Lecture 6
Analysis of electrostatically
actuated micro devices
Some features of nonlinearly coupled electrostatic and
elasto-static and dynamic governing equations of
electrostatic MEMS.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.1
Contents
• Why is electrostatic actuation popular in
MEMS?
• Nonlinearity explained with examples
• Computing the electrostatic force
– Parallel-plate capacitor
– General
• Effects of nonlinearity
– Pull-in, pull-up, hysteresis, etc.
• Squeezed-film damping
– Iso-thermal Reynolds equation
• Design challenges
– Shape optimization
– Topology optimization?
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.2
Why is electrostatic actuation
popular in MEMS?
•
•
•
•
•
•
Ease of fabrication
Ease of actuation
Energy-efficient
High frequency (MHz and even GHz)
Scalability
Easy sensing mechanism (capacitancebased)
Some inconveniences
-- high voltages for large displacements
-- small forces (they move themselves mostly; suited
for sensors)
-- charging and stiction
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.3
Micro-mechanical filters
Electrostatic actuation and sensing is the key to this.
C.T.-C. Nguyen, “Micromechanical Components for Miniaturized Low-Power Communications,” Proc.
1999 IEEE MTT-S Int. Microwave Symposium, RF-MEMS Workshop, Anaheim, CA, June 18, 1999, pp.
48-77.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.4
Why micro-mechanical filters?
Why mechanical filters?
Narrow bandwidth (high selectivity)
Low loss (high Q, 10,000 to 25,000)
Good stability with temperature variation
Passive (no power and clock required)
Why micromechanical filters?
Better performance and cost
Low power consumption
Smaller size (more applications: e.g., cellular phones)
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.5
k
m
b
F sin t
F sin t
Transmission (dB)
mx  bx  kx  F sin t
Transmission (dB)
Working principle of mechanical
filters
Frequency

Frequency

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.6
Schematics of Nguyen’s micro-mechanical
filter and a high-Q disk resonators
“Free-free” beam resonator
Contour disk resonator
Side view
Top view
J. R. Clark, W.-T. Hsu, and C.T.-C. Nguyen, “Measurement Techniques for Capacitively-transduced
VHF-to-UHF Micromechanical Resonators, Proc. of Transducers, 2001, Munich, June 10-14, 2001, pp.
1118-1121.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.7
A bi-directional pump
V sin t
V
Diaphragm
Flow rate
Passive inlet valve
Passive outlet valve
Frequency 
R. Zengerle, J. Ulrich, S. Kluge, M. Richter, and A. Richter, “A Bidirectional Silicon Micropump”,
Sensors and Actuators, A 50, 1995, pp. 81-86.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.8
Electrostatic comb-drive—the
prime mover for MEMS today
Folded-beam suspension
Moving
combs
Shuttle
mass
Misaligned
parallel-plate
capacitor
anchor
Fixed
combs
W. C. Tang, T.-C. H. Nguyen, M. W. Judy and R. T. Howe, “Electrostatic-comb drive of lateral polysilicon
resonators,” Sensors and Actuators, Vol. A 21-23, pp. 328 – 381, 1990.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.9
Computing the electrostatic force
in the parallel-plate capacitor
g
w
l
 0 = permittivity of free space
V = applied voltage
C = capacitance
1
1  0 ( wl ) 2
2
Electrostatic energy
Ee  CV 
V
2
2 g
E
1  0w 2
Force in the length direction
Fl  e 
V
l
2 g
Ee 1  0l 2
Force in the width direction
Fw 

V
w 2 g
Ee
1  0 wl 2
1 0 A 2
Force in the gap
Fg 

V


V
direction
g
2 g2
2 g2
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.10
Computing the electrostatic force
in general 3-D problems
Conductor 1
Conductor 2
2  V2
1  V1
Electric potential =
Electric field = 

