Electromechanics and Mems Toc

Contents

Preface page xiii

1 Introduction 1

1.1 Background 11.2 Some terminology 21.3 Electromechanical systems 31.4 Conclusion 7Problems 7Reference 9

2 Circuit-based modeling 10

2.1 Fundamentals of circuit theory 102.1.1 Motivation 102.1.2 Kirchhoff’s current and voltage laws 112.1.3 Circuit elements 122.1.4 Tellegen’s theorem: power and energy 122.1.5 AC circuits, impedance, and admittance 14

2.2 Circuit models for capacitive devices 152.2.1 Basic RC circuit building block 162.2.2 The series capacitive circuit 172.2.3 The parallel capacitive circuit 182.2.4 Special cases: series and parallel capacitance 202.2.5 Summary 20

2.3 Two-port networks 222.3.1 Impedance and admittance matrices 222.3.2 The transmission matrix 242.3.3 Cascaded two-port networks 252.3.4 Some important two-port networks 272.3.5 The gyrator and the transformer 292.3.6 Embedded networks 302.3.7 Source and impedance reflection 32

2.4 Summary 34Problems 34References 43

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vi Contents

3 Capacitive lumped parameter electromechanics 44

3.1 Basic assumptions and concepts 443.1.1 The lossless electromechanical coupling 453.1.2 State variables and conservative systems 463.1.3 Evaluation of energy function 463.1.4 Force of electrical origin 48

3.2 Coenergy – an alternate energy function 493.2.1 Definition of coenergy 493.2.2 Integral evaluation of coenergy 503.2.3 Evaluation of force of electrical origin 51

3.3 Couplings with multiple ports 523.3.1 Energy conservation relation 533.3.2 System with two electrical and two mechanical ports 54

3.4 Basic capacitive transducer types 563.4.1 Variable-gap capacitors 563.4.2 Variable-area capacitors 583.4.3 Comparison of variable-gap and variable-area actuators 603.4.4 Transducer stroke 623.4.5 The comb-drive geometry 643.4.6 Another variable-area capacitor 65

3.5 Rotational transducers 663.5.1 Modeling rotational electromechanics 673.5.2 Torque of electrical origin 683.5.3 An example 68

3.6 Electrets 713.7 Non-linear conservative electromechanical systems 75

3.7.1 Conservation laws for capacitive devices 753.7.2 Non-linear oscillations and stability 773.7.3 Numerical solutions 793.7.4 Constant charge constraint 803.7.5 Discussion 83


4 Small-signal capacitive electromechanical systems 97

4.1 Background 974.2 Linearized electromechanical transducers 98

4.2.1 Some preliminaries 984.2.2 Linearization in terms of energy and coenergy 99

4.3 Electromechanical two-port networks 1014.3.1 The transducer matrix 1014.3.2 The linear capacitive transducer 103






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4.3.3 Important special cases 1054.3.4 Transducers with angular displacement 1054.3.5 Multiport electromechanical transducers 107

4.4 Electromechanical circuit models 1094.4.1 Analogous variables 1094.4.2 M-form equivalent electromechanical circuit 1104.4.3 N-form equivalent electromechanical circuit 1124.4.4 More about the cascade paradigm 113

4.5 Reconciliation with Neubert 1134.6 External constraints 114

4.6.1 Mechanical constraints 1154.6.2 Electrical constraints 1184.6.3 Fully constrained electromechanical transducers 1184.6.4 Other useful matrix forms 120

4.7 Applications of electromechanical two-port theory 1214.7.1 Application of source and impedance reflection 1214.7.2 A capacitive microphone 1234.7.3 Electromechanical transfer functions 1254.7.4 A comb-drive actuator 1264.7.5 The three-plate capacitive sensor 1274.7.6 Linear model for electret transducer 130

4.8 Stability considerations 1324.8.1 Preliminary look at stability 1324.8.2 General stability criteria 1334.8.3 The pull-in instability threshold 1374.8.4 A physical interpretation of instability 138


