Digital Signal Processing Reference
In-Depth Information
The frequency-dependent capacitance is calculated by dividing
C
quasistatic
by
the quasistatic value of the effective dielectric permittivity and multiplying by
the frequency-dependent dielectric permittivity calculated by equation (6-30a).
The frequency-dependent capacitance is plotted in Figure 6-25c:
ε
∞
+
ε
m
2
C
quasistatic
ε
r,
eff
ln
(ω
2
/ω)
ln
(
10
)
C(ω)
=
−
m
1
where
m
1
=
1 and
m
2
=
11 are chosen to correspond to a frequency range of
10 rad/s to 100 Grad/s.
The conductance,
G(ω)
, is calculated using equation (6-45) and plotted in
Figure 6-25d:
G(ω)
=
tan
|
δ
|
ωC(ω)
(6-45)
where the loss tangent is calculated with (6-30a) and (6-30b):
ε
m
2
−
m
1
−
π/
2
ε
ε
≈
ln
(
10
)
tan
|
δ
=|
ε
m
2
ln
(ω
2
/ω)
ln
(
10
)
ε
∞
+
−
m
1
Step 3:
Calculate the frequency-dependent voltage. Since the transmission line
is perfectly terminated, no reflections will be generated. Therefore, the loss-free
voltage wave will behave as described by equation (6-49) with
v(z)
−
=
0.
=
v
in
e
−
αz
e
−
jβz
=
v
in
e
−
γz
v
out
Therefore, using equation (2-31) to simplify,
=
v
in
e
−
αz
[cos
(
−
βz)
+
j
sin
(
−
βz)
]
v
out
where
γ
is defined by equation (6-47) and
z
is the line length:
γ(ω)
=
α
+
jβ
=
(R
+
j ωL)(G
+
jωC)
(6-47)
Next, the magnitude of the voltage is calculated:
Re
(v
out
)
2
=
+
Im
(v
out
)
2
v
out,mag
and is plotted in Figure 6-26 assuming that
v
in
=
1V.
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