side of sloped combs 1447.11 Static displacement VS DC bias on one side of sloped combs 1457.12 Weakening effect on the resonator due to spring softeningwith DC bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.13 Change in resonant frequency and electrostatic spring soften-ing with DC bias . . . . . . . . . . . . . . . . . . . . . . . . . 1477.14 Schematic for A) DC and B) AC spring modulation usingcommon mode voltages . . . . . . . . . . . . . . . . . . . . . 1477.15 Weakening effect on the resonator due to spring softeningwith common mode DC bias . . . . . . . . . . . . . . . . . . . 1487.16 A) Change in resonant frequency and B) Comparison of elec-trostatic spring modulation using common mode DC voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.17 Impact of parametric gain with 10 V common mode voltageusing shaped comb resonator. Blue curve- with no pump andred curve with parametric pump . . . . . . . . . . . . . . . . 1507.18 Impact of phase on parametric gain . . . . . . . . . . . . . . . 1517.19 SEM shot of the MEMS gyroscope with distinct combs . . . . 1527.20 Resonant modes of the angular sensor with correspondingcomb drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153xixList of Figures7.21 Linear fit of the Force/normalized gap in the linear gap-varying region of the sloped combs . . . . . . . . . . . . . . . 1547.22 3D plot simulation of frequency vs phase vs Displacement,with initial Qx = 100, ?0x=8 kHz . . . . . . . . . . . . . . . 1557.23 Linear fit of the Force/normalized gap in linear region for gapvarying combs . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.24 Simulated, impact of net gain (primary and secondary) on 0ysense displacement . . . . . . . . . . . . . . . . . . . . . . . . 1578.1 Device A: gyroscope for symmetric dynamic pull-in set up;Device B: accelerometer for symmetric dynamic pull-in set up 1628.2 Asymmetric pull in phenomenon, blue curve- snapping of fin-gers to top electrode. 0.1 to 0.2 ms is the meta stable regiondue to damping . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.3 Set up of symmetric pull-in . . . . . . . . . . . . . . . . . . . 1648.4 Normalized potential energy vs normalized displacement. At0.8 of the normalized gap, the oscillations are unstable whichwill lead to pull in. Blue dotted curve is the net gain in energy1668.5 Phase portrait and potential energy curves, highlighting os-cillation, with oscillation at 0.8 normalized gap goes to pullin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.6 Symmetric dynamic pull in behavior. Proof mass initially atoscillation (0.06 ms?2). Depending on the phases betweenelectrostatic forces and external acceleration, pull in possible. 1688.7 Impact of initial acceleration on the proof mass oscillations.Higher initial acceleration leads to pull in (5 ms?2) with com-mon mode voltage of 97.5 V in device A. . . . . . . . . . . . 169xxList of Figures8.8 Spring softening effect seen for both device A and B. Voltagesapplied are the common mode voltages as shown in figure withVDC of 10 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.9 Phase dependent gain VS common mode voltage. At 17 V,two gains possible, with proper phase tuning pull is achievedat 17.5 V (blue dot), when the phases are detuned betweenthe spring modulation and the external acceleration, gain de-creases (shown in red). . . . . . . . . . . . . . . . . . . . . . . 1718.10 Dynamic pull-in characterization (dynamic pull-in voltage vs.frequency) for external acceleration of 0.319 ms?2 . . . . . . 1728.11 Device B- measured dynamic pull-in characterization (Vpi,dynvs. frequency) for various external accelerations . . . . . . . . 1738.12 Measured Vpi,dyn vs equivalent angular acceleration (?) forboth A,B devices . . . . . . . . . . . . . . . . . . . . . . . . . 1749.1 Two mass gyroscope implemented in SOIMUMPS R? technology1789.2 Close up of drive mode and sense mode springs captured byPolytec MSA-500 . . . . . . . . . . . . . . . . . . . . . . . . . 1799.3 Resonant modes of 2 mass gyroscope obtained using PolytecPMA-500 c?. Drive mode (AC-10 V DC-10 V, sense mode(AC-2.2V DC-2.2V) . . . . . . . . . . . . . . . . . . . . . . . . . . 1819.4 A) Quadrature displacement vs DC bias voltage applied onthe combs obtained using Polytec MSA-500 R? B) Image ofquadrature compensation comb . . . . . . . . . . . . . . . . . 1829.5 Image of three coupled resonators with non overlapping combsfor positive and negative spring modulation . . . . . . . . . . 183xxiList of FiguresA.1 A) Electrical representation of MEMS sensor for impedancemeasurement, B) Extracted impedance (real and imaginary)component with the experimental result . . . . . . . . . . . . 201A.2 A) Actuation combs(area varying and shaped combs) realimpedance B) Sensing gap varying combs real impedancemeasurement with spring softening . . . . . . . . . . . . . . . 203A.3 1) Primary and secondary equivalent capacitors with elec-trical nodes for measurements, 2) Variation of resonant fre-quency Vs DC bias for sloped shaped combs(left) and gapvarying combs(right) . . . . . . . . . . . . . . . . . . . . . . . 204A.4 Comparison between parametric amplification response andwithout pumps for both sloped (Gain=2.76 with Ucm= 27.5V ) and gap varying combs (Gain=7 with Ucm =15.2 V ) . . . 206B.1 Polytech equipment with its components used for experiments,snapshot of movement of finger captured by PMA-500, whichhas a movement of 1 ?m . . . . . . . . . . . . . . . . . . . . . 209B.2 Timing diagram for principle of stroboscopic video microscopy 210B.3 Pattern matching for the PMA software . . . . . . . . . . . . 211C.1 Simulink model for parametric resonance . . . . . . . . . . . . 212C.2 Simulink subset model of gyroscope for parametric resonanceon both primary and secondary mode . . . . . . . . . . . . . 213C.3 Simulink model for dynamic pull in . . . . . . . . . . . . . . . 214C.4 Simulink subset model of resonator for dynamic pull in . . . . 215xxiiAcknowledgementsI thank Dr. Edmond Cretu for his vision, guidance and motivation through-out my graduate years in Canada. I thank him for providing with funding formy PhD studies which was supported by AUTO 21. I thank Dr. ShahriarMirabbasi for all the support he provided in this endeavor. I thank mycommittee members Dr. Karen Cheung, Dr. Kenichi Takahata and Dr.Tim Salcudean for reading my dissertation and by providing with correc-tions. I would like to express deepest gratitude to the members of adaptiveMEMS lab, for creating a lively environment. I thank Elie Hanna Sarrafwith whom I have published many research articles. Our interests in MEMSwere complementary to each other?s research goals. I would also like tothank Mr. Jack Shiah, Ms. Wei You, Mr. Miguel Angel Torres, Mr. SiamakMoori and Mr. Andy Tsai for providing with deep intellectual discussions.I also thank my colleagues with whom I worked in Graduate Student So-ciety and Alma Mater Society. The fusion of thought processes and ideasbetween the students of different faculties was very inspiring. I would espe-cially like to thank Dr.Subrata Deb, (Dr). Arvind Saraswat, Dr.RavinderSingh Thakur, Dr. Farid Khan, Mr.Hemant Sharma, (Dr).Karthik Chan-drasekar and Mr.Tamas Weidner for all the support and guidance. Myfriends (Chaupalis) in Vancouver have been a constant source of support. Ithank you all for providing me with a home away from home. My deepestxxiiiAcknowledgementsgratitude to Prof. N. Venkateswaran, Director of WAran Research Founda-Tion (WARFT) who taught me the basics of research. His ever inspiringwords ?Success can be delayed but never denied? are always close to me.I cannot express my gratitude for my parents who have been a constantsource of inspiration and support. Sacrifice and love of my parents can?tbe expressed in words. The smile and love of my brother is what keepsme motivated and I thank him for putting up with my absence for theseyears. I also like to thank Ajahn Bhram and Bodhinyana Monastery for thewonderful talks on youtube.I apologize beforehand to those whose names are kept anonymous. Youremain in my heart and thank you!xxivDedicationThis thesis is dedicated to mummy, papa and Buboon.xxvChapter 1IntroductionOne could say that the end of 20th century was dominated by informationtechnology, which was enabled by the maturation of semiconductor indus-try. The first transistor was built at Bell labs in 1947; transistors havebecome the building block for digital, logic and memory circuit[1]. Transis-tors are used in low power portable radio, which connected the people ofthe world. The integrated circuits (ICs) were the next revolutionary stepin the industry, which became commercially available in early 1960s. Theirinvention led to the new dawn in personal computing, with the inventionof personal computers and super computers. Programma 101 was the firstcommercial desktop computer built in 1965, which in today?s time is a handheld calculator[2]. Building parts of the personal computer, MOS (MetalOxide Silicon)memory i1101 SRAM (Static Random Access Memory) andi1103 (1024 bits) DRAM (dynamic random access memory) chips pavedfoundation for interactive-mode computers[1, 3]. There was an evolution inscience, where a transition from analog electronics to digital electronics be-gan. Unlike analog electronics, where electrical signal was the prime focus,digital electronics focussed on creating, processing and storing of symbols asinformation carriers[4]. In 1970, HP (Hewlett Packard) introduced BASICprogrammable computers that fitted entire an desktop, including keyboard,and printer. Graphical User Interface (GUI) was invented in 1973 by Xe-1Chapter 1. Introductionrox?s Palo Alto Research (PARC), which inspired Apple computers MACand Microsoft Windows[2]. This led to creation of operating systems andmany software applications requiring complex computations. This increasein computation power is made possible by downsizing transistors, fabricat-ing millions of them on IC(increasing processing). Laptop, netbook, tabletPC are some of the new entrants, which have made the usage of comput-ers easy to carry. Meanwhile, the revolution in mobile phone, with largebroadband technologies like 2G and 3G, enabled transmission of large band-width data between one phone to another. Computers are gradually beingreplaced by smart phones for connection to internet, watching live streamingvideos, playing games. As the technology is down scaled, the once second-order effects in transistor models (eg. short channel effects, leakage cur-rents, subthreshold conduction etc.) become significant[4, 5]. Both analogand digital electronics are dependent on technology for higher performance.In the case of analog electronics, there are many specialized software toolstaking into account the second order effects at sub micron level, but theyare very technology dependent. As the feature size is reduced, cost is amajor decision factor in replacing the line of manufacturing equipments.Microelectronics, even with inherent limitations, fosters the developmenttowards MEMS (Micro Electro Mechanical Systems). As the name implies,Micro-electromechanical Systems (MEMS) is the technology that combineselectrical and mechanical systems at micro scale. Photolithography-basedtechnology allows development of mechanical and electrical functions on sil-icon wafers. This patterning of MEMS using photolithography is adaptedfrom the standard IC technology. Mechanical sensors which were macro insize could now be scaled down, aligning with the ideas presented in a fa-mous lecture delivered by Richard Feynman, ?There is plenty of room at the2Chapter 1. Introductionbottom?. Batch processing of MEMS devices, substantially reduces devicesize, weight and cost. There is a new trend, especially in consumer elec-tronics, where MEMS units are adding value to the systems. For instance,in smart phones, accelerometers, gyroscopes and magnetometers are addingvalue with motion processing for various applications like navigation andgames[6, 7]. MEMS actuator units are being used for the lens systems incamera modules[8], micro mirrors are used in projector heads[9], micro flu-idic based ink jet printing heads are used in modern printers[10]. In medicalapplications, MEMS stents are being used for heart patients[11], MEMSbased medical micro needles [12]are aiding in blood sample collections inremote villages of third world countries. The applications are vast and havebecome part of our mainstream life. Fig. 1.1 presents with different MEMSdevices, and the stage of development they are currently in. MEMS gyro-scopes are beginning to mature from the development stage[13].As most of the MEMS technology are Silicon based, the fabrication pro-cesses are tweaked to make MEMS components. A full infrastructure exists:Si foundries,specialized equipment, simulation tools for both electronics andMEMS. Some of the advantages of MEMS sensors in comparison with clas-sical sensors are :1) Miniaturization: saves materials, power, high response time2) Pay per area (many fabless corporations have become success stories!)3) Integration with IC reduces size and cost4) Batch fabrication: cheap5) Foster new applicationsOne such example of integration of MEMS with IC and digital pro-cessing unit is InvenSense c??s MPU 6050[14]. This sensor unit has tri axisgyroscopes and tri-axis accelerometers, packaged in 4x4x0.9mm. Main ap-3Chapter 1. IntroductionFigure 1.1: Development stages of different MEMS devices[13]plications are consumer related (e.g. accelerometer with magnetometer ac-celerometer with gyro), automotive for ESC (Electronic Stability Control)and rollover functions[16]. There is a reduction in footprint and thicknessof sensors with reduction in current consumption down to micro Amperes,while performance increases. Down-scaling of the dimensions has huge po-tential, as it enables energy coupling at microscale level between electricaldomain and the various interface domains(mechanical, thermal, etc.) to-wards the real world[4]. This coupling may give access to parameters thathave been unmeasurable or uncontrollable so far. Fusion sensors in smartphones are one such examples, where magnetometers, gyroscopes and ac-4Chapter 1. Introductioncelerometers provide together 9 degree of freedom [15, 17];linear accelerationin 3 axis (x, y, z), angular rate (roll, pitch, yaw) and magnetic orientations(x, y, z). As most gyroscopes(Silicon based) are based on the measurementof angular rate, further integration of data is required to get the angulartilt. Bias error occurs when there is a output without any rotation im-posed on gyroscope. This error increases with integration in time, unlessis compensated using accelerometer (tilt angle) and magnetometer (head-ing) measurements. This makes the angular results more accurate, thanksto micro scale coupling. Even though MEMS gyroscopes are being widelyused in consumer electronics, they currently lack resolution for military andnavigation applications. In consumer applications emphasis is on sensorsdimension and power consumption and not necessarily on resolution. Forhigh-end navigation and instrumentation applications, displacement sensorsare required to achieve sensitivities below sub-Angstrom displacements [18]and gyroscopes are expected to achieve angular rate sensitivities on the orderof 1 mdeg/h [19].Unlike MEMS gyroscopes, fibre optical and ring gyroscopes are consid-ered ideal for military and navigation purposes. Optical and ring gyroscopesare capable of achieving very high sensitivities (in the range of mdeg/h), butare costly and bulky[19]. A comparison of gyroscope resolutions requiredfor different applications is shown in Fig. 1.2[20]. In the next section, anoverview of the Coriolis effect is presented. A figure of merit for MEMSgyroscopes is described in the subsequent section. This is followed by themotivation of this thesis in a described context. Needs and approaches withbackground work is covered in the subsequent section. Finally, an overviewand thesis organization makes the last section of this current chapter.5Chapter 1. IntroductionFigure 1.2: Resolution vs technology and applications of differentgyroscopes[20]1.1 Coriolis effectThe Coriolis effect is named after Gaspard Gustave Coriolis, who first de-rived the mathematical expression of the Coriolis force in 1835[22]. TheCoriolis effect arises from the fictitious Coriolis force, which appears to acton an object only when the motion is observed in a rotating non-inertialreference frame.?The effect of the Coriolis force is an apparent deflection ofthe path of an object that moves within a rotating coordinate system. The6Chapter 1. Introductionobject does not actually deviate from its path, but it appears to do so becauseof the motion of the coordinate system?[21]. There is no better exampleto understand Coriolis effect other than Earth, 1.3. Earth rotates counterclockwise around its poles. Assume a cannon is fired from north pole to atarget on equator. For a person outside the rotation frame, trajectory of thecannon ball will seem that canon deviates from its intended path; however,a person in the rotating frame (in this case Earth), will observe no apparentdeviation from a straight path.Figure 1.3: Coriolis Effect on the canon ball fired from north pole to equator(not drawn to scale)7Chapter 1. IntroductionIt is a known fact that with increase in latitude, speed of Earth?s rotationdecreases, hence decreasing the Coriolis effect. Einstein in his work (Annalender Physik) explained the variation of oscillations with latitude of a Foucaultpendulum in 1905[22]. The oscillation period of the pendulum was smallerin Nordic countries (near north pole) and higher at tropical countries (nearequator) with infinite oscillations at equators. This is one of the reasons,why many hurricanes occur with increase in latitude from equator. Trop-ical storms from equator rotate and strengthen to become hurricanes dueto increase in Coriolis effect[23]. The direction of deflection due to Corioliseffect depends on the object?s position on Earth. With Equator as refer-ence, objects deflect to the right in Northern hemisphere and to the left inSouthern hemisphere. In order to mathematically describe Coriolis effect itis necessary to understand forces acting on an object in a rotating frame.Let us assume that a disc is rotating from its center, and an object of mass?m? is placed on it, at a distance ?R?from its center. The speed of rotation atwhich the disc rotates is given as ? = d?dt , where ? is the angle of rotation.A centrifugal force will act upon the object, directed away from the rotationaxis, regardless of path of body, given by??F Centrifugal = ?m??? ? (??? ???R ).When the same object moves with a relative velocity V on the same rotatingdisc, a Coriolis force will also act on it, given by??F Coriolis = ?2m??V ???? .The cross product indicates that FCoriolis is perpendicular both to the rela-tive motion V and to the rotational axis ?. Objects with larger velocity willsupport larger Coriolis forces. Coriolis force only changes the direction of theobject and does no work on it. This is why the Coriolis force on a rotatingEarth varies with the sine of latitude ?, FCoriolis = ?2m?sin?V . Since theCoriolis force is perpendicular to V, a body in constant relative horizontalmotion is driven into a circular path, or inertia circle, with radius R = V/2?8Chapter 1. Introductionand a period of ? = pi/?[23]. Thus, larger Coriolis effect is observed awayfrom the equator with an increase in ?. A sensor which can measure therate of rotation of an object is called gyroscope. The name was coined byLe`on Foucault by combining the Greek words ?skopeein? (to see) and gyro(rotation)[21]. The earliest gyroscope utilized a rotating momentum wheelattached to a gimbal structure[24]. This structure can rotate about in allthree axes, while the rotor maintains its spin axis direction regardless of theorientation of the outer frame. This phenomenon of conservation of angularmomentum is used for seeking North pole direction on ships. The gyro-compass, aligned in a north-south axis, maintains its orientation, no matterwhat course a ship takes. It became very popular for navigational purposesin ships and airplanes as it was unaffected by metals used in manufacturingof ships and planes. An image of rotating toy gyroscope is shown along witha gyrocompass used in ships in Fig. 1.4.1.2 Performance figures of meritResolution, drift, zero-rate output (ZRO), and scale factor are impor-tant parameters that determine the performance of a gyroscope[26]. Reso-lution of a gyroscope is the minimum rotation rate that can be detected lim-ited by the noise floor of the system, and is expressed in units of (degree/s/?Hz).Resolution is limited by the equivalent input noise - in the case of themechanical sensor, the dominant component of the noise is the mechano-thermal noise, considered as white noise for the bandwidth of interest[27].White noise is expressed in terms of the standard deviation of equivalentrotation rate per square root of bandwidth of detection [(degree/s)/?Hzor (degree/h)/?Hz] for analog output, or bits/(deg/s) for digital output.9Chapter 1. IntroductionFigure 1.4: Toy gyroscope and the gyrocompass[25]Drift is the minimum change in rotation over the time and has units of(degree/s). The Scale factor is defined as the amount of change in the out-put signal per unit change of rotation rate and is expressed in V/(degree/s).Dynamic range refers to the range of input values over which theoutput is measurable within a prescribed distortion (nonlinearity). It ismeasured as the ratio between the maximum and minimum input rotationrate (full scale rate) that generates a measurable output. Full Scale Range(FSR) defines the upper and lower limits of what a gyroscope can measure.Smaller full-scale range indicates more sensitive output and larger indicatesless precision. Generally, for navigation purposes a smaller full scale rangeis desired while for gaming applications a larger one.For no rotation, the output signal of a gyroscope is a random functionaccumulating the effects of white noise and drift, which is called the ZRO10Chapter 1. Introduction(zero rate output) [28]. Depending on the applications and performancerequired by them, MEMS gyroscopes are broadly classified into rate gradeand navigation grade. Some performance parameters for rate grade gy-roscopes are given as 1) bandwidth > 100 Hz 2) full scale range- 100-1000degree/s and 3) bias drift- 10-1000 degree/h. Similarly, typical parametersfor navigation rate are 1) Bandwidth- 1-10 Hz 2) full scale range-75-300degree/s and 3)bias drift- 0.1-1 degree/h.1.3 MotivationAs seen in the previous section, MEMS gyroscopes have three to four foldlesser resolution compared to fibre optical and ring laser gyroscopes. Navi-gation applications require very high resolution, and MEMS gyroscope areyet to reach that milestone. On the other hand, thanks to miniaturization,rate grade gyroscopes are being held as success in consumer electronics. CanMEMS gyroscope follow the Moore?s law and keep shrinking, while improv-ing performance? According to a research article, performance of MEMS gy-roscope is most affected by down scaling, when compared with other inertialsensors such as accelerometer and pressure sensors. Scaling down of sensingforces is described in 1.1. It is seen from table 1.1 that, in general, scaling ofinertia sensors will reduce the system performance of miniaturized systems(? < 1 is the isotropic scaling factor of the linear dimensions). Moreover,gyroscopes are much more challenging sensors than accelerometers or pres-sure sensors. The absolute magnitude of the Coriolis force sensed is orders ofmagnitude lower than the non-inertial force sensed by any high volume pro-duction MEMS accelerometer. To achieve high performance and low cost,care must be taken during the initial design, such that manufacturing vari-11Chapter 1. IntroductionTable 1.1: Scaling of sensing force for pressure sensor, accelerometer andgyroscope [29]Description Pressure Sensor Accelerometer GyroscopeSensingForce(Fs)Fs = ?2A0P Fs = ?3m0a Fs = 2?4m0v0?Parameters A0=area,P=press. m0=mass,a=accel. v0=vel.,?