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General Inveners, Resonators, and End Couplings 303
The selectivity effects of end coupling are similar to those of inverters.
When the resistance transformations are 10 or greater (e.g., R II > lOR.), then
the end coupling affects selectivity approximation (8.27) like inverters of the
same kind. Lesser resistance ratios produce effects of less than 6 dB/octave. A
good interpolation formula is included in Appendix G. Its derivation is
beyond the scope of the present treatment. Educated guesses between 0 and 6
dB/octave are often satisfactory.
8.3.6. Summary of Inverters, Resonators, and End Couplings. Every lossless,
reciprocal network contains an inverter; it was identified in terms of its
short-circuit y parameters. Inverter Zo= 1/IY211, and YII and Y22 must be
incorporated in adjacent resonators. Inverters affect stopband attenuation
according to the logarithm of the ratio of Zo values at stopband and tune
frequencies; so the original breakpoint attenuation estimate may be used for
any kind of inverter. The trap inverter is particularly useful because it
provides equivalent selectivity with reduced loaded Q's. This effect was
expressed as an added term in the selectivity estimate.
One reason for using minimum loaded-Q values is their direct effect on
dissipative loss. The efficiency of each resonator at the tune frequency was
shown to be a simple function of the loaded-to-unloaded-Q ratio; the product
of all such efficiencies is the overall filter efficiency. The tune frequency model
of a dissipative, direct-coupled filter is just a set of parallel resistances
separated by ideal inverters. The input resistance was shown to be a continued
fraction, and that provided an expression for the ratio of input resistances
with and without dissipation. The change due to dissipation tends to be
greater for an odd number of resonators, but it can be corrected by adjusting
the inverter or end-coupling transformation ratios. It was also shown that
inverter dissipation effects were an order of magnitude less than resonator
effects, and could be safely ignored. An example emphasized the fortuitous
effect of dissipation on stopband selectivity: the lossless estimate is still quite
accurate because it is usually about equal to the combination of input
mismatch loss and efficiency loss in the presence of dissipative elements.
Almost any two-terminal network can be used as a resonator if it is
resonant at the tune frequency and has either an acceptable tune frequency
slope or stopband susceptance. The equivalent lumped-element, prototype
resonator capacitance turned out to be equal to one-half of the resonator slope
versus frequency at WO° The capacitively loaded, short~circuited transmission
line resonator was examined in terms of both its tune frequency slope and
stopband susceptance equivalence to the prototype resonator. The equivalence
was expressed in terms of the ratio of the lumped resonator CK,q to the actual
loading C. across the transmission line stub. This is the same as the ratio of
the equivalent loaded Q to the apparent loaded Q. This type of stub is used in
combline filters. Example 8.4 utilized three such resonators, which were
capacitively coupled.