Biomedical Engineering Reference
In-Depth Information
Integrating.d
U
.for.the.total.work.
U
,
Q Q
C
d
1
2
Q
C
2
1
2
∫
U
=
=
=
CV
2
.
(8.6.2.5)
.
To.ind.the.force.generated.by.a.parallel.plate.actuator,.we.can.use.the.principle.of.virtual.work.by.con-
sidering.the.work.done.when.the.plates.of.a.capacitor.are.moved.a.small.distance.Δ
z
.closer.together.when.
a.constant.voltage.
V
,.set.by.a.battery,.is.applied.between.the.plates..Since.the.plates.have.opposite.charge,.
the.force.between.the.plates.is.attractive..he.decrease.in.the.gap.causes.the.capacitance.of.the.capacitor.to.
increase.by.Δ
C
.and.an.amount.of.charge.Δ
Q
.to.be.transferred.from.the.battery.to.the.capacitor,.increas-
ing.its.stored.energy..We.can.then.balance.the.work.done.by.the.battery.to.transfer.the.charge.to.the.work.
done.by.the.actuator.and.the.potential.energy.stored.in.the.capacitor.when.the.plates.move.closer.together:
∆
W
=
∆
W
+
∆
U
(8.6.2.6)
.
battery
capacitor
capacitor
.
+
1
2
V Q F z
∆
=
∆
V C
2
∆
(8.6.2.7)
.
.
Using.Equation.xx.to.substitute.for.Δ
Q
.at.constant.
V
,
.
Q CV
= → =
∆
Q V C
∆
→
V Q V C
∆
=
2
∆
.
(8.6.2.8)
V
1
2
V C F z
2
∆
=
∆
+
V C
2
∆
(8.6.2.9)
.
.
=
1
2
F z
∆
V C
∆
(8.6.2.10)
.
2
.
=
1
2
2
∆
∆
C
z
F
V
(8.6.2.11)
.
.
We.can.calculate.the.force.by.taking.the.derivative.of.the.capacitance.with.respect.to.the.separation.
between.the.plates:
=
∂
∂
C
z
=
∂
∂
A
A
.
.
(8.6.2.12)
ε
ε
0
0
(
)
z
g
−
z
2
V
g
−
z
0
0
So.that.the.electrostatic.force.is.given.by
1
2
∆
∆
C
z
ε
A V
g
2
.
.
0
F
e
=
V
2
=
(8.6.2.13)
(
)
2
2
−
z
0
8.6.3 Mechanical restoring Force
A.spring.is.typically.used.to.apply.a.mechanical.restoring.force.
F
m
.for.electrostatic.actuators,.as.shown.
in.
Figure.8.24
..he.spring.can.be.linear,.following.Hooke's.Law,
F
m
= −
kz
(8.6.2.14)
.
.
where.
k
.is.the.spring.constant.and.
z
.is.the.distance.to.which.the.spring.is.either.stretched.or.compressed..
As.described.below,.in.some.situations.it.can.be.useful.to.use.a.nonlinear.spring. where.the.restoring.
force.does.not.vary.linearly.with.the.displacement.