5 Capacitive sensing and resonant drive circuits 150

5.1 Introduction 1505.2 Basics of operational amplifiers 1515.3 Inverting amplifiers and capacitive sensing 152

5.3.1 Basic inverting configuration 1535.3.2 One-sided high-impedance (charge) amplifier 1545.3.3 Variable-gap and variable-area capacitors 1575.3.4 Effect of op-amp leakage current 157

5.4 Differential (three-plate) capacitance sensing 1635.4.1 DC feedback for the differential configuration 165

5.5 AC (modulated) sensing 1665.5.1 Capacitive sensor excited by zero-mean sinusoidal voltage 1675.5.2 Two-plate capacitive sensing with AC excitation 169






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5.5.3 Analysis including the feedback resistance Rf 1705.5.4 AM signal demodulation 1725.5.5 Differential AC sensing 1735.5.6 Synchronous demodulation 174

5.6 AC sensors using symmetric square-wave excitation 1755.6.1 Transducers using square-wave excitation 1755.6.2 Three-plate sensing using square-wave excitation 176

5.7 Switched capacitance sensor circuits 1785.7.1 Basics of switched-capacitor circuits 1785.7.2 Simple sensor based on switched capacitance 1795.7.3 Half-wave bridge sensor using switched capacitance 180

5.8 Noise in capacitive MEMS 1835.8.1 Common noise characteristics 1845.8.2 Filtered noise 1855.8.3 Noisy two-ports 1865.8.4 Electrical thermal noise 1865.8.5 Mechanical thermal noise 1895.8.6 1/f amplifier noise 1915.8.7 Effect of modulation on 1/f noise 192

5.9 Electrostatic drives for MEMS resonators 1935.9.1 Mechanical resonators 1945.9.2 Drive electrodes with sinusoidal drive 1945.9.3 Non-harmonic drives 1975.9.4 Sense electrodes 2005.9.5 Harmonic oscillators based on MEMS resonators 2005.9.6 Phase-locked loop drives 2085.9.7 PLL system linearization 210


6 Distributed 1-D and 2-D capacitive electromechanical structures 223

6.1 Introduction 2236.2 A motivating example – electrostatic actuation of a cantilevered

beam 2246.2.1 Problem description 2246.2.2 Derivation of the lumped parameter model 2256.2.3 Evaluation of equivalent spring constant, mass, and mechanical

damping 2286.2.4 Resonance of a cantilevered beam 2296.2.5 Recapitulation of lumped parameter model identification

procedure 233






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6.3 A second look at the cantilevered beam 2356.3.1 Parameterization of distributed capacitance 2366.3.2 Discretized capacitance model for the beam 237

6.4 MDF models for beams 2396.4.1 MDF description 2406.4.2 Application of boundary conditions 2426.4.3 Maxwell’s reciprocity theorem 2436.4.4 Applications of static MDF model 2456.4.5 Modal analysis 2496.4.6 Decoupling of the equation of motion 2526.4.7 Equivalent circuit using modal analysis 2546.4.8 Damping 257

6.5 Using the MDF model for dynamics 2596.6 A first look at plates 261

6.6.1 Equivalent spring constant and mass 2636.6.2 Capacitance 2636.6.3 Resonance 263

6.7 MDF modeling of plates 2656.7.1 Uniform discretization of rectangular 2-D plates 2656.7.2 2-D discretization of circular plates 2676.7.3 2-D example: electrostatic actuation of a circular plate 268

6.8 Additional beam configurations 2776.8.1 Doubly clamped beam 2776.8.2 Simply supported beam 2816.8.3 Vibration isolation of the simply supported beam 2856.8.4 Closure 289


7 Practical MEMS devices 298

7.1 Introduction 2987.2 Capacitive MEMS pressure sensors 299

7.2.1 Basic displacement-based capacitive pressure sensor 2997.2.2 System-level model 3027.2.3 A differential configuration 3047.2.4 Closure 307

7.3 MEMS accelerometers 3077.3.1 Principles of operation 3087.3.2 System transfer function and sensitivity 3107.3.3 Basic construction of an accelerometer 3127.3.4 Mechanical transfer function and mechanical thermal noise 314






