Micro Mechanical Transducers

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1 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

2 Contents ix Micro Mechanical Transducers Pressure Sensors, Accelerometers and Gyroscopes Contents Preface Contents vjj i x Chapter 1. Introduction to micro mechanical transducers Piezoresistive pressure sensors Brief history Working principles Piezoresistive accelerometers Capacitive sensors Features of capacitive sensing Capacitive pressure transducers Capacitive accelerometers Resonant sensors Vibratory gyroscopes Micro gyroscopes Working principle of vibratory gyroscopes Basic principles of micro mechanical transducers 19 References 20 Chapter 2. Basic mechanics of beam and diaphragm structures Stress andstrain Stress Strahl Hooke'sLaw General relations between stress and strain Stress and strain of beam structures Stress, strain and the curvature of beam Displacement of a beam Bending moment and the moment of inertia Moment of inertia for a trapezoid cross section Examples Torsion of beams 41

3 Micro mechanical transducers A Vibration frequency by energy method ^ Spring-mass System 4g The Rayleigh-Ritz method Vibration frequencies of beam structures 2 4 Vibration frequencies of beam by differential equation method Differential equation for free Vibration of a beam :>o Vibration frequencies of a double-clamped beam P/ Vibration with an axial force ^ 2.5. Damped and forced Vibration ^ Damping force Damped Vibration ßg Forced Vibration Resonance Basic mechanics of diaphragms ^ Long rectangular diaphragm 7g Equations for a plate gq Circular diaphragm g Square and rectangular diaphragms Natural Vibration frequencies of diaphragms ^ References Chapter 3. Air damping Viscous flow of a fluid _ Viscosityof a fluid Viscous flow Drag force on a moving object The effects of air damping on micro-dynamics v/ 3 2. Squeeze-film air damping Basic equations for squeeze-film air damping ^ Long rectangular plate 1Q Circular and annular plates Rectangular plate Perforated plate Oscillating beams Effects caused by finite squeeze number 3.3. Slide-filmair damping Basic equations for slide-film air damping J« Couette-flow model Stokes-flow model _ Q Air damping of a comb resonator 89

4 Contents xi 3.4. Damping in rare air Freemolecule model for rare air damping Damping in a vacuum 135 References 137 Chapter 4. Electrostatic driving and capacitive sensing Electrostatic force Force normal to the electrode plate Tangential force to the plate Fringe effects Displacement of elastic structures by electrostatic force Normal displacement Displacement of a cantilever beam-mass structure Torsion bar structure Comb actuator Double-supported beam Step and alternating driving Step driving Alternating driving Capacitive sensing Diode-quad sensing method Opposite driving sensing technique Force-balanced sensing technique Switched capacitor sensing method Effects of electric driving on capacitive sensing Single-sided driving Double-sided driving Double-sided driving with feed-back voltage 194 References 197 Chapter 5. Piezoresistive sensing Metal strahl gauge Piezoresistive effect of Silicon Resistivity tensor Piezoresistive coefficient tensor Piezoresistive coefficient of Silicon Dependence on doping level and temperature Coordinate transformation of tensors of the second rank Coordinate transformation of vector Coordinate transformation of tensors of the second rank

5 XU Micro mechanical transducers 5.4. Coordinate transformation of piezoresistive coefficient General relation of coordinate transformation Simplification by symmetry of Silicon crystal Piezoresistance in an arbitrary coordinate System Piezoresistive sensing elements Piezoresistor Four-terminal sensing dement Sensing elements formed in a diffusum layer Polysilicon piezoresistive sensing elements Piezoresistive effect of polysilicon polysilicon Piezoresistor Four-terminal sensing element Average piezoresistive coefficient Average for specific orientations Completely random distribution Design of polysilicon piezoresistive sensors Factors affecting sensitivity of a polysilicon sensor Design considerations 238 References 239 Chapter 6. Piezoresistive pressure transducers Designs with flat diaphragms Designs with Square diaphragms Designs with rectangular diaphragms Pressure transducers with sculptured diaphragm structure Twin-island structure Beam-diaphragm structure Design of polysilicon pressure transducer Offset voltage and temperature coefficient of offset Offset voltage of pressure transducer Compensation of offset voltage Compensation of temperature coefficient of offset Temperature coefficient of sensitivity Nonlinearity Definitions Nonlinearity of a piezoresistive pressure transducer Nonlinearity caused by the "Balloon effect" Nonlinearity of a piezoresistive effect Calibration of pressure transducers 274 References 279

6 Contents xiii Chapter 7. Piezoresistive accelerometers Cantilever beam accelerometers Basic designs Lateral sensitivities Free Vibration frequency Damping control Over-range protection Quad-beam accelerometer Structure description Mechanical analysis Compensation of lateral sensitivity Damping control and over-range protection Self-testing by an electrostatic force Twin-mass accelerometer Structure description Sensitivity for normal acceleration Displacement and Vibration frequency Lateral sensitivities Comparisons Lateral accelerometers Quad-beam lateral accelerometer Vertical beam accelerometer 315 References 317 Chapter 8. Capacitive pressure transducers and accelerometers Capacitive pressure transducers Hat diaphragm structure Diaphragm-mass structure Open loop capacitive accelerometers Sensitivity Damping control Force-balanced accelerometers Basic principles Bulk micromachined accelerometer Surface micromachined accelerometer Thermo-mechanical noise of mechanical structures 350 References 351

7 XIV Micro mechanical transducers Chapter 9. Resonant sensors and vibratory gyroscopes Resonant pressure transducers Polysilicon beam resonant pressure sensor H-beam resonant pressure sensor Resonant accelerometers ^ Vibratory gyroscopes Basic principles ^*~ Frequency bandwidth 3^ Resolution by thermo-mechanical noise 369 References Subject index 373

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