Coupled Electromechanics

Coupled ElectromechanicsElectromechanics

G. K. [email protected] EngineeringIndian Institute of ScienceBangalore, INDIABangalore, INDIA

November 2014, for ME 237/NE 211 course in IISc

2 Computing the electrostatic force in a parallel‐plate capacitor

0 = permittivity of free space

lw

g V = applied voltageC = capacitance

2 20 ( )1 12 2c

wlESE CV Vg

Electrostatic co‐energy

2012

cl

ESE wF Vl g

Force in the length direction

2012

1 1

cw

ESE lF Vw g

ESE wl A

Force in the width direction

F i th

G.K. Ananthasuresh, Indian Institute of Science

2 20 02 2

1 12 2

cg

ESE wl AF V Vg g g

Force in the gap direction

3 Computing the electrostatic force in general 3‐D problems

Conductor 1

22 V

Conductor 1Conductor 2

V 11 V

Electric potential = Electric field Charge density = charge per unit areaElectric field =

nF

ˆ21 2

eElectrostatic force = Permittivity of the i te e i ediu

g y g p

Surface normal


intervening mediumIt is a surface force (traction).

4Electric field and iso‐potential lines


5Basics of electrostatics

1 212 122

ˆ4

q qF rr

Coulomb’s law

0

12 122 200 0

4

ˆ ˆlim4 4q

r

Qq QE r rq r r

Electric field due to a single charge

work done P

E dlq

E

Electrical potential (voltage)

Electric field is the negative of the gradient of the potential

0r

E

D E E

Electric field is the negative of the gradient of the potential

Density of electric displacement vector

G ’ l

2

v vS V

D dS dV Q

Gauss’s law

Integral form

Diff i l f


2v vD

Differential form

6Computing the electrostatic force

ˆ ˆn D n E Surface charge density

esf

s n D n E

snE

Surface charge density

Normal component of the electric field

2 2s s

sdF Edq dq dA

electric field

2

2 2

2

s

sdFdA

Force along the surface normal2dA

2

ˆsf n

Electrostatic force on the


2esf n

surface per unit area

7

Computing the electrostatic force (contd.)

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

42 02

On the conductorsIn the intervening medium

Pl t ti l th d t ifi dPlus, 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 conductorsneeds to be meshed along with the interior of the conductors.

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

'')()( dSxxxx

Surfaces

the integral equation form:


This is suited for BEM because only conductors’ boundaries need tobe meshed.

8Static equilibrium of an elastic structure under electrostatic force

Charge distribution

‐‐ ‐ ‐ Charge distribution

causes electrostatic force of attraction between conductors

‐‐

‐ ‐‐ ‐ ‐ ‐‐‐+++

++ + + +

+

‐ ‐ ‐‐‐

Electrostatic force

‐‐

++

++

+

+ +++

‐ ‐ ‐ ‐

V

Electrostatic force deforms conductors

+++

VDeformation of conductors causes charges to re‐2 ˆ1 nFEl i f


distribute02

F eElectrostatic force =

9 Coupled governing equations of electro‐ and elasto‐ statics

conductorsalloffor''

)()( sdSxx 'xxSurfaces

0

2 ˆ21

nf te

σ ineverywhere0

u

te

uufnσ

ononˆ

y

0

Tuuε

εEσ

1

:A self‐consistent solution is needed!


uuε 2

solution is needed!

10Start with 1‐dof lumped model…

V

A0= plate area

= permittivity of free space

201 VAk 2

20

21 VA

k Static equilibrium

2

20

0

2V

xgkx

202 xg kx

x0g A cubic equation!


A cubic equation!

11Lumped 1‐dof modeling of coupled electro‐ and elasto‐ static behavior

1 A 2

201 VA

0g

2

20

0

21 V

xgAkx

kx 202

Vxg

Forces

Three solutions

x

ble

ble

able

0)( PE

energy

202 11 VAkxPE

Potential energyStab

Stab

Unst 0

x

Two stable; one unstable; And, one infeasible

x

tential e 022

Vxg

kxPE

22 )( AVPE


Po

30

20

2

2

)()(

xgAVk

xPE

Use to test stability.