=ang.rateations, temperature, linear acceleration variations, and packaging issues areaddressed. Thus, the aim of this thesis is to provide system level solutionsto enhance the performance of the MEMS gyroscopes, such that their per-formance is not limited by technology. Two distinct approaches are possiblefor improving the sensitivity performance of the MEMS gyroscopes: 1) tech-nological, designing and fabricating structures with very thick layers (highaspect ratio) or/and 2) Using system level solutions for increasing the sensi-tivity, for instance by inducing large displacements in the driving mode byhaving larger drive displacement or by externally increasing quality factor.High aspect ratio micro-structures (with large inertial masses) can be for in-stance fabricated in fabrication processes like thick surface micro-machiningprocesses, bulk micro-machining and LIGA (Lithographie, Galvanoformung,Abformung) processes[30]. In 1999, Robert Bosh Gmbh developed a surfacemicro-machined gyroscope with 12 ?m thick poly-silicon layer and achieveda resolution of 0.4 degree/s/?Hz with a bandwidth of 100 Hz [31]. Sam-sung Advanced Institute of Technology demonstrated a resolution of 0.013degree/s/?Hz, with 40 ?m thick bulk micro-machined single crystal silicon12Chapter 1. Introductionsensor with mode decoupling [32]. Ring gyroscopes with 100 ?m and 200?m thicknesses have been reported by British Aerospace Systems [33]andby University of Michigan[34], respectively. With tested technologies avail-able like SOI Tronics R? of MEMSCAP, the highest thickness available inopen fabrication services is 25 ?m (SOIMUMPS R?). Limited thickness con-straints leads to alternate avenues for increasing the sensitivity. Moreover,size of the structure is also limited by the area. Most of the consumerelectronics requires higher performance/ area on the chip. Not only sensordimensions are limited due to this requirement, but also electronics associ-ated for readouts too. Electronic readouts have to be carefully engineeredwith low noise levels and power consumption. System level solutions areable to further improve the sensing performance beyond these technologyrelated constraints. High resolution can be achieved by reducing noise, givenessentially by ?TNE 1 and most importantly ?nm (mechano-thermal equiv-alent). Mechano-thermal equivalent angular rate decreases with increase inthe quality factor and hence higher Qs are desired. Cost is the driving forcebehind any successful sensor design. MEMS sensors require high qualityfactors (Q) for high sensitivity thus requiring hermetically sealed packages,which are costly. According to Analog Devices c?, damping aids neverthe-less in cushioning the fragile sensors from shocks and vibrations. In the caseof ADRXS gyroscope, the mass is a mere 4 ?g, supported by only 1.7 ?mwide flexures over substrate.Study of damping becomes paramount, as damping force is frequencydependent and contributes to frequency dependent noise. An optimizationmechanism is required for matching the noise from electronic and mechanical1 ?TNE is the total noise equivalent rate and is cumulative noise from mechanicalsensor and electronic readout: ?TNE =??2nm + ?2ne13Chapter 1. Introductionstages. Currently, gyroscopes are designed for 10-30 kHz resonant frequency,with only some exceptions, such as bulk acoustic wave gyroscopes, which areoperated in MHz range [50]. As the frequency is inversely proportional tothe mass (? =?k/m), reduction in size will enhance operating frequency ofgyroscope. According to literature, damping forces play an important role inthat frequency range[35, 36], thus design optimization becomes imperative,to achieve good signal to noise ratio (SNR).Currently the estimation of the mechano-thermal equivalent noise usesthe assumption of a constant (frequency-independent) damping coefficient,in contradiction with the frequency-dependent damping models. While thisassumption might be correct for low frequencies, it fails to consider the morecomplex behavior of gas damping at higher operating frequencies, not onlyin gyroscopes but many other inertial sensors. The interaction of the mov-able mass with the surrounding fluid generates both elastic and dampingforce components, both dependent on frequency (and on the amplitude ofmotion for large displacements). This complex behavior shapes the resultingmechano-thermal noise, aspect that can be exploited in the signal-to-noiseratio optimization process of micro-structures used as sensors in our gyro-scope [37, 38]. An image of cost associated with different sections of MEMSdevelopment cycle for automotive and consumer electronics [16] is presentedin Fig. 1.5, along with problems covered in this thesis.1.3.1 Needs and approachSystem level solution is required for increasing the sensitivity of gyroscopesfor a given die area and to achieve higher resolution. Non-deterministicnoises like drift bias, thermal and electronic noise have to be minimized,14Chapter 1. IntroductionFigure 1.5: Cost of MEMS development cycle [16]but cannot always be controlled by designer, due to their random behav-ior. Deterministic errors on the other hand, like quadrature errors, canbe controlled and nullified by using appropriate methodologies. A noise-minimization focused design needs to consider the spectral variation of themechano-thermal noise in the structure design process. While performingnoise analysis on MEMS sensors, electronic noise is also referred to an equiv-alent sensor input signal. The design effort should be to increase the me-chanical signal in the first stage so that successive electronic amplificationis reduced. This in turn reduces the power consumption. For system with15Chapter 1. Introductionnegligible mechano-thermal noise(electronic noise limited), amplification ofsignal results in better SNR. For systems with significant mechano and elec-tronic noises, an optimization methodology is required to match them. Thework in [38] can be extended for gyroscopes, where we showed dependentintegrated noise analysis. The frequency-dependence of the damping andnoise leads to SNR-optimal regions of operation for a given (constant area)sensing structure. Chapter 5 is dedicated to understanding the damping phe-nomenon and the use of this knowledge for a design optimization procedure.Optimization technique has to be implemented and tuned for gyroscope byprobing sensitivity, SNR and frequency varying noise. To achieve high sen-sitivity, a) large drive displacement, b) lower mechano-thermal equivalentangular rate, c) no mismatch between the drive and sense mode frequencyand d) quadrature error free structure are required. Mismatch betweendrive and sense mode can be reduced by applying the concept of stiffnesssoftening, i.e by applying a DC bias. To obtain the large drive displace-ment, lowering of ?TNE and minimizing of quadrature error, parametricamplification can be implemented. In parametric amplification energy ispumped into resonator by varying parameters in time. Example of param-eters that may be varied in time for gyroscopes are its resonance frequency? and stiffness k. This effect is different from regular resonance because itexhibits instability regions [39]. Devices which exploit the parametric effecthave been common in the microwave, electronic and optoelectronic areassince the 1960s, [40, 41]. Parametric excitation has also been used in somemacroscopic vibratory gyroscopes designs [40, 42]. It is only recently thatthe application of parametric excitation to MEMS and NEMS devices hasbeen investigated [40]. Whilst there is an abundance of theoretical researchpapers describing the phenomenon of parametric resonance, there is limited16Chapter 1. Introductionreporting of amplified MEMS/NEMS devices. The seminal work reported in[43] demonstrated parametric amplification with a linear gain of at least 25in micro-cantilevers by using electrostatic excitation and optical detection.Parametric amplification with a gain of 25 in micro-cantilevers through theexploitation of geometrical nonlinearities is reported in [44], with detectionperformed by a piezoresistor. Parametric amplification with a gain of 30has been achieved in micro-disc resonators by optical actuation [45]. Hon-eywell Inc. patented the idea of implementing parametric amplification inMEMS gyroscope in 2002, where the external AC signal pumps energy attwice the resonant frequency to the sense-electrodes [46]. This concept waswell explored by [39]in which they perform a feasibility study of paramet-ric amplification theory on MEMS tuning fork gyroscope. Dr.Turner?s labfrom University of California Santa Barbara has worked extensively on theparametric amplification, from resonators to accelerometers. Only recently,research groups around the world have started to explore parametric am-plification for gyroscopes with feasibility studies. In one of the recent pub-lications [47], authors demonstrated experimentally that by using a para-metric resonance based actuator, the drive-mode signal has a rich dynamicbehavior with a large response in a large bandwidth (1kHz). Parametricamplification in ring gyroscope on both primary and secondary mode wasalso demonstrated in 2008 [48]. Parametric actuation can be used as well toamplify the action of the Coriolis force in the sensing mode of a gyroscope.But to the best of author?s knowledge no structure has yet demonstratedthe parametric pumping in the secondary mode, where Coriolis force in-duces motion. This was the basis of our work in [49], where by applying theexternal force at the sense electrodes at the appropriate phase, sensitivityin the sense mode was increased. The phase sensitivity of parametric para-17Chapter 1. Introductionmetric actuation was used for either amplifying the Coriolis force or for thedamping of the quadrature errors present in sensing mode. Simultaneousapplication of parametric pumping in both drive and sense mode are of keyinterest and again to best of author?s knowledge no one has yet implementedthis so far for non ring gyroscopes. Moreover, to produce optimum drivingto enhance the parametric amplifications, different comb-drive structuresneed to be explored. We have proposed a special novel shaped combs forprimary mode parametric resonance, unlike the conventional fringe-basednon-interdigitated combs. A complete analysis and experimental work hasbeen performed, and a related journal paper is in preparation. To sum up,parametric excitation can be used to amplify desired motions and diminishthe unwanted motions (quadrature errors). Chapter 6 is dedicated to theconcept of parametric resonance,: amplification and damping in MEMS gy-roscope using gap-varying combs. Chapter 7 describes the usage of shapedcombs for the same parametric actuation techniques with implementationon two technologies, SOIMUMPs R? and SOI Tronics R?. The concept of dy-namic pull-in phenomenon for MEMS vibratory gyroscopes is described inthe chapter 8. It is based on the well known static pull-in phenomenon ofhigh-sensitivity MEMS accelerometer, where pull-in voltage is modulatedby external acceleration. This phenomenon can reconcile the small size ofthe proof mass with high performance requirements. In this case, a com-mon mode harmonic voltage is applied on the sensing comb capacitors, ata frequency equal to that of the Coriolis induced motion (close to the reso-nant frequency of the sensing mode). Previous studies have indicated lowervalues of the dynamic pull-in amplitudes for frequencies close to the res-onant frequency of the MEMS device[51, 52]. The variation of dynamicpull-in voltages (AC signal amplitude) with the external acceleration can be18Chapter 1. Introductioncorrelated with the amplitude of the signal to be sensed.1.4 Overview and thesis organizationThe objective of this thesis is to study different nonlinear techniques toimprove SNR (Signal to Noise Ratio) of inertial MEMS sensors. Three dis-tinct aspects are categorically studied in this thesis, a) Noise reduction andoptimization taking into account the spectral behavior of damping, b) Me-chanical signal amplification of the sensor using techniques like parametricamplification and c) Increasing the sensitivity of the sensor by operating itat the border of stability, thus making it sensitive it to the external acceler-ations.The rest of the thesis can be summarized in Fig. 1.6 and is structuredas following:Chapter 2: Background work on gyroscopes is presented. State of the artof MEMS gyroscopes and their applications are presented with their speci-fications requirements.Chapter 3: Modeling and designing of capacitive gyroscopes are presented.Most of the discussion on modeling of gyroscopes covered in this dissertationcan be applied to a wide range of micro-machined inertial sensors.Chapter 4: Fabrication technologies such as SOIMUMPS R? of MEMSCAPand MEMS SOI Tronics R? of TRONICS is presented in this chapter. Asystem level modeling using CoventorWare R? is presented for one of the gy-roscopes.Chapter 5: Frequency dependent noise analysis and damping is covered inthis chapter a semi automatic tool for optimization is presented.Chapter 6: Concept of parametric amplification and damping is applied on19Chapter 1. IntroductionFigure 1.6: Thesis organizationa MEMS gyroscope using gap-varying combs.Chapter 7: A novel shaped combs are designed, modeled and characterizedfor the parametric actuation. These shaped combs have been implementedin two technologies, SOIMUMPS R? and SOI Tronics R? .Chapter 8: The concept of dynamic pull-in phenomenon for MEMS vibra-tory gyroscopes is covered.Chapter 9: Incipient and future work is presented.20Chapter 2Gyroscopes2.1 Classification of gyroscopesA brief review on the existing gyroscopes is presented. Gyroscopes can bebroadly classified in five categories, as seen in Fig. 2.1.Figure 2.1: Classification of gyroscopes[54]21Chapter 2. Gyroscopes2.1.1 Spinning mass gyroscopeWe begin the chapter with a macro gyroscope based on thespinning wheel concept, as illustrated in Fig. 2.2. Spinning motor mountedin a single gimbal has a one degree of freedom. The inner frame is suspendedon the external frame, which has a single degree of freedom and can onlymove orthogonal to the inner frame (input frame). When a force (torque)isapplied to the inner gimbal, either by rotating the external frame or bydirectly applying perpendicular to the inner rotor, it starts to precess2.This precession is generally sensed by a spring. The extension and com-pression of the spring directly corresponds to the external force and thedirection in which the angular rotation is applied. Spinning mass gyroscopeFigure 2.2: Spinning gyroscope with spring[56]were extensively used in aircraft navigation in 1920-40s. Springs, whichwere used for sensing the tilts, were also the limiting factor(lost stiffness).2A constant torque T applied on one axis of the gyroscope, causes drift of angularvelocity ? called as precession rate. T = H d?dt = H ? ?22Chapter 2. GyroscopesThese gyroscopes had a performance of 10 deg/h and navigation of airplanesneeded better performance. Charles Stark Draper, a pioneer engineer, madechanges to this gyroscope by adding a feedback loop to maintain the gimbalat null position, thus removing the need for springs. Gimbal maintains itsnull position due to the restraining torque which provides damping. A servomotor, dampers(dynamic range tuning and shielding against shocks) andtorquers were incorporated. The servo motor produced currents equivalentto angular tilts, such that an equivalent torque was produced to bring therotor to the null position. They were basically the first single degree of free-dom (SDOF) rate integrated gyroscope, which revolutionized navigation inships, submarines and aeroplanes. An image of rate integrated gyroscopewith spinning mass and feedback torquers is shown in Fig. 2.3. Based onthe same principle, dynamically tuned gyroscopes(DTG) were invented in1960s, which were only perfected in 1980s [57]. They have a gimbal config-uration that balances the torques from the support flexures with dynamicspring torques created by gimbal ?flutter? while the gyro rotor spins. Thisconfiguration aids gyro rotor to spin in two axes without any external force.DTG was used for many years in aerospace and military industry and wasincluded in inertial measurement unit in space shuttle. Their performancevaries from .01 deg/h to 30 deg/h for a typical size of 40 mm diameter ?40 mm height[57].2.1.2 Optical gyroscopeOptical Gyroscopes are the highest resolution gyroscopes currently avail-able. As there are no moving parts like a spinning wheel, there are no lossesdue to friction. They can be classified as Ring laser gyroscopes (RLG)23Chapter 2. GyroscopesFigure 2.3: Rate integrated gyroscope with feedback loop[55]and Fibre optics gyroscopes. They are based on the principle of Sagnaceffect. Ring laser gyroscope was first demonstrated in 1963, by setting up aclockwise and anti clockwise resonant light beams, reflected around a closedenclosure by mirrors 2.4. When a rotation is imposed, the beam in thedirection of rotation has a longer path than the other(Sagnac effect). Thephase difference between the light beams gives the angle of rotation. Theyhave been used in commercial flights like Boeing 757, 767 and airbus [58].Because of their robust performance, they are used in stellar aided missiles,while smaller versions of them are used in tactical missiles. One of the dis-advantages of RLG is lock-in [59]. Lock-in is a phenomenon which occurs atsmall rotation rates, where the small frequency difference between the laser24Chapter 2. GyroscopesFigure 2.4: Ring laser gyroscope [59]beams couple, thus locking at a single false value. This error is compensatedby using a dither motor, which vibrates the gyroscope to remove the lockin. Fibre optical gyroscopes were developed based on the same principle toincrease the sensitivity (signal amplification) of RLG, by not making themvery big. The propagation medium is fibre cables, instead of cavity enclo-sure 2.5. They are cheaper, smaller, power efficient and have longer lifetime. As a disadvantage they require relatively complicated phase detectionelectronics.2.1.3 Fluidic gyroscopesEfforts have also been made in using liquids and gas for making fluidic gyroscopes.They are classified as flueric gyroscopes, dual axis rate transducer and25Chapter 2. GyroscopesFigure 2.5: Fibre optical gyroscope[60]magneto-hydrodynamic gyroscope. The flueric gyroscope [54] contains aspherical cavity with holes in walls, with a rotating mass or swirl of gaswithin the cavity. When the device rotates, the direction of the swirl of gasremains fixed in inertial space due to the inertia of the fluid, and the appliedrotation can be detected by monitoring pressure changes inside the cavitythrough the holes in cavity walls. The dual axis rate transducer is based onthe same principle, but instead of gas has liquid mercury as fluid. In the caseof magneto-hydrodynamic gyroscope, there is a permanent magnet, placedbelow a conductive liquid. When the sensor is rotated, due to inertia, theapplied magnetic field and the conductivity of the fluid generates an electriccurrent due to Faraday?s law. This electric current corresponds to the an-gular tilt. They were designed for stabilization purposes but are performingpoorly, as the liquid properties vary with temperature.2.1.4 Non vibratory gyroscopesNon vibratory gyroscopes can be classified as micro-optical and micro-fluidicgyroscopes. Micro-optical gyroscopes can further be classified as resonantoptical and photonic crystal gyroscopes. Micro-optical gyroscopes were mo-26Chapter 2. Gyroscopestivated from the optical gyroscopes but their performance is not high, asthe scaling down of the path length reduces the sensitivity. Further, theirresolution is limited by the shot noise of the detector. These gyroscopeshave shown resolution from 2500 deg/h to few deg/h [58]. Resonant micro-optic gyroscope are also being investigated in many institutions. High qual-ity channels are etched in silicon to produce waveguides for electromag-netic waves, similar to fibre optic gyroscopes. Photonic crystal gyroscopesare based on the Sagnac effect as well. The use a linear array of equallyspaced identical local defects situated within an otherwise perfect photoniccrystal[61]. Perfect cavities, with high Q, will absorb light and propagateslight to cavity defects based on tunneling effects. This effect is similar to thepropagation of electrons in semiconductor medium. These are under studyas a futuristic alternative for small, cheap gyroscopes.2.1.5 Vibratory gyroscopesVibratory gyroscopes are perhaps the most miniaturized gyroscopes. Thissection will elaborate the different vibratory gyroscopes, with an emphasizeto MEMS gyroscopes, which are the concentration of this thesis work.The basic architecture of any vibratory gyroscope comprises an oscillatordriven into linear or angular oscillations. This driven mode of oscillation iscoupled to a secondary vibration mode oscillator, that measures the Corio-lis induced motion due to the combination of drive vibration and externalangular rate. This can be explained by a tuning fork. When a tuning forkis excited by a tong, its two arms are set to vibrations which are 180 degapart. When an angular rotation is applied to the tuning fork, orthogonalto it?s excitation vibration, a resulting vibration is induced which is both27Chapter 2. Gyroscopesorthogonal to the input angular rate and the primary excitation, as shownin Fig. 2.6.Figure 2.6: Tuning fork gyroscopeWhat causes the tongs to vibrate in orthogonal direction is described bythe derivation of the dynamics of a body moving in the rotating referenceframe [21]. The acceleration experienced by a moving body is explainedwith the help of the following definitions:A:Inertial frame (non rotation)B:Rotational frame (non inertial)XA: Position vector relative to inertial frame AXB: Position vector relative to rotation frame B?: orientation vector of rotation frame B relative to frame A?: Angular velocity of rotating frame B, ? = ??X : Position vector of rotating frame B28Chapter 2. GyroscopesImage 2.7 shows the coordinates of a red colored object in both rotating andinertial frames. The time derivative of position vector X, which is definedin both rotating and inertial frames of reference, is given asX?A(t) = X?B(t) + ?? ?XB(t) (2.1)The first and second derivatives of the position vectorX give, the velocityand acceleration of the body in the two reference frames.Figure 2.7: Position vector relative to the inertial frame A and to rotatingreference frame B [21]XA(t) = X(t) +XB(t) (2.2)X?A(t) = X?(t) + X?B(t) + ?? ?XB(t) (2.3)29Chapter 2. GyroscopesX?A(t) = X?(t)+ X?B(t)+ ??? X?B(t)+ ??? (???XB(t))+ ???XB(t)+ ??? X?B(t)(2.4)In the above equations X?(t) corresponds to the linear acceleration A, X?B(t)corresponds to the acceleration of the object with respect to rotating frameand X?B corresponds to the velocity vB of the object with respect to frameof rotation. The above equation can be rewritten asaA = A+ aB + ???XB + ?? (??XB) + 2?? vB (2.5)Where, (A+aB + ???XB) is the local acceleration, and (?? (??XB))is centripetal force. The Coriolis acceleration which causes the fictitiousinertial force when observed in the rotating frame, is given as 2?? vB. Theforce acting on the object, when a rotation is imposed on it can be obtainedmy multiplying the above equation with its mass of the moving object.For a constant angular rate input ?? = 0, and for very small angular rate,centripetal force (?? (??XB)) and ???XB become negligible compared tothe Coriolis force 2?? vB. Hence the Coriolis induced motion is sensed forobtaining the angular rate; the corresponding rotation angle is obtained bya supplementary integration. Coriolis acceleration is proportional to the ve-locity of the object in motion, undergoing rotation. Direction of the motiondue to this acceleration is given by vector multiplication. An object movingin x direction, undergoing rotation about z axis, will thus get displaced in ydirection when observed in the frame of rotation, as seen in the case of tuningfork. In the two degrees of freedom mass-spring damper system, the forcesacting on the system are the spring and damping forces Fspring and Fdampingin the x and y directions, and the actuation force Felectrostatic(generally) inthe x direction, as shown in Fig. 2.8.30Chapter 2. GyroscopesFigure 2.8: Two degrees-of-freedom mass spring damper system undergoingrotationFelectrostatic + 2m?z y? = mx?