chang liu mass uiuc electrostatic sensors and actuators chang liu
TRANSCRIPT
Chang Liu MASSUIUC
Electrostatic Sensors and Actuators
Chang Liu
Chang Liu MASSUIUC
• 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 m thick– 150 mm (6 in) diameter x 750 m thick– 200 mm (8 in) diameter x 1000 m thick
Single crystal silicon and wafers
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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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Stress and Strain
• Definition of Stress and Strain– The normal stress (Pa, N/m2)
– The strain
– Poisson’s ratio
A
F
00
0
L
L
L
LL
x
z
x
y
Chang Liu MASSUIUC
Hooke’s Law E
A
F
L
X
E: Modulus of Elasticity, Young’s Modulus
The shear stress
The shear strain
The shear modulus of elasticity
The relationship
12
EG
G
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General Relation Between Tensile Stress and Strain
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• 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
Chang Liu MASSUIUC
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 23-28 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.)
2/10
Kdy
Chang Liu MASSUIUC
General Stress-Strain Relations
654 ,,,, TTTxyxzyz
654 ,,,, TTTxyxzyz
6
5
4
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
6
5
4
3
2
1
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
T
T
T
T
T
T
321 ,,,, TTTzzyyxx
CT
TSC: stiffness matrix
S: compliance matrix
PaCSi11
100, 10
8.000000
08.00000
008.0000
00066.164.064.0
00064.066.164.0
00064.064.066.1
For many materials of interest to MEMS, the stiffness can be simplified
Chang Liu MASSUIUC
Flexural Beam Bending
• Types of Beams; Fig. 3.15• Possible Boundary Conditions
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Longitudinal Strain Under Pure Bending
EI
My
EI
Mt
2max
Pure Bending; The moment is constant throughout the beam
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Deflection of Beams
)(2
2
xMdx
ydEI
EI
Fld
EI
Fl
3,
2
3
max
2
max
EI
Mld
EI
Ml
2,
2
maxmax
EI
Fld
192
3
max
EI
Fld
12
3
max
Appendix B
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Finding the Spring Constant
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Outline
• Basic Principles– 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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Mechanics of Micro Structures
Chang LiuMicro Actuators, Sensors, Systems Group
University of Illinois at Urbana-Champaign
Chang Liu MASSUIUC
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
Felectric
Fmechanical
Km
xKF mmechanical
• Importance of resonant freq.– Limits the actuation speed– lower energy consumption at Fr
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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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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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Examples
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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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A Clamped-Clamped Beam
Fixed-guided
Fixed-guided
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A Clamped-Free Beam
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One-end Supported, “Clamped-Free” Beams
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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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Force Constants for Fixed-Free Beams• Dimensions
– length, width, thickness– unit in m.
• Materials– Young’s modulus, E– Unit in Pa, or N/m2.
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Modulus of Elasticity
• Names– Young’s modulus– Elastic modulus
• Definition
• Values of E for various materials can be found in notes, text books, MEMS clearing house, etc.
LLAF
Ex
x
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Large Displacement vs. Small Displacement• Small displacement
– end displacement less than 10-20 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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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.
EI
Fl
2
212
3wt
EI
Fl
3
3
3
3
33 4
3
3l
Ewt
l
EI
EIFl
Fkm
42
52.3
l
EIg
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Zig-Zag Beams
• Used to pack more “L” into a given footprint area on chip to reduce the spring constant without sacrificing large chip space.
Saves chipreal-estate
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An Example
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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.
Chang Liu MASSUIUC
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.
m
k
21
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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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Basic Principles
• Sensing– 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)
dA
Parallel plate configuration
Interdigitated finger configuration
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Examples
• Parallel Plate Capacitor• Comb Drive Capacitor
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Parallel Plate Capacitor
– 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.
dA
V
QC
AQE /
d
A
dAQQ
C
Fringe electric field(ignored in first orderanalysis)
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Fabrication Methods
• Surface micromachining• Wafer bonding• 3D assembly
Flip andbond
Movablevertical plate
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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
C
QCVE
22
2
1
2
1
2
2
1V
d
C
d
EF
d
A
dAQQ
C
d
CVV
d
A
d
EF
22
2 2
1
2
1
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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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Capacitive Accelerometer
• Proof mass area 1x0.6 mm2, and 5 m thick.
• Net capacitance 150fF• External IC signal processing
circuits
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Deformable Mirrors for Adaptive Optics
• 2 m surface normal stroke • for a 300 m square mirror, the displacement is 1.5 micron at
approximately 120 V applied voltage• T. Bifano, R. Mali, Boston University
(http://www.bu.edu/mfg/faculty/homepages/bifano.html)
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BU Adaptive Micro Mirrors
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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 300-350 oC.
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“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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Mirror-10 deg
Mirror+10 deg
Hinge
YokeCMOS
Substrate
Digital Micromirror Device (DMD)
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Perspective View of Lateral Comb Drive
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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.
])(2
[ 00ptot c
d
xxtNC
200
Vd
tNF
x
N=4 in above diagram.
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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.
)(
)(
0
0
0
0
fsr
fsl
Cxx
ltNC
Cxx
ltNC
Chang Liu MASSUIUC
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 234-236.
