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Electrostatic Sensors and Actuators
Chang Liu
Chang Liu
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UIUC
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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Miller indices
•
A popular method of designating crystal planes (hkm) and orientations
<hkm>
–
–
–
–
•
<hkm> designate the direction normal to the plane (hkm)
–
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Identify the axial intercepts
Take reciprocal
Clear fractions (not taking lowest integers)
Enclose the number with ( ) : no comma
(100), (110), (111)
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Stress and Strain
• Definition of Stress and Strain
– The normal stress (Pa, N/m2)

F
A
– The strain

L  L0 L

L0
L0
– Poisson’s ratio

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y z

x x
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Hooke’s Law
  E
E: Modulus of Elasticity, Young’s Modulus
The shear stress

The shear strain

F
A
X
L
The shear modulus of elasticity
G


The relationship
G
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E
21  
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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
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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;
 y   0  Kd 1/ 2
• 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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General Stress-Strain Relations
 xx ,  yy ,  zz  T1 , T2 , T3
 yz , xz , xy  T4 , T5 , T6
T1  C11
T  C
 2   21
T3  C31
 
T4  C41
T5  C51
  
T6  C61
C12 C13
C22 C23
C32 C33
C42 C43
C52 C53
C62 C63
C14
C24
C34
C44
C54
C64
C15
C25
C35
C45
C55
C65
C16   1 
C26   2 
C36   3 
 
C46   4 
C56   5 
 
C66   6 
T  C
C: stiffness matrix
  ST
S: compliance matrix
For many materials of interest to
MEMS, the stiffness can be
simplified
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 yz , xz ,  xy  T4 , T5 , T6
CSi,100
0
0
1.66 0.64 0.64 0
0.64 1.66 0.64 0
0
0 

0.64 0.64 1.66 0
0
0  11

10 Pa
0
0
0
0
.
8
0
0


 0
0
0
0 0.8 0 


0
0
0
0 0.8
 0
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Flexural Beam Bending
• Types of Beams; Fig. 3.15
• Possible Boundary Conditions
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Longitudinal Strain Under Pure Bending
Pure Bending; The moment is constant throughout the beam
My

EI
 max
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Mt

2 EI
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Deflection of Beams
d2y
EI 2  M ( x)
dx
 max
Fl 2
Fl 3

, d max 
2 EI
3EI
d max
Fl 3

12 EI
d max 
Fl 3
192EI
 max
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Appendix B
Ml
Ml 2

, d max 
EI
2 EI
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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
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Micro Actuators, Sensors, Systems Group
University of Illinois at Urbana-Champaign
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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
Fmechanical
Felectric
Km
Fmechanical  K m x
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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
–
–
–
–
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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 fixedguided 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 mm.
• 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
F
x
E
 A
 x L
L
• Values of E for various materials can be found in notes, text
books, MEMS clearing house, etc.
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Large Displacement vs. Small Displacement
• Small displacement
– end displacement less than 1020 times the thickness.
– Used somewhat loosely
because of the difficulty to
invoke large-deformation
analysis.
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• Large deformation
– needs finite element computeraided 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)
3
– I= wt
for rectangular crosssection
12
• Maximum angular displacement
Fl 2
2 EI
Fl 3
3EI
• Maximum vertical displacement under F is
• Therefore, the equivalent force constant is
F
3EI Ewt 3
km 
 3 
Fl 3
l
4l 3
3EI
• Formula for 1st order resonant frequency
– where
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
is the beam weight per unit length.
3.52 EIg
2 l 4
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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 chip
real-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.
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Resonant frequency of typical spring-mass system
• Self-mass or concentrated mass being m
• The resonant frequency is
1
2
k
m
• 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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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)
A
Interdigitated finger configuration
d
Parallel plate configuration
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Examples
• Parallel Plate Capacitor
• Comb Drive Capacitor
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Parallel Plate Capacitor
A
Fringe electric field
(ignored in first order
analysis)
C
d
Q
V
E  Q / A
C
Q
A

Q
d d
A
– 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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Fabrication Methods
• Surface micromachining
• Wafer bonding
• 3D assembly
Flip and
bond
Movable
vertical plate
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Forces of Capacitor Actuators
• Stored energy
1
1 Q2
2
E  CV 
2
2 C
• Force is derivative of energy with F  E  1 C V 2
d 2 d
respect to pertinent dimensional
variable
• Plug in the expression for capacitor C  Q  A
Q
d
A
• We arrive at the expression for force
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d
E
1 A 2
1 CV 2
F

V 
d
2 d2
2 d
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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.
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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 mm thick.
• Net capacitance 150fF
• External IC signal processing
circuits
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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
(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
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• Torsional parallel plate
capacitor support
• Two stable positions (+/10 degrees with respect
to rest)
• All aluminum structure
• No process steps entails
temperature above 300350 oC.
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“Digital Light” Mirror Pixels
 Mirrors are on 17 mm
center-to-center spacing
 Gaps are 1.0 mm nominal
 Mirror transit time is
<20 ms from state to state
 Tilt Angles are minute at
±10 degrees
 Four mirrors equal the
width of a human hair
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Digital Micromirror Device (DMD)
Mirror
-10 deg
Mirror
+10 deg
Hinge
Yoke
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CMOS
Substrate
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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:
2 0t ( x  x0 )
Ctot  N [
 cp ]
d
F
x0