+ cxx?+ kxx (2.6)Fy ? 2m?zx? = my? + cyy? + kyy (2.7)The Coriolis term in x direction (2m?z y?) is negligible when compared to theelectrostatic force, hence is not considered. In the y direction, (?2m?zx?)is the Coriolis induced force which needs to be sensed. Fy is the totalnet external force, comprised of parasitic and external forces. Generally,quadrature errors induce undesired motion in y direction. They are gener-31Chapter 2. Gyroscopesally filtered out, from the Coriolis induced motion, as they have a 90 degphase shift relative to each other. Vibratory gyroscopes are classified asdegenerate or non-degenerate modes. When the resonant mode of drive di-rection and sense direction are identical, then the gyroscope is termed ashaving degenerate modes. This type of gyroscopes measure absolute angle.The non-degenerate gyroscopes have distinct resonant frequencies for driveand sense modes, and measure angular rate. Vibrating shell and solid gyro-scopes are typical degenerate types, whereas all other vibratory gyroscopesare non degenerate type. In vibrating shell gyroscope, standing waves aregenerated on the shell, which precess when rotation is imposed on the sensor.The shell of this sensor is generally made up of quartz material.Hemispherical resonator gyroscope (HRG) is one such example as shownin Fig. 2.9.Figure 2.9: Hemispherical resonator gyroscope[62][84]Sub-deg/h resolution performance has been reported for gyroscopes andused in space inertial reference units (SIRU). They have demonstrate a sta-bility of .003 deg/hr and a resolution of .00006 deg/h/?Hz[62]. Ring gyro-scopes are degenerate type which have been implemented in MEMS technol-ogy based on the principle of vibrating wine glass. The phenomenon of pre-32Chapter 2. Gyroscopescession of waves (Wine-glass modes) arise in several shell-like structures, in-cluding common ring gyroscope designs, which are fabricated using isotropicmaterials[63]. University of Michigan, Georgia Tech and Southampton uni-versity have implemented many versions of ring gyroscopes. They are excitedusing the electrostatic forces, causing the ring to vibrate elliptically. Whenthe rotation is imposed, a portion of energy is transferred from the drivemode to sense mode, causing the elliptical modes to rotate. The amplitudeof radial displacement of the secondary flexures mode is proportional to ro-tation. This displacement thus corresponds to the external rotation rate.An image of ring gyroscope is presented in Fig. 2.10. Generally, ring gyro-scopes have many electrodes, placed at the periphery of the ring, to cancelthe mismatch between the flexural driven mode and the sense mode.Figure 2.10: Ring gyroscope resonant modes [64]Non-degenerate MEMS gyroscopes are classified as in Fig. 2.11 [64],where either a single mass or combination of masses have several DOFs.Based on the application, either large bandwidth gyroscope or high sensi-33Chapter 2. GyroscopesFigure 2.11: Classification of non degenerate MEMS gyroscopestive, low bandwidth gyroscopes are designed. In applications such as spacenavigation, where robustness is the key, temperature changes, shocks andpressure changes should not vary the sensitivity of the angular sensor. Forthese sensors, the resonant frequency of the sense mode of the gyroscope istuned in between the drive resonant modes. A research team at the Uni-versity of California Irvine, has worked a lot in this area and has publishedseveral research articles on this concept two-mass vibrating gyroscopes arethe most popular choice; they are based on the tuning fork mechanismdescribed previously in this section. Sensitivity to linear acceleration is can-34Chapter 2. Gyroscopesceled by oscillating the two masses(drive mode) out of phase, as shown inFig. 2.12. When one external rotation is imposed, the phases of the sensemodes is orthogonal to each other and differential capacitance changes arepicked off by the sensing electrodes.Figure 2.12: Prototype of tuning fork gyroscope [64]Table 2.1 summaries the present states of MEMS gyroscopes.Table 2.1: MEMS gyroscope prototypes in literatureYear/Ref. Institution Sens.axis Resolution Technology1994/1 U.Mich[65] Z 0.5deg/s/?Hz Ring gyrofabricatedusing electro-forming35Chapter 2. GyroscopesTable 2.1 ? continued from previous pageYear/Ref. Institution Sens.axis Resolution Technology2001/2 U.Mich[66] Z .01deg/s/?Hz Ring gyrofabricated us-ing HARPS(80?m thick-ness)2001/3 CMU[67] X,Y&Z0.5deg/s/?Hz Two mass,drive sus-pended gyro-scope madeusing thinfilm CMOSprocess.Device thick-ness is 5?m.Copper waspreviouslyused, whichcaused exces-sive curling.Aluminiumshowed bet-ter results.36Chapter 2. GyroscopesTable 2.1 ? continued from previous pageYear/Ref. Institution Sens.axis Resolution Technology2003/4 CMU[68] X&Y .02deg/s/?Hzat 5HzPost CMOSmicro ma-chiningemployed.Uses combi-nation of 1.8? m thin filmstructuresand 60?mfor springs inin-plane andout of planestiffness re-spectively.Curling isremovedwhich wasa problembefore.37Chapter 2. GyroscopesTable 2.1 ? continued from previous pageYear/Ref. Institution Sens.axis Resolution Technology2004/5 Georgia Tech[69] Z .01deg/s/?Hzat 12HzTuning forkgyroscopefabricated inSOI technol-ogy (40?mthickness)2006/6 METU[70] Z .097deg/s/?Hz Single massgyroscopefabricated inSOIMUMPs R?(60?m thick-ness).2006/7 UC Irvine[71] Z 0.64deg/s/?Hzat 50HzRobust gy-roscope wasdemonstratedwith 1 DOFdrive modemass and2-DOF sense-mode mass.This wasoperated atair.38Chapter 2. GyroscopesTable 2.1 ? continued from previous pageYear/Ref. Institution Sens.axis Resolution Technology2008/8 UC Irvine[72], Z 0.09deg/s/?Hz This designwas theimprove-ment overtheir pre-vious massdistributedgyroscope. Ithad higheroperationalfrequency(2.5kHz ) andbandwidthover 250Hz.39Chapter 2. GyroscopesTable 2.1 ? continued from previous pageYear/Ref. Institution Sens.axis Resolution Technology2010/9 Georgia Tech[73] Z 0.006deg/s/?HzBulk acous-tic wavegyroscopehas been re-ported whichoperates in6MHz range.Gyroscopehas a hightquality factor(Q 100,000).2011/10 UC Irvine[74] Z 0.27 ? 10?4deg/s/?HzRate gyro-scope hasdynamicallybalanced 4mass (QMG),which sup-pressessubstrateenergy dissi-pation, whichmaximizes Q.40Chapter 2. Gyroscopes2.2 State of the art MEMS gyroscopes inautomotive and consumer marketMEMS gyroscopes have been part of the automotive industry since 1995,when they were first used in luxury cars for ESC(Electronic stability con-trol). ESC helps in reduction of fatal accidents by preventing roll overs.MEMS gyroscopes and accelerometers are used for sensing the roll overangles, deploying airbags and aiding in control. They also aid in automati-cally activating the individual brakes (asymmetric actuation), when the cargets out of control due to over steering. These gyroscopes are required towith-stand temperature variations, vibrations, shocks etc, while not com-promising on performance. Syster Donner c? were the pioneers in the fieldwho introduced the tuning fork gyroscope packaged in quartz casing[75].Their gyroscope, unlike other sensors from Robert Bosch GmbH and Ana-log Devices c?, uses piezoelectric actuation and sensing mechanism. Boschdeveloped their first MEMS gyroscope for ESC application based on bulkmicro fabrication and then moved to surface micro machining[76]. SMI540is their latest surface micro machined gyroscope, an evolution of MM series;the system includes as well an accelerometer and a signal processing ASICpacked together in a molded plastic housing[78]. Analog Devices c? came upwith the ADRXS, fully integrated gyroscopes. These sensors have full scalerange of 150 deg/s to 300 deg/s; fabricated using surface micro machining,and have two to four independent polysilicon structures[75]. The specifi-cation for MEMS gyroscopes in the automotive applications is depicted inTable. 2.2.As seen in the table, the emphasis is more on the robustness of the gyro-scope rather than the resolution. High resolution sensors could be used for41Chapter 2. GyroscopesTable 2.2: Requirement for ESC application[79]Performance RequirementRange (?75to? 300)Total bias offset(deg/s) ?3.0Operating temp -40 to +80Scale factor (%) ?3.0Scale factor (% of full range) ?3.0Offset over temperature range(deg/s) ?0.25 to ?5inertial navigation eliminating the expensive GPS-controlled inertial navi-gation platforms. New generations of MEMS sensor clusters can be usedfor various applications, exploiting the electro-mechanical coupling at microscale. A three degree freedom monitoring of the car trajectory is one suchexample, as illustrated in 2.13. It could be used as a stand-alone modulefor adaptive road lightning system, GPS navigation and imminent collisionwarning systems. The developments in software and GPS precision needto be matched by similar developments at the level of the correspondingembedded systems. MEMS gyroscopes have recently become very popularin consumer electronics. Nintendo Wii video game used the first 3 axis gy-roscope, ITG 3200 (digital), in 2009[77]. ST Microelectronics soon followedwith a 3 axis gyroscope (analog), LYPR540AH, which was part of hand setphones[77]. Gyroscopes are not only used for gaming applications, but alsobeing used in camera phone modules. Currently, the phone makers are fo-cussing to design cameras with 8 mega pixel or higher resolution. Opticalimage stabilization becomes crucial as hand jitters in any light condition42Chapter 2. GyroscopesFigure 2.13: MEMS inertial navigation platform for carsimpacts the picture quality. Gyroscopes are helping the image stabiliza-tion control, making the picture quality par with DSC (digital still camera).Their small sizes and thin chips allows them to be included in the smartphones. In case of the minimally invasive surgery (MIS), where cathetersare introduced through a small slit in the body, an IMU system would fur-ther enhance the control of surgeon. As the organs pulsate, an accuratefeedback can be obtained. The critical requirements of MEMS gyroscopesare small size, lower current consumption and fast response time, withoutcompromising the sensitivity and resolution. Bosch Sensortec GmbH, re-cently launched the smallest three axis MEMS gyroscope, BMG 160, of size43Chapter 2. GyroscopesTable 2.3: Sensitivity and full-scale-range for consumer gyroscopes[83]Application Sensitivity Full scale range (FSR)Image stabilization 20 mV deg/s 20-30 deg/sNavigation 4-15mV deg/s 50-67 deg/s3-D remote 2 mV deg/s 500 deg/sGame controller 0.5 mV deg/s 2000 deg/sSurgical tools 500 deg/sTremor compensation 150 deg/s3x3x.9mm [80], competing with IMU 3050 of InvenSense c? of 4x4x.9mm[81]and L3G4200D of ST Microelectronics c? of 4x4x1.1mm size [82], all consum-ing a current less than 10 mA. These chips contain both MEMS sensors andthe readout and control circuitry. Table. 2.3 presents the specification fordifferent consumer applications.MEMS gyroscopes applications are evolving fast and are already con-tributing to improving the quality of life. Sensor fusion is a key ingredientto evolve further complex applications.44Chapter 3Modeling of vibratoryMEMS gyroscopeThis chapter describes the mathematical modeling of a single mass gyro-scope.3.1 Resonance characterizationThe structure of a MEMS vibratory gyroscope contains two resonant modes,whose coupling is modulated by the external angular rate. The genericequation of motion for one resonant mode is :Fext = mx?+ cx?+ kx (3.1)The above equation can be reformulated into a more convenient form, toemphasize on the main parameters. If we divide by the mass of the movablestructure, we get:aext = x?+?0Qx?+ ?20x (3.2)Here, ?0 =?km is the natural resonant frequency of the system, and Q =?kmc =12? is the quality factor. Quality factor is the number of cycles aresonator takes to dissipate stored energy, generally approximated as Q =energy storedenergy lost per cycle for Q >> 1. Depending on the damping, resonators are45Chapter 3. Modeling of vibratory MEMS gyroscopeclassified as underdamped (Q > 0.5), critically damped (Q = 0.5) andoverdamped(Q < 0.5). A good insight is obtained by the transfer functionin the frequency domain(s = j?) as seen inF [s] = m(s2 +?0Qs+ ?2)X[s] (3.3)The displacement of the resonator can be written as|x(j?)| =|F |/m?(?20 ? ?2)2 + (??0/Q)2(3.4)The static (dc level) displacement (? = 0) is |x(0)| = |F (0)|/k, while atresonance x(j?0)=Q|F (j?0)|k . The phase of the displacement x relative tothe excitation force F is given as:?(?) = tan?1(???0/Q?20 ? ?2) (3.5)Frequency response and the transient analysis for the resonator are shownin Fig. 3.1 and Fig. 3.2 respectively.As seen in the Fig. 3.1, for Q >> 1, the peak of amplitude is at theresonance frequency; for a constant spectral force amplitude, |x(j?0)| =Q|x(0)| there is no lag between the force F and displacement x in quasistaticmode. At resonance the phase is ?(?0) = ?pi/2, thus displacement lags theforce exactly by 90 deg. Designer can optimize the design for the resonatordepending on the application. As seen in Fig. 3.2, a maximum flat responsein the frequency domain is obtained at Q = 0.707; on the other hand, atQ = 0.5, then the transient time is minimum. Gyroscopes require highquality factors for sensing the minute displacements caused due to Coriolisacceleration. Thus an estimate for displacements, for Q  10 is presentedin Tab. 3.1[85].46Chapter 3. Modeling of vibratory MEMS gyroscopeFigure 3.1: Bode plot for different quality factorsTable 3.1: Mechanical gain of the resonator force to displacements [85]angular frequency gain(displacement) ? gain(displacement)0 1m?021k?021(m?20)? 1?0.7+ 0.5Q j?43(1k )?0 1(m?20)? 1??j 1Q?1c?02?0 1(m?20)? 1??3+ 2Q j?131k47Chapter 3. Modeling of vibratory MEMS gyroscopeFigure 3.2: Step response for different quality factor (damping) values3.2 Dynamics of gyroscopeA system level schematic of a vibratory MEMS gyroscope is presented inFig. 3.3. As described earlier, the primary mode(0x) of the gyroscope is ex-cited by applying an electrostatic force; the Coriolis acceleration will trans-fer part of the vibration energy into the secondary mode (0y). However,quadrature mode errors also get coupled due to fabrication imperfections.Fquadx = ?kxyy and Fquady = ?kyxx should also be included in the dynamicmodel, where kyx and kxy are the cross coupling stiffness in 0x and 0y modes.Primary mode actuation and secondary mode sensing are achieved by capaci-tive comb fingers. When operated at atmospheric pressure, the air molecules48Chapter 3. Modeling of vibratory MEMS gyroscopewill generate additional forces acting upon the movable mass, damping theoscillations. Thus the noise forces should also be correlated with the respec-tive damping coefficients, to establish a thermodynamic equilibrium betweenthe microstructure and the surrounding fluid. The dynamical equations ofFigure 3.3: System level schematic of a vibratory MEMS gyroscope(2-DOF) of the gyroscope taking into account quadrature mode and noisedue to damping is describes as following;Felectrostatic + Fnoise,x(cx, t) = mx?+ cxx?+ kxx+ kyxy (3.6)?2m?zx?+ Fnoise,y(cy, t) = my? + cyy? + kyy + kxyx (3.7)The equivalent spectral characterization is achieved by applying Laplacetransformation to eq 3.1 and 3.2.Felectrostatic[s] + Fnoise,x[s] = (ms2 + cxs+ kx)X[s] + kyxY [s] (3.8)?2m?zsX[s] + Fnoise,y[s] = (ms2 + cys+ ky)Y [s] + kxyX[s] (3.9)49Chapter 3. Modeling of vibratory MEMS gyroscope3.2.1 Drive mode oscillationsGeneration of momentum is necessary for the gyroscopic effect, as Corioliseffect is connected to the conservation of momentum, as described in chapter1 and chapter 2. The scale factor of the gyroscope is directly proportionalto the drive mode oscillation amplitude, hence designers are required tooptimize the drive loop for increased sensitivity gyroscopes. The drive modeamplitude is obtained as:|x(?)| =|F (j?)|/kx?(1? (?2yx+?2?20x))2 + ( ?Qx?0x )2(3.10)In the above equation ?yx is the quadrature component coupling the sensemode to drive mode. Generally, sense mode displacements are more thantwo magnitude smaller than the drive mode. As a minute quadrature errorfrom sense mode is coupled to a large drive mode displacements, it can beneglected. The phase of the drive mode steady state is :?0x(?) = tan?1 ??/Qx?0x1? (?2yx+?2?20x)(3.11)Here?0x =?kxm(3.12)?yx =?kyxm(3.13)Qx =m?0xcx(3.14)At resonance ? = ?0x; neglecting the small quadrature error, drive modephase becomes -90deg and amplitude is given as:x(?0x) = QxFm?20x(3.15)50Chapter 3. Modeling of vibratory MEMS gyroscopeIn order to drive the proof mass, area-varying combs are used for lateraldisplacement (0x) mode. Net capacitance for area varying combs with Nnumber of fingers is :Cx = (h(loverlapx + x)d0x+ Cf )N (3.16)Here,  is the permittivity, loverlapx is the overlap length of the capacitivefingers, d0x is the gap between the fingers and Cf is the parasitic capaci-tance (fringing field effects). Parasitic capacitance remains constant withthe change in overlap area from loverlapx to loverlapx + x. Change in capac-itance with respect to the lateral displacement leads to, for a single fingerpair:dCxdx=hd0x(3.17)Linearized actuation is the most common actuation scheme. AC voltageswith 180 degrees phase apart are applied on the left and right fixed combs,and a DC bias on the proof mass as shown in Fig. 3.4. The net force istherefore:Fnet,0x =12dCxdx[(VAC + VDC)2 ? (?VAC + VDC)2] = 2dCxdxVDCVAC (3.18)Fnet,0x = 2Nhd0xVDCVAC (3.19)3.2.2 The Coriolis responseThe sense mode oscillator is a 1-DOF resonator as well, which amplifies themechanical response to the Coriolis force. The equation 3.9, will lead tosense mode displacement and phase:? y(?) ?= ?z? 2x?0x ??20y?(1? (?20x?20y))2 + ( ?0xQy?0y )2+ ? ?xyx ? (3.20)51Chapter 3. Modeling of vibratory MEMS gyroscopeFigure 3.4: Linearized actuation scheme?0y(?) = ?tan?1(?0x/(Qy?0y)1? (?20x?20y)2) + ?0x(?) (3.21)Here?0y =?kym(3.22)?xy =?kxym(3.23)Qy =m?0ycy(3.24)52Chapter 3. Modeling of vibratory MEMS gyroscopeFor achieving maximum gain, sense-mode is desired to be tuned near thedrive mode resonance(?0x = ?0y). The phase of the sense mode becomes-90 deg from the drive velocity, and sensitivity( y?z ) becomes:y(?0y)?z=2Qyx?0y+?xyx?z(3.25)Optical stroboscopic method aids in obtaining the resonant mode of thesense mode. Resonant mode is obtained by applying a frequency sweep onthe gap varying combs as seen in Fig. 3.5. The 0y displacement correspond-ing to the sense mode is usually measured by using the gap-varying combs.The capacitance of the gap-varying combs with N fingers is with N fingersis, for a differential (top-bottom) structure:Cy? = hNloverlapyd0y ? y(3.26)Cy+ = hNloverlapyd0y + y(3.27)The net force applied on the gap-varying combs in the case of a differentialvoltage actuation is:Fnet,0y =12hNloverlapy(d0y ? y)2(VAC + VDC)2 ?12hNloverlap(d0y + y)2(?VAC + VDC)2(3.28)Gap-varying combs are used for measuring the y-displacement by detect-ing the current through the capacitor, given byi = CdVdt+ VdCdt(3.29)In inertial sensors the capacitance changes(dCdt 6= 0) due to movement offingers; and voltages applied are generally constant(dVdt = 0). The overalldifferential change in capacitance for a given displacement y is:4Cy = hNloverlapyd0y ? y? hNloverlapyd0y + y(3.30)53Chapter 3. Modeling of vibratory MEMS gyroscopeFigure 3.5: Actuation scheme using gap varying combsFor small y displacements(d0y  y), we get4Cy ?= 2hNloverlapyd20yy (3.31)Thus the current flowing through the capacitor becomesi = 2V hNloverlapyd20yy (3.32)3.2.3 Mode matching and ?fTo achieve optimized displacements in the sense mode of gyroscope, fre-quency tuning is essential. With 1% change in the resonant frequency, the54Chapter 3. Modeling of vibratory MEMS gyroscopesensitivity drops from 40 db to 33.8 db and for 5% change to 25.8 db asseen in Fig. 3.6. This graph was obtained by assuming, Q0x = 100 andFigure 3.6: Impact of frequency mismatchQ0y = 30. The fall in sensitivity is higher for sensors with very high qualityfactors. From Eq.3.22, the resonant frequency of sense mode can be tunedby reducing the stiffness ?kk , (0.001) for 1% and by (.0025) for 5% changes.Electrostatic spring softening is widely used for tuning the secondary reso-nant mode, by exploiting the nonlinearity in gap varying capacitors.y? +?0yQyy? + ?20yy =12hNloverlapym(d0y ? y)2V 2 +FCoriolism+FQuadraturem(3.33)55Chapter 3. Modeling of vibratory MEMS gyroscopeTaylor?s series can be applied on the electrostatic force, assuming smalldisplacements(d0y >> y):?(d0y ? y)2??d20y(1 +2d0yy +3d20yy2 + ...) (3.34)Here, ? = 12mhNloverlapyV2. In the above equation, term correspondingto y is an equivalent elastic-like force electrostatic spring. Its coefficient isequivalent to a negative stiffness constant that reduces the overall springcoefficient of the system. Depending on the amplitude of the bias voltageapplied, resonant frequency can be reduced. Neglecting the y2 term (forsmall y displacements), equation becomes:y? + (?0yQy?2?d30y)y? + (?20y ??d20y)y =FCoriolism+FQuadraturem(3.35)3.3 Capacitive sensing methodologyAn image of capacitive sensing methodology based on synchronous demod-ulation method is presented in Fig. 3.7. This sensing methodology is widelyused in industry and is explained in detail in[21]. For the ease of the compu-tations Vdsin(?d)t is used as the actuation signal, where Vd is the amplitudeof the voltage in volts and ?d is the driving angular frequency in rad/s.A carrier sinusoidal signal (Vc) is imposed on the structure (sensing mass),which is the common mode of the differential capacitive bridge in the sensemode.Vc(Carrier) = vcsin(?c)t[21] (3.36)here ?c = 2pifc and fc >>> fd. The sense capacitances for the top andbottom gap-varying sections can be modeled as:C1 = Cbase + Cy+sin?dt[21] (3.37)56Chapter 3. Modeling of vibratory MEMS gyroscopeFigure 3.7: Schematic of the Sense mode readout of the gyroscopeC2 = Cbase ? Cy?sin?dt[21] (3.38)The capacitances Cy+ and Cy? are the amplitudes of the capacitance changedue to Coriolis response. The sense currents flowing through the amplifiers(top as I+ and bottom as I?) are:I+ =d[Vc(t)C1(t)]dt(3.39)I? =d[Vc(t)C2(t)]dt(3.40)57Chapter 3. Modeling of vibratory MEMS gyroscopeCurrents flowing through the trans-impedance are amplified with a gainKnet and converted to Vs+ and Vs?:Vs+ = Knet?cCnetvccos?ct+KnetvcCnet2(?(?c??d)sin(?c??d)t+(?c+?d)sin(?c+?d)t)[21](3.41)Vs? = Knet?cCnetvccos?ct?KnetvcCnet2(?(?c??d)sin(?c??d)t+(?c+?d)sin(?c+?d)t)[21](3.42)The differential amplifier output is therefore:Vnet = Knet(Vs? ? Vs+)[21] (3.43)Vnet = KnetvcCnet(?(?c??d)sin(?c??d)t+(?c+?d)sin(?c+?d)t) (3.44)The next step is to demodulate Vnet signal with the high frequency carriersignal, sin(?c)t.Vdemod = Knet12vcCnet[(?c+?d)cos(?d)t?cos(2?c+?d)t?(?c??d)[cos(?d)t?cos(2?c??d)t](3.45)In the above equation, Cnet = |Cy+| = |Cy?