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Electromechanics and MEMS

Book Description

Offering a consistent, systematic approach to capacitive, piezoelectric and magnetic MEMS, from basic electromechanical transducers to high-level models for sensors and actuators, this comprehensive textbook equips graduate and senior-level undergraduate students with all the resources necessary to design and develop practical, system-level MEMS models. The concise yet thorough treatment of the underlying principles of electromechanical transduction provides a solid theoretical framework for this development, with each new topic related back to the core concepts. Repeated references to the shared commonalities of all MEMS encourage students to develop a systems-based design perspective. Extensive use is made of easy-to-interpret electrical and mechanical analogs, such as electrical circuits, electromechanical two-port models and the cascade paradigm. Each chapter features worked examples and numerous problems, all designed to test and extend students' understanding of the key principles.

Table of Contents

  1. Coverpage
  2. Electromechanics and MEMS
  3. Title page
  4. Copyright page
  5. Contents
  6. Preface
  7. 1 Introduction
    1. 1.1 Background
    2. 1.2 Some terminology
    3. 1.3 Electromechanical systems
    4. 1.4 Conclusion
    5. Problems
    6. Reference
  8. 2 Circuit-based modeling
    1. 2.1 Fundamentals of circuit theory
      1. 2.1.1 Motivation
      2. 2.1.2 Kirchhoff's current and voltage laws
      3. 2.1.3 Circuit elements
      4. 2.1.4 Tellegen's theorem: power and energy
      5. 2.1.5 AC circuits, impedance, and admittance
    2. 2.2 Circuit models for capacitive devices
      1. 2.2.1 Basic RC circuit building block
      2. 2.2.2 The series capacitive circuit
      3. 2.2.3 The parallel capacitive circuit
      4. 2.2.4 Special cases: series and parallel capacitance
      5. 2.2.5 Summary
    3. 2.3 Two-port networks
      1. 2.3.1 Impedance and admittance matrices
      2. 2.3.2 The transmission matrix
      3. 2.3.3 Cascaded two-port networks
      4. 2.3.4 Some important two-port networks
      5. 2.3.5 The gyrator and the transformer
      6. 2.3.6 Embedded networks
      7. 2.3.7 Source and impedance reflection
    4. 2.4 Summary
    5. Problems
    6. References
  9. 3 Capacitive lumped parameter electromechanics
    1. 3.1 Basic assumptions and concepts
      1. 3.1.1 The lossless electromechanical coupling
      2. 3.1.2 State variables and conservative systems
      3. 3.1.3 Evaluation of energy function
      4. 3.1.4 Force of electrical origin
    2. 3.2 Coenergy – an alternate energy function
      1. 3.2.1 Definition of coenergy
      2. 3.2.2 Integral evaluation of coenergy
      3. 3.2.3 Evaluation of force of electrical origin
    3. 3.3 Couplings with multiple ports
      1. 3.3.1 Energy conservation relation
      2. 3.3.2 System with two electrical and two mechanical ports
    4. 3.4 Basic capacitive transducer types
      1. 3.4.1 Variable-gap capacitors
      2. 3.4.2 Variable-area capacitors
      3. 3.4.3 Comparison of variable-gap and variable-area actuators
      4. 3.4.4 Transducer stroke
      5. 3.4.5 The comb-drive geometry
      6. 3.4.6 Another variable-area capacitor
    5. 3.5 Rotational transducers
      1. 3.5.1 Modeling rotational electromechanics
      2. 3.5.2 Torque of electrical origin
      3. 3.5.3 An example
    6. 3.6 Electrets
    7. 3.7 Non-linear conservative electromechanical systems
      1. 3.7.1 Conservation laws for capacitive devices
      2. 3.7.2 Non-linear oscillations and stability
      3. 3.7.3 Numerical solutions
      4. 3.7.4 Constant charge constraint
      5. 3.7.5 Discussion
    8. 3.8 Summary
    9. Problems
    10. References
  10. 4 Small-signal capacitive electromechanical systems
    1. 4.1 Background
    2. 4.2 Linearized electromechanical transducers
      1. 4.2.1 Some preliminaries
      2. 4.2.2 Linearization in terms of energy and coenergy
    3. 4.3 Electromechanical two-port networks
      1. 4.3.1 The transducer matrix
      2. 4.3.2 The linear capacitive transducer
      3. 4.3.3 Important special cases
      4. 4.3.4 Transducers with angular displacement
      5. 4.3.5 Multiport electromechanical transducers
    4. 4.4 Electromechanical circuit models
      1. 4.4.1 Analogous variables
      2. 4.4.2 M-form equivalent electromechanical circuit
      3. 4.4.3 N-form equivalent electromechanical circuit
      4. 4.4.4 More about the cascade paradigm