N 0 t 2
V
d
N=4 in above diagram.
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– 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.
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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.
Csl  N (
Csr  N (
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 0lt
Cf )
x0  x
 0lt
x0  x
Cf )
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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
Lateral comb drive banks
Mechanical
springs
Gear train
Optical shutter
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• http://www.mdl.sandia.gov/mic
romachine/images11.html
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Sandia Gears
• Use five layer
polysilicon to increase
the thickness t in lateral
comb drive actuators.
Mechanical springs
Position
limiter
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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
•
•
•
•
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Large angle
Long distance
Low voltage
Linear movement
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1x4 Optical Switch
•
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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 Mirrors
with Large Displacement Angle
Torsional mechanical spring
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R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &Act
Workshop, Hilton Head Island, 2000.
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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.
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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
Fmechanical
x
If top plate
moves downward, x<0.
Felectric
Km
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, Fmechanical  Km x
• At equilibrium, the two forces, electrical force and mechanical
restoring force, must be equal. Less the plate would move under
Newton’s first law.
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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
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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Electrical And Mechanical Forces
V3
V2
Equilibrium:
|electric force|=|mechanical force|
V3>V2>V1
V1
X0
Km
X0+x1
fixed
X0+x2
X0+x3
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Force Balance Equation at Given Applied Voltage V
 km x 
• 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
V increases
spacing (x0+x).
AV 2
2 x  x0 
2
km
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• 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
VPI
Equilibrium:
|electric force|=|mechanical force|
X=-x0/3
X0
Km
fixed
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Positive
feedback
-snap, pull in
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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.
F
CV 2
ke 
ke  2
x
d
• When voltage is below the pull-in voltage, the magnitude of Ke
and Km are not equal at equilibrium.
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Review of Equations Related To Parallel Plate
• The electrostatic force is
E
1 A 2
1 CV 2
F

V 
2
d
2d
2 d
• The electric force constant is
1
A 2 A V 2
V2
K e   (2) 3 V 
C 2
2
2
d
d d
d
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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.
Ke  K m
• This is on top of the condition that the magnitude of Fm and Fe
are equal.
2

2
k
x
(
x

x
)
 2km x( x  x0 )
m
0
– Force balance yields V 

A
C
2
2
CV
– Plug in expression of V2 into the expression for Ke, ke  2
d
• we get
 2k m x
CV 2
ke 

( x  x0 ) 2 ( x  xo )
– This yield the position for the pull-in condition, x=-x0/3.
Irrespective of the magnitude of Km.
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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
2
4
x
V p2  0 k m
9C
• At pull in, C=1.5 Co
• Thus,
Vp 
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2 x0
km
.
3 1.5C0
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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 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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Counteracting Pull-in Effect: Variable Gap Capacitor
Existing Tunable Capacitor
Counter
capacitor plate
Suspension
spring
Tuning range: 88%
(with parasitic capacitance)
d0
Actuation
electrode
Capacitor
plate
Actuation
electrode
NEW DESIGN
Variable Gap Variable Capacitor
Suspension
spring
d0
Actuation
electrode
Chang Liu
<(1/3)d0
Counter
capacitor
plate
Capacitor
plate
Actuation
electrode
MASS
UIUC
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.
Chang Liu
MASS
UIUC
Step 1: Find mechanical force constants
• Calculate force constant of one beam first
– use model of left end guided, right end fixed.
3
– Under force F, the max deflection is d  Fl
12 EI
– The force constant is therefore
F 12EI Ewt 3 120 109 10 106  (1106 )3
Km   3  3 
 0.01875N / m
d
l
l
(400 106 )3
– 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
K m  0.0375 N / m
Chang Liu
MASS
UIUC
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
8.85  1012 ( F / m)  (400  106 ) 2
C0 
 7.083  1013 F
6
2  10
• Find the pull-in voltage value
2 x0
km
2  2 106
0.0375
Vp 

 0.25(volts)
13
3 1.5C0
3
1.5  7.083 10
• When the applied voltage is 0.4 volt, the beam has been pulledin. The displacement is therefore 2 mm.
Chang Liu
MASS
UIUC
What if the applied voltage is 0.2 V?
• Not sufficient to pull-in
• Deformation can be solved by solving the following equation
 2km x( x  x0 ) 2  2km x( x  x0 )
V 

A
C
2
• or
v 2A
x  2 x0 x  x x 
0
2k m
3
2
2
0
x 3  4  10  6 x 2  4  10 12 x  7.552  10 19  0
• How to solve it?
Chang Liu
MASS
UIUC
Solving Third Order Equation ...
• To solve
x 3  ax 2  bx  c  0
• Apply y  x  a / 3
• Use the following definition
 a2
a
ab
p
 b, q  2( )3 
c
3
3
3
3
 p q
Q    
 3   2
A3
• The only real solution is
•
2
q
q
 Q,B  3
 Q
2
2
y  A B
a
x  A B 
3
Chang Liu
MASS
UIUC
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 -0.54 mm.
Chang Liu
MASS
UIUC