Charge density = charge per unit area
1  2nˆ
Electrostatic force = Fe 
2 
It is a surface force (traction).
Surface normal
Dieletric constant of
the intervening medium
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.11
Computing the electrostatic force
(contd.)
Governing equations to solve for the charge density in
the differential equation form:
 2  4
On the conductors
 2  0
In the intervening medium
Plus, potentials on the conductors are specified.
This is suited for FEM but sufficient intervening medium also
needs to be meshed along with the interior of the conductors.
Governing equations to solve for the charge density in
the integral equation form:
 ( x)
 ( x)  
dS '
Surfaces x  x '
This is suited for BEM because only conductor boundaries need
to be meshed.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.12
Static equilibrium of an elastic
structure under electrostatic force
-
- -- +
+
+
- -- +
+
+++ - +
++
- - ++
-++
+
++
+
+
V
1  2nˆ
Electrostatic force = Fe 
2 0
Charge distribution
causes electrostatic
force of attraction
between conductors
-
Electrostatic force
deforms conductors
Deformation of
conductors causes
charges to redistribute
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.13
Coupled governing equations of
electro- and elasto- statics
 ( x) 

Surfaces
 ( x)
x  x'
dS ' for s of all conductors
1  2nˆ
f te 
2 0
  σ  0 everywhere in 
σnˆ  f te on 
u  u 0 on  u
A self-consistent
solution is needed!
σ  E:ε
1
ε   u  u T
2


Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.14
Start with 1-dof lumped model…
V
A = plate area
 0 = permittivity of free space
k
kx
x
g0
1 0 A
2
V
2 g 0  x 2
Static equilibrium
1 0 A
2
kx 
V
2 g 0  x 2
A cubic equation!
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.15
Lumped 1-dof modeling of coupled
electro- and elasto- static behavior
Forces
g0
kx
1 0 A
2
V
2 g 0  x 2
1 0 A
2
kx 
V
2 g 0  x 2
Three solutions
Unstable
Stable
Stable
Potential energy
x
Potential energy
x
PE 
 ( PE )
0
x
1 2 1 0 A
kx 
V2
2
2 g 0  x 
Two stable; one unstable; And, one infeasible
 0 AV 2 to test stability.
 2 ( PE )
k
Use
2
x
( g 0  x) 3
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.16
Pull-in phenomenon
Condition for critical stability
V1 < V2 < V3
Potential energy
g0 / 3
g0
3
 0 AV 2
k
(
g

x
)
 2 ( PE )
2
0

k


0

V

x 2
( g 0  x) 3
0 A
k ( g 0  x)
g0
1 0 A
2
kx 
V 
x
2
2 g 0  x 
2
3
x
Vpullin 
8 k g 30
27  0 A
x
2 g0 / 3
Vpull
in
Vpull
in
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.17
With a dielectric layer: pull-up and
hysteresis
Vpull in
td 
8k 
 g 0  

27 0 A 
r 
3
Dielectric layer
x
Vpull up
g0
td
Pull-up voltage is found by
equating the forces of spring
and electrostatics at x  g 0 .
Gilbert, J. R., Ananthasuresh, G. K., and Senturia,
S. D., “3-D Modeling and Simulation of Contact
Problems and Hysteresis in Coupled
Electromechanics,” presented at the IEEE-MEMS96 Workshop, San Diego, CA, Feb. 11-15, 1996.
g0
 td
2k

g 0 
0 A  r



2
x
Vpullup Vpullin V
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.18
Distributed modeling of the
electrostatically actuated beam
Finite element method
FEM or FDM could be used to
solve the nonlinear equation:
+++++++++
V
 0 wV 2
d 4u
EI 4 
0
2
dx
2( g 0  u )
+
V
Finite difference method
+ + +++ + +
+
Include the effects of
residual stress as well:
 0 wV 2
d 4u
d 2u
EI 4   0 wt 2 
0
2
dx
dx
2( g 0  u )
A correction due to fringing field (edge and corner effects) is also included.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.19
Solving the general 3-D problem
Boundary element method for the
integral equation of electrostatics
 ( x) 
 ( x)