    5. 4.5 Reconciliation with Neubert
    6. 4.6 External constraints
      1. 4.6.1 Mechanical constraints
      2. 4.6.2 Electrical constraints
      3. 4.6.3 Fully constrained electromechanical transducers
      4. 4.6.4 Other useful matrix forms
    7. 4.7 Applications of electromechanical two-port theory
      1. 4.7.1 Application of source and impedance reflection
      2. 4.7.2 A capacitive microphone
      3. 4.7.3 Electromechanical transfer functions
      4. 4.7.4 A comb-drive actuator
      5. 4.7.5 The three-plate capacitive sensor
      6. 4.7.6 Linear model for electret transducer
    8. 4.8 Stability considerations
      1. 4.8.1 Preliminary look at stability
      2. 4.8.2 General stability criteria
      3. 4.8.3 The pull-in instability threshold
      4. 4.8.4 A physical interpretation of instability
    9. 4.9 Summary
    10. Problems
    11. References
  11. 5 Capacitive sensing and resonant drive circuits
    1. 5.1 Introduction
    2. 5.2 Basics of operational amplifiers
    3. 5.3 Inverting amplifiers and capacitive sensing
      1. 5.3.1 Basic inverting configuration
      2. 5.3.2 One-sided high-impedance (charge) amplifier
      3. 5.3.3 Variable-gap and variable-area capacitors
      4. 5.3.4 Effect of op-amp leakage current
    4. 5.4 Differential (three-plate) capacitance sensing
      1. 5.4.1 DC feedback for the differential configuration
    5. 5.5 AC (modulated) sensing
      1. 5.5.1 Capacitive sensor excited by zero-mean sinusoidal voltage
      2. 5.5.2 Two-plate capacitive sensing with AC excitation
      3. 5.5.3 Analysis including the feedback resistance Rf
      4. 5.5.4 AM signal demodulation
      5. 5.5.5 Differential AC sensing
      6. 5.5.6 Synchronous demodulation
    6. 5.6 AC sensors using symmetric square-wave excitation
      1. 5.6.1 Transducers using square-wave excitation
      2. 5.6.2 Three-plate sensing using square-wave excitation
    7. 5.7 Switched capacitance sensor circuits
      1. 5.7.1 Basics of switched-capacitor circuits
      2. 5.7.2 Simple sensor based on switched capacitance
      3. 5.7.3 Half-wave bridge sensor using switched capacitance
    8. 5.8 Noise in capacitive MEMS
      1. 5.8.1 Common noise characteristics
      2. 5.8.2 Filtered noise
      3. 5.8.3 Noisy two-ports
      4. 5.8.4 Electrical thermal noise
      5. 5.8.5 Mechanical thermal noise
      6. 5.8.6 1∕f amplifier noise
      7. 5.8.7 Effect of modulation on 1∕f noise
    9. 5.9 Electrostatic drives for MEMS resonators
      1. 5.9.1 Mechanical resonators
      2. 5.9.2 Drive electrodes with sinusoidal drive
      3. 5.9.3 Non-harmonic drives
      4. 5.9.4 Sense electrodes
      5. 5.9.5 Harmonic oscillators based on MEMS resonators
      6. 5.9.6 Phase-locked loop drives
      7. 5.9.7 PLL system linearization
    10. 5.10 Summary
    11. Problems
    12. References
  12. 6 Distributed 1-D and 2-D capacitive electromechanical structures
    1. 6.1 Introduction
    2. 6.2 A motivating example – electrostatic actuation of a cantilevered beam
      1. 6.2.1 Problem description
      2. 6.2.2 Derivation of the lumped parameter model
      3. 6.2.3 Evaluation of equivalent spring constant, mass, and mechanical damping
      4. 6.2.4 Resonance of a cantilevered beam
      5. 6.2.5 Recapitulation of lumped parameter model identification procedure
    3. 6.3 A second look at the cantilevered beam
      1. 6.3.1 Parameterization of distributed capacitance
      2. 6.3.2 Discretized capacitance model for the beam
    4. 6.4 MDF models for beams
      1. 6.4.1 MDF description
      2. 6.4.2 Application of boundary conditions
      3. 6.4.3 Maxwell's reciprocity theorem
      4. 6.4.4 Applications of static MDF model
      5. 6.4.5 Modal analysis
      6. 6.4.6 Decoupling of the equation of motion
      7. 6.4.7 Equivalent circuit using modal analysis
      8. 6.4.8 Damping
    5. 6.5 Using the MDF model for dynamics
    6. 6.6 A first look at plates
      1. 6.6.1 Equivalent spring constant and mass
      2. 6.6.2 Capacitance
      3. 6.6.3 Resonance
    7. 6.7 MDF modeling of plates
      1. 6.7.1 Uniform discretization of rectangular 2-D plates
      2. 6.7.2 2-D discretization of circular plates
      3. 6.7.3 2-D example: electrostatic actuation of a circular plate
    8. 6.8 Additional beam configurations
      1. 6.8.1 Doubly clamped beam
      2. 6.8.2 Simply supported beam
      3. 6.8.3 Vibration isolation of the simply supported beam
      4. 6.8.4 Closure
    9. 6.9 Summary
