MECHANICALLY AMPLIFIED CAPACITIVE STRAIN SENSOR by JUN GUO Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Adviser: Dr. Wen H. Ko Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY May, 2007 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. Dedication To my parents. TABLE OF CONTENTS Mechanical Amplified Capacitive Strain Sensor Table of Contents ......................................................................................................................... 1 List of Tables ................................................................................................................................ 7 List of Figures .............................................................................................................................. 9 Acknowledgements .................................................................................................................. 15 Abstract ................................................................................................................................. 16 Chapter 1 Introduction to Strain Measurement ................................................. 18 1.1. Introduction and motivation............................................................................................ 18 1.2. Thesis organization ......................................................................................................... 23 Chapter 2 A Review of Micromachined Strain Gages ........................................ 25 2.1. Introduction..................................................................................................................... 25 2.2. Metallic-foil strain gage.................................................................................................. 25 2.2.1 Principle ........................................................................................................ 26 2.2.2 Gage factor and sensitivity............................................................................ 28 2.2.3 Instrumentation ............................................................................................. 29 2.2.4 Resolution of a typical metallic-foil strain gage........................................... 30 2.3. Piezoresistive strain gage................................................................................................ 32 2.3.1 Piezoresistive effect ...................................................................................... 32 1 2.3.2 Longitudinal and transverse piezoresistive coefficients ............................... 33 2.3.3 Impurity concentration (doping level) and temperature effect on piezoresistance coefficients....................................................................................... 35 2.3.4 Resolution and issues.................................................................................... 36 2.4. Resonant silicon micromachined strain gage.................................................................. 37 2.4.1 Principle ........................................................................................................ 38 2.4.2 Excitation and detection scheme................................................................... 39 2.4.3 Resolution: minimum detectable frequency ................................................. 40 2.5. Capacitive strain gage..................................................................................................... 42 2.5.1 Principle and configurations ......................................................................... 42 2.5.2 Performance of capacitive strain gage .......................................................... 44 2.6. Summary......................................................................................................................... 46 Chapter 3 Capacitive Strain Sensor Employing a Buckled Beam Mechanical Amplifier ........................................................................................................................ 48 3.1. Introduction..................................................................................................................... 48 3.2. Mechanical amplifier ...................................................................................................... 49 3.2.1 Background................................................................................................... 50 3.2.2 Principle ........................................................................................................ 51 3.3. Analytical modeling........................................................................................................ 52 3.3.1 Derivation of the curvature function under an applied force........................ 53 2 3.3.2 Calculate the mechanical gain ...................................................................... 56 3.3.3 Small input force simplification and the structure nominal mechanical gain 59 3.3.4 Large deflection compensation ..................................................................... 60 3.3.5 Calculate the buckling beam equivalent stiffness in x-direction .................. 60 3.4. Finite-Element-Analysis verification.............................................................................. 61 3.4.1 FEA verification of nominal mechanical ...................................................... 61 3.4.2 FEA verification of equivalent stiffness ....................................................... 62 3.4.3 FEA verification of mechanical gain corresponding to large displacement input 63 3.5. Nonlinearity of the buckled beam amplifier ................................................................... 65 3.6. Fundamental noise analysis ............................................................................................ 67 3.6.1 Resistive thermal noise estimation................................................................ 68 3.6.2 Mechanical thermal noise (Brownian motion noise) estimation .................. 70 3.7. The maximum stress of the structure and locations........................................................ 73 3.8. Device fabrication........................................................................................................... 75 3.9. Fabricated device ............................................................................................................ 77 3.10. Testing............................................................................................................................. 80 3.10.1 Test fixture.................................................................................................... 80 3.10.2 Device functionality test under microscope.................................................. 83 3.10.3 Capacitive readout circuitry.......................................................................... 84 3 3.10.3.1 C/V characteristics using MS3110®.................................................. 84 3.10.3.2 C/V characteristics using developed interface electronics................ 88 3.11. Conclusions..................................................................................................................... 93 Chapter 4 Sensor Backing Design ........................................................................ 95 4.1. Strain transmission ratio of a solid backing......................................................................95 4.1.1 Transmission ratio definition .......................................................................... 95 4.1.2 Strain transmission ratio of a solid backing.................................................... 96 4.2. Transmission ratio of a folded-spring backing................................................................102 4.2.1 Folded-spring backing structure analysis...................................................... 103 4.2.2 Anchor location and package gain................................................................ 106 4.2.3 Package gain discussions .............................................................................. 109 4.2.3.1 The maximum package gain for piezoresistive sensor design.......... 109 4.2.3.2 The maximum displacement for capacitive sensor design ................111 4.3. Overall sensor module backing design ........................................................................... 116 4.4. Summary and conclusions .............................................................................................. 119 Chapter 5 Sensing Structure Design .................................................................... 120 5.1. Structure modification ....................................................................................................120 5.2. Capacitive output sensitivity calculation ........................................................................122 5.3. Fringe capacitance effect on comb