Digital Signal Processing Reference
In-Depth Information
However, the convention for an
ABCD
matrix assumes that
i
2
is flowing out of
port 2, and a
Z
matrix assumes that it is flowing into port 2. Consequently, the
sign of
i
2
must be changed, which results in the definition of
Z
12
in terms of
ABCD
parameters:
AD
−
BC
C
Z
12
=
(9-44b)
The terms
Z
21
and
Z
22
are derived using a similar procedure.
i
2
=
0
v
2
i
1
Z
21
=
v
1
=
Av
2
i
1
=
Cv
2
1
C
Z
21
=
(9-44c)
i
1
=
0
v
2
i
2
Z
22
=
v
1
=
Av
2
+
Bi
2
0
=
Cv
2
+
Di
2
D
C
Z
22
=
(9-44d)
The final relationship between a two-port
Z
-matrix and the
ABCD
matrix is
shown as
A
C
AD
−
BC
C
Z
11
Z
22
Z
12
=
(9-45)
Z
21
1
C
D
C
To derive the transformation of the
ABCD
to
S
-parameters, the results of (9-45)
are substituted into equation (9-34). The final solution is summarized in (9-46),
where
Z
n
is the termination impedance at the ports, which are all assumed to be
equal.
B
CZ
n
)
B
+
Z
n
(D
+
A
+
CZ
n
)
−
Z
n
(D
−
A
+
BC)
B
+
Z
n
(D
+
A
+
CZ
n
)
2
Z
n
(AD
−
S
11
S
12
=
(9-46)
S
21
S
22
2
Z
n
B
+
Z
n
(D
+
A
+
CZ
n
)
B
−
Z
n
(A
−
D
+
CZ
n
)
B
+
Z
n
(D
+
A
+
CZ
n
)
Similar techniques are used to derive the transformation of
S
into
ABCD
param-
eters. The complete sets of transformations are summarized in Table 9-3.
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