Digital Signal Processing Reference
In-Depth Information
l
1
l
2
v
s
R
s
R
t
z
01
z
02
T
2
T
3
Γ
1
Γ
2
Γ
3
Γ
4
0
v
a
t
d
v
c
v
b
v
B
2
t
d
v
A
′
v
d
v
e
3
t
d
v
g
v
f
v
C
4
t
d
v
B
′
v
h
v
i
Figure 3-33
Lattice diagram for two cascaded transmission lines with different
impedance values and identical lengths (
l
1
=
l
2
).
Figure 3-33 also depicts how a lattice diagram can be used to solve for multiple
reflections on a transmission-line system with a series of transmission lines with
more than one characteristic impedance. Note that the transmission lines in this
example are of equal length (
l
1
=
l
2
), which simplifies the problem because the
reflections from each section will be in phase. For example, in Figure 3-33,
the transmitted portion of
v
e
adds directly to the reflection,
v
f
. When the two
transmission lines are of different lengths, the reflections from one section will
not be in phase with the reflections from the other section, which complicates
the diagram drastically. When the signal reaches the termination, the reflection
is governed by the reflection coefficient looking into the termination resistance
at the load (
4
):
R
t
−
Z
02
R
t
+
Z
02
4
=
The portion of the signal reflected off the termination resistor will travel back
toward the source and experience another reflection when it reaches the junction
between transmission lines,
Z
01
−
Z
02
3
=
Z
01
+
Z
02
where
3
is the reflection looking into line 1 from line 2. Part of the signal will
be reflected back toward the load as calculated by
3
and part transmitted toward
the source, as dictated by the transmission coefficient
T
3
:
T
3
=
1
+
3
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