Surfaces
x  x'
Finite element method
for the differential
equation of elastostatics
dS '
Discretize the boundary surfaces into n panels.
K U  Fe 
Charge on kth panel
n
qi
da'
pk  

i 1 ai paneli x ' xk
Area of kth panel
Potential on kth panel
f on panel k 
Assemble to get:
p  Pq
( qk / ak ) 2
 Cp  q
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
0
nˆ k
Slide 6.20
Solution approaches
Relaxation
-- iterate between the elastic and electrostatic domains.
-- converges except in the vicinity of pull-in voltage; but slow.
Surface Newton
-- compute sensitivities of surface nodes.
-- use a Newton step to update those nodes.
-- then, re-compute electrostatic force and internal deformations.
Direct Newton
-- compute all derivatives to update charges and deformations.
 R M
 U

 R E
 U
R M
q
R E
q

 U 
R M 
    

  q 
RE 

Residuals in mechanical
and electrical domains
For example, see: G. Li and N. R. Aluru, “Linear, non-linear, and mixed-regime analysis of
electrostatic MEMS,” Sensors and Actuators, A 91, 2001, pp. 279-291, and references therein.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.21
What about dynamic behavior?
Vdynamic pull-in = dynamic pull-in voltage
Frequency =

Potential energy
g0
V
t
Lumped 1-dof model
x
mx  kx 
 0 AV 2
2( g 0  x) 2
Beam model
 0 wV 2
d 4u
 wt u  EI 4 
0
2
dx
2( g 0  u )
V 2  (Vdc  Vac sin t ) 2  Vdc2  2VdcVac sin t  Vac2 sin 2 t
Will contain a
So, the response will show two resonance at two frequencies.
2 term!
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.22
Damping: squeezed film effects
Squeezed-film damping
V
Lumped 1-dof model
mx  bx  kx 
 0 AV
Beam model
2
2( g 0  x) 2
 0 wV 2
d 4u
 wt u  bu  EI 4 
0
2
dx
2( g 0  u )
How do you obtain b ?
Use isothermal, compressible, narrow gap Reynolds equation to model
the film of air beneath the beam/plate/membrane.
It is widely used in lubrication theory.
By analyzing this equation, we can extract the essence of damping as a
lumped parameter – the so called “macromodeling”.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.23
Modeling squeezed film effects:
isothermal Reynolds equation
Pressure distribution in
the 2-D x-y plane
Gap varies in the x-y plane for a deformable
structure (beam,plate, membrane)
  p ( x, y ) g ( x, y ) 
1

  p( x, y ) g ( x, y )3 p( x, y )
t
12


Viscosity of air
For lumped 1-dof modeling, we have a rigid plate. So,
g
does not depend on ( x,
 p( x, y) g  g 3
g3


   p( x, y) p( x, y)  
t
12
12
y ).
1 2 2

  p ( x, y) 
2

Assume further that pressure distribution is the same along the length of the
plate so that it becomes a one dimensional problem.
  p( y ) g  g


t
12
3
1 2 2


p
(
y
)


2

Assumed pressure distribution
x
y
S. D. Senturia, Microsystems Design, Kluwer, 2001.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.24
Behavior with small displacements
 p( y ) g  g 3

Linearize
t
12
p  p0  p
1 2 2


p
(
y
)

 around ( p0 , g 0 ):
2

g  g 0  g
y
p
g
, gˆ 
Also, use non-dimensional variables:   , pˆ 
w
p0
g0
g 02 p0  2 pˆ gˆ
g 02 p0  2 pˆ g
pˆ





width
2
2
2
2
t 12 w 
t 12 w 
g0
ˆ ( , t )  ~
p( ) e t
Separation of spatial and temporal components: p
g 02 p0  2 ~
p
g
~

 p  
2
2
12 w 
g 0e t
Assume a sudden velocity impulse to the plate. Then, for t > 0, this term is zero.
(with displacement x  x0)
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.25
Behavior with small displacements (contd.)




