Biomedical Engineering Reference
In-Depth Information
F
m
+V
g
0
F
e
z
-V
FIGuRE 8.24
Parallel.plate.electrostatic.actuator.with.a.mechanical.spring.that.provides.a.restoring.force.
F
m
.in.
opposition.to.the.attractive.electrostatic.force.
F
e
..he.initial.gap.is.
g
0
.
Mechanical and electrical force balance
F
m
=
F
e
20
F
F
e
(
V
=
1)
F
e
(
V
=
2)
F
e
(
V
=
3)
F
e
(
V
=
4)
V
pi
18
16
14
12
10
8
6
4
2
0
Displacement
FIGuRE 8.25
Graphical.solution.for.balancing.the.mechanical.and.electrical.forces..he.capacitor.has.an.initial.
gap.
g
0
.equal.to.2.1.μm.
he.force.balance.between.the.electrostatic.force.that.pulls.the.released.plate.down.toward.the.ixed.
counter. electrode. and. the. mechanical. restoring. force. that. pulls. holds. them. apart. can. be. determined.
graphically.as.shown.in.Figure.8.25..he.electrostatic.force.is.shown.for.a.few.diferent.voltages..here.
are.two.solutions.for.low.voltages,.and.no.solutions.for.the.highest.voltage.shown..At.a.critical.voltage,.
called.the.pull-in.voltage,.there.is.only.a.single.solution..he.electrostatic.force.for.this.single.solution.
is.shown.as.a.dashed.line..If.the.voltage.is.increased.further,.the.nonlinear.electrostatic.force.is.greater.
than.the.mechanical.spring.force.and.the.two.plates.pull-in.and.touch..At.the.critical.voltage,.the.electri-
cal.and.mechanical.forces.are.equal:
F
=
F
(8.6.2.15
)
.
.
m
e
ε
0
A V
g
2
.
.
kz
=
(8.6.2.16)
(
)
2
2
−
z
0
he.slopes.of.the.electrical.and.mechanical.forces.are.also.equal:
d
d
F
z
d
d
F
z
m
e
(8.6.2.17)
=
.
.
V
2
.
.
k
=
ε
0
A
(8.6.2.18)
(
)
3
g
−
z
0
