1. Electrostatic Sensors and Actuators
Chang Liu
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2. Single crystal silicon and wafers
To use Si as a substrate material, it should be pure Si in a single crystal form
The Czochralski (CZ) method: A seed crystal is attached at the tip of a puller, which slowly pulls up to form a larger crystal
100 mm (4 in) diameter x 500 mm thick
150 mm (6 in) diameter x 750 mm thick
200 mm (8 in) diameter x 1000 mm thick
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3. Miller indices
A popular method of designating crystal planes (hkm) and orientations <hkm>
Identify the axial intercepts
Take reciprocal
Clear fractions (not taking lowest integers)
Enclose the number with ( ) : no comma
<hkm> designate the direction normal to the plane (hkm)
(100), (110), (111)
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4. Stress and Strain Definition of Stress and Strain
The normal stress (Pa, N/m2)
The strain
Poisson’s ratio
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5. E: Modulus of Elasticity, Young’s Modulus
Hooke’s Law
E: Modulus of Elasticity, Young’s Modulus
The shear stress
The shear strain
The shear modulus of elasticity
The relationship
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6. General Relation Between Tensile Stress and Strain
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7. The behavior of brittle materials (Si) and soft rubber used extensively in MEMS
A material is strong if it has high yield strength or ultimate strength. Si is even stronger than stainless steel
Ductility is a measure of the degree of plastic deformation that has been sustained at the point of fracture
Toughness is a mechanical measure of the material’s ability to absorb energy up to fracture (strength + ductility)
Resilience is the capacity of a material to absorb energy when it is deformed elastically, then to have this energy recovered upon unloading
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8. Mechanical Properties of Si and Related Thin Films
거시적인 실험데이터는 평균적인 처리로 대개 많은 변이가 없는데 미시적인 실험은 어렵고 또 박막의 조건 (공정조건, Growth 조건 등), 표면상태, 열처리 과정 때문에 일관적이지 않음
The fracture strength is size dependent; it is times larger than that of a millimeter-scale sample
Hall Petch equation;
For single crystal silicon, Young’s modulus is a function of the crystal orientaiton
For plysilicon thin films, it depends on the process condition (differ from Lab. to Lab.)
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9. General Stress-Strain Relations
C: stiffness matrix
S: compliance matrix
For many materials of interest to MEMS, the stiffness can be simplified
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10. Flexural Beam Bending Types of Beams; Fig. 3.15
Possible Boundary Conditions
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11. Longitudinal Strain Under Pure Bending
Pure Bending; The moment is constant throughout the beam
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12. Deflection of Beams
Appendix B
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13. Finding the Spring Constant
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14. Outline Basic Principles Applications examples
capacitance formula
capacitance configuration
Applications examples
sensors
actuators
Analysis of electrostatic actuator
second order effect - “pull in” effect
Application examples and detailed analysis
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15. Mechanics of Micro Structures
Chang Liu
Micro Actuators, Sensors, Systems Group
University of Illinois at Urbana-Champaign
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16. Mechanical Variables of Concern
Force constant
flexibility of a given device
Mechanical resonant frequency
response speed of device
Hooke’s law applied to DC driving
Importance of resonant freq.
Limits the actuation speed
lower energy consumption at Fr
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17. Types of Electrical-Mechanical Analysis
Given dimensions and materials of electrostatic structure, find
force constant of the suspension
structure displacement prior to pull-in
value of pull-in voltage
Given the range of desired applied voltage and the desired displacement, find
dimensions of a structure
layout of a structure
materials of a structure
Given the desired mechanical parameters including force constants and resonant frequency, find
dimensions
materials
layout design
quasistatic displacement
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18. Analysis of Mechanical Force Constants
Concentrate on cantilever beam (micro spring boards)
Three types of most relevant boundary conditions
free: max. degrees of freedom
fixed: rotation and translation both restricted
guided: rotation restricted.
Beams with various combination of boundary conditions
fixed-free, one-end-fixed beam
fixed-fixed beam
fixed-guided beam
Fixed-free
Two fixed-
guided beams
Four fixed-guided beams
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19. Examples
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20. Boundary Conditions
Six degrees of freedom: three axis translation, three axis rotation
Fixed B.C.
no translation, no rotation
Free B.C.
capable of translation AND rotation
Guided B.C.
capable of translation BUT NOT rotation
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21. A Clamped-Clamped Beam
Fixed-guided
Fixed-guided
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22. A Clamped-Free Beam
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23. One-end Supported, “Clamped-Free” Beams
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24. Fixed-Free Beam by Sacrificial Etching
Right anchor is fixed because its rotation is completely restricted.
Left anchor is free because it can translate as well as rotate.