12Pull‐in phenomenon

1V 2V< 3V< xgkVAVkPE 302

20

2 )(0)(

Condition for critical stability

gy

3/0g AgV

xgk

x 0

023

0

02

)(0)(

)(

)(1 0020 gxxgkVAkx

0g

al energ 322 2

0

xVxg

kx

x

Potentia

Agk

V inpull0

3

278

0

P

3/2 Vx


3/2 0g inpullVinpullV

13 With a dielectric layer: pull‐up and hysteresis

3

inpull 0278

dtg

AkV

Dielectric layer027 rA

2

x0g

t0

up pull 0

2

r

dtgAkV

dt x

0gPull‐up voltage is found by

ti th f f i dequating 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


uppullV inpullV V, g

Problems and Hysteresis in Coupled Electromechanics,” presented at the IEEE‐MEMS‐96 Workshop, San Diego, CA, Feb. 11‐15, 1996.

14Pull‐in phenomenon used in a display device

www.iridigm.com (a QualComm acquisition)

Interference‐modulation by electrostatic actuation of vertically moving membranes.


15Iridigm became mirasol…


16 Distributed modeling of the electrostatically actuated beam

Finite element method Finite difference method

V

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

0)(2 2

0

20

4

4

ug

wVdx

udEI

V++ +

++ ++

++ Include the effects of

residual stress as well:V residual stress as well:

0)(2 2

0

20

2

2

04

4

ug

wVdx

udwtdx

udEI


)(2 0 ugdxdx

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

17

Solving the general 3‐D problem

Boundary element method for the integral equation of electrostatics

Finite element method for the differential equation of elastostatics

'')()( dSxxxx

Surfaces

elastostatics

Charge on kth panel

Discretize the boundary surfaces into n panels. eFUK

ipanel k

n

i i

ik xx

daaqp

''

1

Area of kth panel

Potential on kth panel

Assemble to get:2

l k( / ) ˆk k

kq a

f n


qPp qpC

gon panel k 2 kf n

18Solution approaches

Relaxation‐‐ iterate between the elastic and electrostatic domains.‐‐ converges except in the vicinity of pull‐in voltage; but slow.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.p p‐‐ then, re‐compute electrostatic force and internal deformations.

Direct Newton‐‐ compute all derivatives to update charges and deformations.

MM

RUR

UR Residuals in mechanical

and electrical domains

E

M

EE RR

qU

qR

UR

qU and electrical domains


qUFor 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.

19Comb‐drive example

(a) Tang, W. C., Nguyen, C., Judy, M. W., and Howe, R. T., “Electrostatic Comb‐Drive of LateralPolysilicon Resonators,” Sensors and Actuators A, 21 (1), 1990, pp. 328‐331


(b) http://mems.sandia.gov/scripts/images.asp (Sandia National Laboratories, New Mexico, USA)

20Schematic of the comb‐drive


21Lumped mechanical stiffness

3

3

3

33

12 EbhFl

bhFl

EIFl

3 3Ebh Etwk

1212

12 EbhbhEEI

3EbhFk

3 3kl l

3

l2Etwk


3lk

total 3k

l

22Electric potential solution


FEMLAB (old COMSOL) result

23Close‐up of the electric potential

FEMLAB (old COMSOL) result


24Deformation of the comb‐drive


25What about dynamic behavior?

in-pulldynamicV = dynamic pull‐in voltage

0gFrequency =

ergy

0g

tV

L d 1 d f d l

ntial ene

x2

0

20

)(2 xgAVkxxm

Lumped 1‐dof model

Poten 0 )(g

02

04

wVudEIuwt

Beam model

0)(2 2

0

04

ugdxEIuwt

tVtVVVtVVV acacdcdcacdc 22222 sinsin2)sin(


acacdcdcacdc )(

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

26Damping: squeezed film effects

S d fil d iV Squeezed‐film damping

20 AVkxxbxm 02

20

4

4

wVudEIubuwt

Lumped 1‐dof model Beam model

20 )(2 xg

kxxbxm

)(2 20

4 ugdx

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


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”.

