Slide 6.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lecture

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 6.1

Lecture 6Analysis of electrostatically actuated micro devicesSome features of nonlinearly coupled electrostatic and elasto-static and dynamic governing equations of electrostatic MEMS.


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?


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 (capacitance-

based)Some inconveniences-- high voltages for large displacements-- small forces (they move themselves mostly; suited for sensors)-- charging and stiction


Micro-mechanical filters

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.

Electrostatic actuation and sensing is the key to this.


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)


Working principle of mechanical filters

FrequencyT

rans

mis

sion

(d

B)

Tra

nsm

issi

on (

dB

)

Frequency

tF sin

tF sin

tFkxxbxm sin

m

b

k


Schematics of Nguyen’s micro-mechanical filter and a high-Q disk resonators

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.

Side viewContour disk resonator“Free-free” beam resonator

Top view


A bi-directional pumpV

Diaphragm

Passive inlet valve Passive outlet valve

tV sin

Frequency

Flow

rate

R. Zengerle, J. Ulrich, S. Kluge, M. Richter, and A. Richter, “A Bidirectional Silicon Micropump”, Sensors and Actuators, A 50, 1995, pp. 81-86.


Electrostatic comb-drive—the prime mover for MEMS today

anchorShuttlemass

Folded-beam suspension Movingcombs

Fixedcombs

Misaligned parallel-plate capacitor

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.


Computing the electrostatic force in the parallel-plate capacitor

22

022

0

20

20

202

2

1

2

1

2

1

2

1

)(

2

1

2

1

Vg

AV

g

wl

g

EF

Vg

l

w

EF

Vg

w

l

EF

Vg

wlCVE

eg

ew

el

e

l

wg

Electrostatic energy

Force in the length direction

Force in the width direction

Force in the gap direction

0 = permittivity of free spaceV= applied voltageC= capacitance


Computing the electrostatic force in general 3-D problems

Electric potential = Electric field =

n

Fˆ

2

1 2

eElectrostatic force = Dieletric constant of the intervening medium

Charge density = charge per unit area

Surface normal

22 V

Conductor 1Conductor 2

It is a surface force (traction).

11 V


Computing the electrostatic force (contd.)

42 02

On the conductorsIn the intervening medium

Plus, potentials on the conductors are specified.

Governing equations to solve for the charge density in the differential equation form:

This is suited for FEM but sufficient intervening medium also needs to be meshed along with the interior of the conductors.

''

)()( dS

xx

xx

Surfaces

Governing equations to solve for the charge density in the integral equation form:

This is suited for BEM because only conductor boundaries need to be meshed.


Static equilibrium of an elastic structure under electrostatic force

V

+ ++

++

+

++++

+

--

--

--

- -

--

-

-

++ ++++

+

++

++

--- - - - -

--

--

- Charge distribution causes electrostatic force of attraction between conductors

Electrostatic force deforms conductors

Deformation of conductors causes charges to re-distribute

0

2 ˆ

2

1

n

F eElectrostatic force =


Coupled governing equations of electro- and elastostatics

conductorsalloffor''

)()( sdS

xx

xx

Surfaces

0

2 ˆ

2

1

n

f te

T

u

te

uuε

εEσ

uu

fnσ

σ

2

1

:

on

onˆ

in everywhere0

0

A self-consistent solution is needed!


Start with 1-dof lumped model…

V

2

20

0

2

1V

xg

Akx

2

20

0

2

1V

xg

A

kx

x

k

0g

A

0= plate area

= permittivity of free space

Static equilibrium

A cubic equation!


Lumped 1-dof modeling of coupled electro- and elasto- static behavior

2

20

0

2

1V

xg

Akx

kx

2

20

0

2

1V

xg

A

x

Forc

es

Three solutions

Two stable; one unstable; And, one infeasible

x

Pote

nti

al en

erg

y

2

0

02

2

1

2

1V

xg

AkxPE

Potential energyS

tab

le

Sta

ble

Unst

able

0g

0)(

x

PE

30

20

2

2

)(

)(

xg

AVk

x

PE

Use to test stability.


Pull-in phenomenonPote

nti

al en

erg

y

3/0g

1V 2V< 3V<

A

xgkV

xg

AVk

x

PE

0

302

30

20

2

2 )(0

)(

)(

32

)(

2

1 0022

0

0 gx

xgkV

xg

Akx

Condition for critical stability

3/2 0g inpullV

x

x

A

gkV inpull

0

3

27

80

inpullV

0g


With a dielectric layer: pull-up and hysteresis

x0g

dt

Dielectric layer

3

0inpull 027

8

r

dtgA

kV

uppullV

2

0up pull 0

2

r

dtgA

kV

inpullV

x

V

0gPull-up voltage is found by equating the forces of spring and electrostatics at .

0gx 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-MEMS-96 Workshop, San Diego, CA, Feb. 11-15, 1996.


V

+ + + + + + + + +

V++ +

++ ++

++

Distributed modeling of the electrostatically actuated beam

0)(2 2

0

20

4

4

ug

wV

dx

udEI

Include the effects of residual stress as well:

0)(2 2

0

20

2

2

04

4

ug

wV

dx

udwt

dx

udEI

FEM or FDM could be used to solve the nonlinear equation:

Finite element method Finite difference method

A correction due to fringing field (edge and corner effects) is also included.


