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Four Important Passband Shapes 315
that the Zo values of the inverters do not affect selectivity, so that all of these
and the terminating resistance(s) may be set equal to unity without loss of
generality. The superfluous inverter next to the load will introduce a 90-degree
phase shift, but will have no other effect. Each typical subsection has a chain
matrix defined by
10][0 '1] [0 '1]
[ Y 1 j I J
o
= j I Jy . (8.92 )
The typical resonator is shown in Figure 8.18, where GKKis approximately
unity for low dissipation. To a good approximation under these assumptions,
Y=QL( ~u +jF). (8.93)
Thus the argument of the Chebyshev polynomials of the second kind is
y=A+ D=jY. Note that Y is complex, and so is jY. Thus (8.90) yields the
overall ABCD matrix of a somewhat dissipative, direct-coupled filter ending
in one superfluous inverter:
where the polynomial argument is
TN=[ -PN-l(y) jPN(y) ]
jPN(y) jYPN(y) - P N -ley) , (8.94)
y=jY (8.95)
for Y in (8.93).
The result in (8.94) may be used to obtain the frequency response of both
doubly and singly terminated minimum-loss filters. According to Beatty and
Kerns (1964) or (3.68), the transducer-gain scattering parameter normalized to
I ohm may be expressed in terms of ABCD parameters:
2
SZI= A+B+C+D'
Therefore, the transducer gain for a doubly terminated network is
(8.96)
where
L= IOlog lo -
[M N ['
- dB, (8.97)
4
(8.98)
using (8.94). This may be evaluated recursively using (8.91); however, the M N
values in (8.98) may be expanded as polynomials in complex Y, as tabulated
in Table 8.9.
Figure 8.27 shows the N = 4 response in the passband and stopband,
respectively. Note the QL/Qu parameter. Two design graphs for fourresonator,
doubly terminated, minimum-loss filters are provided in Appendix
F. Note that it is convenient to use QLF as the independent variable.