Transcript Class 8

MEMS
Class 6
Modeling of MEMS Devices
Mohammad Kilani
The Scaling of MEMS Devices
20 mm
2 mm
1 mm
Isometric Scaling
10 mm
How does electrostatic force change?
How does electrostatic force change relative to other forces?
How does response time change?
The Scaling of MEMS Devices
20 mm
2 mm
1 mm
Isometric Scaling
10 mm
L S, W S, H S
A  L W  S  S  S 2
V  L W  H  S  S  S  S 3
Water Bug
Weight = V  S3
Surface Tension = kA  S2
Man : 2m
Bug: 2 mm
Water bug uses surface
tension to walk on water
S = 1/1000
Surface Tension / Weight  S2/S3 = S-1 = 1000
The Scaling of Forces
20 µm
2000 µm
10 μm
Isometric Scaling
1000 μm
S = 1/100
Weight = gm = gV  S3
Electrostatic Force = kA  S2
Electrostatic Force / Weight  S2/ S3 = S-1 = 100
1
 F1  S 
F   2 
 2   S 
 F3  S 3 
   4
 F4  S 
The Scaling of Work or Mechanical Energy
1
 F1  S 
F   2 
 2   S 
 F3  S 3 
   4
 F4  S 
W  F D
1
1
2
W 1  S  S  S 
W   2   1   3 
 2   S   S   S 
W 3  S 3  S 1  S 4 
   4   1  5 
W 4  S  S  S 
Gravitational force  S3
Wok of gravity  S4
If S = 1/100 the gravitational energy required to move an object
from the bottom to the top of the machine under consideration
decreases by (1/100)4  1/100,000,000.
Drop an ant from ten times his height, and he walks away.
Please do not try this with a horse!
The Scaling of Acceleration
1
 F1  S 
F   2 
 2   S 
 F3  S 3 
   4
 F4  S 
aF m
1
3
2
a1  S  S  S 
a   2   3   1 
 2   S  / S   S 
a3  S 3  S 3  S 0 
   4  3  1 
a4  S  S  S 
A predominance of the forces we use in the microdomain scale as S2.
For these forces, the acceleration scales as S 1. If S = 1/100, a
increases by a factor of 100.
Small systems tend to accelerate very rapidly.
The Scaling of Velocity
adx  vdv , v  2 ax
1
 F1  S 
F   2 
 2   S 
 F3  S 3 
   4
 F4  S 
v  2xa  x 0.5a 0.5
0.5
1
0.5
v 1  S  S  S 
v   0.5   0.5   0 
 2   S   S   S 
v 3  S 0.5  S 0  S 0.5 
   0.5   0.5   1 
v 4  S  S  S 
For the case where the force scales as S2, velocity scales as S-0.5.
If S = 1/100, the velocity increases by a factor of 10.
Small things tend to be fast.
The Scaling of Response Time
The time needed to travel a certain distance, x is given from the
relation
t
t t
x  vd     ad  d 
0
1
 F1  S 
F   2 
 2   S 
 F3  S 3 
   4
 F4  S 
0 0
Assume
constant a
2x
t
 x 0.5a 0.5
a
1 2
x  at
2
0.5
1
1.5
t 1  S  S  S 
t   0.5   0.5   1 
 2   S   S   S 
t 3  S 0.5  S 0  S 0.5 
   0.5   0.5   0 
t 4  S  S  S 
For the case where the force scales as S2, transit time t scales
as S1. If S = 1/100, the transit time decreases by a factor of 100.
Small things tend to be fast.
The Scaling of Mechanical Power
The time needed to travel a certain distance, x is given from the
relation
P  F v
1
0.5
0.5
 P1  S  S  S 
P   2   0   2 
 2   S   S   S 
 P3  S 3  S 0.5  S 3.5 
   4  1   5 
 P4  S  S  S 
For the case where the force scales as S2, power scales as S2.
If S = 1/100, the power decreases by a factor of 10000.
Small things have very small mechanical power.
The Scaling of Magnetic Forces
1. Constant current density
o
L
F
I aIb
2
d
Ib
Ia
L
I   J .dA  JA
Ia  S 2, Ib  S 2,
F S
4
The Scaling of Magnetic Forces
2. Constant heat flow through the surface of the wire
Q
  S 0
As
Ib
 L 
Q P I R I 

A
 e 
2
Q
I
L
0
 [S ] 
 I 2 [S 3 ]
As
A s Ae
2
Ia
L
I  S 1.5
F S 3
2
The Scaling of Magnetic Forces
3. Constant temperature rise of the wire
Ib
Ia
L
F S
2
The Scaling of Magnetic Forces
4. Wire and permanent magnet
Ia
L
1. Constant current density: F S3
2. Constant heat flow: F S2.5
3. Constant temperature rise: F S2
The Scaling of Different Forces