| (assuming the magnitudesof capacitance variations on both sides are equal. By further simplifyingthe above equation and removing the 2?c, 2?c ? ?d and 2?c + ?d throughfiltering, Vdemod becomes:Vdemod = Knet12VcCnet?dcos(?d)t (3.46)This is further demodulated with the drive signal frequency(Vdsin(? +d)t) to get a voltage readout of the Coriolis-induced displacement.Voutput = KnetVcCnet?d (3.47)58Chapter 3. Modeling of vibratory MEMS gyroscope3.4 System level MEMS design issuesWe have described so far the dynamics of the primary(drive) and secondary(sense)modes of the gyroscope, together with common actuation and sensing mech-anisms. The importance of the mode matching for achieving an optimumsensitivity was also discussed. In this section, different noise sources orimperfections are presented.3.4.1 Quadrature errorThe fundamental cause of the quadrature errors in gyroscopes is the slightvariation in the average widths of suspension. Broken fingers of drive combsalso result in tilted displacements. As the name suggests, the displacementcaused by variations of suspension has a 90 degree phase difference relativeto sense mode Coriolis displacement. Quadrature displacements are presenteven when no rotation is applied to the sensor. To estimate the undesiredsignal a, rate equivalent quadrature error in the sense mode is computed bytaking the ratio of quadrature force(FQuadrature) to Coriolis force(FCoriolis)for a unit angular rate.?Q =FQuadratureFCoriolis?z= kxyx? sin?0xt2m?0xx? cos?0xt(3.48)The above equation can be simplified to?Q =kxykx?0xsin(?d)t2cos(?d)t(3.49)For a cross coupling stiffness kxy = .01kx for a gyroscope with ?0x = 8kHz,this results in a quadrature error of 14318 deg/s. As described before,larger displacements in drive mode are desired for higher sensitivity (x >1?m). For a drive mode displacement x = 1?m, with a misalignment of59Chapter 3. Modeling of vibratory MEMS gyroscopestructure due to fabrication imperfection of ? = 1deg, they would lead to10 nm quadrature displacement(y = xtan?), much greater than the Coriolisinduced displacements (generally in pico meters), as shown in Fig. 3.8. Thiserror gets amplified further with larger drive mode displacements.Figure 3.8: Quadrature displacementFrom a system designer?s perspective, quadrature error compensation isthe highest priority. Front end electronics needs to be designed taking intoaccount large levels of this error, which could be higher than the dynamicrange of the gyroscope operation range. It affects the Signal- to-Noise Ra-tio(SNR) and hence the resolution. Furthermore, stiffness variation due totemperature and time drift will affect the long term stability. A part of the60Chapter 3. Modeling of vibratory MEMS gyroscopechapter 6 is dedicated to a novel methodology for reducing quadrature er-ror, based on parametric actuation for damping. In general, dedicated fingerstructures with tunable DC biases are used for compensating the quadra-ture force. The asymmetric tilting induced by static electrostatic forcesgenerated by dedicated gap-varying combs can compensate, to a certain de-gree, for the off-diagonal coefficients in the stiffness matrix (responsible forthe quadrature errors). Synchronous demodulation is also used for separat-ing the quadrature error from the Coriolis induced motion, exploiting theirphase difference.3.4.2 Mechano-thermal (Brownian) noiseFor a gyroscope operated at atmospheric pressure, damping is another lim-iting factor. A spring mass resonator stores kinetic energy in the mass andpotential energy in the spring. Each degree of freedom has on average 12kBTthermal energy, so the spring mass system (2 degree of freedom) has a ther-mal energy equal to kBT , where kB is the Boltzmann constant and T is thetemperature. According to the equipartition theorem, the thermal energyis equally divided between potential and kinetic energies. The noise spec-tral density is associated with damping (loss mechanism within the system),with a noise force given as Fn =?4KBTcBW , where c corresponds to thedamping coefficient and BW is the bandwidth. For a resonator, in our casethe 0x mode, Fnoise, (cx, t) is computed as:Fnoise, (cx, t) =?4KBTcxBW (3.50)61Chapter 3. Modeling of vibratory MEMS gyroscopeanoise is obtained by dividing by the movable mass m on both sides. Usingcx =m?0xQx, we get :anoise =?4KBT?0xBWmQxm/s2 (3.51)For the sense mode gyroscope sense mode, the mechano-thermal noisecan be analyzed in terms of an equivalent noise angular rate ?nm:?2m?nmx? =?4KBTcyBW (3.52)This equation can be reformulated:?nm =?KBT?0yBW?20xmx2Q0yrad/s (3.53)Here, the mechano-thermal noise is computed based on the assumptionthat damping is constant(frequency independent). A frequency-dependentmodel will be analyzed in Chapter 5. This refinement is important for theoptimization of a gyroscope design.3.4.3 Electronic noise equivalent rateThe statistically independent noise generated by the electronic readout cir-cuit needs to be added to the mechanical noise, for an overall noise evalu-ation. A readout circuit with an equivalent input spectral noise voltage Vn(constant spectrum amplitude within the operating bandwidth) will lead toan equivalent input noise angular rate ?ne:?ne =d20y?20yCparasiticVnVpQ0xQeff?BWrad/s[94] (3.54)The net noise equivalent rate is given by statistically adding the mechano-thermal and electronic noises:?TNE =??2ne + ?2nmrad/s (3.55)62Chapter 3. Modeling of vibratory MEMS gyroscopeMost of the equivalent electronic noise is due to the front stage of the readoutcircuit. The subsequent stages of amplification have usually a negligiblecontribution to the overall noise. The total input referred noise voltage atthe input of the the readout circuit is:Vrms,in =?V 2rms,mech(out)A2mech+V 2rms,elec(out)A2mechA2elec(3.56)Here, the Amech and Aelec are the amplification gain factor for mechanicalsensor and electronic readout respectively.In the case of MEMS inertial sensors, the mechanical gain is the firststage of the amplification, which should be as high as possible. The elec-tronic amplification through the readout will impact less the performance,as seen from eq.3.56. Parametric amplification techniques, as described inChapter 6, make possible to amplify the mechanical signals before enteringinto the electronic readout subsystem. This will therefore contribute to abetter signal-to-noise ratio, and implicitly help achieving a better resolutionfor vibratory MEMS gyroscopes.63Chapter 4Fabrication technology andmodeling inConventorWare R?During the course of this PhD, three gyroscopes were designed, two inSOIMUMPS R? process and one in SOI Tronics R? process. In this chaptera brief description of these processes and considerations required in usingthem for design of inertial sensors are covered. Step by step implementationof modeling, using CoventorWare R? is covered for a single mass gyroscopeusing SOIMUMPS R? process. Images of inertial sensors fabricated duringthis PhD work are shown in Fig. 4.1.64Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.1: Fabricated inertial sensors during the period of this PhD. A)Decoupled gyroscope. B) Single mass gyroscope. C) Tronics based gyro-scope (vacuum packaged). D) Two mass gyroscope. E) Resonator designsimilar to design B) with slope shaped combs and F) Coupled resonators65Chapter 4. Fabrication technology and modeling in ConventorWare R?4.1 SOIMUMPS R?SOIMUMPS R? process stands for Silicon on Insulator Multi-User MEMSProcesses[86]. This process was designed for producing highly planar sur-faces, for devices which require good reflective index such as micro-mirrors.From a perspective of designing a gyroscope, large device thickness is de-sired for high sensitivity. SOIMUMPS R? provides two options of devicethicknesses, 10 ?m and 25 ?m. Further, unlike other fabrication processes(POLYMUMPS R?, SOI Tronics R?, etc.), back etching is carried out to re-move the silicon substrate. This eliminates the usage of holes in the proofmass which reduces the net mass and modifies the damping behavior (airtraps in holes). Air damping reduces the net quality factor of sensors. Con-sidering all these factors, SOIMUMPS R? becomes a desirable fabricationprocess for MEMS inertial sensors. This process has a minimum featuresize of 2 ?m and possible minimum gap between two silicon features is 2?m. This process can be called as a 4 mask level SOI patterning and etchprocess. This process is illustrated in Fig. 4.2 and has following steps:Step 1: 100 nm n type double sided polished SOI wafer is taken.Step 2: Wafer is doped by depositing a photo silicate glass (PSG) layer andannealing it for an hour at 1050 degrees.Step 3: Wet chemical etching removes the PSG layer.Step 4: Pad metal for interconnects is patterned through lift off.Step 5: Si is lithographically patterned with second mask level SOI. DRIE(Deep Reactive Ion Etching) is used for etching the silicon(for instancecombs and beams). Special SOI Tronics R? recipe allows no undercutting.Step 6: Front side protection material is applied to top surface of Si layer.Bottom side of the wafers are coated with photo resist and second layer66Chapter 4. Fabrication technology and modeling in ConventorWare R?(Trench) is lithographically patterned. Oxide in the bottom side of the sili-con is removed using the wet etch, defined by trench mask.Step 7: Front side protection material is removed by dry etch process. HF(Hydrofluoric acid) process removes the top surfaces using a vapor of HF.Step 8: Shadow mask is applied on the front side and blanket metal is pat-terned. Using a through hole metal can be patterned on the Substrate.Metal on the shadow mask (undesired area) is evaporated using the e-beam.After the evaporation of the metal, shadow mask is removed, leaving the pat-terned metal only on the desired area on substrate. This metal is generallyused as the ground for the whole chip to reduce the parasitic effects.The final structure, obtained from Solid modeler in CoventorWare R?using the SOIMUMPs R? process is illustrated in Fig. 4.3.67Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.2: Fabrication steps in SOIMUMPS R? [86]4.2 SOI Tronics R?SOI Tronics R? is a HARM (high-aspect-ratio micro-structures) process. Un-like SOIMUMPS R?, the device thickness is 60 ?m in TRONICS and devicesare vacuum packaged (< 1 mbar)[87]. The minimum feature size is 4 ?mand possible gap between two silicon structures is 3 ?m. This process, unlikeSOIMUMPS R? requires holes for etching process. Vacuum packaging aids68Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.3: SOIMUMPS R? gyroscope structure in Solid Modelerin reducing the air damping. This process is illustrated in Fig. 4.4 and theprocess is presented as following (Steps 1-4 for the bottom device):Step 1: SOI wafer is doped by p-type material. Alignment markers areetched.Step 2: Metallization (METAL1) is carried out. Depending upon the place-ments of bond pads a protective material is used for etching.Step 3: Silicon devices are patterned on the SOI. DRIE (Deep reactive ionetching) is carried out for patterning SOI and with SiO and where Si O2 actas etch stop layers.Step 4: Sacrificial layer is etched by HF and the wafer is rinsed and dried.For the top cap which is placed on the device structure for vacuum enclo-sure, following steps are required :Step 5: Silicon wafer doped with P type material is used.Step 6: Thermal oxidation is carried out for oxide layers with 2.6 ?mStep 7: Cavity is patterned for the area where sealing is carried out.69Chapter 4. Fabrication technology and modeling in ConventorWare R?Step 8: Oxide is etched on the back side and metal cap is deposited andpatterned.Step 9: Metal 2 is finally deposited and patterned.Alignment markers guide in placing the top and bottom wafers for pack-aging. The final structure obtained from Clewin R? layout editor and SolidFigure 4.4: Fabrication steps in SOI Tronics R?modeler in CoventorWare R? using the Tronics process is shown in Fig. 4.5.70Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.5: Tronics Layout and Solid modeler image(Bottom side)4.3 Design simulationsThis section describes the system level design flow using the MEMS softwarenamely CoventorWare R?. This software has three basic components: 1) Ar-chitect (where the basic design of the MEMS structure is shaped), which ispowered by SABER c? . 2) Designer (Layout editor, where the layout can beexported from the Saber or can be created using the different templates).3) Analyzer is the FE tool of the CoventorWare R?, which is powered byABACUS. Architect of CoventorWare R? is user friendly in two aspects 1) itliterally can be used as pencil on paper design methodology. A brief ideacan be easily put on the Architect and fundamental analysis results (reso-nant frequency, stiffness) can be visualized. After gaining faith in the basicstructure, the macro models can be net-listed and exported to SCENE3D71Chapter 4. Fabrication technology and modeling in ConventorWare R?or FEA for further optimization. Fig. 4.6 illustrates the typical flow chartfor modeling in CoventorWare R?.Figure 4.6: Flow chart for system level modeling in CoventorWare R?Dynamical modeling of the single mass gyroscope based on SOIMUMPS R?process is presented in the following subsections. This design was fabricatedand used extensively for proving nonlinear concepts.4.3.1 Dynamic responseFor a single mass gyroscope a crab leg suspension aids in orthogonal move-ments. Saber architect is invoked based on the SOIMUMPS R? process.Material properties and process files are imported from the foundry tem-plates. A rigid plate and four crab legs are connected through mechanical72Chapter 4. Fabrication technology and modeling in ConventorWare R?Parameter DimensionsProof mass area 800?m? 800?mCrab leg beam (lb2? wb2) 420?m? 10?mCrab leg beam (lb1? wb1) 360?m? 8?mTable 4.1: Dimensions of proof mass and suspensionsbusses as shown in Fig. 4.7.Figure 4.7: Four crab legs and beams mechanically connected for AC anal-ysisThe drive mode specifications are specified in Tab. 4.1. For a purelytranslational modes, the boundary condition of the beam connected to therigid plate in this example, can be assumed like a fixed guided end configu-ration. For a beam length l as the x dimension, width w as the y dimensionand h device thickness as the z dimension, where E is Young?s modulus,73Chapter 4. Fabrication technology and modeling in ConventorWare R?stiffness is analytically modeled as [58]:kx =Ewhl(4.1)ky =Ehw3l3(4.2)kz =Eh3wl3(4.3)An image of resonant modes for a crab leg is shown in Fig. 4.8, wherebeam-2 (with dimensions lb2 and wb2) has freedom in 0x direction andbeam-1 (with dimensions lb1 and wb1) in 0y direction. In order to seeFigure 4.8: Crab leg beam dimensions and FEA resonant modesthe resonant frequency for different modes, an AC sweep is performed by74Chapter 4. Fabrication technology and modeling in ConventorWare R?applying linear accelerations in three degrees of freedom. Resonant modesin all three degrees of freedom are shown in Fig. 4.9. The resonant modesof the sensor can be tuned by varying either length or width of beams. Inthe Fig. 4.9, width of beam-2 is varied from 12 ?m to 8 ?m to obtain tunedresonance. For now the exact resonant frequency is not computed, as massof combs have to be taken into account.Figure 4.9: Dynamic response and impact of beam width on the resonancebehavior4.3.2 Drive mode oscillationsFor drive mode, area varying combs are selected with the configuration asseen in Tab. 4.2. Area varying combs are connected with proof mass andbeams using the electrical and mechanical ports, specified by the macro-75Chapter 4. Fabrication technology and modeling in ConventorWare R?Table 4.2: Dimensions of area varying combs(Actuation)Parameter DimensionsNumber of fingers 34 each sideLength 100?mOverlap Length 60?mGap between fingers 5?mWidth 5?mmodel. Differential actuation scheme is applied on the combs such thatVACsin?dt+ vDC acts on one side of fixed combs and ?VACsin?dt+ vDC onthe other side of fixed combs. ?d is the resonant frequency obtained from theAC sweep, amplitudes of AC and DC voltages depend on the displacementdesired. A simple amplitude demodulator is connected to the electrical portsof the sensor. An integrator and diode with low pass filter is connected suchthat, when the input voltage swings high, current flows through the diodeand capacitor is charged at the same time. For low input voltage swings,diode is reverse biased and is disconnected from the output; thus capacitordischarges through resistor. The RC low pass filter finally smooths out thepeaks of the carriers oscillations in the output. DC analysis shows that forthe given comb parameters, static capacitance is 83.32fF for each side. Atransient analysis for drive mode is presented with signals at different stagesas seen in figures, Fig. 4.10 and Fig. 4.11.76Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.10: Architect connections for transient analysis for the drive mode4.3.3 Sense mode oscillationsTo detect motion in sense mode, gap varying combs are selected with theconfiguration as shown in Tab. 4.3 and are connected to the schematic usingmechanical and electrical ports.77Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.11: Transient analysis results for drive mode oscillationsTable 4.3: Dimensions of gap varying combs(Sensing)Parameter DimensionsLength 60?mNumber of fingers 84 each sideOverlap Length 40?mGap between fingers(Sensing small) 6?mGap between fingers(Sensing large) 9?mWidth 5?mBased on the capacitive sensing mechanism, covered in Chapter 3, section3.2, synchronous demodulation detection is set up as shown in Fig. 4.12. It is78Chapter 4. Fabrication technology and modeling in ConventorWare R?Modes Frequency(Hz) Generalized mass (m) Stiffness1st mode 8.005 kHz 4.810?8kg 121.3 N/m2nd mode 8.413 kHz 4.8 10?8kg 133.98N/m3rd mode 11.420 kHz 4.8 10?8kg 246.88 N/mTable 4.4: Resonant modes and the stiffness for different modes obtainedfrom Saber architect AC plotsworthwhile to obtain the static capacitances of the gap varying combs, whichis obtained from DC analysis in Saber Architect. The static capacitance ofgap varying combs is 0.332pF for each side. In Fig 4.13, AC response showsthat for imposed harmonic motion along x axis(yellow line), for angularvelocity imposed along the z direction, there is a displacement in the ydirection (green line) due to the Coriolis coupling and no displacement in thez direction (black line). Thankfully there is no coupling in z direction withcurrent beam dimensions, which may not always be the case requiring furtheroptimization. As analytical model shows that stiffness kz is proportional tothe cube of device thickness (h) as seen from equation 4.7. As the widthof the beams are smaller than the thickness, cross coupling in z directiondoesn?t occur. By applying VAC=25 V and vDC=15 V to actuation combs,drive mode displacement of 797.58 nm is obtained, which for a 1 deg/sangular velocity, couples to 6.28 nm displacement in sense mode (0y). Theresonant modes for drive and sense frequency obtained from Saber architectAC plots is shown in Tab. 4.4.To simulate the combined model of a single mass gyroscope in time do-main, transient analysis is carried out by applying the differential voltages atresonant frequency of drive mode. Carrier frequency of 70 kHz, much higher79Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.12: Complete architect model with sensing schemethan the resonant modes is applied on to the sense capacitances. Clock isconnected to the reference frame at ?z for applying ramp of angular velocity.Output for drive mode and sense mode displacements are shown with thefinal demodulated stage in Fig. 4.14. Using the appropriate feed back resis-tors and capacitances, larger output voltage/deg can be reproduced. Thesimulated resolution of the gyroscope is 2.78 aF/deg/s,.80Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.13: AC analysis response of the final structure with angular velocityalong z direction4.3.4 Mode analysis using FE analysis and final parametertuningTo see the accuracy of the macro model based Saber simulations, FiniteElement Analysis is carried out on the structure to obtain the resonantmodes. The simulations are done for coarse and extra coarse meshes. The81Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.14: Transient analysis response of the final structure with angularvelocity along z directionresult for 6 modes is shown in Fig. 4.15A comparison for FE analysis for extra coarse and coarse mesh is pre-sented in Tab. 4.3.4.82Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.15: 6 resonant modes obtained from FE analysis of MEMech (blue-no displacement, red- maximum displacementExtra Coarse MeshModes Frequency(Hz) Generalized mass (m) Stiffness1st mode 8.244 kHz 4.68410?8kg 125.55 N/m2nd mode 8.58 kHz 4.818 10?8kg 139.9 N/m3rd mode 10.39 kHz 4.67 10?8kg 198.82 N/mCoarse Mesh1st mode 8.143 kHz 4.43910?8kg 115.8 N/m2nd mode 8.46 kHz 4.792 10?8kg 135.20N/m3rd mode 10.3 kHz 4.68 10?8kg 195.8 N/m83Chapter 4. Fabrication technology and modeling in ConventorWare R?Table 4.5: FEA Mech MM comparison for coarse and extra coarse meshesAs seen from the FE Analysis, the resonant modes of the gyroscopes arestill mistuned and has a bandwidth of (? 320 Hz). To achieve matchedresonant peaks, length of the beams lb2 is varied from 420 ?m to 400 ?mand Saber architect AC analysis is rerun, to see the impact on the resonantmodes. In Fig. 4.16, resonant modes of drive and sense is matched for lb2 as407 ?m at 8380 Hz. This variation in stiffness can be achieved by applyingDC bias to the proof mass as electrostatic force causes spring softening.This is visualized by applying the voltages and running AC analysis. Theresonant modes are tuned by applying a DC bias of VDC=5 V at proof-massas seen in Fig. 4.16.Figure 4.16: a)Impact of variation of length (lb2) on resonant modes b)Frequency tuning using DC bias84Chapter 4. Fabrication technology and modeling in ConventorWare R?4.4 Analytical modeling of dampingIn inertial sensors, air molecules trapped between the fingers of combs, proofmass and substrate leads to damping in oscillations. There are two typesof damping, slide film and squeeze film damping. Both damping are pres-sure dependent. Air molecules are characterized by their mean free path(distance before a molecule collides). Air molecules in atmospheric pressureacts like viscous fluid (they easily collide with each other due to their largenumbers and smaller mean free path 65 nm), leading to high damping. Asthe pressure is reduced, the air molecules are reduced, which allows themto easily move and their mean free path increases (comparable to devicedimension). This region is generally called as molecular region. Here indi-vidual gas molecules interact with the device unlike viscous fluid, thus lessdamping.4.4.1 Slide film dampingSlide film damping occurs when two sliding plates of Area (A), separatedwith a gap g0, have a frictional force acting on them. This frictional forceis resultant of air molecules hindering the motion. In order to correctlymodel damping (gas rarefaction effects), effective viscosity (?eff )of the gasis required. For a newtonian gas, damping coefficient is given as:cslide = ?effAg0(4.4)Where ?eff :?eff =?1 + 2Kn + 0.2K0.788n e?Kn10(4.5)85Chapter 4. Fabrication technology and modeling in ConventorWare R?For the structure designed, in the 0x mode, both gap varying and areavarying combs undergo sliding mode damping. For atmospheric pressure Pa= 105 Pa, with ?eff = 1.5919 ? 