Scaling  1
Many MEMS devices were constructed based on technologies of existing macroscale
machines. However, some of them need drastic modifications to operate when shrinking
down in sizes. Scaling techniques can provide guidelines to estimate the consequences of
going small. This section is going to introduce the basic scale laws to facilitate the design
of a micro device.
1.
Area, Volume and Mass
The most direct impacts of decreasing the size of a device are on its area, volume and
mass. A device can be expressed in a spherical polar coordinate system. The outer surface
of a finite potion of this device can be defined as
R
g
r
s
)
,
(
θ
φ
=
(1)
where
g(
φ,θ
)
is a geometrical function of
φ
and
θ
that ranges from 0 to 1, and
R
is the
maximum value of
r
s
reached by the surface.
φ
is the polar angle and
θ
is the azimuthal
angle.
Figure 1 Spherical polar coordinate system
The area of an infinitesimal surface area element,
dA
, is defined by
(
29
(
29
�
°
°
°
°
°
°
°
°
°
°
°
°
°
°
+
+
=
2
1
2
2
2
1
2
2
2
sin
f
f
f
q
f
q
q
gd
d
r
d
g
d
r
R
dA
(2)
1
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Assuming the surface encloses the origin, the surface area can be obtained by integrating
(2) as
(
29
=
=
+
+
=
π
φ
π
θ
φ
θ
φ
φ
θ
0
2
1
2
2
2
1
2
2
2
0
2
sin
d
d
g
r
g
r
R
A
(3)
The enclosed finite volume,
dV
is defined by
dr
d
d
r
dV
θ
φ
φ
σιν
2
=
(4)
Therefore, the total volume enclosed by the outer surface can be written as
(
29
φ
θ
φ
θ
φ
π
φ
π
θ
π
φ
π
θ
d
d
g
R
dV
V
R
r
sin
,
3
/
3
0
2
0
3
0
0
2
0
a
a
a
a
a
=
=
=
=
=
=
=
(5)
For a simple sphere around the origin,
g(
φ,θ
)
=1 (i.e.
r
s
=R
), and
0
=
=
φ
θ
ρ
ρ
, so
the surface area is
2
0
2
0
2
4
sin
R
d
d
R
A
π
φ
θ
φ
π
φ
π
θ
=
=
=
=
The volume is
3
3
4
R
V
π
=
.
Observations from the analyses above lead to the following conclusion.
The total
enclosed surface area is
A = R
2
x
(a geometric integral that depends only on shape) = R
2
x
(G
A
)
And the total enclosed volume is
V = R
2
x
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 Spring '08
 Han
 Force, Mass, Trigraph, Polar coordinate system

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