No Slide Title Senturia, Chapter 6, ... ÊPotential undesired electrostatic actuation ... X-axis...
Transcript of No Slide Title Senturia, Chapter 6, ... ÊPotential undesired electrostatic actuation ... X-axis...
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