27 Modeling squeezed film effects: isothermal Reynolds equation

1)()(

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

(beam,plate, membrane)

),(),(),(12

1),(),( 3 yxpyxgyxpt

yxgyxp

Viscosity of air

For lumped 1‐dof modeling, we have a rigid plate. So, g does not depend on . ),( yx

)(1)()(),( 2233 gggyxp

),(21

12),(),(

12),( 22 yxpgyxpyxpg

tgyxp

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

)(1)( 22

3

ypggyp yAssumed pressure distribution


)(212

ypt x y

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

28Behavior with small displacements

)(

21

12)( 22

3

ypgt

gyp

Linearize around :),( 00 gp

gggppp 00

Also, use non‐dimensional variables: ˆ,ˆ, ggppy

2

2

20

20

2

2

20

20 ˆ

12ˆˆ

12ˆ

ggp

wpg

tgp

wpg

tp

Also, use non dimensional variables:00

,,g

gp

pw

width01212 gwtwt

Separation of spatial and temporal components: teptp )(~),(ˆ22

teggpp

wpg

0

2

2

20

20 ~~

12


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

29Behavior with small displacements (contd.)

nnnn BAppp

wpg cossin~0~~

12 2

2

20

20

020

212pg

w nn

w12 00 pg

Boundary conditions and velocity‐impulse assumption give:

nn nwnpgn 2

220

20

4

,...5,3,1;12

;

n

tn

nenng

xAodd0

0 )sin(4

1 8Force on the plate =

n

tsq

negxwlpdtpwlptf

odd22

0

00

1

00 n

8),(ˆ)(


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

30Finally, getting to lumped approximation…

)(11961196)(33

ssXwlxwlsF

)(

1n1n)(

odd43

040

odd43

04 ssXsg

xsgsF

n

n

n

n

sq

)(1

)(1

196)( 30

4

3

ssXsbssXsg

wlsFsq

For only.1n

11cc

396 wlb 0

20

2 pg

Transfer function for general displacement input!

30

4gb

212 wc

Damping Cut‐off

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


coefficient frequency

cbC 1

31What does it mean mechanically?

396 wlb

k

208wpbk csq

30

4gb

x x0

2gcsq m

Thus, squeezed film effect creates two effects:qViscous damping + “air‐spring”Further analysis indicates that at low frequencies, damping d i t d i i t hi h f i


dominates, and air‐spring at high frequencies.See S. D. Senturia, Microsystems Design, Kluwer, 2001, for details.

32Move up to beam modeling…

)()}({)(1)},({),,( 30

0 tyxptxugtyxptxugtyxp

),,()},({),,(12 0 tyxptxugtyxp

t

0)(),()(),( 20

42/

twVtxudEIdytyxptxuwt

w 0)},({2

),,( 20

42/

2

txugdx

EIdytyxpt

wtw

Note that this is still al h l d 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 di ti i th d i ti d i t t i ti


discretizing the dynamic equation, and integrate in time using the Runge‐Kutta method.

33FD solution of Reynolds equation

3( )12 .( )Pg g P Pt

Solution submitted by N. Sajinu for ME 237 course at IISc.

t223 2 2 2

2 2

3 . .12 12 12

P g P P Pg P P Pg P g P g P Pt x y x x x x y y g t


34

A typical beam’s response with squeezed film effect

The transverse deflection of the mid‐point of a fixed‐fixed beam under (Vdc+Vac) voltage input under the squeezed film effect:

pointmidu pointmid

G.K. Ananthasuresh, Indian Institute of Sciencet

35What about this problem now?

VDiaphragm tV sin

Passive inlet valve Passive outlet valve

rate

Frequency

Flow


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

36Main points

MEMS are systems that tightly integrate many energetic phenomena which makesmany energetic phenomena, which makes their modeling non‐trivial.

Coupled multi‐physics equations need to be Coupled multi‐physics equations need to be solved.

Reduced order lumped “macro” models are Reduced order lumped macro models are useful for design and system‐level simulation


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Coupled Electromechanics