Digital Signal Processing Reference
In-Depth Information
If the loss parameters of equation (6-47) are
small
compared to
jωL
total
and
jωC
, which is the case for most practical transmission lines, (6-47) can be
approximated [Johnk, 1988]:
R
C
L
+
G
L
+
jω
√
LC
1
2
γ
≈
α
+
jβ
=
(6-48a)
C
R
C
L
+
G
L
1
2
α
≈
(6-48b)
C
β
≈
ω
√
LC
(6-48c)
As described in Section 3.2.4, the voltage propagating on a loss-free transmission
(
α
=
0) line can be written
v(z)
=
v(z)
+
e
−
jzw
√
LC
+
v(z)
−
e
jzw
√
LC
(3-29)
However, for a lossy transmission line, the voltage is described by multiplying
equation (3-29) by the decay factor
e
−
αz
:
v(z)
=
v(z)
+
e
−
α
+
jw
√
LC
z
+
v(z)
−
e
α
+
jw
√
LC
z
+
v(z)
−
e
γz
(6-49)
Equation (6-49) describes the voltage propagating on a lossy transmission line.
=
v(z)
+
e
−
γz
Example 6-4
Calculate the frequency-dependent voltage at the output of a 0.5-m
transmission line that is perfectly terminated in its characteristic impedance with
the following properties:
L
ext
10
−
7
H/m,
C
quasistatic
10
−
10
F/m,
=
2
.
5
×
=
1
.
5
×
10
7
S/m,
10
−
7
H/m,
10
−
6
m,
σ
=
5
.
8
×
µ
=
4
π
×
l
=
0
.
5m,and
w
=
100
×
ε
r,
eff
=
3
.
32, and tan
δ
=
0
.
0205 at 1 GHz.
SOLUTION
Step 1:
Calculate the frequency-dependent parameters of the conductor. The
resistance can be calculated approximately using equation (5-12), and is plotted
in Figure 6-25a.
πµf
σ
l
w
R
ac
=
The inductance is calculated with equations (5-20) and (5-30) and plotted in
Figure 6-25b.
L
total
=
L
internal
+
L
external
(5-20)
R
ac
ω
L
internal
=
(5-30)
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