Digital Signal Processing Reference
In-Depth Information
Incident wave
V
R
=
Z
0
Reflected wave
Z
0
R
l
z
I
z
0
= −
=
Γ (
z
= −
I)
Γ (
z
=
0)
Figure 9-1
The reflection looking into a network is dependent on the distance between
the point where the reflection is being evaluated and the impedance discontinuity.
The reflection coefficient looking into a network can be derived from equa-
tions (6-49) and (3-102):
v(z)
=
v(z)
+
e
−
γz
+
v(z)
−
e
γz
Let
v(z)
+
=
v
i
and
v(z)
−
=
v
r
. Then
v(z)
=
v
i
e
−
γz
+
v
r
e
γz
=
v
i
(e
−
γz
+
0
e
γz
)
=
v
i
e
−
γz
[1
+
(z)
]
(9-1)
where
v
r
e
γz
R
l
−
Z
0
v
i
e
−
γz
=
0
e
2
γz
R
l
+
Z
0
e
2
γz
(z)
≡
=
Equation (9-1) describes the reflection coefficient looking into a transmission
line with characteristic impedance
Z
0
, length
z
, termination impedance
R
l
, and
propagation constant
γ
.
In Section 3.5 the concept of lattice diagrams was introduced to demonstrate
how time-domain signals propagate on transmission lines. An important concept
demonstrated was that the period of transmission-line “ringing” was dependent on
the electrical length of the line. In frequency-domain analysis, the same principles
apply; however, it is more useful to calculate the frequency when the reflection
coefficient is either maximum or minimum, which is dependent on both the
electrical length of the structure and the frequency of the input stimulus. To
demonstrate this concept, consider the case of a loss-free transmission line as
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