10?5, sliding mode damping coefficient iscomputed as 1.7256 ?Ns/m.4.4.2 Squeeze film dampingAn extensive literature is available on squeeze film damping analysis basedon MRE (Modified Reynolds Equation). A common concern in squeeze-filmtheory is the correct use of the continuum hypothesis considering that thedevice scale continues to be reduced [36]. At a certain limit the typical inter-molecular distances are comparable to the device dimensions and the use ofcontinuum fluid equations can not describe appropriate flow behavior. Forgases, the Knudsen number (Kn) relates the gas specific mean free path (?)and the film thickness (g):Kn =?g(4.6)For small Knudsen numbers (Kn < 0.01) the flow is continuous and thecontinuum fluid equations apply. At ambient pressure, the mean free pathof an air particle is small (0.068 ?m) when compared to the typical gaps ofthe considered MEMS device (Knudsen number is equal to 0.014). Thus,the continuum hypothesis is appropriate and the laminar flow (Reynoldsnumber is equal to 0.03) can be modeled using NavierStokes equation assuggested by [88]. From a macro-modeling perspective, an across-throughequivalent representation (generalized network modeling) could be used tomodel the combined damping and elastic interaction of the air with the86Chapter 4. Fabrication technology and modeling in ConventorWare R?movable plate. For instance, the air damping and spring force component(for small displacements) could be expressed as:Fdg=64?PApi6g0?m2 + (n/?)2(mn)2[(m2 + (n/?)2)2 + ?2pi2 ](4.7)Fsg=64?2PApi8g0?1(mn)2[(m2 + (n/?)2)2 + ?2pi2 ](4.8)In equation 4.7 and 4.8 assumption is that m and n are odd numbers[21].P is pressure, A(w? l) is area of the plate, g0 is the initial gap between theplates. ? corresponds to lw of the plates and ? corresponds to squeezenumber which is described as:? =12?effw2??g20(4.9)?eff is approximated as?1+9.638K1.59n, here ? is viscosity. For the gyro-scope designed, effective viscosity is estimated as 1.289410?5 and squeezefilm damping coefficient as 37.38 ?Ns/m. An image of frequency varyingdamping coefficient obtained from the damping force in MATLAB is pre-sented in Fig. 4.1787Chapter 4. Fabrication technology and modeling in ConventorWare R?Figure 4.17: Squeeze film damping variation with frequency of the designedgyroscope88Chapter 5Frequency dependent noiseanalysis and damping inMEMS inertial sensors 3Micro-machining technology has made possible the fabrication of very sen-sitive micro-scale structures like accelerometers and gyroscopes, used in dif-ferent military, biomedical, aerospace and automotive applications. Thesesensitive micro-sensors have suspended masses moving in gas or liquid, witha large variety of frequency ranges, depending on their applications. Thequality factor (Q), defines the sensitivity of many vibratory sensors andeven the feasibility of their applications[89]. When inertial sensors are op-erated in atmospheric pressure, mechano-thermal damping influences theperformance over combination of several mechanisms, such as thermo-elasticdamping, surface loss, dissipation through support(anchor loss), which dom-inate at vacuum ([90],[91]). These works also show that sensitivity is gen-erally limited by the mechano-thermal noise generated by the interactionof the movable structure with the surrounding fluid, of a certain viscosity.3A version of this chapter has been published. Reprinted from [38] with permission.2009, Springer. Sharma, M, Cretu, E. Frequency dependent noise analysis and dampingin MEMS inertial sensors. Journal of Microsystem technology 1/200989Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsAn accurate modeling of the frequency-dependent noise behavior is there-fore essential in order to reach the sensitivity limits of MEMS-based sensingmicro-systems, for it can suggests optimum operating frequency regions.Frequency-dependent damping behavior is well analyzed in several papers([92]; [93]; [91]), where the physical phenomena related to the gas flow be-tween parallel plates are analyzed and corresponding compact analytical arederived. Nevertheless, to the best of our knowledge, there are no papers an-alyzing the impact of these damping models on the mechano-thermal noise.Existing scientific literature ([97];[95];[96]) analyzes the mechanothermalnoise in microstructures based on the assumption of a constant (frequency-independent) damping coefficient, in contradiction with the frequency-dependentdamping models. While this assumption might be correct for low frequen-cies, it fails to consider the more complex behavior of gas damping at higheroperating frequencies, as usual for a large category of resonating micro-devices. The interaction of the movable mass with the surrounding fluidgenerates both elastic and damping force components, both dependent onfrequency (and on the amplitude of motion for large displacements). Thiscomplex behavior shapes the resulting mechano-thermal noise, aspect thatcan be exploited in the signal-to-noise ratio optimization process of mi-crostructures used as sensors. While even state-of-the art design tools likeCoventorWare and MEMS Pro neglect the impact of this noise shaping onnoise analysis, there are nevertheless detailed models of the combined elastodamping action of the gas upon the movable plate in the case of squeeze filmdamping ([91]; [98]), used for tuning the frequency and transient responses.The present work presents a unitary, integrated approach for damping andnoise analysis in MEMS devices, using the same behavioral models for bothnoise and damping analyses.90Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensors5.1 Design criteria for frequency varying noiseSqueeze-film mechanism is generally dominant relative to slide-film dampingin most MEMS structures [98], and it is the focus of the present analysis.From a macro modeling perspective, an across-through equivalent repre-sentation (generalized network modeling) [36] could be used to model thecombined damping and elastic interaction of the air with the movable plateas seen in Fig. 5.1. A complex elasto-damping behavior of the fluid struc-ture interaction translates, due to thermodynamic equilibrium, into a similarfrequency shaping of the equivalent noise force. It is therefore necessary toinclude such frequency dependency into the noise analysis in order to obtaina proper signal-to noise ratio (SNR) estimation and optimization. Froma noise analysis viewpoint, air-structure interaction has two direct conse-quences and damping coefficient is written as c(?) = b(j?) + j?kd(j?), asillustrated in Fig. 5.1:(1) A frequency-dependent noise force term is associated with the loss mech-anism due to b(j?).(2) A frequency-dependent elastic interaction, represented as an equivalentinductance L(j?) = 1/kd(j?), which will shape the transmission of the in-put noise to the equivalent output displacement noise. While their influencemight be negligible in the low frequency range, their magnitude becomesequal and then surpasses the magnitude of the elastic spring force as fre-quency.Equivalent electrical representations of the air elasto-damping effectshelp in obtaining a correct simulation and understanding of the dynamics.It is necessary to extend this approach to (a) noise analysis (as the noiseis intrinsically related to the damping behavior) based on the frequency-91Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.1: Macro model of resonator with frequency varying noise([38])dependent behavior of air-structure interaction and (b) SNR optimizationprocedure, where design parameters are varied such that a maximum per-formance will be obtained. To maximize the sensitivity of an inertial sensor,the proof mass is usually made as large as possible, within the limits ofthe given technological constraints. This will maximize both the sensitivityto external inertial forces and SNR. The displacement of the mass is themain source of damping, and lets the choice of the spring constants as opti-mization parameter, such that the natural resonance frequency is optimallychosen.5.2 Implementation of noise model on out ofplane structureThe design and optimization flow procedure can be summed up as:Step1: Design of the structure.Step2: Finite Element Analysis (FEA) for damping.Step3: Macro-model extraction/generation (Y (j?)).92Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsStep4: Symbolic/Numeric optimization with respect to S/N.Step5: Check for optimization.It is implemented as a combination of several design tools: the structuredesign and finite element analysis (FEA) of the damping are performedusing CoventorWare R?, while the macro model extraction and the combinednumeric/symbolic optimizations are presently done using Mathematica c?.The procedure will be presented for the resonator structure illustrated inFig. 5.2. The structure has a surface area of 143600 ?m2, with a gap of 1.6?m from substrate and stiffness in 0z mode as 215 N/m.Layout geometry is designed based on the given design rule set, followedby extensive finite element simulations (using MemMech module withinCoventorWare R?) of air structure interaction for small vertical displace-ments of the mass. The result of the (time-consuming) FEA step gives thefrequency-dependency damping and the equivalent gas damping and springconstants, b(j?) and kd(j?), as shown in figures, Fig. 5.3 and Fig. 5.4.Here, b(j?) and kd(j?) can be best described for a resonator in mo-tion(oscillating) in z direction with Z0 as steady state oscillation as:Fdamping(?) = b(?)vz(t)?|Fdamping(j?)b(j?)?|= Z0 (5.1)Fspring(?) = Kd(?)z(t)?|Fspring(j?)Z0|= kd(?) (5.2)In the Fig. 5.4, the damping and spring forces are normalized (dividedby spring stiffness). At 7.2 kHz, the stiffness offered by air damping isequivalent to the spring stiffness. As the frequency increases, the air behavesmore like a spring and becomes the major hindering force. Based on thesqueeze film damping theory, squeeze number indicates the region wherecompressible gas has significant damping behavior. The Fig. 5.5 illustrates93Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.2: FE analysis result for out of plane resonator for dampingstudy([38])the cut of frequency (Damping and spring forces are equal) of the resonatorat ? = 20, obtained from Eq.4.9.The results are then exported to Mathematica c?as a list of values forb(j?), kd(j?), corresponding to the simulated frequency points. Mathematica c?isthen used to create smooth interpolating functions for both b(j?) and kd(j?),which are used in the subsequent steps. To estimate the influence of b(j?)and kd(j?) on the overall performance, both an equivalent input accelera-tion noise and an equivalent output displacement noise are computed and94Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.3: FE analysis result of damping for out of plane resonator fordamping study([38])compared with the common white-noise assumption. The equivalent inputspectral acceleration noise is given by:anoise =?4KBTb(j?)mms2?Hz(5.3)The Fig. 5.6 illustrates the variation with frequency of the equivalent in-put acceleration noise, compared with the frequency-independent model; inthe last case, the low frequency value of the damping coefficient (as ex-tracted from finite element simulations) was extended over the entire fre-quency range (red dotted line curve). The equivalent input noise limits thesensitivity of the device to external input inertial effects. It is therefore obvi-ous from the previous figure that the white-noise model will overestimate thenoise, and does not lead to a potential noise optimization of the resonant95Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.4: Normalized damping and stiffness forces extracted from FEAnalysis([38])structure. Considering the frequency-dependent behavior of the dampingcoefficient leads to the identification of reduced flat noise frequency-ranges,better suited for sensing external inertial effects.The SNR is therefore frequency-dependent; if given in terms of inputacceleration, it has the expression:SN=| Aext(j?) || Anoise(j?) |=m | Aext(j?) |?4KBTb(j?)(5.4)The second air-structure interaction term, the elastic force component,will also play a role in tuning the resonant behavior of the structure. Its96Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.5: Normalized damping(red) and stiffness(blue) forces vs Squeezenumber (?)effect is observed in the expression of the output signal. The displacementof the movable part, Z(j?), has two components, one corresponding to theinput signal (the external acceleration in this example), and the other onerepresenting the equivalent output noise:Z(j?) =mAext(j?) +?4KbTb(?)ks + kd + j?b(?)? ?2m(5.5)Znoise(j?) =?4KbTb(?)ks + kd + j?b(?)? ?2m(5.6)The spectral behavior of the displacement noise is presented in Fig. 5.6b,where the results processed from finite element simulations are compared97Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.6: a Equivalent input acceleration noise, b equivalent displacementnoise([38])again with the white-noise approximation (dotted line curve). An opti-mum system design will match the mechano-thermal noise and the equiva-lent input electrical noise at the interface between the mechanical structureand electrical readout, in order to obtain a balanced performance/cost ra-tio. Therefore, for a capacitive readout scheme, the equivalent displacement98Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsnoise is to be translated into an equivalent capacitance variation noise. Thedesign of the readout circuit must be done in such a way that the resultingequivalent electric input noise is in the same range as the mechano-thermalone. The frequency-dependent components are useful only for AC small-signal analysis, and cannot be used for time domain simulations (transientsimulations in Spice). In consequence, Mathematica c?was used to gener-ate, from list of simulated FEA points, an equivalent lumped-componentrepresentation of the squeeze-film damping behavior, which can be directlymapped into a Spice-like macro model, useful for both time- and frequency-based analysis. The generation procedure combines the FEA data with thetheoretical insight given by the analytical theory of squeeze-film damping.The combined effect of the elastic and damping action of the air results ina direct complex admittance representation, Yd(j?) = b(j?) ? jkd(j?)/?,whose frequency dependent magnitude is illustrated in Fig. 5.7a.To map this frequency variation to an equivalent lumped elements modelwith parallel R?L branches, as suggested by theory, an equivalent R(?)?L(?) series model is firstly computed from Yd(j?).Rair(?) =?2b(?)b2(?)?2 + k2d(?)(5.7)Lair(?) =kd(?)b2(?)?2 + k2d(?)(5.8)Zair(?) = Rair(?) + j?Lair(?) (5.9)Interpolated functions are used for b(?),kd(?). In a second step, this initialfrequency-dependent R ? L series model is approximated with a series ofthree parallel R ? L branches with frequency-independent components, as99Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.7: a Equivalent air admittance, b lumped component modelapprox-imation with frequency-independent components([38])shown in Fig. 5.7b. The entire mapping procedure is performed using thepowerful language of Mathematica c?.The resulting model can be afterwardsused in general Spice-based simulators, for the analysis and tuning of the100Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsperformance in both frequency and time domains. The equivalent constantresistor elements R11, R13, R31 are the only dissipative components asso-ciated with intrinsic noise generators, whose corresponding amplitudes aregiven by the Nyquist relation. The approach presented here allows the op-timization of a resonator sensor. The frequency-dependent noise analysisoffers a more accurate description and further insight than the white noiseapproximation. In the case of inertial resonant sensors (e.g. vibratory an-gular rate sensors), it suggests the optimum frequency range of operationin order to achieve both a large output signal and a low noise. The massof the movable element is typically taken as large as possible (within thedesign rules of the technology), to maximize the inertial sensing. This letsthe mechanical spring constant as a design parameter to be optimized, fortuning an optimum resonant mode of operation. To sum up, a compara-tive illustration of different noise analysis and optimization approaches ispresented in Tab. 5.15.3 Implementation of noise model on gyroscope(in-plane movement)The noise model as described in the above sections is applied to the designof a gyroscope fabricated in SOI Tronics R?-MUMPS technology.5.3.1 Step 1The designed gyroscope, as seen in Fig. 5.8, has two sets of movable masses(primary and secondary), to maximize the decoupling between the drivingand sensing modes. The primary mass (inner) is used for actuation, while101Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsthe sensing uses the displacement of the four outer masses. The elasticsuspensions are designed such that the structure has matching resonantfrequencies for the primary and secondary modes, at 23.101 kHz. To achievea large displacement actuation and linearity, the capacitances associatedwith the driving mode rely on area variation, leading to slide film damping.The sensing capacitances used for the detection of external angular rate aredesigned instead for maximum sensitivity; their gap-varying mode leads toa dominant squeeze-film damping of the secondary mode motion.Figure 5.8: Gyroscope structure in analysis with area and gap varyingcombs([38])5.3.2 Step 2 FE analysisDamping analysis is carried out for both the primary and secondary modes,to characterize how the air damping and spring equivalent forces vary with102Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsfrequency. Primary mode analysis shows that the slide film damping hasnegligible spring force component, while the influence of damping force be-comes significant only beyond 109 Hz as seen in Fig. 5.9, well outside anyoperating regime.Figure 5.9: Slide film damping-actuation (spring force-green negligible,damping force-red increases at high frequency)([38])The squeeze-film damping phenomenon in the case of the sensing modesleads instead to significant air-mass interaction effects, both in terms ofdamping and elastic forces, as shown by the FEA and MATLAB code(referEq.4.7-4.9) results illustrated in Fig. 5.10. The damping and spring forcesobtained from the analytical model based on the theory gives a rough es-103Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorstimate. To estimate, the percentage of error, frequency varying dampingcoefficient(refer Eq.5.3) is plotted in Fig. 5.11. The plots indicate that an-alytical modeling of the damping force has an error of 16%, hence makingFEA analysis pertinent for the analysis!Figure 5.10: Squeeze film damping- spring and damping forces comparisonbetween FE analysis(straight lines) and analytical modeling (circles)Nevertheless, modeling of the damping and spring forces only present afast estimate, only suggesting that the overall mechano-thermal noise effectwill be dominated by the damping in the sensing capacitances.104Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.11: Squeeze film damping- damping coefficient comparison betweenFE analysis(dashed lines) and analytical modeling (circles)5.3.3 Macro-model extractionThe dominant noise source will appear in the gyroscope equation for thesensing mode (Eq.3.9). The consequences are easier to analyze if one con-siders driving the structure with a harmonic electrostatic force, such thatit induces a displacement of amplitude X0 (chosen as 10 ?m) and angu-lar frequency ?0 (coincident with the matched primary- secondary resonantfrequency value). This reduces the analysis of dynamics to Eq.3.9, whilethe solution of Eq.3.8 becomes x(t) = X0cos(?0t). Equations in frequency105Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsdomain are represented as :(ms2 + cys+ ky)Y [s] =2msX[s]?z(s2 + ?20)+ noise (5.10)Rearranging the above equation yields:Y [s]?z=2msX[s](s2 + ?20)(ms2 + cys+ ky)+noise(ms2 + cys+ ky)(5.11)The damping coefficient cy has two component (1) stiffness and (2) dampingforces and could be modeled as cy = kd + jb(j?) the kd term is obtainedfrom the FEA for a discrete frequency values and so is the frequency varyingdamping coefficient (refer to Fig. 5.10). The displacement noise is analyzedby choosing the damping and stiffness values for a given resonant frequencyas seen in Fig. 5.10. For the noise optimization the net displacement in they direction is given as the:Youtput = ?zsensitivity + Ynoise (5.12)Here sensitivity should be high and displacement noise should be as low aspossible to obtain the optimized structure. Ynoise can be computed as:Ynoise =?4KBTb(j?)BWks + kd + j?b(?)? ?2m(5.13)It is interesting to plot the frequency varying noise displacement and thewhite noise displacement with respect to frequency, (1) to observe the com-parison in noise levels and (2) to get a hint on optimization with respectto noise level and sensitivity. The Fig. 5.12 shows the comparison in noiselevels of white noise approximation and the frequency varying noise plot.The constant damping noise approximation shows the overestimation of dis-placement noise near the designed resonance frequency of sense mode ofgyroscope (23 kHz). On the other hand at higher frequencies near 105 Hz106Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.12: Comparison of displacement noise, red-white noise displace-ment and blue-frequency varying noise displacementfrequency dependent equivalent displacement noise is higher compared toconstant damping based noise displacement estimation.Mechano-thermal noise further dips with increase in frequency. It be-comes tempting to design a structure at higher frequency, however, it can-not be concluded until the variation between the SNR and Sensitivity isobserved with respect to resonant frequency. An optimization occurs whenSNR/Sensitivity and noise levels are probed together. The current anal-ysis has to be extended to take into account the SNR and sensitivity forcorrectly optimizing the MEMS design which paves for future work. The107Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsFigure 5.13: Comparison of mechano thermal equivalent rate based on con-stant damping (red) and frequency dependent damping (blue) with elec-tronic equivalent rate noise (green)integrated noise analysis with frequency dependent damping analysis givesmore insights in optimizing the noise level of MEMS structures. This couldfurther aid in tuning the electronics in accordance with the MEMS struc-ture. Reduction in noise floor is one of the important criteria in the design ofgyroscope and its electronics for the aerospace applications as also observedby a research group [99]. Our study helps design and model resonant struc-tures, to operate at optimized frequency, such that the noise floor is as lessas possible without affecting the sensitivity. Moreover as mentioned in the108Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsmotivation of this thesis work, gyroscopes are recently being scaled from thekHz to MHz region to reduce the impact of mechano-thermal damping. Inthis region there is a reduction in mechanical equivalent noise rate (mechano-thermal noise). Designers set the operating frequency of gyroscope to thepoint where the mechanical equivalent noise equals the electronic noise com-ponent, so that total noise equivalent is minimum [94]. There is still a needto understand where the true optimization occur, to test this we check bothmechanical (constant damping and frequency dependent damping) and elec-tronic noise component. Based on the equations 3.53 and 3.54 shown inChapter 3, both mechanical and electronic equivalent noises are plotted inFig. 