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

Contents ix

Micro Mechanical Transducers — Pressure Sensors, Accelerometers and Gyroscopes

Contents

Preface vjj

Contents ix

Chapter 1. Introduction to micro mechanical transducers 1

§1.1. Piezoresistive pressure sensors 1 §1.1.1. Brief history 1 §1.1.2. Working principles 3

§1.2. Piezoresistive accelerometers 5 §1.3. Capacitive sensors 9

§1.3.1. Features of capacitive sensing 9 §1.3.2. Capacitive pressure transducers 10 §1.3.3. Capacitive accelerometers 11

§1.4. Resonant sensors 13 §1.5. Vibratory gyroscopes 15

§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

§3.2.3. Circular and annular plates §3.2.4. Rectangular plate . §3.2.5. Perforated plate 1 4

§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

§4.3. Step and alternating driving 165 §4.3.1. Step driving 165 §4.3.2. Alternating driving 171

§4.4. Capacitive sensing 177 §4.4.1. Diode-quad sensing method 178 §4.4.2. Opposite driving sensing technique 181 §4.4.3. Force-balanced sensing technique 182 §4.4.4. Switched capacitor sensing method 185

§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

References 239

Chapter 6. Piezoresistive pressure transducers 241 §6.1. Designs with flat diaphragms 241

§6.1.1. Designs with Square diaphragms 241 §6.1.2. Designs with rectangular diaphragms 244

§6.2. Pressure transducers with sculptured diaphragm structure 247 §6.2.1. Twin-island structure 247 §6.2.2. Beam-diaphragm structure 252

§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

Contents xiii

Chapter 7. Piezoresistive accelerometers 281 §7.1. Cantilever beam accelerometers 281

§7.1.1. Basic designs 281 §7.1.2. Lateral sensitivities 286 §7.1.3. Free Vibration frequency 289 §7.1.4. Damping control 291 §7.1.5. Over-range protection 294

§7.2. Quad-beam accelerometer 295 §7.2.1. Structure description 295 §7.2.2. Mechanical analysis 297 §7.2.3. Compensation of lateral sensitivity 300 §7.2.4. Damping control and over-range protection 302 §7.2.5. Self-testing by an electrostatic force 303

§7.3. Twin-mass accelerometer 306 §7.3.1. Structure description 306 §7.3.2. Sensitivity for normal acceleration 307 §7.3.3. Displacement and Vibration frequency 309 §7.3.4. Lateral sensitivities 311 §7.3.5. Comparisons 313

§7.4. Lateral accelerometers 313 §7.4.1. Quad-beam lateral accelerometer 314 §7.4.2. Vertical beam accelerometer 315

References 317

Chapter 8. Capacitive pressure transducers and accelerometers 319 §8.1. Capacitive pressure transducers 319

§8.1.1. Hat diaphragm structure 319 §8.1.2. Diaphragm-mass structure 322

§8.2. Open loop capacitive accelerometers 325 §8.2.1. Sensitivity 326 §8.2.2. Damping control 329

§8.3. Force-balanced accelerometers 333 §8.3.1. Basic principles 333 §8.3.2. Bulk micromachined accelerometer 339 §8.3.3. Surface micromachined accelerometer 342

§8.4. Thermo-mechanical noise of mechanical structures 350 References 351

XIV Micro mechanical transducers

Chapter 9. Resonant sensors and vibratory gyroscopes 353 §9.1. Resonant pressure transducers 353

§9.1.1. Polysilicon beam resonant pressure sensor 353 §9.1.2. H-beam resonant pressure sensor 356

§9.2. Resonant accelerometers ^59 §9.3. Vibratory gyroscopes 36

§9.3.1. Basic principles ^*~ §9.3.2. Frequency bandwidth 3 ^ §9.3.3. Resolution by thermo-mechanical noise 369

References

373 Subject index