HANDBOOK OF SENSORS AND ACTUATORS 8 Micro Mechanical Transducers Pressure Sensors, Accelerometers and Gyroscopes Min-Hang Bao Department of Electronic Engineering Fudan University Shanghai, China ELSEVIER Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris • San Diego - San Francisco - Singapore - Sydney - Tokyo
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§1.5.1. Micro gyroscopes 15 §1.5.2. Working principle of vibratory gyroscopes 16
§1.6. Basic principles of micro mechanical transducers 19 References 20
Chapter 2. Basic mechanics of beam and diaphragm structures 23 §2.1. Stress andStrain 24
§2.1.1. Stress 24 §2.1.2. Strahl 25 §2.2.3. Hooke'sLaw 28 §2.1.4. General relations between stress and strain 30
§2.2. Stress and strain of beam structures 32 §2.2.1. Stress, strain and the curvature of beam 32 §2.2.2. Displacement of a beam 33 §2.2.3. Bending moment and the moment of inertia 34 §2.2.4. Moment of inertia for a trapezoid cross section 35 §2.2.5. Examples 36 §2.2.6. Torsion of beams 41
Micro mechanical transducers
A 44 §2.3. Vibration frequency by energy method ^
§2.3.1. Spring-mass System 4 g
§2.3.2. The Rayleigh-Ritz method 82 3 3 Vibration frequencies of beam structures •••••••
§2 4 Vibration frequencies of beam by differential equation method 55 §2 4 1 Differential equation for free Vibration of a beam :>o §2.4.2. Vibration frequencies of a double-clamped beam P/ §2.4.3. Vibration with an axial force ^
§2.5. Damped and forced Vibration ^ §2.5.1. Damping force 6 5
§2.5.2. Damped Vibration ß g
§2.5.3. Forced Vibration 7 2
§2.5.4. Resonance 7 5
§2.6. Basic mechanics of diaphragms ^ 5
§2.6.1. Long rectangular diaphragm 7 g
§2.6.2. Equations for a plate g Q
§2.6.3. Circular diaphragm g 2
§2.6.4. Square and rectangular diaphragms §2.6.5. Natural Vibration frequencies of diaphragms ^
References
89 Chapter 3. Air damping . 8 9
§3.1. Viscous flow of a fluid _ §3.1.1. Viscosityof a fluid §3.1.2. Viscous flow - 9 6
§3.1.3. Drag force on a moving object - — §3.1.4. The effects of air damping on micro-dynamics v/
§3 2. Squeeze-film air damping • §3.2.1. Basic equations for squeeze-film air damping ^ §3.2.2. Long rectangular plate 1 Q 6
§3.2.6. Oscillating beams §3.2.7. Effects caused by finite squeeze number
§3.3. Slide-filmair damping §3.3.1. Basic equations for slide-film air damping J « §3.3.2. Couette-flow model §3.3.3. Stokes-flow model _Q
§3.3.4. Air damping of a comb resonator
Contents xi
§3.4. Damping in rare air 133 §3.4.1. Freemolecule model for rare air damping 133 §3.4.2. Damping in a vacuum 135
References 137
Chapter 4. Electrostatic driving and capacitive sensing 139 §4.1. Electrostatic force 140
§4.1.1. Force normal to the electrode plate 140 §4.1.2. Tangential force to the plate 142 §4.1.3. Fringe effects 144
§4.2. Displacement of elastic structures by electrostatic force 147 §4.2.1. Normal displacement 147 §4.2.2. Displacement of a cantilever beam-mass structure 153 §4.2.3. Torsion bar structure 155 §4.2.4. Comb actuator 158 §4.2.5. Double-supported beam 162
§4.5. Effects of electric driving on capacitive sensing 187 §4.5.1. Single-sided driving 188 §4.5.2. Double-sided driving 191 §4.5.3. Double-sided driving with feed-back voltage 194
References 197
Chapter 5. Piezoresistive sensing 199 §5.1. Metal strahl gauge 199 §5.2. Piezoresistive effect of Silicon 201
§5.2.1. Resistivity tensor 201 §5.2.2. Piezoresistive coefficient tensor 202 §5.2.3. Piezoresistive coefficient of Silicon 203 §5.2.4. Dependence on doping level and temperature 204
§5.3. Coordinate transformation of tensors of the second rank 206 §5.3.1. Coordinate transformation of vector 206 §5.3.2. Coordinate transformation of tensors of the second rank... 210
XU Micro mechanical transducers
§5.4. Coordinate transformation of piezoresistive coefficient 214 §5.4.1. General relation of coordinate transformation 214 §5.4.2. Simplification by symmetry of Silicon crystal 215 §5.4.3. Piezoresistance in an arbitrary coordinate System 216
§5.5. Piezoresistive sensing elements 219 §5.5.1. Piezoresistor 219 §5.5.2. Four-terminal sensing dement 222 §5.5.3. Sensing elements formed in a diffusum layer 227
§5.6. Polysilicon piezoresistive sensing elements 229 §5.6.1. Piezoresistive effect of polysilicon 229
§5.6.1.1. polysilicon Piezoresistor 230 §5.6.1.2. Four-terminal sensing element 231
§5.6.2. Average piezoresistive coefficient 232 §5.6.2 1. Average for specific orientations 232 §5.6.2.2. Completely random distribution 236
§5.6.3. Design of polysilicon piezoresistive sensors 237 §5.6.3.1. Factors affecting sensitivity of a polysilicon sensor ... 237 §5.6.3.2. Design considerations 238
§6.3. Design of polysilicon pressure transducer 254 §6.4. Offset voltage and temperature coefficient of offset 256
§6.4.1. Offset voltage of pressure transducer 256 §6.4.2. Compensation of offset voltage 257 §6.4.3. Compensation of temperature coefficient of offset 260
§6.5. Temperature coefficient of sensitivity 262 §6.6. Nonlinearity 265
§6.6.1. Definitions 265 §6.6.2. Nonlinearity of a piezoresistive pressure transducer 268 §6.6.3. Nonlinearity caused by the "Balloon effect" 270 §6.6.4. Nonlinearity of a piezoresistive effect 272
§6.7. Calibration of pressure transducers 274 References 279