5.13. Red dashed lines indicate the mechanical equivalent rate noisebased on the constant damping, blue dashed lines indicate the mechanicalrate noise based on the frequency dependent noise and finally electronic noiserate is indicated in green line. One can see that the points where electronicnoise equals the mechanical noise is different in both the cases. Actual opti-mization occurs at 0.278 MHz which is approximately 0.25 MHz less thanthe point (0.5316 MHz) obtained from constant damping analysis. In theterms of noise there is an overestimation of noise at 0.5316 MHz as 0.01475deg/s/?Hz using constant damping based estimation, where as noise at0.278 MHz is 0.007719 deg/s/?Hz. This erroneous estimation based onconstant damping estimation can cause high noise sensor, thus frequencydependent based noise analysis is essential.5.3.4 Concluding remarksA noise analysis of a resonant MEMS structure, taking into account theelasto-damping behavior of the airstructure interaction is presented. Af-109Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorster defining the geometry of the structure, extensive finite analysis simu-lations are used to extract the frequency dependence of the air dampingand spring coefficients. The resulting list of values are then post-processedin Mathematica c?,and interpolating functions are generated and used for acombined symbolic and numeric mechano-thermal noise analysis. The com-bined finite element and symbolic analysis give a powerful design tool inthe hands of the designers, which can generate in a semi-automatic fashionreduced order macro-models to be used in Spice-based simulators for thor-ough noise simulation. Compared with the standard procedure of assumingan equivalent white spectral mechano-thermal noise, the present analysisbrings the advances in the squeeze-film damping theory into the realm ofnoise-based optimization of micro-mechanical resonators. It also presents amixed design flow, combining FEA with mixed numerical and symbolic pro-cessing algorithms (implemented in a computer algebra program), and withSpice-based analysis of the generated macro models. However, the currentanalysis has to be extended to take into account the SNR and sensitivity forcorrectly optimizing the MEMS design, which paves for future work.110Chapter 5. Frequency dependent noise analysis and damping in MEMS inertial sensorsTable 5.1: Features of white noise analysis and frequency model with lumpedmodelWhite noise model(Yd = b)Yd(j? = b(j?)? jkd(j?)? Lumped model1) The noise shapingdue to kd is ignored aswhite noise1) The model showsthat resistor and induc-tor are frequency vary-ing. Good for frequencyvarying noise analysisbut not for time domainanalysis1) Macro-model can beused for both frequencyand time domain analy-sis2) Fails to consider themore complex behaviorof gas damping as theoperating frequency in-creases, which is thecase for resonators andresonating sensors2) Considers complexbehavior damping at allfrequencies2) Inductor is used fornoise shaping and canbe observed for spec-trum of frequencies3) Signal-to-noise ra-tio optimization proce-dures exploit spectralnoise shaping of theequivalent input noisesource and tunes me-chanical suspensions tooptimal value.111Chapter 6Parametric resonance:amplification and damping inMEMS gyroscopes 4This chapter reports on parametric amplification and damping employedin a MEMS gyroscope. As parametric amplification is phase dependent,with appropriate phases, we can either amplify the mechanical oscillationsor reduce the unwanted oscillations (quadrature error oscillations). This isin line with the recent work [100] which, showed improvement in SNR onring-gyroscope, by taking advantage of the phase dependence of paramet-ric amplification and the orthogonality of the Coriolis force and quadratureforcing. Major contribution of this chapter are :1)Implementation of both parametric amplification and damping in MEMSgyroscopes.2)We show improvement in SNR with experimentally measured parametricamplification up to 2.5 for our device.4A version of this chapter has been published. Reprinted from [49] with permission.2012, Elsevier. Sharma, M, Sarraf,E, Baskaran, R and Cretu, E. Parametric resonance:Amplification and Damping in MEMS gyroscopes. Journal of Sensors and ActuatorsPhysical A. 4/2012112Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesThis chapter is structured as follows. Section 6.1, describes the device struc-ture. Analysis of Duffing oscillator and parametric oscillator is carried outwith brief background. Comparison between the two nonlinearities is alsodiscussed. Section 6.2, describes the parametric amplification in MEMS gy-roscopes. Section 6.3 and 6.4 describes the experimental setup and resultswith conclusions in section 6.5.6.1 Device structure: nonlinearity in MEMScoupled resonatorsA proof mass suspended by crab leg fixtures ensures the two orthogonaldegrees of freedom (driven/sense) of the MEMS gyroscope. The structure isimplemented in SOIMUMPs R? technology, with a minimum gap size of 2 ?mand a thickness of 25 ?m for the structural layer. The Fig. 6.1 shows distinctsets of actuating and sensing comb fingers. The horizontal actuation (0x) isachieved by applying AC voltages between the moving and the fixed fingersof the actuators (with a 5 ?m gap) and a DC voltage to the proof mass.When an external angular rate ?z rotates the sensor about 0z axis, theresultant Coriolis force will induce a proportional motion in the sense mode(0y). The 0y displacement is measured using asymmetric comb drives, withgaps of 6 ?m and 9?m, intentionally left large to facilitate the parametricamplification experiments. The dynamic behavior of the structure obeys thedifferential equations (3.6) and (3.7), system level block diagram depictingit in Fig. 3.3. Both equations (3.6) and (3.7) are linearized and should beconsidered with nonlinearity to get the thorough understanding. As thispaper deals with the parametric amplification in secondary mode only, allour calculations from here on will be targeting the secondary mode motions.113Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.1: Gyroscope fabricated in SOIMUMPs R? (25 ?m) technology.Red/Blue markers-Actuation/Sensing combsWe re-write, equation (3.7) with cubic nonlinearity (k3y) and summing theforces on the write hand side as F . We treat the secondary mode of thegyroscope as a normal oscillator and get equation (6.1)my? + cyy? + kyy + kxyx+ k3yy3 = Fcos(?t+ ?) (6.1)114Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesNormalizing equation (6.1)[ [101]] 5??y + 2???y + ?y? + ?y?3 = Fcos(?t+ ?) (6.2)Where? = (ky + k3y), ? =cy2m?n, ?2n =?m, ? =k3yy20?,? =??n,? = ?nt, ??y =y??2ny0, ??y =y??ny0, y? =yy0(6.3)Where,? denotes the differentiation with respect to non-dimensional time? . This transformation leads to Duffing?s equation as seen in equation 6.2.Depending upon the cubic nonlinearity (?), we either see the amplitudeskew towards left (? < 0), spring softening or towards right (? > 0), springhardening. Solution for this equation can be approximated by iteration orperturbation methodology [102]. We obtain, the solution for the equationsusing the iteration method, with an assumption of a periodic solution forequation 6.2 as, y0 = Acos(?t+ ?) :[(?? ?2)A+34?A3]2 + ?2A2?2 = F 2 (6.4)Duffing?s approximation: The coefficient A (Amplitude of the oscillation) isapproximated as same as what it would have been with linear spring. Weplot the Amplitude (normalized) Vs Frequency (detuned) in Fig. 6.2 andFig. 6.3, to show the impact of cubic nonlinearity on Duffing?s oscillator.Simulations were ran with assumption (? = ? ? 0.3, ? = 0.2) for F =0, 1, 2, 3, 4.The Loci of the tangents are also shown with the dotted lines. Theseare obtained by differentiating the equation 6.4. They form the outline5The quadratic terms are canceled by employing common mode actuation schemes.Only odd terms remain.115Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.2: Simulation results of the Duffing Oscillator with normalizedforce(F=01,2,3,4.) Positive cubic nonlinearity(? > 0)of the region in which the response curves turn over on themselves. Thismeans, there are more than one solution for a given frequency. For largeamplitude parametric amplifiers with displacements greater than 110ththecapacitive gap the cubic term should be considered, as considered before inthe Duffing?s equation. For parametric resonance, the homogenous nonlinearMathieu equation is described by:?2y??2+ ??y??+ [? + 2?cos(2?)]y + [?3 + 2??3cos(2?)]y3 = 0 (6.5)The significance of the parameters (?, ?, ?, ?, y, ??3, ?3) is described in moredepth in [103]. Fig. 6.4a and b shows the effects on the dynamics of the pa-rameter space for the two cases of a linear parametric system ( 6.4a), and a116Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.3: Simulation results of the Duffing Oscillator with normalizedforce(F=01,2,3,4.) Negative cubic nonlinearity(? < 0)parametric system with cubic nonlinearity ( 6.4b), respectively. Operatingat the normalized natural frequency, ? = 1, and increasing the paramet-ric term ? causes the linear system to transition from stable to unstablebehavior. Cubic parametric system transitions from stable to unstable andback to stable behavior is also observed under the same operating condition.When an external harmonic actuation force is added the equation becomesa Mathieu Hill?s equation Eq. (6.5) becomes in such a case:my?+cyy?+[ky+?ksin(2?t)]y+[k3y+?k3ysin(2?t)]y3 = Fcos(?t+?) (6.6)117Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.4: Dynamic characterization of Mathieu equation with stable andunstable points. [105] [103] [104]Normalizing Eq.(6.6) [104], [106], with ?kym?2 as perturbation parameter:? = ?t,kym?2= 1 + ??2, k3 =k3ym?2, c0 =cym?, f =Fm?2(6.7)118Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesSubstituting this into Eq.(6.6), we get:d2ydt2+ y = ?(cydydt+ [??2 + sin(2?)]y+ [k3 + ?ksin(2?)]y3? fcos(? + ?))(6.8)Eq. (6.8) is further separated into two time scales: slow time (?) andstretched time (?), while y is approximated through a power series (ignoringhigher order terms): ? = 2?? = ? ; y = y0 + y1 + ...dyd?= 2?y??+ ?y??(6.9)?2y??2= 4?2y??2+ 4?2y????+ 2?2y??2(6.10)Substituting the set of equations from (6.9) and (6.10) into (6.8) and as-suming that the general form of the (steady-state) solution corresponds toharmonic oscillation with slowly varying time dependent amplitude:y0 = A(?)cos(?2) +B(?)sin(?2) (6.11)Introducing the values for y0, y?0, y?0 into the equation, the values of the slowlyvarying coefficients A and B can be determined and the steady-state solutioncomputed [103]:yss =FQky[cos?sin(?t)1 +Qy ?k2ky+sin?cos(?t)1?Qy ?k2ky] (6.12)Eq.(6.12) shows the phase dependence of the displacement. Depending uponthe phase difference (?) between the stiffness modulating signal and the ac-tuation force f , the gain could be positive or negative. Fig. 6.5 illustratesthe Duffing type and parametric response experimentally measured for ourdevice. The normalized cubic nonlinearity (?) was separately estimated as119Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopes2.2x10?3, based on a least-square fit from the experimental static displace-ment characteristic. In the case of parametric curves, the so-called detuningfactor is ? = 2?, while for Duffing it is ? = ?. The Duffing curves (1, 2,and 3) show a skew towards left due to a small negative cubic nonlinearity.In the case of parametric curves we see sudden jumps (after the bifurcationpoint), as predicted by the theory. The resonators are required to operatebelow this bifurcation point, such that an optimal gain is obtained. Thetheoretical stability limit computed using the perturbation method indicatea common mode ac voltage amplitude of 17 V as stability boundary, whileexperimentally a value of 15.7 V was obtained, as shown in Fig. 6.6 (thatillustrates the so-called Arnold?s tongue). To show its significance, we haveperformed up and down frequency sweeps (using 17.3 V common mode acvoltage amplitude), and one such experimental result is shown in Fig. 6.7.Blue points represent the sweep-up frequency variation, while the green onesare associated with the sweep-down frequency change. We see that hystere-sis exists in both situations, cause by the small cubic nonlinearity [ [104]].6.2 Parametric amplification in MEMSgyroscopes (secondary mode)Parametric amplification is a nonlinear effect, due to periodic modulationof the equivalent spring constant (in our case through common-mode acvoltages), and it can be applied in principle to both the driven and sensingmodes (with dedicated capacitive combs). If one assumes harmonic actu-ation at resonance in 0x mode and considers the nonlinear form given by120Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.5: Experimental results obtained for Duffing oscillator(1,2,3) andparametric amplification on the secondary mode of the MEMS gyroscope.Parametric resonance differs from Duffing?s due to presence of instabilityregionMathieus equation instead of (3.6), one has:x?+cxmx?+kxm(1 +?kxkxsin(?pt+ ?p))x =Factuationmsin(?xt+ ?0x)?kyxmy(6.13)121Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.6: Stability points VS the parametric resonance frequency. Pointsindicate the experimental results and line is the theoretical fitHere the pumping frequency ?p = 2?0x, while ?x,?p are the phases of drivingforce and of the parametric pump, respectively. ?kx is the amplitude ofthe modulation factor for the spring constant. To attain maximum gain,the phase of the parametric pump should be orthogonal to the drive phase[106] [107]. A similar parametric pumping scheme is required in the sensingmode, where the phase of the electro-mechanical pump is tuned such thatthe energy is pumped into the resonator synchronously with the Coriolis122Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.7: Hysteresis phenomenon observed in the secondary mode ofMEMS gyroscope. Frequency sweep up is blue line and sweep down ingreen line. Distinct jumps occur as shown with black arrow marks.force (a lag of 90? relative to the driving force).y?+cymy?+kym(1+?kykysin(?pt+?p?90?))y = 2x??zsin(?xt+?0x?90?))?kxymx(6.14)The sign of the Coriolis force has been incorporated as a phase change inEq.(6.12). Assuming the system is actuated at its resonance, the steadystate solution of Eq.(6.12) is given by:yss =FCoriolisky(1?Qy?ky2ky)sin(?xt+ ?x ? 90?)? kxyxss (6.15)123Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopes6.3 Experimental setupTo experimentally prove the technique, the Coriolis force term is mimickedby a voltage controlled electrostatic force in the structure shown in Fig. 6.8.As mentioned before, there are three independent bond pads for actuationand sensing combs. One set of the comb drives is used to mimic a Coriolisforce (the blue connections in Fig. 6.8. Other two sets of the comb drives (redconnections in Fig. 6.8) are used for parametric pumping, using commonmode ac voltages. Existing defects like missing fingers in the fabricateddevices (Fig. 6.9) generate the equivalent of quadrature signal components.A common mode AC voltage is applied on the differential gap-varyingcapacitances and used to pump energy from electrical to mechanical domain.The resulting electrostatic force results in:Felectrostatic =C02(11? (y(t)d )2?11 + (y(t)d )2)u2CM (6.16)Here, C0 is the capacitance at rest position and uCM is the common modevoltage. For y  d0, where d0 is the zero-voltage gap, the net effect of theelectrostatic force is a modulation by uCM of the equivalent spring constantky:my? + cyy? + (ky ??ky(uCMV0)2)y = FCoriolis + FQuadrature (6.17)For an harmonic common mode voltage of the same frequency as thedriving mode actuation voltage, uCM (t) = V0cos(2?yt + ?), the relativephase ? is the essential parameter for the electromechanical parametric cou-pling.124Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.8: Device concept and working principle6.4 Experimental resultsThe experimental validation measured optically the induced displacementsin the proof mass, for various parametric pumping conditions. PolytecPMA-500 c? equipment was used for the test and characterization of thegyroscope structures. Both the scanning laser-Doppler vibrometry and thevideo-stroboscopic planar motion analyzer modules were used to extract theparameters related to the driven and sensing resonant modes. The mea-surements led to a stiffness coefficient value ky = 132.23 N/m, for a sense125Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.9: Sense comb drives with gap-varying fingersmode resonant frequency at 8.57 kHz. The equivalent of a Coriolis forcegenerated by ?z = 1?/s is generated electrically by inducing the 0y motionof the proof mass, while a common mode voltage of adjustable phase delayand amplitude is used for parametric pumping. Fig. 6.10 illustrates the de-pendence of the net parametric gain (amplification of Fc effect) on the uCMamplitude.A maximum gain factor of 25 was experimentally measured before reach-ing dynamic instability in 0y motion. Parametric coupling increases theselectivity of the resonant mechanical structure, leading to a larger equiv-alent quality factor and a lower noise level, as illustrated in Fig. 6.11 and126Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.10: Experimental (points) and theoretical (line) parametric gainvs. pump voltage amplitudeFig. 6.12. Noise estimation was done for both the equivalent 0y resonatorand for the MEMS gyroscope. The mechano-thermal noise depends on thedamping coefficient and the operating bandwidth.The Fig. 6.11 shows the measurements indicating the reduction of thenoise force (Fn,y =?4KBTcyBW ) with a higher parametric gain, for aconstant measurement bandwidth (BW). For a gain of 5, the equivalentinput acceleration noise diminishes from 0.033 ms?2 to 0.022 ms?2. Theincreased spectral selectivity will also contribute, in the case of the gyro-127Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.11: Influence of the parametric amplification on the equivalentbandwidthscope, to a reduction of the equivalent noise input angular rate (given byEq.(11a)). For a gain of 5, ?z,n reduces from 0.0046 (deg/s?Hz) to 0.0026(deg/s?Hz). Fig. 6.13 shows the dependence of the parametric pumpingon the relative phase between the driving force and the common mode ACvoltage. A reduction of the normalized gain up to 0.6 of its value in the ab-sence of parametric pumping was measured. According to theory, the phasedependence of the gain is given by [106] [107]:G(?) =?(cos?1 +Qy ?k2ky)2 + (sin?1?Qy ?k2ky)2 (6.18)Consequently, the parametric gain is maximum at ?= 90 deg and reaches128Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.12: Equivalent input angular rate noise (blue) and equivalent out-put displacement due to the mechano-thermal noise (red) vs. parametricgaina minimum at ?=180 deg, facts validated experimentally. The model pa-rameters extracted from the experimental measurements were used in com-plementary Simulink simulations, which have shown a good matching withthe data. Numerical simulations indicate an attenuation of 0.8 of the motioncomponent when the phase difference is set to ?= 180 deg. This paramet-ric damping mode is extremely useful for the attenuation of quadratureerror component. The signal-to-noise ratio (SNR) obtained for differentparametric gains is illustrated in Fig. 6.14, where we see the relative im-provement in SNR compared to non-parametric actuation case. Theoreticalresults are thus matched by the experimental measurements. Some spectral129Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.13: Phase dependence of the normalized gain (experimental andsimulated)components of the mechano-thermal noise might also be amplified by theparametric amplification mechanism, but the strong phase sensitivity willensure that the dominant amplified term will be the Coriolis-induced signal.6.5 Concluding remarksThe minute capacitive variations in MEMS inertial sensing, coupled withthe effects of both random (noise) and deterministic (quadrature errors) sig-nal components require an amplification of the mechanical vibration signalbefore the conversion into an electrical one. It is shown, validated by mea-surements, that mechanical motion in the sense mode can be amplified orattenuated, depending on specific phase synchronization conditions between130Chapter 6. Parametric resonance: amplification and damping in MEMS gyroscopesFigure 6.14: SNR vs. normalized gain (experimental and theoretical fit).the mechanical motion and the common-mode AC actuation. Parametricgains in the range 5 to 25 have been experimentally measured on the teststructures. The technique can be therefore applied either for the amplifi-cation of the Coriolis induced motion, or for a reduction of the quadratureerror signal.131Chapter 7Novel sloped combs forparametric resonance5In this chapter slope-shaped combs are designed, modeled and experimen-tally verified for parametric amplification of micro-displacements. As weknow that the electrostatic force generated by area-varying combs is inde-pendent of the displacement;, so it cannot generate an equivalent stiffnessmodulation through electrical control. As a result, parametric amplificationconcepts cannot be directly applied to the driving combs of the MEMS an-gular rate sensors, except if we consider and model second order effects, likefringe-fields variations with the relative position of the fixed and movableplates. Shaped comb drive is presented, which produces nonlinear force dueto its geometry as seen in Fig. 7.1Introducing the angle ? and notation Tan[?] = wLs . Then the expressionof the capacitance C[x], for the sloped part of the movable fingers is derived(neglecting the fringe field effect) as following:y[x] = xTan[?] = wxLs(7.1)5A version of this chapter has been published. Reprinted from [124] with permis-sion. 2013, Elsevier. Sharma, M, Sarraf,E, and Cretu, E. Shaped combs and parametricresonance in MEMS inertial sensors. Proceedings of IEEE SENSORS(accepted). 11/2013132Chapter 7. Novel sloped combs for parametric resonanceFigure 7.1: Shaped comb drive with dimension definitions and surface elec-tric potential (Simulation)d1 = d0 ?W, forx ? Ls (7.2)dC[x] =hdxd[x]=hdxd0 ? xTan[?](7.3)dC[x]dx=hd0 ? xTan[?]= hd011? xLswd0(7.4)Total capacitance (C[x]) and net electrostatic force (f [x]) is thus givenbyC[x] = hd0? Z011? zLswd0dZ =hTan[?]log11? xTan[?]d0=hLswln(11? xLswd0)(7.5)f [x] =?12dC[x]dxV 2 = ?12V 2hd011? xLswd0(7.6)Net force by the sloped shape comb is gap dependent, and the equivalentstiffness (neglecting the higher orders) can be obtained from the Taylor?s133Chapter 7. Novel sloped combs for parametric resonanceseries expansion as:f(x) =?12V 2hd0(1 +xLswd0+ (xLs)2(wd0)2 + ...) (7.7)As seen from Eq.