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Mechanics of Micro Structures
Chang Liu
Micro Actuators, Sensors, Systems Group
University of Illinois at Urbana-Champaign
Chang Liu
MASS
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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MASS
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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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Calculate spring constant
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Vertical Translational Plates
12 EI Ewt 3
k 3  3
l
l
k
12 EI Ewt 3
 3
l3
l
Ewt 3
(a)k  2 3
l
Ewt 3
(b)k  4 3
l
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Torsional Deflections
• Pure Torsion; Every cross section of the bar is identical
 max
Tr0

J
1
J  r04
2
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Intrinsic Stress
• Many thin film materials experience internal stress even when
they are under room temperature and zero external loading
conditions
• In many cases related to MEMS structures, the intrinsic stress
results from the temperature difference during deposition and
use
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Intrinsic Stress
The flatness of the membrane is
guaranteed when the membrane
material is under tensile stress
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Intrinsic Stress
• There are three strategies for minimizing undesirable intrinsic
bending
– Use materials that inherently have zero or very low intrinsic stress
– For materials whose intrinsic stress depends on material processing
parameters, fine tune the stress by calibrating and controlling
deposition conditions
– Use multiple-layered structures to compensate for stress-induced
bending
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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 cross section
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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Electrostatic Sensors and Actuators
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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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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
•
1
Stored energy U  CV
2
2
1Q 2

2 C
• Force is derivative of energy with F  U  1 C V
respect to pertinent dimensional
d
2 d
variable
• Plug in the expression for capacitor C  Q  A
Q
d
A
•
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2
d
U
1 A

V
We arrive at the expression for force F 
2
d
2d
2
1 CV

2 d
2
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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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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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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 ) Eq.(*)
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,
A
(2 /3)d
2 x0
km
Vp 
.
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
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<(1/3)d0
Counter
capacitor
plate
Capacitor
plate
Actuation
electrode
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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 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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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
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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.
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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?
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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
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)
Chang Liu
MASS
UIUC
Chang Liu
MASS
UIUC
Chang Liu
MASS
UIUC
BU Adaptive Micro Mirrors
Chang Liu
MASS
UIUC
Optical Micro Switches
• Texas Instrument DLP
Chang Liu
• 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.
MASS
UIUC
“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
Chang Liu
MASS
UIUC
Digital Micromirror Device (DMD)
Mirror
-10 deg
Mirror
+10 deg
Hinge
Yoke
Chang Liu
CMOS
Substrate
MASS
UIUC
Perspective View of Lateral Comb Drive
Chang Liu
MASS
UIUC
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.
Chang Liu
– 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.
MASS
UIUC
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 (
Chang Liu
 0lt
Cf )
x0  x
 0lt
x0  x
Cf )
MASS
UIUC
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.
Chang Liu
MASS
UIUC
Sandia Electrostatically driven gears
- translating linear motion into continuous rotary motion
Lateral comb drive banks
Mechanical
springs
Gear train
Optical shutter
Chang Liu
• http://www.mdl.sandia.gov/mic
romachine/images11.html
MASS
UIUC
Sandia Gears
• Use five layer
polysilicon to increase
the thickness t in lateral
comb drive actuators.
Mechanical springs
Position
limiter
Chang Liu
MASS
UIUC
More Sophisticated Micro Gears
Chang Liu
MASS
UIUC
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.
Chang Liu
MASS
UIUC
Three Phase Motor Operation Principle
Chang Liu
MASS
UIUC
Starting Position -> Apply voltage to group A
electrodes
Chang Liu
MASS
UIUC
Motor tooth aligned to A -> Apply voltage to Group
C electrodes
Chang Liu
MASS
UIUC
Motor tooth aligned to C -> Apply voltage to Group
B electrodes
Chang Liu
MASS
UIUC
Motor tooth aligned to B -> Apply voltage to Group
A electrodes
Chang Liu
MASS
UIUC
Motor tooth aligned to A -> Apply voltage to Group
C electrodes
Chang Liu
MASS
UIUC
Example of High Aspect Ratio Structures
Chang Liu
MASS
UIUC
Some variations
•
•
•
•
Chang Liu
Large angle
Long distance
Low voltage
Linear movement
MASS
UIUC
1x4 Optical Switch
•
Chang Liu
John Grade and Hal Jerman, “A large deflection electrostatic actuator for
optical switching applications”, IEEE S&A Workshop, 2000, p. 97.
MASS
UIUC
Actuators that Use Fringe Field - Micro Mirrors
with Large Displacement Angle
Torsional mechanical spring
Chang Liu
R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &Act
Workshop, Hilton Head Island, 2000.
MASS
UIUC
Curled Hinge Comb Drives
Chang Liu
MASS
UIUC
Other Parallel Plate Capacitor - Scratch Drive
Actuator
• Mechanism for realizing
continuous long range
movement.
Chang Liu
Scratch drive invented by H. Fujita of Tokyo University.
The motor shown above was made by U. of Colorado, Victor Bright.
MASS
UIUC