Digital Signal Processing Reference
In-Depth Information
10
d
peaks
8
6
b
base
h
tooth
4
2
80
100
0
20
40
60
Position,
µ
m
Figure 5-19
Example of a surface profilometer measurement of a rough copper sample
showing peak heights that range from 0.7 to 8
.
5
m. The flat surface is assumed to be at
µ
0
.
5
m.
µ
[Mathworld, n.d.]. The spheroid volume is in turn equal to one-half the volume
of a sphere to calculate the radius of a hemisphere with the same volume as the
hemispheroid-shaped surface protrusion:
3
h
tooth
b
base
2
2
r
e
=
(5-59)
where
b
base
is the tooth base width,
h
tooth
the tooth height, and
r
e
the radius of
a hemisphere with equivalent tooth volume. The base area of the hemisphere,
A
base
, is then calculated:
=
π
b
base
2
2
A
base
(5-60)
The square tile area of the surrounding flat plane is calculated based on the
distance between peaks:
=
d
peaks
A
tile
(5-61)
If the RMS values of
d
peaks
,
h
tooth
, and
b
base
values are calculated, the surface
shown in Figure 5-16b and measured in Figure 5-19 can be represented as the
equivalent surface in Figure 5-20.
A comparison between the correction factor calculated from the Hammerstad
model (5-48) and the hemispherical model (5-58) with the modified equivalent
volume is shown in Figure 5-21. Note that the hemisphere model saturates at
a much higher value than Hammerstad, which will always saturate at a value
of 2. The implementation shown in (5-58) causes a nonphysical discontinuity
at the frequency where the model transitions from dc to ac behavior, which is
the point where the surface of the flat plane with an area equal to the base
of the hemisphere has more loss than the hemisphere. Some engineers may
be concerned that the discontinuity may induce nonphysical glitches into the
time-domain responses. However, simulations of pulses as fast as 30 Gb/s were
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