Consider the beam only moves in 2D plane (paper plane). No out-of-plane translation or rotation is encountered.
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25. Force Constants for Fixed-Free Beams
Dimensions
length, width, thickness
unit in mm.
Materials
Young’s modulus, E
Unit in Pa, or N/m2.
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26. Modulus of Elasticity Names Definition
Young’s modulus
Elastic modulus
Definition
Values of E for various materials can be found in notes, text books, MEMS clearing house, etc.
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27. Large Displacement vs. Small Displacement
end displacement less than times the thickness.
Used somewhat loosely because of the difficulty to invoke large-deformation analysis.
Large deformation
needs finite element computer-aided simulation to solve precisely.
In limited cases exact analytical solutions can be found.
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28. Force Constants for Fixed-Free Beams
Moment of inertia I (unit: m4)
I= for rectangular crosssection
Maximum angular displacement
Maximum vertical displacement under F is
Therefore, the equivalent force constant is
Formula for 1st order resonant frequency
where is the beam weight per unit length.
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29. Zig-Zag Beams Saves chip real-estate
Used to pack more “L” into a given footprint area on chip to reduce the spring constant without sacrificing large chip space.
Saves chip
real-estate
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30. An Example
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31. Order of Resonance
1st order: one node where the gradient of the beam shape is zero;
also called fundamental mode.
With lowest resonance frequency.
2nd order: 2 nodes where the gradient of the beam shape is zero;
3nd order: 3 nodes.
Frequency increases as the order number goes up.
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32. Resonant frequency of typical spring-mass system
Self-mass or concentrated mass being m
The resonant frequency is
Check consistency of units.
High force constant (stiff spring) leads to high resonant frequency.
Low mass (low inertia) leads to high resonant frequency.
To satisfy both high K and high resonant frequency, m must be low.
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33. Quality Factor
If the distance between two half-power points is df, and the resonance frequency if fr, then
Q=fr/df
Q=total energy stored in system/energy loss per unit cycle
Source of mechanical energy loss
crystal domain friction
direct coupling of energy to surroundings
distrubance and friction with surrounding air
example: squeezed film damping between two parallel plate capacitors
requirement for holes: (1) to reduce squeezed film damping; (2) facilitate sacrificial layer etching (to be discussed later in detail).
Source of electrical energy loss
resistance ohmic heating
electrical radiation
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34. Basic Principles Sensing Actuation Two major configurations
capacitance between moving and fixed plates change as
distance and position is changed
media is replaced
Actuation
electrostatic force (attraction) between moving and fixed plates as
a voltage is applied between them.
Two major configurations
parallel plate capacitor (out of plane)
interdigitated fingers - IDT (in plane)
Interdigitated finger configuration
Parallel plate configuration
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35. Examples
Parallel Plate Capacitor
Comb Drive Capacitor
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36. Parallel Plate Capacitor
Fringe electric field
(ignored in first order
analysis)
Equations without considering fringe electric field.
A note on fringe electric field: The fringe field is frequently ignored in first-order analysis. It is nonetheless important. Its effect can be captured accurately in finite element simulation tools.
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37. Fabrication Methods Flip and bond Surface micromachining Wafer bonding
3D assembly
Flip and
bond
Movable
vertical plate
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38. Forces of Capacitor Actuators
Stored energy
Force is derivative of energy with respect to pertinent dimensional variable
Plug in the expression for capacitor
We arrive at the expression for force
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39. Relative Merits of Capacitor Actuators
Pros
Nearly universal sensing and actuation; no need for special materials.
Low power. Actuation driven by voltage, not current.
High speed. Use charging and discharging, therefore realizing full mechanical response speed.
Cons
Force and distance inversely scaled - to obtain larger force, the distance must be small.
In some applications, vulnerable to particles as the spacing is small - needs packaging.
Vulnerable to sticking phenomenon due to molecular forces.
Occasionally, sacrificial release. Efficient and clean removal of sacrificial materials.
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40. Capacitive Accelerometer
Proof mass area 1x0.6 mm2, and 5 mm thick.
Net capacitance 150fF
External IC signal processing circuits
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41. Deformable Mirrors for Adaptive Optics
2 mm surface normal stroke
for a 300 mm square mirror, the displacement is 1.5 micron at approximately 120 V applied voltage
T. Bifano, R. Mali, Boston University (
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42. MASS
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43. MASS
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44. BU Adaptive Micro Mirrors
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45. Optical Micro Switches
Texas Instrument DLP
Torsional parallel plate capacitor support
Two stable positions (+/- 10 degrees with respect to rest)
All aluminum structure
No process steps entails temperature above oC.
