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General Inverters, Resonators, and End Couplings 299
Note that the resonator loaded Q computed on this basis will be useful as a
passband parameter, but not for determining dissipation. (The equivalent
parallel resistance due to transmission line loss may be obtained as described
in Section 4.3.2.) The 90-degree resonators have repeating passbands at odd
harmonics of the tune frequency, because of periodic resonances.
The capacitive loading of the short-circuited transmission lines in Figure
8.20a spaces the recurring resonances in a nonperiodic manner. Suppose that
B(wo)=O=B(w,); from (8-46) and (8.47),
Yo . ewo ew,
-=wotan-=w,tan- (8.51)
C k
v v
Therefore, a second resonance at w, is related to tune frequency Wo by
WI tan eo
-c:---'-:-~ = 0 (8.52)
Wo tan[(w,/wo)Ool .
This transcendental equation may be solved by secant search. For example,
when the resonator is 45 degrees long at the tune frequency w o , then WI =
4.2915wo. Resonators may be as short as 10 degrees, which increases the
second resonance to about 17.5 times the tune frequency resonance. Incidentally,
a useful relationship of transmission line length in free space is
ef
0 0 = 32.81 degrees, (8.53)
where e is in inches and f is in MHz. A graph of (8.52) solutions is shown in
Figure 8.21.
20
17.5
14
12
10
9
B
7
6
5
4
3
1\
\\
Qf,
ao - 32.81 degrees
\
\
I\.
r'-..
~ r--
o 10 20 30 40 50 60 70 80 90 100 110
8 0 (degrees) ~
Figure 8.21. Frequency ratios of recurring pass bands versus short·circuited resonator length at
the tune frequency.