EEL5225: Principles of MEMS Transducers (Fall 2003)1

EEL5225: Principles of MEMS Transducers (Fall 2003)Instructor: Dr. Hui-Kai Xie

Transducers

Today:ElectrostaticCapacitive

Reading: Senturia, Chapter 6, pp. 125-138

Last lecturePiezoresistivePressure sensor

Lecture 22 by H.K. Xie 10/15/2003

EEL5225: Principles of MEMS Transducers (Fall 2003)2

Capacitive Transducers

Electrostatic TransducerSensorActuator

AdvantagesReciprocal

– sensor and actuator in same deviceNegligible temperature dependenceHigh accuracy

ChallengesSmall signal magnitudeEffect of parasitic capacitancePotential undesired electrostatic actuation

MEMS Applicationsaccelerometersgyroscopesactuatorsvoltage controlled capacitance

EEL5225: Principles of MEMS Transducers (Fall 2003)3

Capacitive Transducers

Geometrical configurationsParallel plate

– Vertical– Parallel

Interdigitated comb finger

– Transverse comb– Longitudinal comb

– Vertical comb

Anchor

EEL5225: Principles of MEMS Transducers (Fall 2003)4

Parallel Plate

platesof areaS

x

fixedplates

x=0

x=x0

movableplates

x0 ∆x

1 1

0 0 00

0 0 0 0

00

0

The capacitance can be expressed as:

( ) ( )( ) 1 1 ,( )

where is the capacitance at rest,

: gap at rest, and x(t): gap change.

S S S x t x tC t Cx x x t x x x

SCx

x

ε ε ε

ε

− − ∆ ∆

= = = − = − − ∆

=

∆

EEL5225: Principles of MEMS Transducers (Fall 2003)5

Capacitive Transducers

0 0

The voltage, ( ), is related to the charge on the parallelplate of the capacitor, ( ), through the capacitance, ( ).

( ) ( ) ( )( ) 1 ( )

( ) "behavior at rest" + "electromechanical

V tQ t C t

Q t Q t x tV tC t C x

V t

∆= = −

= coupling"

2* 2

0 0 0 0

By first principles, we find the electrostatic force frompotential energy stored in this capacitor:

12 2

Q QV V

P PQ QW QdV VCdV CV W VdQ dQC C

= = = = = = =∫ ∫ ∫ ∫

EEL5225: Principles of MEMS Transducers (Fall 2003)6

Parallel PlateCharging Capacitor at Fixed Gap

Q2 2

0 0 2 2

Q Q

PQ Q Q xW VdQ dQC C Sε

= = = =∫ ∫ CV

2 2 2*

2 2 2PQ CV SVWC x

ε= = =

Lifting up one electrode at Fixed Charge2

( , )2

PW QF x Qx Sε

∂= =

∂0

( )x

PW F x dx= ∫

Electrostatic force:

Note: Electrostatic force always tries to narrow the gap.

+QF(x)

x-Q

FE(x)2

( , )2

PE

W QF F x Qx Sε

∂= = =

∂

EEL5225: Principles of MEMS Transducers (Fall 2003)7

Parallel Plate

Lifting up one electrode at Fixed Voltage* ( , ) ( , )W V x QV W Q x= −

Electrostatic force:

Note: Electrostatic force always tries to narrow the gap.

+VF(x)

x0

* ( , ) ( , )where ( , )dW V x QdV VdQ dW Q x

dW Q x VdQ Fdx⇒ = + −

= + FE(x)

* ( , ) ( )dW V x QdV F x dx⇒ = −* ( , )( )

V

W V xF xx

∂⇒ = −

∂

*21( , )

2P

EW dCF F x V Vx dx

∂= = − = −

∂

EEL5225: Principles of MEMS Transducers (Fall 2003)8

Parallel Plate

x

fixedplatex=0

x=x0

movableplate

Cm =1/k

The sum of the mechical and electrical forces is0 M EF F F= + =

EEL5225: Principles of MEMS Transducers (Fall 2003)9

Electrostatic Spring Softening

2 22

22 20 02

00

The electrostatic force opposes motion in the x-direction as follows:

212 2

2 1E

S SV SV xF Vx x xxx

x

ε ε ε ∆= − = − ≈ − +

∆−

If ∆x << x0

,0,0 ,0

0 0

22or 1 EE E E

FxF F F xx x

∆≈ − + = − − ∆

Electrostatic spring softening effect,0

,00

,0,0

0

2

2or

Etotal E M E m

Etotal E m

FF F F F x k x

xF

F F k xx

= + = − − ∆ + ∆

− + − ∆

2

,03

0 0

2 Ee

F SVkx x

ε= − = −

Equivalent electrostatic spring constant:

EEL5225: Principles of MEMS Transducers (Fall 2003)10

Pull-In

( )2

022net E m mSVF F k x k x xx

ε = + ∆ = − −

0 0Example: 1 , 1 , 1 / , 0.54

E

PI

x um C pFk N m V V

= == =

FM

FEFE,0

2

3

Consider the effect of a small perturbation in the gapspacing, x+ x, on the net force, F:

or netnet net m

V

F SVF x F k xx x

δ δ

εδ δ δ δ

⇒

∂ = = − ∂

∆x

net

2 2

,min3 3

F must oppose x to avoid collapse (pull-in),

which requires: or m mSV SVk kx x

δ δ

ε ε

⇒

> =

( )2

,min 0 min2max

Thus, 02net mSVF k x xx

ε ⇒ = = − −

min 0

30

0 PI

2Then we obtain = 3

82So, pull-in occurs at at which V3 27PI

x x

kxx xSε

⇒

⇒ = =

EEL5225: Principles of MEMS Transducers (Fall 2003)11

Capacitive TransducersPosition Sensing

ac input voltage– parasitic electrostatic force

capacitive dividerneed to match Cr to C0 to minimize offsetoutput proportional to ∆x