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7.3.5 Selection of the electrode types 3187.3.6 Force-feedback configuration 3197.3.7 Higher-order effects 322

7.4 MEMS gyroscopes 3257.4.1 A qualitative description of mechanical gyroscopes with some

historical notes 3267.4.2 Rotating reference frames 3287.4.3 A simple z axis rate vibratory gyroscope 3307.4.4 Other examples of MEMS-based gyroscopes 3367.4.5 Background material 3397.4.6 Torsional-vibration gyroscope 3427.4.7 Higher-order effects 3457.4.8 Closure 346

7.5 MEMS energy harvesters 3477.5.1 Basic principle of capacitive energy harvesting 3477.5.2 Power considerations and efficiency 3497.5.3 Multiple resonators 3517.5.4 Capacitive energy harvesters with bias voltage 3567.5.5 Practical electrostatic energy harvesters 357


8 Electromechanics of piezoelectric elements 372

8.1 Introduction 3728.2 Electromechanics of piezoelectric materials 373

8.2.1 Piezoelectric phenomenology 3738.2.2 Piezoelectric properties 3758.2.3 The L-type piezoelectric transducer 3778.2.4 The T-type piezoelectric transducer 3798.2.5 Shear mode piezoelectric transducer 3808.2.6 Summary 381

8.3 Two-port models for piezoelectric systems 3838.3.1 General transformer-based two-port network model 3838.3.2 External constraints 384

8.4 Piezoelectric excitation of a cantilevered beam 3868.4.1 Force couple model 3868.4.2 Optimal placement of piezoelectric element 3888.4.3 Excitation of higher-order resonant modes 389

8.5 Sensing circuits for piezoelectric transducers 3908.5.1 The charge amplifier 3918.5.2 Two-port piezo sensor representation 392

8.6 Summary 395






Contents xi

Problems 396References 398

9 Electromechanics of magnetic MEMS devices 399

9.1 Preliminaries 3999.1.1 Organization and background 4009.1.2 Note to readers 400

9.2 Lossless electromechanics of magnetic systems 4009.2.1 State variables and conservative systems 4019.2.2 Evaluation of magnetic energy 4029.2.3 Force of electrical origin 4039.2.4 Coenergy formulation 4039.2.5 Magnetic non-linearity 4049.2.6 Multiport magnetic systems 405

9.3 Basic inductive transducer geometries 4069.3.1 Variable-gap inductors 4099.3.2 Variable-area inductors 4119.3.3 Nature of magnetic system constraints 4119.3.4 A magnetic transducer with two coils 413

9.4 Rotational magnetic transducers 4169.4.1 Electromechanics of rotating magnetic transducers 4169.4.2 Rotating magnetic actuator 417

9.5 Permanent magnet transducers 4199.6 Small-signal inductive electromechanics 421

9.6.1 M-form transducer matrix based on W ′m(x, i) 421

9.6.2 N-form transducer matrix based on Wm(x, λ) 4229.6.3 Linear circuit models for magnetic transducers 4249.6.4 External constraints 4259.6.5 Linear two-port transducers with external constraint 4269.6.6 Cascade forms 427

9.7 Two-port models for magnetic MEMS 4279.7.1 Current-biased magnetic transducers 4289.7.2 Variable-gap and variable-area transducers 4309.7.3 A magnetic MEMS resonator 4309.7.4 A permanent magnet actuator 434

9.8 Stability of magnetic transducers 4359.8.1 Use of small-signal analysis 4359.8.2 Constant current and constant flux limits 4369.8.3 General stability criteria 4369.8.4 Variable-gap and variable-area devices 437

9.9 Magnetic MEMS sensors 4379.9.1 DC biased current-bridge sensor 4389.9.2 Linear variable differential transformer sensor 441






xii Contents


Appendix A Review of quasistatic electromagnetics 454Appendix B Review of mechanical resonators 473Appendix C Micromachining 498Appendix D A brief review of solid mechanics 523

Index 552






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Electromechanics and Mems Toc