Digital Signal Processing Reference
In-Depth Information
Fig. 6.2
Typical
π
-matching network
associate a quality factor:
R
opt
X
C1
Q
a
=
,
(6.1)
R
L
X
C2
,
Q
b
=
(6.2)
where
X
C1
and
X
C2
are the reactances of the capacitors
C
1
and
C
2
.
For a given overall transformation factor, the components of the matching net-
work can be determined, for instance, by choosing a value for
Q
b
. With
Q
b
and
(
6.2
), the value of
C
2
is determined.
R
t
is found by the downward transformation
factor (
m
b
):
Q
b
+
m
b
=
1
,
(6.3)
R
L
m
b
R
t
=
.
(6.4)
The reactance of
L
1b
is determined through
R
t
and
Q
b
with
X
L1b
=
Q
b
R
t
.
(6.5)
The reactance of
L
1a
is determined by the upward transformation factor,
R
t
, and
Q
a
with
R
opt
R
t
m
a
=
,
(6.6)
m
a
−
Q
a
=
1
,
(6.7)
X
L1a
=
Q
a
R
t
,
(6.8)
whereas the reactance associated with
C
1
can be found by replacing
Q
a
and
R
opt
in (
6.1
).
Inductor
L
1
is the sum of
L
1a
and
L
1b
and, hence, all the three components are
determined. An important parameter of a matching network is its loaded quality
factor. According to Sun and Fidler in [
27
, (10)], its definition for a
π
-network is
Q
a
+
Q
b
Q
0
=
.
(6.9)
2
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