(7.7), a larger slope of the fingers (w/Ls) is desirable inorder to enhance the effective electrostatic stiffness, kelslope. The DC com-ponent (and the higher order even power components) of the electrostaticforce can be eliminated by symmetric (common-mode left-right) actuation.The shaped fingers used for the parametric amplification in the present workhave a similar geometry with one of the configurations used in work [108].The authors have shown there that, through the spring softening effect, theywere able to tune the resonant frequency of a structure from 4.2 kHz to closeto 0 Hz, with a lowest stable resonant frequency of 165 Hz (for 79.1 VoltsDC actuation on the sloped combs). The high level of equivalent stiffnessmodulation obtained in that case suggested the use of such shaped combsfor parametric amplification. For comparison, the case of non-interdigitatedfingers, in order to better understand their usage for tuning the equivalentelastic stiffness. Adam et.al [109] presented spring softening mechanism formovable fingers, centered between the fixed comb fingers. The electrostaticspring softening relies on the fringe field effects. They termed this config-uration of fingers as transverse-reduction actuator as seen in Fig. 7.2. In acomplementary way, if the horizontal position of the fixed fingers is alignedwith that of the movable ones, the fringe field effect will generate a netrestoring force dependent on the horizontal (0x) displacement, equivalentto a positive spring constant. This transverse augmentation actuation wasillustrated by Adam et al [109], who have further shown that the net elec-trostatic force generated by these fingers depend on the ratio of the fingerswidths and the gap distance between the moving and fixed fingers. The lin-134Chapter 7. Novel sloped combs for parametric resonanceFigure 7.2: Non inter-digitated combs with dimension definitions and sur-face electric potentialearly dependent electrostatic force on displacement means that these fringefield effects can be exploited in parametric amplification techniques. Wehave performed finite element analysis in COMSOL Multiphysics c?, in or-der to compare the sloped combs with non-interdigitated combs with respectto their actuation effectiveness. For a fair comparison, sloped comb driveswith the same geometry as the fabricated ones have been simulated, andtheir electro-mechanical effect compared to that of non-interdigitated fin-gers having similar dimensions. Movable combs are supported by crab legsuspensions with length and width of 200 ?m and 10 ?m respectively; theactuation is generated by a varying DC voltage applied between the fixedand movable combs. The length of the fixed fingers is 150 ?m, while thatof the movable fingers (for both non-interdigitated and sloped fingers) is172.5 ?m, for a total width of the movable fingers of 125 ?m. The slopeof shaped combs used in the simulation is 1. The dimensions d0 and d1(earlier defined in Fig. 7.1 for the sloped combs) are 11.5 ?m and 4 ?m,respectively. In the case of non-interdigitated fingers, d0 is 2 ?m (described135Chapter 7. Novel sloped combs for parametric resonancein Fig. 7.2), corresponding to the minimum gap allowed by most fabricationrules. For a reference 10 V DC voltage actuation, the induced electrodedisplacement was about 1.6 times higher for the sloped combs comparedto the non-interdigitated fingers, as seen in Fig. 7.3 and Fig. 7.4. Themaximum electrical field for sloped combs is about 10% higher than fornon-interdigitated combs, as seen in Fig. 7.5 and Fig. 7.6. Simulation re-sults confirm therefore that, for the same applied voltages, shaped combsproduce higher displacement than the non-interdigitated combs, or in otherwords they generate larger electrostatic forces. They also have a strongerposition dependence (and therefore a higher stiffness modulation) than theirnon-interdigitated alternative.7.1 Device structure and behaviorcharacterizationA structure similar to one used in previous chapter is fabricated using theSOIMUMPS R? technology, with exception to comb drives. Sloped shapedcombs for non linear analysis along with area varying combs for systemcharacterization is designed. Area varying combs, due to their inherentlinear properties act as actuation combs, while common mode voltages areapplied to the shaped combs to modulate springs. Structure has a massof 36.9 nKg, with simulated resonant modes in 0x as 8.299 kHz, 0y as8.65 kHz and 0z mode as 10.48 kHz. Designed static capacitance of areavarying combs is 5.63 pF and for shaped combs is 3.762 pF . An image ofthe structure obtained from Polytec and FE Analysis resonant modes arepresented in Fig. 7.7 and Fig. 7.8136Chapter 7. Novel sloped combs for parametric resonanceFigure 7.3: 0y displacement obtained with a 10 V DC, for sloped shapedfingers7.1.1 Static measurementsCharacterization of the shaped combs is done by applying DC bias to oneside of the shaped combs and grounding the proof mass and other set ofcombs to ground as shown in Fig. 7.9A.Voltages are steadily increased to obtain the 0x displacement. Displace-ments of sensor is obtained by choosing the time domain of FFT option inPolytec MSA-500. A small AC signal is applied with sweeping frequency is137Chapter 7. Novel sloped combs for parametric resonanceFigure 7.4: 0y displacement obtained with a 10 V DC, for non interdigitatedfingersbelow the resonant frequency range. Results showing 0x displacements forapplied voltages is presented in Fig. 7.10. At about 140 V, resonator actshighly nonlinear and is unstable and reaches to pull in. One such exampleis shown in Fig. 7.11. Oscillations are larger in vertical direction than inhorizontal direction, leading to pull in as seen in the spikes.138Chapter 7. Novel sloped combs for parametric resonanceFigure 7.5: Surface electric field obtained with a 10 V DC actuation voltage,for slope shaped fingers7.2 Nonlinearity measurementsTo investigate the nonlinearity in the structure, one end of the shaped combsis applied a constant AC signal with a DC bias, sweeping near the resonantfrequency as seen in Fig. 7.9B. The other end of the shaped comb is appliedwith a varying DC bias. To ensure cubic nonlinearities are eliminated, smallactuating electrostatic forces are applied AC=8 sin(?t) V and DC= 8 V .DC bias on the other set of shaped combs are varied from no bias to 40 V.139Chapter 7. Novel sloped combs for parametric resonanceFigure 7.6: Surface electric field obtained with a 10 V DC actuation voltage,for non interdigitated fingersA shift in the resonant frequency (reduction) is seen with increase in DCbias. A reduction from 8347 Hz to 8338 Hz is observed from 0 to 40 V asseen in Fig. 7.12. Electrostatic spring stiffness can be obtained by:2?ff=?kkmech(7.8)Resonant frequency of the resonator at no bias is used to obtain thekmech. From Fig. 7.12, resonant frequency at no bias is 8347.6 Hz and cor-140Chapter 7. Novel sloped combs for parametric resonanceFigure 7.7: Image of resonator with sloped shaped and area varying combsresponding stiffness obtained is 101.5395 N/m. The Fig. 7.13 shows theimpact of resonant frequency and corresponding electrostatic spring modu-lation with the different DC biases. According to the results, single endedDC bias tuning can produce 0.223 % stiffness modulation.7.3 Common mode analysis spring softening andparametric amplificationFor parametric resonance, spring modulation is desired with common modeactuation scheme. The net force acting on the sensor due to common mode141Chapter 7. Novel sloped combs for parametric resonanceFigure 7.8: FE analysis results for resonant modesvoltages can be expressed as:Fleft ? Fright =?12V 2hd011? xwLsd0??12V 2hd011 + xwLsd0= 0 (7.9)142Chapter 7. Novel sloped combs for parametric resonanceFigure 7.9: A) DC bias characterization for shaped combs and B)Schematicfor half actuation and DC bias tuning schemeDifferential actuation scheme is applied on the area varying combs, whilecommon mode varying DC bias is applied on the shaped combs to inves-tigate the spring softening effect. The Fig. 7.14A shows the experimentalsetup. Benefit of using common mode approach is that the DC component(and the higher order even power components) of the electrostatic forcecan be eliminated (Taylor?s series approximation) as seen in equation 7.9.The electrostatic stiffness can be approximated for the common mode typeactuation as:kelcommon = ?V2hd0(wlsd0)N/m (7.10)Spring softening due to varying common mode DC voltages is obtained143Chapter 7. Novel sloped combs for parametric resonanceFigure 7.10: Static displacement VS DC bias on one side of sloped combsusing the Polytec MSA-500 c?, planar motion analyzer(PMA). Differentialactuation voltages applied on the area varying combs are same as previousconfiguration. Spring softening from 8347.6 Hz to 8320 Hz is obtained forDC= 40 V . The Fig. 7.15 shows the bode plots obtained from the PMA.Impact of resonant frequency due to common mode DC bias and corre-144Chapter 7. Novel sloped combs for parametric resonanceFigure 7.11: Static displacement VS DC bias on one side of sloped combssponding electrostatic spring softening modulation is plotted in Fig. 7.16Aand B.Spring softening effect of common mode DC bias tuning is approximatelydouble of the single ended DC bias tuning. From the experimental fits asseen in Fig. 7.16B and Fig. 7.13 electrostatic stiffness modulation experimen-tally obtained is double kmech?kelslope,halfkmech = 0.22% andkmech?kelslope,commonkmech=0.57% respectively. Theoretically, results are verified as nonlinearity willdouble for common mode biasing scheme as shown in equation 7.10. Ana-145Chapter 7. Novel sloped combs for parametric resonanceFigure 7.12: Weakening effect on the resonator due to spring softening withDC biaslytical model is based on the assumption that shaped comb fingers are placedin the limit of their slope. Fabricated structure on the other hand are in-ter digitated. Next generation of combs can further enhance this nonlineareffects.For parametric resonance, similar actuation scheme on area varyingcombs is employed with an exception of AC common mode voltages on the146Chapter 7. Novel sloped combs for parametric resonanceFigure 7.13: Change in resonant frequency and electrostatic spring softeningwith DC biasFigure 7.14: Schematic for A) DC and B) AC spring modulation usingcommon mode voltage147Chapter 7. Novel sloped combs for parametric resonanceFigure 7.15: Weakening effect on the resonator due to spring softening withcommon mode DC biasshaped combs. This set up is shown in Fig. 7.14B. Based on the same mod-ulating principle described in previous chapter, parametric pump is applied.For a common mode AC voltage UCM=10 V , parametric gain of 1.35 isobserved as seen in Fig. 7.17. From experimental and simulation estimationthis gain is achieved for 0.1% stiffness modulation.An impact of phase on the parametric gain is shown in Fig. 7.18.148Chapter 7. Novel sloped combs for parametric resonanceFigure 7.16: A) Change in resonant frequency and B) Comparison of elec-trostatic spring modulation using common mode DC voltage7.4 Parametric amplification in MEMSgyroscopes both in (primary and secondarymode)In this section, a gyroscope fabricated in TRONICS technology is tested forparametric resonance using both sloped combs(for primary) and gap varying149Chapter 7. Novel sloped combs for parametric resonanceFigure 7.17: Impact of parametric gain with 10 V common mode voltageusing shaped comb resonator. Blue curve- with no pump and red curve withparametric pumpcombs (for secondary). A proof mass is suspended by crab leg fixtures sothat it ensures the two orthogonal degrees of freedom (driven/sensing) ofthe MEMS gyroscope. The structure is fabricated in TRONICS technology,with a minimum gap size of 3 ?m between the active combs and a thicknessof 60 ?m for the structural layer. A set of about 1208 holes (10?mx10?m)in the proof mass helps in its etch-release process. The Fig. 7.19 shows thedistinct groups of actuating and sensing comb fingers.Two different types of combs are used for the actuation: (a) area-varyingcombs and (b) sloped combs. The horizontal actuation (0x) is achievedby applying AC voltages between the moving and the fixed fingers of theactuators (with a 7 ?m gap) and a DC voltage to the proof mass. When150Chapter 7. Novel sloped combs for parametric resonanceFigure 7.18: Impact of phase on parametric gainan external angular rate ?z rotates the sensor about 0z axis, the resultantCoriolis force will induce a proportional motion in the sense mode (0y). The0y displacement is measured using asymmetric comb drives, with large gapsof 3.5 ?m and 12.5 ?m, intentionally left wider than the minimum allowedgaps in order to facilitate the parametric amplification experiments.Resonant modes obtained from FEA simulations are 7.8 kHz for 0x di-rection, 8.88 kHz for 0y direction and 9.487 kHz for 0z direction as seen inFig. 7.20. The resonant mode at 0z is 100x smaller than in plane motions,as the thickness of the beams are much larger than their widths. Stiffnesscoefficients are (kx =570 N/m) and (ky=748 N/m). The net, zero displace-ment capacitance of one side of actuation is 0.642 pF and for sensing is 1.6pF . The capacitances of sensing combs are higher than the area varyingactuation combs, as a section of these combs will be utilized for parametric151Chapter 7. Novel sloped combs for parametric resonanceFigure 7.19: SEM shot of the MEMS gyroscope with distinct combsamplification in sense direction; meanwhile extra combs are incorporatedwith the area varying comb drives as shaped slope comb drives for paramet-ric amplification in 0x direction. ?kx is the amplitude of the modulationfactor for the spring constant. In order to produce the modulation factor, acommon mode voltage is applied on the sloped combs of the structure, suchthat the DC and the other even harmonic components of the net electrostaticforce dependence on displacement will cancel out, as described previously.For small displacements, the mechanical cubic nonlinearity of the elasticsuspension springs can be neglected, hence the cubic modulation too. Thesimulation curves of electrostatic forces ( FV 2 ) vs. the normalized gap (xd0)are illustrated in Fig. 7.21, together with the linear ?kx/V 2 in Fig. 7.21.152Chapter 7. Novel sloped combs for parametric resonanceFigure 7.20: Resonant modes of the angular sensor with correspondingcomb drivesThe linear stiffness modulation is 2.210?3V ?2.Depending upon the phase (?) between the parametric modulation andthe actuation force Factuation, the parametric gain can be positive or nega-tive. Maximum amplification occurs when the phase difference is zero, witha maximum gain achieved near the resonance, as seen in Fig. 7.22, wheredisplacement, frequency and phases are plotted.A parametric pump, using gap-varying comb drives, with common modevoltages, will produce a stiffness modulation (?ky) for enhancing the weak153Chapter 7. Novel sloped combs for parametric resonanceFigure 7.21: Linear fit of the Force/normalized gap in the linear gap-varyingregion of the sloped combssignal. The Fig. 7.23shows the simulation curves of the normalized elec-trostatic force (F/V 2) vs. the normalized gap (y/dy); the computed linearcoefficient ?ky/V ?2 is 0.077V ?2.As the Coriolis-induced motion is directly proportional to the 0x (drivenmode) velocity, the total parametric gain, considering the parametric ampli-fication for both the driven and sensing modes, will be the effective product154Chapter 7. Novel sloped combs for parametric resonanceFigure 7.22: 3D plot simulation of frequency vs phase vs Displacement, withinitial Qx = 100, ?0x=8 kHzof their partial effects, as seen in Eq.7.11.G(?) =?(cos(?)1 +Qx ?k2kx)2 + (sin(?)1?Qx ?k2kx)2??(cos(?? 90?)1 +Qy ?k2ky)2 + (sin(?? 90?)1?Qy ?k2ky)2(7.11)The partial gains will depend not only on the common mode voltages, butalso on the quality factors of the two resonant modes. For instance, thestiffness modulation obtained by gap-varying combs (secondary mode) isstronger than that induced by the sloped combs (in the primary mode), butthe quality factor of the primary (driven) mode, Qx, is larger than that of the155Chapter 7. Novel sloped combs for parametric resonanceFigure 7.23: Linear fit of the Force/normalized gap in linear region for gapvarying combssecondary mode (Qy), due mainly to the different air damping mechanisms.A desired net gain can be therefore set by the appropriate biasing of theindividual pumps. The Fig. 7.24 depicts the impact of net gain on the sensedisplacement (y/deg/s), as obtained from Eq.7.11.7.5 Concluding remarksIt is shown that sloped combs are suitable candidates for parametric ampli-fication techniques, better than non-interdigitated combs, when large dis-placements are necessary (like the case of the driven mode of resonant gyro-scopes). We have designed, modeled and experimentally shown that a larger156Chapter 7. Novel sloped combs for parametric resonanceFigure 7.24: . Simulated, impact of net gain (primary and secondary) on 0ysense displacementstiffness modulation of the springs is possible than in the case of fringe field-related techniques. Sloped shaped combs were tested in two technologiesnamely SOIMUMPS R? and SOI TRONICS R? (refer to appendix A for ex-perimental work on TRONICS). A complete system may therefore use adouble parametric amplification technique, on both primary and secondarymodes (with appropriate phase tuning of their respective common modevoltages), in order to get larger gains.157Chapter 8Concept of dynamic pull-inMEMS gyroscope 6This chapter analyzes numerically and experimentally the dependence ofthe dynamic pull-in voltage amplitude on the values of externally-inducedaccelerations (e.g. Coriolis accelerations in the case of vibratory gyroscopes).We have investigated the nonlinear dynamic behavior on two different devicetypes, a micro-gyroscope (A) and a micro-accelerometer (B), both fabricatedin the SOIMUMPS R? process (25 ?m thick structural layer). Experimentalmeasurements on the MEMS structures have been performed using PolytecMSA-500 equipment for analyzing the mechanical motion. They indicatethat the dynamic pull-in voltages reduce from 100 V to 56 V for device Aand from 21.77 V to 17.3 V for device B, for an equivalent acceleration of0.319 ms?2 , when the structures are actuated at their resonance frequency.If the induced acceleration is translated into an equivalent angular rate,?equivalent, modulating the Coriolis-induced motion, the dynamic pull-involtages vary from 57.612 V to 56.5 V for device A type and from 20.9 Vto 10.75 V for device B type, for a change of 1 rad/s to 5 rad/s in ?equivalent.6A version of this chapter has been published. Reprinted from [123] with permission.2011, Elsevier. Sharma, M, Sarraf,E, Cretu, E. Novel dynamic pull-in MEMS gyroscope.Procedia Engineering. 2011158Chapter 8. Concept of dynamic pull-in MEMS gyroscope8.1 IntroductionNonlinearity behavior is exhibited by most of the MEMS structures, due toactuation (electrostatic forces in gap varying comb drives), stiffness [101]and damping [37]. There is a presence of vast literature on understand-ing the stiffness nonlinearity and generating larger amplitudes by varyingstiffness in time (electrostatic modulation) [113], [114]. This phenomenonis popularly known as parametric amplification. Generally high voltage isneeded to realize the novel nonlinear based MEMS inertial sensors as thereis always a limit on high the voltage can be raised before a device is caughtwith the pull in instability [115]. Pull-in phenomenon has been exploited inhigh-sensitivity MEMS accelerometers, in order to reconcile the small sizeof the proof mass with high performance requirements. In one of the ap-proaches, for instance, the external acceleration is indirectly measured fromthe pull-in time of the device [116]. When the resonators are operatedat resonance or at its integer there is a phenomenon called dynamic pullin [117]. The present paper proposes the use of a similar dynamic pull-inphenomenon for MEMS vibratory gyroscopes, where the value of the (dy-namic) pull-in voltage, Vpi,dyn, is modulated by the external angular rateto be measured. In this case, a common mode harmonic voltage is appliedon the sensing comb capacitors, at a frequency equal to that of the Coriolisinduced motion (close to the resonant frequency of the sensing mode). Previ-ous studies have indicated lower values of the dynamic pull-in amplitudes forfrequencies close to the resonant frequency of the MEMS device [ [51], [52]].The variation of dynamic pull-in voltages (AC signal amplitude) with theexternal acceleration can be correlated with the amplitude of the signal tobe sensed. In the case of MEMS gyroscopes, the frequency and phase of159Chapter 8. Concept of dynamic pull-in MEMS gyroscopeCoriolis induced motion in the sensing mode is predetermined. By imposinga common mode AC signal with appropriate frequency, phase, and ampli-tude, MEMS gyroscopes can be biased near their border of stability, suchthat they lose stability (snap to pullin) when an external acceleration has aspecific frequency, phase and amplitude. Dynamic pull-in is directly relatedto the kinetic energy of the system, which, in turn, is affected by transienteffects and disturbances. Understanding of frequency varying damping isessential, for inertial sensors which are operated in atmospheric pressure.As mechano-thermal damping influences the performance over combinationof several mechanisms, such as thermo-elastic damping, surface loss, dissipa-tion through support (anchor loss), which dominate at vacuum [ [35], [36]].These works also show that sensitivity is generally limited by the mechano-thermal noise generated by the interaction of the movable structure withthe surrounding fluid, of a certain viscosity. In this work, we present a newperspective for MEMS gyroscopes, where in we study the impact of externalaccelerations on the dynamic pull-in behavior. The modulation of dynamicpull in with external acceleration is experimentally shown for two inertialsensors.8.2 Pull-in mechanismTo understand pull-in mechanism (static and dynamic), gyroscope as de-scribed in Chapter 6 is taken (device A). This senors has stiffness coefficientsas kx= 135 N/m, ky= 123 N/m in x and y direction respectively and mass of48.67 nKg. It is notified to the reader that not all MEMS chips(gyroscope)had the same resonant frequencies. For the pull-in analysis, gyroscope de-vice had resonant frequency of 8.1 kHz and 8.3 kHz instead of 8.5 kHz and160Chapter 8. Concept of dynamic pull-in MEMS gyroscope8.85 kHz in x and y direction as used in chapter 6. The damping coefficientin 0y direction is 57 ?Nms?1. An accelerometer (device B) is consideredwith stiffness k=49.56 N/m, damping coefficient, c of 2.57 ?Nms?1 andwith a mass of 56.3 ?g . An image of these two sensors is shown in Fig. 8.1.8.2.1 Static asymmetric pull-in phenomenonTo explain this phenomenon, a DC bias on the gap varying comb drives(independently) is applied. This leads to increase in the electrostatic forceand when it dominates the mechanical force, pull in is observed (snappingof fingers) as seen in Fig. 8.2. This can be observed for both comb drives,damping adds the resistive force along with the stiffness in flexures. Thistype of pull-in is known as asymmetric pull-in [118]. Net forces acting onthe proof mass is given by Eq.3:Fmass + Fstiffness + Fdamping = Felectrostatic (8.1)Here, Fmass = my?, Fstiffness = kyy3+kyy, Fdamping = Cyy? and Felectrostatic =C0u22(1? yd )2 . The potential energy added in the system can be computed by tak-ing the electrostatic forces and energy stored in the springs and inertia.V? (y) =12y2ky +12x2kxy +14y4k3y ?C0u22(1? y(t)d )2(8.2)Neglecting the higher terms of y and quadrature stiffness, pull-in voltageof this set up is given as, Vpullin =?827d0?kyC0[118]. In the structure shownin Fig. 8.2, asymmetric pull in voltage is 69.3 V.161Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.1: Device A: gyroscope for symmetric dynamic pull-in set up;Device B: accelerometer for symmetric dynamic pull-in set up162Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.2: Asymmetric pull in phenomenon, blue curve- snapping of fingersto top electrode. 0.1 to 0.2 ms is the meta stable region due to damping163Chapter 8. Concept of dynamic pull-in MEMS gyroscope8.2.2 Static symmetric pull-in phenomenonWhen a common mode DC bias is applied to both comb drives simultane-ously, the electrostatic forces tend to cancel each other out. However, dueto fabrication imperfections, nonlinearities in stiffness, damping, and whitenoise, snapping of fingers can occur in either direction, refer to Fig. 8.3 forthe set up. Total energy of such a system can be:V? (y) =12y2ky +12x2kxy +14y4k3y ?C02(1(1? y(t)d )2?1(1 + y(t)d )2)u2 (8.3)Neglecting the nonlinear terms and quadrature stiffness term in the aboveequation, symmetric pull in is given as Vpullin =?12d0?kyC0[118]. Theequation suggests that symmetrically driven structures have 30% larger pullin voltages when compared to asymmetric drive. In the structure (DeviceA) the symmetric pull in voltage is 97.8 V.Figure 8.3: Set up of symmetric pull-in164Chapter 8. Concept of dynamic pull-in MEMS gyroscope8.2.3 Resonant pull-in and symmetric dynamic pull-inThe previous pull in phenomenon had the proof mass initially at zero dis-placement. When oscillators are resonating at their natural frequency, pullin can be achieved by increasing the potential energy of the system. InFig. 