00 0

0 0 0

0

0 0 0out 0

0 0 0 0 0

Ssense capacitor= 1x

If , then we have

V if 1.2 2 2

S

ref

ii i

CxC C C xx x x

C CC x x x x VxV V x xC C x x x x x

ε ∆= ≅ + = + ∆ − ∆

=

∆ ∆ ∆≅ = ≅ ⋅ ∆

+ ∆ + ∆

Vi

CS Vout

CrBuffer

-Vi

( )out

The output of the capacitor divider is:

V 2s r si i i

s r s r

C C CV V VC C C C

−= − + =

+ +

EEL5225: Principles of MEMS Transducers (Fall 2003)12

Capacitive Sensor

Transverse comb Flexture

Anchor

Fixed Plates

Ref. Analog Devices ADXL-50

EEL5225: Principles of MEMS Transducers (Fall 2003)13

Capacitive Sensor

Transverse comb

Thickness=t

LCs1 Cs2

S1 S2

10

20

where C and C are given by:

S fringe

S fringe

LtC N Cx x

LtC N Cx x

ε

ε

= + +

= + −

x0+x

EEL5225: Principles of MEMS Transducers (Fall 2003)14

Capacitive Sensor

Transverse comb for senseVout

Vi -Vi

Cs1

1 0

2 0

x=0 00

0

0

0

For small displacements,

|

|

where C|

S x

S x

fringe

out i

CC C xxCC C xx

NLtC Cx

CC sensitivityx x

xV Vx

ε

=

=

∂≅ −

∂∂

≅ +∂

= = +

∂= ≅

∂

⇒ ≅

Cs2

Differential Capacitive Bridge

EEL5225: Principles of MEMS Transducers (Fall 2003)15

Electrostatic Actuator

Transverse comb for actuation

Differential force (x=0)

( ) ( )

( ) ( )

1 2

2 20 0

2 200 0

0

0 0

0

1212

2

Differential force is proportional to voltage, .

x x

x x

x

x

F F FdC V V V VdxC V V V Vx

C V Vx

V

∆ = −

= − − − − −

≅ − − − − −

=

x

+V0 -V0+Vx

F1 F2

∆F

EEL5225: Principles of MEMS Transducers (Fall 2003)16

Electrostatic Actuator

Electrostatic spring (Vx=0)

1 2

2 220

0 0

20 0

20

( )

1 12

2

eldk F Fdx

SVddx x x x x

C Vx

ε

= −

= − + −

≅ −

x

+V0 -V0Vx=0

F1 F2

Electrostatic Softening Effect

EEL5225: Principles of MEMS Transducers (Fall 2003)17

Capacitive Transducers

Lateral comb

x00

2 20

2

( )

( )12 2

N ote: non-linear w ith V .2

N ote that the spring constant, 0 !

p

pE

Ee

t x xCd

t x xW CV Vd

W tF Vx d

Fkx

ε

ε

ε

+=

+= =

∂= =

∂∂

= =∂

d

No electrostatic softening effect for longitudinal actuation!

( )22 2

2 2 2

If sin

( sin ) ( 2 sin sin )2 2

( 0.5 2 sin 0.5 cos 2 )2

dc ac

dc ac dc dc ac ac

dc ac dc ac ac

V V V tt tF V V t V V V t V td dt V V V V t V td

ωε εω ω ω

ε ω ω

= +

= + = + +

= + + −Second harmonic

x

EEL5225: Principles of MEMS Transducers (Fall 2003)18

Vertical Comb

Z-axis sensingX-axis sensing

stator statorrotor

z

xy

C1 C2

Vm+ Vm-

Vs

C2

C1 Vm+

Vm-

C1 ≠ C2 (at zero displacement)C1 = C2 (at zero displacement)

EEL5225: Principles of MEMS Transducers (Fall 2003)19

Vertical Comb

Maxwell 2D Field Simulation

C1

C2

C2-C1C2+C1

Z (µm) Z (µm)

Cap

acit

ance

(aF

/fin

ger/

mm

)

Nor

mal

ized

dif

f. c

apac

itan

ce

C1 and C2 have high nonlinearityHowever, their normalized difference has wide linear rangeA large offset exists

EEL5225: Principles of MEMS Transducers (Fall 2003)20

Vertical Comb

Wiring for x-axis actuation But we can make wiring like this

Fz

Fz

V

z

xy FxFx

V

2

21 VdxdCFx =

• Total of 25 different combinations

2

21 VdzdCFz =

EEL5225: Principles of MEMS Transducers (Fall 2003)21

Vertical Comb

xz

C2

C1

V2

V1

Z-axisspring

Z-axiscomb

C1

C2

dzdC1

dzdC2

Cap

acita

nce

(aF/

µm)

Z-axis displacement (µm)-4 -2 0 2-3 -1 1 3 40

5

10

15

20

-5

-2.5

0

2.5

5

Cap

acita

nce

grad

ient

(aF/

µm2 )

H. Xie, et al, MSM 2000, San Diego

Z-ax

is d

ispl

acem

ent (

µm)

Applied voltage, V1 (V)

0

100

200

300

400

0 4 8 12

Experimental data

Simulation