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296 Direct-Coupled Filters
~I---+T()-----;'-;;-/4~) i () '14 >-h
J_GOO ~~fG-+-'"..L----.----~~ G~
Figure 8.19.
An N=3 dissipative, direct-coupled filter at tune frequency woo
Figure 8.12a shows that the network at the tune frequency has the form
shown in Figure 8.19, because all the resonators are parallel resonant. Inverter
relationship (8.9) in its admittance form enables the expression of the input
conductance:
G IIG 22
G;n=Gld + G G G I(G G)'
2d+ 22 33 3d+ 33
(8.38)
The input conductance of the lossless network is G II' Divide both sides of
(8.38) by Gil' and introduce G 22 and G 33 so as not to disturb the equality; this
yields
G m G ld GIIG22/GIIG22
-=-+--=-----'-'"=''''::---'7.",.::;'-=:--
Gil Gil G2d G22G"/G22G33
- + -=--"T","",~=?--~-
G 22 G Jd /G 33 +G 33 /G JJ
(8.39)
The left-hand side is inverted to express a resistance ratio, and the loaded and
unloaded Q's of (8.3) and (8.34) are incorporated. The result is the continued
fraction
R;n(lossless)
(8.40)
R;n(lossy)
which always ends with I. The relative Q, QLK' is defined as
QLK
A
QLK=-Q .
oK
(8.41 )
The continued fraction expansion in (8.41) will be computed by a recursive
procedure in Section 8.6. In general, the direct-coupled-filter input resistance
tends to change very little as a result of dissipation when there is an even
number of resonators. This is due to the "seesaw" effect that each inverter has
on its load and input resistances [see (8.7)]. When the inverter's load resistance
increases, the inverter input resistance decreases, and vice versa. Whatever the
change from the desired input resistance, it may be restored simply by
changing some inverter Zo value (top-coupling reactance). It is also possible to
restore the match by adjusting the input end coupling, when it is employed as
desired in Section 8.3.5.
To compare the effect of dissipation on resonators and inverters, consider
the dissipative resonator in Figure 8.18. At tune frequency wo, the input