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290 Direct-Coupled Filters
It was noted that classical bandpass filters, like that in Figure 6.31 (Section
6.5.1), may be modified by employing one capacitive and one inductive
Norton transformer (Section 6.5.3), so that a filter having a direct-coupled
appearance is obtained. Even though it appears to have been obtained using
one L and one C inverter, the design is not direct coupled because it violates
the node-resonance rule. However, such Norton transformer applications are
possible for odd N, and there is no passband or stopband distortion in these
cases.
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8.3. General Inverters, Resonators, and End Couplings
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Design of practical filters requires substantial departure from the ideal case.
Inverters may be realized as apertures in waveguide walls, resonators may
depart from lumped-element frequency behavior and dissipate energy, and
acceptable impedance levels for these may require end couplings.
This section develops the fact that all lossless passive networks contain an
inverter with some residual admittances that must become parts of adjacent
resonators. The trap top-coupling network will be developed from this principle,
and its remarkable ability to improve stopband selectivity will be demonstrated.
It will be shown- that resonator dissipation affects tune frequency input
impedance much more than inverter dissipation. An expression for input
impedance with dissipation will be derived, and a means for compensating for
the change will be described. A reasonable amount of dissipation will not
seriously affect the stopband attenuation estimate (8.27), because it has
offsetting effects. It will also be shown that any resonant two-terminal network
may be viewed as an ideal resonator to a first-order approximation, namely
with the same resonance frequency and slope versus frequency as the lumped
prototype.
End couplings can be L sections, radio frequency (rf) transformers, or
direct connections to terminating resistors. Dissipation will be considered for
L sections in a treatment that is only a slight extension of Section 6.1. The rf
transformer may be realized as actual windings. However, the resonator is
often a coaxial or waveguide cavity, and the transformer is just a wire loop
that provides coupling to the magnetic field in the cavity. A basis for these
more general situations will be provided.
8.3.1. Inverters in Admittance Parameters. An equivalent circuit for the
defining admittance parameters was given in Section 7.3.1 (Figure 7.10). The
defining equations in (3.79) and (3.80) for short-circuit y parameters show that
the equivalent circuit in Figure 8.13 is valid for the reciprocal case where
Y12=Y21'
Equating Z = - I!Y21 and Y = Y21 in (8.22) shows that the inverter characteristic
admittance, Yo, is Yo= B when Y21 =jB, i.e" when Y21 is an imaginary