Digital Signal Processing Reference
In-Depth Information
The ratio of
v
1
and
i
2
is used to calculate
B
:
B
=
Z
0
e
γl
−
e
−
γl
2
=
Z
0
sinh
γl
(9-42c)
v
2
=
0
i
1
i
2
D
=
1
Z
0
i
2
=
e
γl
+
e
−
γl
2
1
/
2
Z
0
e
γl
−
−
1
/
2
Z
0
1
Z
0
e
−
γl
i
1
=
i(z
=
l)
=
=
The ratio of
i
1
and
i
2
is used to calculate
D
:
e
γl
+
e
−
γl
2
D
=
=
cosh
γl
(9-42d)
Therefore, the
ABCD
matrix for a lossy transmission line.
AB
CD
cosh
γl
Z
0
sinh
γl
1
Z
0
sinh
γl
lossy transmssion line
=
(9-43a)
cosh
γl
Using the same procedure, the
ABCD
parameters for a loss-free line can easily
be derived where
γ
=
jβ
:
AB
CD
cos
βl
jZ
0
sin
βl
j
Z
0
sin
βl
loss-free transmssion line
=
(9-43b)
cos
βl
Table 9-2 depicts the relationship between common circuits and the
ABCD
parameters. These common forms are useful for extracting equivalent circuits
from
S
-parameter measurements. Of course, a methodology is needed to convert
S
-parameters into an
ABCD
matrix, which is covered in the next section.
Relationship Between ABCD and S-Parameters
To take advantage of the rela-
tionships between the
ABCD
matrix and common circuit forms, it is necessary
to determine the relationship between the
ABCD
and
S
-parameters. The most
straightforward derivation is first to define the transformation of
ABCD
parame-
ters into a two-port
Z
-matrix and then use equation (9-34) to get the
S
-parameters.
Beginning with the definition of
Z
11
from equation (9-8),
i
2
=
0
v
1
i
1
=
Z
11
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