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302 Direct-Coupled Filters
ptoduct based on the teactances in Example 8.4 was 150.6130; the effective
Q L
of each resonator is one-third more than that in its effect on the secondharmonic
selectivity. Thus the loaded-Q ptoduct must be increased by 2.3704,
or 7.4963 dB, before applying selectivity estimate (8.27); in this case, there are
two capacitive inverters and no inductive inverters. Program A8-1 evaluates
(8.27) to estimate a 43.58-dB second-harmonic attenuation. Analysis of the
combline filter in Figure 8.22 by using values ftom Table 8.2 yields an actual
attenuation of 43.18 dB at the second harmonic (100 MHz).
8.3.5. End Coupling. The combline filtet in Example 8.4 was left with an
input resonator having an extremely low Zo0 Increasing the parallel resistance
at node I ftom 50 to 150 would triple ZOI to the acceptable value of 35.52
ohms. A transformer can be placed between the source and node I if the first
resonator incorporates the equivalent parallel inductance it presents (see
Figure 8.1). The required coupling coefficient is
K= (8.57)
where X p
and X, are the primary and secondary reactances at W
o (see Figure
8.lOa). The positive, parallel inductance to be combined into the adjacent
resonator is
R II
X pl = RII/X,-Rg/X p '
(8.58)
where K'« I has been assumed.
An alternative means for increasing the end-node parallel resistance is the
use of an L section ftom Section 6.1.2. Any element adjacent to the resonator
must become a part of it. However, in this case and in the case of the
transformer, the rule that "all L's and C's touching a node must resonate"
applies only before these coupling sections are combined into the end resonator(s).
Element dissipation effects have been considered in Section 8.3.3, where
the input resistance disturbance was quantified. It may be offset by the input
L section, which also may have dissipative elements, especially inductors. It is
straightforward to account for the Q" of the top-coupling inductor Lgi in
Figure 8.2. The design relationships for Qu» I are
[
2]1/'
R ,, -2R g
R ,, -2Rg
XgI = 2Q" + Rg(R ,, - R g)+ ( 2Qu) ,
.R ll
X p I = .,,--------'-'---------:-;-1/"" '
[RII/(R,+X,I/Qu)-I]
(8.59)
(8.60)
where X,I is the top-coupling inductor reactance, and X.I parallels C,. As
before, this coupling may be used on either end or on both ends of the filter.