2
g 02 p0  2 ~
p
12

w
n
~
~
  p  0  p  An sin  n   Bn cos  n   n 
2
2
12 w 
g 02 p0
Boundary conditions and velocity-impulse assumption give:
g 02 p0 n 2 2
 n  n ;  n 
; n  1,3,5,...
2
12 w 
x0
4
 n t
An  
sin(
n

)
e

g 0 odd n n
1
Force on the plate =
x0
8  nt
f sq (t )  p0 wl  pˆ (t ,  ) d   p0 wl
 2 2e
g 0 odd n n 
0
Take the Laplace transform (continued on the next slide).
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.26
Finally, getting to lumped approximation…



3 
96 l w 
1
1

Fsq ( s ) 
4 3    4
 g 0  odd n n 1  s

n


96 l w3 1
Fsq ( s) 
 4 g 03 1  s
sX ( s) 
c
96 l w3
b
 4 g 03
Damping
coefficient


3
96

l
w

 x 
4 3
 0

g0



b
1
s
sX ( s )
c
 2 g 02 p0
c 
12 w2
Cut-off
frequency




1
1

  4
 odd n  n 1  s
n




sX ( s )



For n  1 only.
Transfer function for general
displacement input!
Rb
x(t ), X ( s )
C 1
f sq (t ), Fsq ( s )
bc
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.27
What does it mean mechanically?
96 l w3
b
 4 g 03
k
x
m
8wp0
k sq  bc  2
 g0
x
Thus, squeezed film effect creates two effects:
Viscous damping + “air-spring”
Further analysis indicates that at low frequencies,
damping dominates, and air-spring at high frequencies.
See S. D. Senturia, Microsystems Design, Kluwer, 2001, for details.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.28
Move up to beam modeling…
  p( x, y, t ) {g 0  u ( x, t )}
1

  p ( x, y, t ) {g 0  u ( x, t )}3 p ( x, y, t )
t
12


 0 wV 2 (t )
u ( x, t ) w / 2
d 4 u ( x, t )
 wt
  p( x, y, t ) dy  EI

0
2
4
2
t
dx
2{g 0  u ( x, t )}
w / 2
Solve these two coupled equations.
Note that this is still
a parallel-plate
approxmation!
An approach
Use FDM for pressure equation and FEM or FDM for
discretizing the dynamic equation, and integrate in time
using the Runge-Kutta method.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.29
A typical response
The transverse deflection of the mid-point of a fixedfixed beam under (Vdc+Vac) voltage input under the
squeezed film effect:
umid  point
t
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.30
What about this problem now?
V sin t
V
Diaphragm
Flow rate
Passive inlet valve
Passive outlet valve
Frequency 
A problem involving three energy domains that are strongly coupled.
Furthermore, the fluids part is non-trivial.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.31
Shape optimization: example of
electrostatic comb-drive
Straight-finger comb-drive
(Ye and Mukherjee, Cornell)
Linear
Quadratic
(Made with Cornell’s SCREAM process)
Need to compensate the nonlinearities caused by the foldedbeam suspension.
Cubic
Variable force comb-drives with curved stationary fingers
W. Ye and S. Mukherjee, “Optimal design of three-dimensional MEMS with applications to electrostatic
comb drives,” International Journal for Numerical Methods in Engineering, Vol. 45, pp. 175-194, 1999.
(Figures
provided
W. Ye)A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Stiff Structures, Compliant
Mechanisms,
andby
MEMS:
Slide 6.32
Synthesis with electrostatic
actuation
Shape-optimized comb-fingers to compensate suspension’s nonlinearities.
(Ye and Mukherjee, Cornell)
(Made with Cornell’s SCREAM process)
Ye and Mukherjee used BEM for
discretizing both the electrostatic and
elasto-static governing equations.
(Figures
provided
W. Ye)A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Stiff Structures, Compliant
Mechanisms,
andby
MEMS:
Slide 6.33
How about topology optimization?
Introducing new holes (i.e., topology variation) certainly
helps elastic behavior but it complicates the
electrostatics problem.
To use the “smoothening” (between 0 and 1) approach,
every spatial point should be able to assume the states
of empty space, a conductor, or a dielectric.
Fringing field effect becomes harder to deal with as
new holes get introduced.
Can be done but there are issues to be resolved…
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.34
Main points
• Electrostatic force is THE most widely
used actuation in MEMS – for many good
reasons.
• Very interesting nonlinear behaviors.
• Analysis of couple electrostatic and
elastostatics (and dynamics) is non-trivial.
• Squeezed film effect causes damping as
well as air-spring stiffness.
• Shape optimization makes more sense but
topology optimization can certainly be
attempted.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 6.35