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304 Direct-Coupled Filters
Unloaded, 90-degree shorted stubs are used in interdigital filters; their
slope equivalence was obtained. In that case, the passband reoccurred at odd
harmonics of the tune frequency. It was shown that shortening the line by
capacitive loading produces nonperiodic passband reoccurrences tha\ are far
removed from the tune frequency for short stubs. A graph of this effect was
furnished.
The combline filter in example 8.4 produced two stub Zo values that were
impractically low. The interior node's parallel resistance can be raised to
increase the resonator Zo without affecting the selectivity. The need for end
coupling to raise the input (and/or output) node's parallel resistance level was
made evident. Both the transformer and the L-section end couplings were
described; the latter included an inductor unloaded Qu in the matching
formulas provided. The viewpoint for properly combining the end-coupling
reactance with the resonator was discussed. Also, the effect on selectivity was
described as being similar to the effect on the same type of inverters,
especially when the end-coupling resistance ratio exceeded 10: 1. An example
of end-coupling design is included in Section 8.6.
8.4. Four Important Passband Sbapes
It was sbown in Section 8.1.3 that the selectivity function of the prototype
network was a function of only the resonator loaded-Q values. When inverters
have some frequency dependence in tbe Zo or 0 parameters, some distortion
occurs, but it may be negligible for bandwidths of less than 20%. The
stopband effects of inverter Zo are easily anticipated in terms of asymptote tilt
in a simple breakpoint analysis. This section concentrates on the loaded-Q
distribution. Although loaded-Q values may be selected arbitrarily, it is often
useful to have closed formulas for passband characteristics, even if they are
only approximately realized by the network.
Four response shapes will be considered; the first three belong to a closely
related family based on elliptic loci in the complex frequency plane. These are
the Chebyshev (equal-ripple or overcoupled), Butterworth (maximally flat or
critically coupled), and Fano (undercoupled) shapes. Fano's undercoupled
response shape is not related to his broadband-matching metbod in Chapter
Six: it was suggested in connection with video amplifier design. The fourth
shape results from choosing all loaded-Q values to be equal. The resonator
efficiency relationsbip from Section 8.3.3 will show that tbis produces minimum
loss when given a loaded-Q product, i.e., fixed stopband attenuation. It
also produces a network having minimum sensitivity, and serves as a simple
norm for all other cases.
Transducer functions were described in Section 3.2.4, and their zeros were
identified as the natural frequencies of resistively terminated LC networks.
The problem set for Chapter Three included botb Chebyshev and Butterworth