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46. “Digital Light” Mirror Pixels
Mirrors are on 17 m center-to-center spacing
Gaps are 1.0 m nominal
Mirror transit time is <20 s from state to state
Tilt Angles are minute at ±10 degrees
Four mirrors equal the width of a human hair
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47. Digital Micromirror Device (DMD)
-10 deg
Mirror
+10 deg
Hinge
CMOS Substrate
Yoke
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48. Perspective View of Lateral Comb Drive
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49. Lateral Comb Drive Actuators
Total capacitance is proportional to the overlap length and depth of the fingers, and inversely proportional to the distance.
Pros:
Frequently used in actuators for its relatively long achievable driving distance.
Cons
force output is a function of finger thickness. The thicker the fingers, the large force it will be.
Relatively large footprint.
N=4 in above diagram.
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50. Transverse Comb Drive Devices
Direction of finger movement is orthogonal to the direction of fingers.
Pros: Frequently used for sensing for the sensitivity and ease of fabrication
Cons: not used as actuator because of the physical limit of distance.
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51. Devices Based on Transverse Comb Drive
Analog Device ADXL accelerometer
A movable mass supported by cantilever beams move in response to acceleration in one specific direction.
Relevant to device performance
sidewall vertical profile
off-axis movement compensation
temperature sensitivity.
* p
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52. Lateral comb drive banks
Sandia Electrostatically driven gears - translating linear motion into continuous rotary motion
Lateral comb drive banks
Mechanical
springs
Gear train
Optical shutter
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53. Sandia Gears Mechanical springs Position limiter
Use five layer polysilicon to increase the thickness t in lateral comb drive actuators.
Mechanical springs
Position
limiter
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54. More Sophisticated Micro Gears
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55. Actuators that Use Fringe Electric Field - Rotary Motor
Three phase electrostatic actuator.
Arrows indicate electric field and electrostatic force. The tangential components cause the motor to rotate.
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56. Three Phase Motor Operation Principle
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57. Starting Position -> Apply voltage to group A electrodes
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58. Motor tooth aligned to A -> Apply voltage to Group C electrodes
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59. Motor tooth aligned to C -> Apply voltage to Group B electrodes
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60. Motor tooth aligned to B -> Apply voltage to Group A electrodes
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61. Motor tooth aligned to A -> Apply voltage to Group C electrodes
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62. Example of High Aspect Ratio Structures
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63. Some variations Large angle Long distance Low voltage Linear movement
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64. 1x4 Optical Switch
John Grade and Hal Jerman, “A large deflection electrostatic actuator for optical switching applications”, IEEE S&A Workshop, 2000, p. 97.
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65. Torsional mechanical spring
Actuators that Use Fringe Field - Micro Mirrors with Large Displacement Angle
Torsional mechanical spring
R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &Act
Workshop, Hilton Head Island, 2000.
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66. Curled Hinge Comb Drives
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67. Other Parallel Plate Capacitor - Scratch Drive Actuator
Mechanism for realizing continuous long range movement.
Scratch drive invented by H. Fujita of Tokyo University.
The motor shown above was made by U. of Colorado, Victor Bright.
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68. Analysis of Electrostatic Actuator
What happens to a parallel plate capacitor when the applied voltage is gradually increased?
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69. An Equivalent Electromechanical Modelx
If top plate
moves down-
ward, x<0.
Note: direction
definition of
variables
This diagram depicts a parallel plate capacitor at equilibrium position. The mechanical restoring spring with spring constant Km (unit: N/m) is associated with the suspension of the top plate.
According to Hooke’s law,
At equilibrium, the two forces, electrical force and mechanical restoring force, must be equal. Less the plate would move under Newton’s first law.
Gravity is generally ignored.
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70. Mechanical Spring Cantilever beams with various boundary conditions
Torsional bars with various boundary conditions
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71. Electrical And Mechanical Forces
If the right-hand plate moves
closer to the fixed one, the magnitude
of mechanical force increases linearly.
Equilibrium:
|electric force|=|mechanical force|
If a constant voltage, V1, is applied
in between two plates, the electric force
changes as a function of distance. The
closer the two plates, the large the
force. X0 x
Equilibrium
position Km
fixed
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72. Electrical And Mechanical ForcesV3 V2
V3>V2>V1
Equilibrium:
|electric force|=|mechanical force| V1 X0 Km
X0+x1
fixed
X0+x2
X0+x3
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73. Force Balance Equation at Given Applied Voltage V
The linear curve represents the magnitude of mechanical restoring force as a function of x.
Each curve in the family represents magnitude of electric force as a function of spacing (x0+x).
Note that x<0. The origin of x=0 is the dashed line.