    10. Problems
    11. References
  13. 7 Practical MEMS devices
    1. 7.1 Introduction
    2. 7.2 Capacitive MEMS pressure sensors
      1. 7.2.1 Basic displacement-based capacitive pressure sensor
      2. 7.2.2 System-level model
      3. 7.2.3 A differential configuration
      4. 7.2.4 Closure
    3. 7.3 MEMS accelerometers
      1. 7.3.1 Principles of operation
      2. 7.3.2 System transfer function and sensitivity
      3. 7.3.3 Basic construction of an accelerometer
      4. 7.3.4 Mechanical transfer function and mechanical thermal noise
      5. 7.3.5 Selection of the electrode types
      6. 7.3.6 Force-feedback configuration
      7. 7.3.7 Higher-order effects
    4. 7.4 MEMS gyroscopes
      1. 7.4.1 A qualitative description of mechanical gyroscopes with some historical notes
      2. 7.4.2 Rotating reference frames
      3. 7.4.3 A simple z axis rate vibratory gyroscope
      4. 7.4.4 Other examples of MEMS-based gyroscopes
      5. 7.4.5 Background material
      6. 7.4.6 Torsional-vibration gyroscope
      7. 7.4.7 Higher-order effects
      8. 7.4.8 Closure
    5. 7.5 MEMS energy harvesters
      1. 7.5.1 Basic principle of capacitive energy harvesting
      2. 7.5.2 Power considerations and efficiency
      3. 7.5.3 Multiple resonators
      4. 7.5.4 Capacitive energy harvesters with bias voltage
      5. 7.5.5 Practical electrostatic energy harvesters
    6. 7.6 Summary
    7. Problems
    8. References
  14. 8 Electromechanics of piezoelectric elements
    1. 8.1 Introduction
    2. 8.2 Electromechanics of piezoelectric materials
      1. 8.2.1 Piezoelectric phenomenology
      2. 8.2.2 Piezoelectric properties
      3. 8.2.3 The L-type piezoelectric transducer
      4. 8.2.4 The T-type piezoelectric transducer
      5. 8.2.5 Shear mode piezoelectric transducer
      6. 8.2.6 Summary
    3. 8.3 Two-port models for piezoelectric systems
      1. 8.3.1 General transformer-based two-port network model
      2. 8.3.2 External constraints
    4. 8.4 Piezoelectric excitation of a cantilevered beam
      1. 8.4.1 Force couple model
      2. 8.4.2 Optimal placement of piezoelectric element
      3. 8.4.3 Excitation of higher-order resonant modes
    5. 8.5 Sensing circuits for piezoelectric transducers
      1. 8.5.1 The charge amplifier
      2. 8.5.2 Two-port piezo sensor representation
    6. 8.6 Summary
    7. Problems
    8. References
  15. 9 Electromechanics of magnetic MEMS devices
    1. 9.1 Preliminaries
      1. 9.1.1 Organization and background
      2. 9.1.2 Note to readers
    2. 9.2 Lossless electromechanics of magnetic systems
      1. 9.2.1 State variables and conservative systems
      2. 9.2.2 Evaluation of magnetic energy
      3. 9.2.3 Force of electrical origin
      4. 9.2.4 Coenergy formulation
      5. 9.2.5 Magnetic non-linearity
      6. 9.2.6 Multiport magnetic systems
    3. 9.3 Basic inductive transducer geometries
      1. 9.3.1 Variable-gap inductors
      2. 9.3.2 Variable-area inductors
      3. 9.3.3 Nature of magnetic system constraints
      4. 9.3.4 A magnetic transducer with two coils
    4. 9.4 Rotational magnetic transducers
      1. 9.4.1 Electromechanics of rotating magnetic transducers
      2. 9.4.2 Rotating magnetic actuator
    5. 9.5 Permanent magnet transducers
    6. 9.6 Small-signal inductive electromechanics
      1. 9.6.1 M-form transducer matrix based on W′m(x, i)
      2. 9.6.2 N-form transducer matrix based on Wm(x, λ)
      3. 9.6.3 Linear circuit models for magnetic transducers
      4. 9.6.4 External constraints
      5. 9.6.5 Linear two-port transducers with external constraint
      6. 9.6.6 Cascade forms
    7. 9.7 Two-port models for magnetic MEMS
      1. 9.7.1 Current-biased magnetic transducers
      2. 9.7.2 Variable-gap and variable-area transducers
      3. 9.7.3 A magnetic MEMS resonator
      4. 9.7.4 A permanent magnet actuator
    8. 9.8 Stability of magnetic transducers
      1. 9.8.1 Use of small-signal analysis
      2. 9.8.2 Constant current and constant flux limits
      3. 9.8.3 General stability criteria
      4. 9.8.4 Variable-gap and variable-area devices
    9. 9.9 Magnetic MEMS sensors
      1. 9.9.1 DC biased current-bridge sensor
      2. 9.9.2 Linear variable differential transformer sensor
    10. 9.10 Summary
    11. Problems
    12. References
  16. Appendix A Review of quasistatic electromagnetics
  17. Appendix B Review of mechanical resonators
  18. Appendix C Micromachining
  19. Appendix D A brief review of solid mechanics
  20. Index