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Sandia Electrostatically driven gears- translating linear motion into continuous rotary motion
• http://www.mdl.sandia.gov/micromachine/images11.html
Lateral comb drive banks
Gear train
Optical shutter
Mechanicalsprings
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Sandia Gears • Use five layer polysilicon to increase the thickness t in lateral comb drive actuators.
Positionlimiter
Mechanical springs
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More Sophisticated Micro Gears
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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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Three Phase Motor Operation Principle
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Starting Position -> Apply voltage to group A electrodes
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Motor tooth aligned to A -> Apply voltage to Group C electrodes
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Motor tooth aligned to C -> Apply voltage to Group B electrodes
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Motor tooth aligned to B -> Apply voltage to Group A electrodes
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Motor tooth aligned to A -> Apply voltage to Group C electrodes
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Example of High Aspect Ratio Structures
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Some variations
• Large angle • Long distance• Low voltage• Linear movement
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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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Actuators that Use Fringe Field - Micro Mirrorswith Large Displacement Angle
R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &ActWorkshop, Hilton Head Island, 2000.
Torsional mechanical spring
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Curled Hinge Comb Drives
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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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Analysis of Electrostatic Actuator
What happens to a parallel plate capacitor when the applied voltage is gradually increased?
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An Equivalent Electromechanical Model
• 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.
Felectric
Fmechanical
Km
xKF mmechanical
x
Note: directiondefinition of variables
If top platemoves down-ward, x<0.
Gravity is generally ignored.
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Mechanical Spring
• Cantilever beams with various boundary conditions• Torsional bars with various boundary conditions
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Electrical And Mechanical Forces
fixed
Km
X0
Equilibrium:|electric force|=|mechanical force|
If the right-hand plate movescloser to the fixed one, the magnitudeof mechanical force increases linearly.
If a constant voltage, V1, is applied in between two plates, the electric forcechanges as a function of distance. Thecloser the two plates, the large the force.
Equilibriumposition
x
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Electrical And Mechanical Forces
fixed
Km
X0
Equilibrium:|electric force|=|mechanical force|
V3
V2
V1
V3>V2>V1
X0+x3
X0+x2
X0+x1
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Force Balance Equation at Given Applied Voltage V
20
2
2 xx
AVxkm
km
V increases
• 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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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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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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A threshold point
fixed
Km
X0
Equilibrium:|electric force|=|mechanical force|
VPI
X=-x0/3
Positivefeedback-snap, pull in
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Mathematical Determination of Pull-in VoltageStep 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.
x
Fke
2
2
d
CVke
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Review of Equations Related To Parallel Plate
• The electrostatic force is
• The electric force constant is
d
CVV
d
A
d
EF
22
2 2
1
2
1
2
2
2
22
3)2(
2
1
d
VC
d
V
d
AV
d
AKe
Chang Liu MASSUIUC
Mathematical Determination of Pull-in VoltageStep 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.
me KK
C
xxxk
A
xxxkV mm )(2)(2 0
202
)(
2
)( 20
2
o
me xx
xk
xx
CVk
2
2
d
CVke
Chang Liu MASSUIUC
Mathematical Determination of Pull-in VoltageStep 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,
mp kC
xV
9
4 202
.5.13
2
0
0
C
kxV mp
Chang Liu MASSUIUC
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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Counteracting Pull-In EffectLeveraged 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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Counteracting Pull-in Effect: Variable Gap CapacitorExisting Tunable Capacitor
Capacitor plate
Actuationelectrode
Actuationelectrode
Suspensionspring
Counter capacitor plate
NEW DESIGN
Capacitor plate
Actuationelectrode
Actuationelectrode
Suspensionspring
Counter capacitor
plate
Variable Gap Variable Capacitor
d0
d0
<(1/3)d0
Tuning range: 88% (with parasitic capacitance)
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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 m, w=10 m, and t=1 m.
• The gap x0 between two plates is 2 m.
• The area is 400 m by 400 m. • Calculate the amount of vertical
displacement when a voltage of 0.4 volts is applied.
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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
EI
Fld
12
3
mNl
Ewt
l
EI
d
FKm /01875.0
)10400(
)101(1010101201236
3669
3
3
3
mNKm /0375.0
Chang Liu MASSUIUC
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 m.
)(25.010083.75.1
0375.0
3
1022
5.13
213
6
0
0 voltsC
kxV mp
FmF
C 136
2612
0 10083.7102
)10400()/(1085.8
Chang Liu MASSUIUC
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?
C
xxxk
A
xxxkV mm )(2)(2 0
202
010552.7104104
02
2
1912263
220
20
3
xxx
k
Avxxxxx
m
Chang Liu MASSUIUC
Solving Third Order Equation ...
• To solve
• Apply• Use the following definition
• The only real solution is•
023 cbxaxx
3/axy
33
23
32
2,
2
23
3)
3(2,
3
BQq
A
qpQ
caba
qba
p
3
aBAx
BAy
Chang Liu MASSUIUC
Calculator … A Simple Way Out.
• Use HP calculator, – x1=-2.45x10-7 m– x2=-1.2x10-6 m– x3=-2.5x10-6 m
• Accept the first answer because the other two are out side of pull-in range.
• If V=0.248 Volts, the displacement is -0.54 m.