Solving the general 3-D problem

Boundary element method for the integral equation of electrostatics

''

)()( dS

xx

xx

Surfaces

ipanel k

n

i i

ik xx

da

a

qp

'

'

1

Potential on kth panel

Charge on kth panel

Area of kth panel

Discretize the boundary surfaces into n panels.

qPp qpC

Assemble to get:

Finite element method for the differential equation of elastostatics

kkk aqnf ˆ

)/(

0

2

kpanelon

eFUK


Solution approachesRelaxation

-- 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.

E

M

EE

MM

R

R

q

U

q

R

U

Rq

R

U

R 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.


What about dynamic behavior?Pote

nti

al en

erg

y

in-pulldynamicV = dynamic pull-in voltage

x

0g

tV

20

20

)(2 xg

AVkxxm

0)(2 2

0

20

4

4

ug

wV

dx

udEIuwt

Lumped 1-dof model

Beam model

Frequency =

tVtVVVtVVV acacdcdcacdc 22222 sinsin2)sin(

Will contain a term!2So, the response will show two resonance at two frequencies.


Damping: squeezed film effects

V Squeezed-film damping

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”.

20

20

)(2 xg

AVkxxbxm

0)(2 2

0

20

4

4

ug

wV

dx

udEIubuwt

Lumped 1-dof model Beam model

How do you obtain ?b


Modeling squeezed film effects: isothermal Reynolds equation

),(),(),(12

1),(),( 3 yxpyxgyxpt

yxgyxp

Pressure distribution in the 2-D x-y plane Gap varies in the x-y plane for a deformable

structure (beam,plate, membrane)

Viscosity of air

For lumped 1-dof modeling, we have a rigid plate. So, gdoes not depend on . ),( yx

),(2

1

12),(),(

12

),( 2233

yxpg

yxpyxpg

t

gyxp

Assume further that pressure distribution is the same along the length of the plate so that it becomes a one dimensional problem.

)(2

1

12

)( 223

ypg

t

gyp

xy

Assumed pressure distribution

S. D. Senturia, Microsystems Design, Kluwer, 2001.


Behavior with small displacements

)(2

1

12

)( 223

ypg

t

gyp

Linearize around :),( 00 gp

gggppp 00

02

2

20

20

2

2

20

20 ˆ

12

ˆˆ

12

ˆ

g

gp

w

pg

t

gp

w

pg

t

p

Separation of spatial and temporal components:teptp )(~),(ˆ

teg

gp

p

w

pg

0

2

2

20

20 ~

~

12

Also, use non-dimensional variables:00

ˆ,ˆ,g

gg

p

pp

w

y

width

Assume a sudden velocity impulse to the plate. Then, for t > 0, this term is zero. (with displacement )0xx


Behavior with small displacements (contd.)

nnnn BAppp

w

pgcossin~0~

~

12 2

2

20

20

020

212

pg

w nn

Boundary conditions and velocity-impulse assumption give:

n

tn

nn

nenng

xA

nw

npgn

odd0

0

2

220

20

)sin(4

,...5,3,1;12

;

Force on the plate = n

tsq

neg

xwlpdtpwlptf

odd22

0

00

1

00 n

8),(ˆ)(

Take the Laplace transform (continued on the next slide).


Finally, getting to lumped approximation…

)(1

1

n

196

1

1

n

196)(

odd43

04

3

0odd

430

4

3

ssXsg

wlx

sg

wlsF

n

n

n

n

sq

)(1

)(1

196)(

30

4

3

ssXs

bssX

sg

wlsF

cc

sq

For only.1n

30

4

396

g

wlb

20

20

2

12 w

pgc

Damping coefficient

Cut-off frequency

Transfer function for general displacement input!

bR

cbC 1

)(),( sXtx )(),( sFtf sqsq


What does it mean mechanically?

x

k

x0

208

g

wpbk csq

m

30

4

396

g

wlb

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.


Move up to beam modeling…

),,()},({),,(12

1)},({),,( 30

0 tyxptxugtyxpt

txugtyxp

0)},({2

)(),(),,(

),(2

0

20

4

42/

2/2

txug

twV

dx

txudEIdytyxp

t

txuwt

w

w

Note that this is still a parallel-plate approxmation!

Solve these two coupled equations.

An approachUse FDM for pressure equation and FEM or FDM for discretizing the dynamic equation, and integrate in time using the Runge-Kutta method.


A typical responseThe transverse deflection of the mid-point of a fixed-fixed beam under (Vdc+Vac) voltage input under the squeezed film effect:

pointmidu

t


What about this problem now?

VDiaphragm

Passive inlet valve Passive outlet valve

tV sin

Frequency

Flow

rate

A problem involving three energy domains that are strongly coupled. Furthermore, the fluids part is non-trivial.


Shape optimization: example of electrostatic comb-drive

Straight-finger comb-drive

Variable force comb-drives with curved stationary fingers

Linear

Quadratic

CubicNeed to compensate the non-linearities caused by the folded-beam suspension.

(Figures provided by W. Ye)

(Made with Cornell’s SCREAM process)

(Ye and Mukherjee, Cornell)

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.


Synthesis with electrostatic actuation

(Ye and Mukherjee, Cornell)

(Figures provided by W. Ye)

Shape-optimized comb-fingers to compensate suspension’s nonlinearities.

(Made with Cornell’s SCREAM process) Ye and Mukherjee used BEM for discretizing both the electrostatic and elasto-static governing equations.


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…


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.

Documents

Slide 6.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lecture