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74. Determining Equilibrium Position Graphically
At each specific applied voltage, the equilibrium position can be determined by the intersection of the linear line and the curved line.
For certain cases, two equilibrium positions are possible. However, as the plate moves from top to bottom, the first equilibrium position is typically assumed.
Note that one curve intersects the linear line only at one point.
As voltage increases, the curve would have no equilibrium position.
This transition voltage is called pull-in voltage.
The fact that at certain voltage, no equilibrium position can be found, is called pull-in effect.
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75. Pull-In Effect
As the voltage bias increases from zero across a pair of parallel plates, the distance between such plates would decrease until they reach 2/3 of the original spacing, at which point the two plates would be suddenly snapped into contact.
This behavior is called the pull-in effect.
A.k.a. “snap in”
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76. fixed A threshold point Equilibrium:
VPI
Equilibrium:
|electric force|=|mechanical force|
X=-x0/3 X0 Km
Positive
feedback
-snap, pull in
fixed
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77. Mathematical Determination of Pull-in Voltage Step 1 - Defining Electrical Force Constant
Let’s define the tangent of the electric force term. It is called electrical force constant, Ke.
When voltage is below the pull-in voltage, the magnitude of Ke and Km are not equal at equilibrium.
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78. Review of Equations Related To Parallel Plate
The electrostatic force is
The electric force constant is
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79. Mathematical Determination of Pull-in Voltage Step 2 - Pull-in Condition
At the pull-in voltage, there is only one intersection between |Fe| and |Fm| curves.
At the intersection, the gradient are the same, I.e. the two curves intersect with same tangent.
This is on top of the condition that the magnitude of Fm and Fe are equal.
Force balance yields
Plug in expression of V2 into the expression for Ke,
we get
This yield the position for the pull-in condition, x=-x0/3. Irrespective of the magnitude of Km.
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80. Mathematical Determination of Pull-in Voltage Step 3 - Pull-in Voltage Calculation
Plug in the position of pull-in into Eq. * on previous page, we get the voltage at pull-in as
At pull in, C=1.5 Co
Thus,
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81. Implications of Pull-in Effect
For electrostatic actuator, it is impossible to control the displacement through the full gap. Only 1/3 of gap distance can be moved reliably.
Electrostatic micro mirros
reduced range of reliable position tuning
Electrostatic tunable capacitor
reduced range of tuning and reduced tuning range
Tuning distance less than 1/3, tuning capacitance less than 50%.
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82. Counteracting Pull-In Effect Leveraged Bending for Full Gap Positioning
E. Hung, S. Senturia, “Leveraged bending for full gap positioning with electrostatic actuation”, Sensors and Actuators Workshop, Hilton Head Island, p. 83, 2000.
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83. Counteracting Pull-in Effect: Variable Gap Capacitor
Existing Tunable Capacitor
Tuning range: 88%
(with parasitic capacitance)
Counter
capacitor plate
Suspension
spring d0
Actuation
electrode
Capacitor
plate
Actuation
electrode
NEW DESIGN
Variable Gap Variable Capacitor
Suspension
spring
Counter
capacitor
plate
<(1/3)d0 d0
Actuation
electrode
Capacitor
plate
Actuation
electrode
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84. Example
A parallel plate capacitor suspended by two fixed-fixed cantilever beams, each with length, width and thickness denoted l, w and t, respectively. The material is made of polysilicon, with a Young’s modulus of 120GPa.
L=400 mm, w=10 mm, and t=1 mm.
The gap x0 between two plates is 2 mm.
The area is 400 mm by 400 mm.
Calculate the amount of vertical displacement when a voltage of 0.4 volts is applied.
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85. Step 1: Find mechanical force constants
Calculate force constant of one beam first
use model of left end guided, right end fixed.
Under force F, the max deflection is
The force constant is therefore
This is a relatively “soft” spring.
Note the spring constant is stiffer than fixed-free beams.
Total force constant encountered by the parallel plate is
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86. Step 2: Find out the Pull-in Voltage
Find out pull-in voltage and compare with the applied voltage.
First, find the static capacitance value Co
Find the pull-in voltage value
When the applied voltage is 0.4 volt, the beam has been pulled-in. The displacement is therefore 2 mm.
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87. What if the applied voltage is 0.2 V?
Not sufficient to pull-in
Deformation can be solved by solving the following equation or
How to solve it?
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88. Solving Third Order Equation ...
To solve
Apply
Use the following definition
The only real solution is
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89. Calculator … A Simple Way Out.
Use HP calculator,
x1=-2.45x10-7 mm
x2=-1.2x10-6 mm
x3=-2.5x10-6 mm
Accept the first answer because the other two are out side of pull-in range.
If V=0.248 Volts, the displacement is mm.
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