Digital Signal Processing Reference
In-Depth Information
the profile within the tile size is calculated. The sphere radius
a
=
0
.
8
m. From
µ
377
/
√
ε
=
377
/
√
4
209
.
3m
−
1
,
η
=
√
µ
0
/ε
0
ε
=
Example 5-4,
k
=
=
188
.
5
,
and
δ(
5 GHz
)
=
√
2
/
2
πf µ
0
σ
≈
0
.
935
m.
µ
The scattering coefficient
is calculated from equation (5-51a). For
this
example, it can be shown that
β
(1) is negligible, so it is ignored.
2
j
3
(ka)
3
1
−
(δ/a)(
1
+
j)
α(
1
)
=−
1
+
(δ/
2
a)(
1
+
j)
10
−
6
0
.
935
×
−
+
j)
1
10
−
6
(
1
"
2
j
3
[209
.
3
(
0
.
8
0
.
8
×
10
−
6
)
]
3
=−
×
10
−
6
0
.
935
×
1
+
10
−
6
(
1
+
j)
2
·
0
.
8
×
10
−
12
10
−
12
=−
1
.
93
×
−
j
1
.
05
×
The Huray surface roughness correction factor is calculated with (5-66) at 5 GHz:
P
flat
+
P
N
spheres
P
flat
K
Huray
=
+
n
=
1
Re
2
η(
3
π/k
2
)(α(
1
)
+
β(
1
))
n
(µ
0
ωδ/
4
)A
tile
(µ
0
ωδ/
4
)A
tile
=
20 Re
2
η(
3
π/k
2
)α(
1
)
(µ
0
ωδ/
4
)A
tile
(µ
0
ωδ/
4
)A
tile
+
=
10
−
13
10
−
13
8
.
15
×
+
7
.
8
×
=
=
1
.
95
10
−
13
8
.
15
×
Therefore, the Huray model predicts that the series resistance of a transmission
line manufactured according to the roughness profile defined above would be
approximately 1.95 times higher than the same transmission line constructed
with a smooth conductor at a frequency of 5 GHz.
Figure 5-25 shows a comparison of the insertion losses of a transmission line
constructed with rough copper modeled using the Huray equation compared to
measured results. Note that the Huray model correctly predicts the shape of the
insertion loss curve with less than 1.5 dB of error at 30 GHz. Simulations show
that if 23 spheres are used in this example, the model fits the measured results
almost exactly [Olufemi, 2007]. However, aside from detailed statistical analysis
of the SEM photographs, there is no known deterministic methodology to arrive
at the perfect fit. The methodology presented allows the use of a profilometer
and produces reasonable wideband results.
Figure 5-26 shows the Huray equation (5-65) constructed with twenty 0.8-
m
spheres as calculated in Example 5-5, compared to the Hammerstad and hemi-
sphere models for the roughness profile assumed in Examples 5-4 and 5-5. Note
that the Hammerstad equation saturates at 2, which does not provide enough loss
µ
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