8.4, potential energy of such oscillation is shown, with red and blackcurves as the energy variation with respect to the deflection per half cycleof the oscillation. Blue curve, depicts the net gain in the energy. Near 60%(0.6) normalized deflection of the gap there is addition of potential energywhich leads to larger amplitudes. We plot the phase portrait along with thepotential energy, showing stable oscillation near 60 % (0.6) deflection. Greencolor curves are the basin of attraction; blue dotted lines depict the null-clines. At about 80% of the gap, sensor loses its stability and snapping canbe seen in Fig. 8.5. Net addition (electrostatic)and loss of energy(damping)per half cycle is [120]:V?add = C0u2CMy(t)d1? (y(t)d )2(8.4)V?lost =pi2?cy(y)y(t) (8.5)In the case of MEMS gyroscopes, there needs to be a primary mode ofoscillation, to sense the Coriolis induces motion. In order to achieve pullin in secondary mode; the phase of the electrostatic force would have tomatch with external acceleration (in our case Coriolis induced acceleration).Further, frequency of the bias should be carefully manipulated. For a com-mon mode AC voltage (UCM = V0cos(?yt+?)) applied to the sense combs,with the resonant frequency, there is a spring modulation which occurs atdouble the resonant frequency (parametric resonance), refer previous chap-165Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.4: Normalized potential energy vs normalized displacement. At0.8 of the normalized gap, the oscillations are unstable which will lead topull in. Blue dotted curve is the net gain in energyFigure 8.5: Phase portrait and potential energy curves, highlighting oscil-lation, with oscillation at 0.8 normalized gap goes to pull in166Chapter 8. Concept of dynamic pull-in MEMS gyroscopeters on parametric resonance. Depending upon the direction of the Coriolisacceleration, pull in occurs at either comb drives. The Fig. 8.6, depicts thetransient motion of proof mass due to initial acceleration (0.06 ms?2) andcommon mode voltage of (58 V ). Assuming the phases and the resonant fre-quency of the common mode voltage is matched with the Coriolis inducedmotion, the pull in voltage will still be dependent on the external initialcondition (amplitude of the Coriolis induced motion). The Fig. 8.7 depictsthat AC common mode voltage of 97.5 V, leads to pull in, when the externalacceleration is 5 ms?2. For the external acceleration less than those values,there is an increase in the amplitude of the sense motion (parametric gain).Both structures were operated at atmospheric pressure, in a frequencyrange were the elastic behavior of the air is negligible relative to its dampingaction. The theoretical analysis and the experimental results show that forboth structures the quadratic spring coefficients are dominant, as given bythe Eq. [27]:ky =38k3yk1yf0 ?512k22yk21yf0 < 0 (8.6)Electrostatic spring softening is the dominant effect observed for bothstructures when actuated with unipolar voltages on the gap varying combs,as seen in Fig. 8.8. The nonlinear springs of the system can be computedby expanding the electrostatic forces. The resonant frequency of deviceA(sense mode) is about 8.5 kHz and for the device (accelerometer) is about4.75 kHz.As seen from Eq.6.18, displacement is phase dependent (?) and willdetermine if the pull in will occur. When the phases are not tuned, negativegain is possible and the oscillations of the structure can be brought to zero167Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.6: Symmetric dynamic pull in behavior. Proof mass initiallyat oscillation (0.06 ms?2). Depending on the phases between electrostaticforces and external acceleration, pull in possible.displacement. However, by further increasing the voltage, pull in still canbe achieved, which will be independent of the external acceleration. As theintend of this study is to see the impact of external acceleration on the pullin we, apply electrostatic force with phase and frequency correlating to theCoriolis acceleration. To illustrate the dependence of phase to the pull in wepresent the result in Fig. 8.9. With the increase in common mode voltagewith tuned frequency and phase, there is increase in the gain (shown inblack). At about 17 V, if the phases are tuned, pull in achieved (shownin blue), if the phases are detuned (to 0), there is a drastic reduction in168Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.7: Impact of initial acceleration on the proof mass oscillations.Higher initial acceleration leads to pull in (5 ms?2) with common modevoltage of 97.5 V in device A.gain (shown in red). These tests were carried on device B, with externalacceleration of 0.4253 ms?2 .8.3 Experimental setupTo experimentally prove the technique, the Coriolis force term is mimickedby a voltage controlled electrostatic force in the structure shown in Fig. 8.1.Both the devices have independent bond pads for actuation and sensingcombs. One set of the comb drives is used to mimic a Coriolis force (theblue connections in Fig. 8.1). Other two sets of the comb drives (red con-nections in Fig. 8.1) are used for parametric pumping, using common modeAC voltages. A common mode AC voltage is applied on the differential gap-169Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.8: Spring softening effect seen for both device A and B. Voltagesapplied are the common mode voltages as shown in figure with VDC of 10V.varying capacitances and used to pump energy from electrical to mechanicaldomain as shown in Eq.6.18.8.4 Experimental resultsThe experimental validation measured optically the induced displacementsin the proof mass, for various parametric pumping conditions. PolytecPMA-500 c? equipment was used for the test and characterization of thegyroscope structures. Both the scanning laser-Doppler vibrometry and thevideo-stroboscopic planar motion analyzer modules were used to extract theparameters related to the driven and sensing resonant modes. The zero-voltage resonant frequency (f0) is 8.5 kHz for device A and 4.75 kHz fordevice B, respectively. The Fig. 8.10 shows the variation of dynamic pull-in amplitude for both devices, with a minimum value corresponding to the170Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.9: Phase dependent gain VS common mode voltage. At 17 V, twogains possible, with proper phase tuning pull is achieved at 17.5 V (bluedot), when the phases are detuned between the spring modulation and theexternal acceleration, gain decreases (shown in red).resonant frequency of the devices. As the common mode voltage is applied,electrostatic modulation occurs at double the frequency as seen in Eq.6.18.The pull-in voltage will be lowest for spring modulation occurring at thisfrequency. The experimental results were validated by the Simulink simu-lations, which included both the spring and damping nonlinearity. The re-sults also highlight that pull in voltages were also relatively lower, for springmodulation occurring at resonant frequency. Fig. 8.11 illustrates the depen-dence of the dynamic pull-in voltage for device B on both the amplitude171Chapter 8. Concept of dynamic pull-in MEMS gyroscopeand frequency of the Coriolis equivalent accelerations. As the amplitudeof the external acceleration is increased, dynamic pull in voltages decrease.Fig. 8.12 shows the dependence of Vpi,dyn on the magnitude of ?equivalent,for both device types.Figure 8.10: Dynamic pull-in characterization (dynamic pull-in voltage vs.frequency) for external acceleration of 0.319 ms?28.5 Concluding remarksPull-in phenomenon has been exploited in high-sensitivity MEMS accelerom-eters, in order to reconcile the small size of the proof mass with high perfor-mance requirements. We extend this approach to MEMS gyroscopes. We172Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.11: Device B- measured dynamic pull-in characterization (Vpi,dynvs. frequency) for various external accelerationshave studied the influence of external acceleration (Coriolis induced accel-eration) on the dynamic pull-in of the MEMS gyroscopes. We showed that,dynamic pull in voltage, modulates with the external acceleration. This phe-nomenon could be further explored, along with robust feedback loop controlfor high performance sensor design.173Chapter 8. Concept of dynamic pull-in MEMS gyroscopeFigure 8.12: Measured Vpi,dyn vs equivalent angular acceleration (?) forboth A,B devices174Chapter 9Conclusion and future workThis thesis work presented system level issues and solutions for inertialMEMS sensors. MEMS gyroscopes were the main theme of the research,which suffer the most in sensitivity with down scaling. Three distinct prob-lems were researched:1) Damping analysis and its impact on the noise analysis.2) Parametric amplification and damping in MEMS gyroscopes.3) Concept of dynamic pull-in in MEMS gyroscopes.Contribution of this thesis, chapter wise is highlighted below:Chapter 1: Limitation of current MEMS gyroscopes, needs and methodsare highlighted. Basics of Coriolis force is presented.Chapter 2: Classification of gyroscopes with a brief description is pre-sented. Research achievements in some of the leading institutes is presented.Application and performance specifications of MEMS gyroscope present inindustry is highlighted.Chapter 3: Complete analytical modeling of single mass gyroscope ispresented.Chapter 4: Fabrication techniques used in this thesis is presented. Mod-eling using SABER architect and FE Analysis is covered. A complete ap-proach from building macro-models to implementing FE Analysis for modalanalysis and damping analysis is covered.175Chapter 9. Conclusion and future workChapter 5 : Analytical model of frequency dependent spring and damp-ing forces are compared with FE Analysis. Compared with the standardprocedure of assuming an equivalent white spectral mechano-thermal noise,the present analysis brings the advances in the squeeze-film damping theoryinto the realm of noise-based optimization of micro-mechanical resonators.It is seen that influence of spring forces are dominant in the higher order offrequencies (MHz ranges). As the inertial sensors are scaled down, opera-tional resonant frequency gets larger. The damping and spring forces willneed to be considered in that range for the optimization of design structures.Chapter 6: Noise theory dictates that for larger signal to noise ratio(SNR), amplification in first stage is most crucial. In MEMS inertial sen-sors, amplification in mechanical structure (first stage) will be dominant,compared to electronics amplification stages (second stage) in read out. Onthose lines, parametric amplification and damping techniques are employedon the secondary and primary modes of gyroscope. Phase relationship be-tween quadrature induced displacements and Coriolis induced displacementsallows selectively amplifying the desired displacement and damping unde-sired motion. In chapter 6 concept of parametric resonance is applied on thesecondary modes using gap varying combs. Parametric amplification gainsupto 25 and damping of 0.8 is experimentally verified. For the applicationof parametric amplification in the primary mode a novel shaped combs arestudied.Chapter 7: Unlike the area varying combs which don?t offer any elec-trostatic spring modulation, gap varying shaped combs can be used forspring modulation. Analytical model for the shaped combs is presentedwhich is experimentally verified. A comparison between fringe field capaci-tors (non interdigitated) and shaped combs is also presented using COMSOl176Chapter 9. Conclusion and future workMultiphysics c?tool.Chapter 8: Concept of dynamic pull in studied on the gyroscope, whichhas been previously exploited on accelerometers in various research groups.In gyroscope, secondary mode is applied with common mode AC voltagesand dynamic pull in voltages for various external accelerations is experimen-tally obtained. It is shown that, dynamic pull in voltage, modulates withthe external acceleration. This technique along with robust control systemcan produce very sensitive inertial sensors.Based on the current research theme following future paths are summa-rized:1) Impact of parametric amplification on mechano-thermal noise can bestudied.2) Shaped combs can be further optimized for their nonlinear behavior. Con-formal mapping techniques have been presented for studying fringe fields andcan be extended to shaped combs to achieve more accurate models.3) In the current thesis so far a single mass gyroscope was designed andmodeled. Intent of this thesis work is to implement all the above mentionedtechniques on a robust structure. A two mass gyroscope is designed and fab-ricated using SOIMUMPS c? technology. This design was motivated from adesign made by a student at TU Delft, co-supervised by Dr. Edmond Cretu.Design is modified to meet the fabrication requirements of the SOI Tronics R?MUMPS technology. The gyroscope has a dimension of 3mm?3mm?25?m.This design incorporates specialized springs, gap varying combs for quadra-ture tuning and parametric amplification. An image of fabricated structureis shown in Fig. 9.1 and close up for drive and sense spring in Fig. 9.2.Experimentally verified resonant frequency for drive mode and sensemode of two mass gyroscope is presented in Fig. 9.3. Parameters of the177Chapter 9. Conclusion and future workFigure 9.1: Two mass gyroscope implemented in SOIMUMPS R? technologygyroscope is summarized in Tab. 9.1. Quadrature error compensation usingthe gap varying combs is presented in Fig. 9.4A. A DC bias is applied onthe specialized comb shown in Fig. 9.4B.The sense mode of the gyroscope has high damping resulting in poorquality factor in atmospheric pressure. Large number of gap varying combs444 in each side with a minimum gap of 2 ?m adds significant amountof squeeze film damping 4.8279 ? 10?4Ns/m. Large number of combs aidin high sensitive measurements increasing the net capacitance and changein capacitance. Quality factors can be significantly increased by reducingthe pressure. Estimated sensitivity at atmospheric pressure is 0.08 deg/s.178Chapter 9. Conclusion and future workFigure 9.2: Close up of drive mode and sense mode springs captured byPolytec MSA-500Further experimental results based on parametric amplification can lead tomuch better sensitivity.4) The previous work carried out by Greg Reynan from our lab, wasbased on two mass coupled resonator, employing gap varying combs forapplying asymmetry[53]. The system proved the resonant frequency de-pendence on disorder in the system and also showed that the eigenvectorsensitivity to disorder was at least an order of magnitude greater than thefrequency sensitivity. According to his findings, sensitivity can be furtherincreased, by increasing the coupling proof mass. In order to achieve ampli-fied mode localization of energy, spring softening and spring hardening at179Chapter 9. Conclusion and future workParameter Drive mode Sense mode Experiment.verifiedf 2.5kHz 3.6kHz Yesmass 1.467 ?10?7Kg1.2468?10?7KgNok 34.783 N/m 27.21 N/m YesQ(atmosphericpressure)30 5 YesActuationCapacitance(static)0.63pF (?2) NoSensing Ca-pacitance(static)9.43pF (?2) NoQuadraturecombs (static)0.47pF (?2) NoTable 9.1: Parameters of two mass gyroscopeterminal proof masses are required (for a three mass coupled resonators).We employ this by employing a unique resonator design with non overlap-ping fingers for inducing asymmetry in the system, which in turn leads tothe localization of energy in one of the resonators. An image of three masscoupled resonator is shown in Fig. 9.5.180Chapter 9. Conclusion and future workFigure 9.3: Resonant modes of 2 mass gyroscope obtained using PolytecPMA-500 R?181Chapter 9. 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The mechan-ical velocity of the movable mass(m) is in this alternative model equivalentto a current, while the mechanical force is equivalent to the electrical volt-age [110], [111]. The variable capacitors coupled mechanically to eachmechanical degree of freedom excite the proof mass of the device, establish-ing a bi-directional electromechanical interaction [112]. In the small sig-nal approximation, the electromechanical couplings (for electrical actuationand capacitive sensing) can be represented with the help of the equivalenttransformers illustrated in Fig. A.1. Vac is applied on the excitation combsand converted into the actuation force exciting the proof mass. The re-sulting velocity of motion is then transduced to electrical current throughthe receiving electromechanical port (sensing combs). The ground termi-200Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopenals of the transformers in the electrical domains are connected to Vbias. InFig. A.1, ? = CVbias, where C is the constant capacitance of the excitationcombs. The electrostatic stiffness has to be included as capacitor (Ce) in theequivalent circuit, whose value depends on the type of the movable combsstructure.Figure A.1: A) Electrical representation of MEMS sensor for impedancemeasurement, B) Extracted impedance (real and imaginary) componentwith the experimental result201Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopeAll the required parameters reflect into a net impedance Z(?)as:Lm =m?2, Rm =?kmQ?2, Cm =?2k,Ce =??2ke(A.1)Z(?) =1Xcp +1Rm+Xcm?Xce+XLm,Xcp =1j?cp, Xcm =1j?cmXce =1j?cm, Xlm = j?Lm (A.2)The total impedance reflected into the electrical domain needs to take intoaccount as well the parasitic capacitances Cp existent in the measurementsetup, added as extra parameters in the curve fitting algorithm implementedon Agilent 4294A. The Fig. A.1B illustrates such resulting curve fittingresults, for the 0y resonant mode. Unlike the area-varying combs, the slopedcombs showed variation in their resonant frequency with the applied DCbias. The parameter identification of the measured characteristics resultedin Qx = 2.8 (for Vbias= 10.3V), to vary to Qx = 1.76 (for a 30V biasing,for the area-varying and for sloped combs of the driven mode). A similarimpedance analyzer procedure followed by curve fitting led to Qy = 1.95 forthe sensing (gap-varying combs) mode, for Vbias = 10.3V refer Fig. A.2.The electrostatic stiffness effect was experimentally checked through thevariation of the secondary mode resonance frequency when Vbias was variedVbias= 10.3 V changed the resonant frequency of secondary mode from 7.88kHz to 7.79 kHz. Measurements have been performed on a total number of25 fabricated dies - the resulting low quality factors are potentially due toboth an increased damping in the mechanical resonators (compared to theinitial finite element simulations, based on TRONICS process specification)and to extra electrical losses in the measurement setup. The results ofthe spring softening effects obtained for sloped and gap-varying combs areillustrated in Fig. A.3 and Fig. A.2.202Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopeFigure A.2: A) Actuation combs(area varying and shaped combs) realimpedance B) Sensing gap varying combs real impedance measurement withspring softening203Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopeFigure A.3: 1) Primary and secondary equivalent capacitors with electricalnodes for measurements, 2) Variation of resonant frequency Vs DC bias forsloped shaped combs(left) and gap varying combs(right)204Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopeA function generator synchronized in phase with the impedance analyzerwas used to experimentally check the parametric amplification. Commonmode voltages are applied (through the function generator) to the slopedcombs (D in Fig. A.3)1, for the primary mode, while normal impedancemeasurements were performed on the area-varying combs (B in Fig. A.3)1.The equivalent equation of motion including the parametric pump is givenby:mx?+ cxx?+ (kx ??kx(UcmV0)2)x = F (A.3)Where UCM = V0cos(?xt + ?), where relative phase ? between the actu-ation and the readout is an essential parameter for the electro-mechanicalparametric coupling. Similarly, the exercise is repeated for the sense mode,where, one set of gap varying combs (E) are used for parametric pump, usingcommon mode voltages and remaining sets are used for impedance measure-ments. Real impedances, obtained with parametric amplification for bothprimary and secondary resonances are illustrated in Fig. A.4. We see thatgain of 2.75 is obtained, using sloped combs, with Ucm =27.5 V, for primaryresonance. Similarly, gain of 7 is obtained, using gap varying combs, withUcm = 15.2 V, for secondary resonance.For sloped combs, stable gains up to 5 was achieved with 40V and for gapvarying combs up to 11.75, with 15.4V for real impedances. The mechanicalresonator lost its stability, for higher voltages in their respective directions,causing pull in.Further experimentations are required to conclusively verify the impactof parametric amplification. Real impedances at least show an increase withapplication of parametric pump.205Appendix A. Experimental results of parametric amplification for SOI Tronics R? gyroscopeFigure A.4: Comparison between parametric amplification response andwithout pumps for both sloped (Gain=2.76 with Ucm= 27.5 V ) and gapvarying combs (Gain=7 with Ucm=15.2 V )206Appendix BCharacterization usingoptical planar motionanalyzerPolytec Planar Motion Analyzer (PMA-500, Polytec GmbH), as seen inFig. B.1 can obtain both the time and frequency domain based measure-ment (non contact) using velocity and/or displacement sensors. It is verycommon to measure out of plane motion of the microstructure using LDV(Laser Doppler Vibrometry) based on the doppler frequency shift/phase ina direction along the axis of the laser beam. Now a days stroboscopic prin-ciple is being used to get the in plane measurement ([127]). For stroboscopicmeasurements, device placed under the microscope is exposed with a cam-era. Fast motion of the device under probe is recorded with a period whichis shorter than the minimum exposure time of the camera. During measure-ments, the driving signal, the LED-strobe flashes, and the camera exposureare accurately synchronized as described in Fig. B.2A periodic motion of a device is frozen by an illumination source such asa strobed light (flash) with certain duration (flash duration, ns). The mov-ing device is easily captured at its own exact positions using illuminationand digital imaging. Short light pulses record the position of the measured207Appendix B. Characterization using optical planar motion analyzerdevice at precise phase angles (shots per period). A camera shot is a se-quence of flashes within a camera exposure time. The whole fast motionof the device is measured by shifting the timing of those pulses at phaseangle increments by recording the total number of shots. For each shot, thetotal number of flashes (periods) is illuminated for a specified flash duration(ns) in each flash. The time between two shots is the cycle duration of thecamera-framing rate. To obtain the bode plot and FFT measurement, sys-tem uses PMA software. Measurements are based on pixel deviations, whichare extracted and shown as displacement values with a subpixel resolution.The measurement system then calculates the captured images by employingthe image processing techniques. For supporting the pattern matching, thetwo regions such as search pattern (object of interest) and search region ofinterest (ROI) are specified and located as shown in Fig. B.3.By doing this and employing motion capturing algorithm, system is ableto compute the ?x and ?y between the object of interest and region ofinterest. By using the phase shifting technique, phase term which is thefunction of x, y and t can be computed. Thus, this measurement processcan detect motion at amplitudes with a resolution in the nanometer scale.208Appendix B. Characterization using optical planar motion analyzerFigure B.1: Polytech equipment with its components used for experiments,snapshot of movement of finger captured by PMA-500, which has a move-ment of 1 ?m209Appendix B. Characterization using optical planar motion analyzerFigure B.2: Timing diagram for principle of stroboscopic videomicroscopy,([126],[127])210Appendix B. Characterization using optical planar motion analyzerFigure B.3: Pattern matching for the PMA software,([126],[127])211Appendix CSimulink modelsFigure C.1: Simulink model for parametric resonance212Appendix C. Simulink modelsFigure C.2: Simulink subset model of gyroscope for parametric resonanceon both primary and secondary mode213Appendix C. Simulink modelsFigure C.3: Simulink model for dynamic pull in214Appendix C. Simulink modelsFigure C.4: Simulink subset model of resonator for dynamic pull in215

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