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300 Direct-Coupled Filters
The ratio of the two capacitors (and related loaded Q's) in Figure 8.20,
based on slope equivalence at the tune frequency, may be found from (8.48)
and (8.49):
(8.54)
For example, when 8 0 = 7T /4, CKoq/Ck= 1.285. The loaded Q of the resonators
in Figure 8.20 is proportional to the resonating capacitors. Therefore, the
effective loaded Q that determines passband behavior is 28.5% greater than
the apparent loaded Q determined by tune frequency reactance in (8.3). It will
be shown that the increased loaded Q decreases the passband width.
Example 8.4. Design an N = 3 combline filter using capacitive coupling
between nodes and 45-degree resonators. Make the effective loaded-Q values
and resistance levels equal to those in Figure 8.12. The desired circuit
arrangement is shown in Figure 8.22.
The effective loaded-Q values will be 1.285 times the values on a reactance
basis, as previously noted for 45-degree resonators. Dividing the Q L values in
Figure 8.12 by 1.285 yields Q L1 = 4.2229 = QL3 and Qu = 8.4458. Synchronous
node reactance XK= RKK/QLK' so that R ll = 50, R 22 = 75, and R3J = 100 lead
to Xl = 11.8403, X n = 8.8801, and XIII = 23.6805 ohms, respectively. The shorted-stub
input impedance is jZo tan 8, so that ZOk = XK when 8 = 45 degrees. At
50 MHz, the corresponding synchronous capacitances are: C I
= 268.84, C n
=358.45, and C III = 134.42 pF. The values of X 12 and X 23 are the same as
shown in Figure 8.12, so that C'2=51.98 and C 23 =36.76 pF. There are two
negative inverter capacitances to subtract from ell' and only one each from
C 1 and C III . The resulting topology code (corresponding to the analysis
scheme in Chapter Four) for the combline filter in Figure 8.22 is shown in
Table 8.2.
The analysis at 50 MHz showed that the lossless network was tuned to
Zi"=50.0125+jO.0073 ohms. The 3-dB loss frequencies were 46.05 and 55.55
MHz; this is an 18.78% bandwidth, compared to the ideal, maximally flat
3-dB bandwidth of 18.43%.
Clearly, ZOI and Z02 are impractically low because the physical range of
transmission line characteristic impedances is about 20-120 ohms. Parallel
c" II III
"
1\ 1\
Zoo Z", Zoo
';50 Q -:=C 1
c =c, c, loon
0, 0,
"
Figure 8.22.
A combline filter using two capacitive inverters.