Actuators & the Coil Gun
Outline
Review of Electrostatic Actuators
-Energy method for Electrostatics
Rail Gun
-Energy method for Magnetostatics
Coil Gun
∂λ
fx = i
∂x
1
TRUE or FALSE?
1.  Since W = ½ CV2, the force always acts to
decrease capacitance to go to lower energy.
2.  If the gap is decreased, the magnetic flux will
increase.
3. The energy stored in the electric field is ½ εoE2
and has units of Joules.
2
Attractive Force Between Parallel Plates
+ +
+ + + ++ + + ++++ +
-- --- - -- - -
-- - - -- -
-
CAPACITOR PLATES ARE PULLED TOWARDS EACH OTHER
(FORCE ACTS TO INCREASE C)
3
Linear Electrostatic Actuator
+
+ + + + +
+
+ + + + + + +
+
+ + + ++ + + + +
- - -
- - -
- - - - - - - -
- - - -
- - - - - - - -
CONSTANT FORCE ALONG
DIRECTION OF MOTION
(FORCE ACTS TO INCREASE C)
4
Gap Closing Electrostatic Actuators
o Q2 w
f (y) =
2 C2 d
Moderate force
Weak force
5
MEMS Actuator Mirror
Hinges
Yoke
Landing Tip
6
Rail Gun Force … Using Energy Conservation
Perfectly conducting plates and slider
K
f
x
Depth w
L(x) = μo Gx/w … one-turn solenoid
i = Kw
1 ∂L
1
f = i2
= μo K 2 Gw
2 ∂x
2
7
G
Rail Gun Force -Energy Conservation Again
Perfectly conducting plates and slider
H, B
K
v
x
W =
1
μo H 2 dV =
2
f
Depth w
1 2
B /μo dV ; B = μo H
2
1
= μo K 2 xGw (Recall that H = K )
2
1
… (Keep λ constant)
= λ2 /(μo xG)
2
1
−∂W
f=
= μo K 2 Gw
∂x
2
8
G
Coil Gun
How can we use a
solenoid to propel a
bullet if the force is
always into the solenoid?
9
Consider this energy model:
R
Back voltage v is due to motion
dx
dx dλ
= vi =
i
Pm = fm u = fm
dt
dt dx
dλ
Then, force is
fm = i
dx
simply:
L
+
i
-
v=
dx dλ
dt dx
Some caution here: this is the force due to the interaction
with the coil -- It doesnt mean there may not be other
forces!
To find back voltage: run the system open circuited
This will give the right back
voltage; If you know how fast
the magnet is going, you get
flux change
+
v
-
10
+
-
v=
dx dλ
dt dx
Permanent
Ws =
i(λ)dλ =
=
fm
dλ
=i
dx
11
λf
0
fx dx
0
λf
∂λ
i dx
∂x
Key Takeaways
Energy method for calculating Forces
calculated at constant flux linkage
Coil Gun
fx = i
∂λ
∂x
Wmechanical =
fm dx = M ghthrow
Force acts to increase Capacitance
o Q2 w
f (y) =
2 C2 d
Moderate force
Weak force
12
MIT OpenCourseWare
http://ocw.mit.edu
6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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