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Prototype Network 277
and Wo is the midband geometric mean frequency according to (6.85) in
Section 6.5.1. It is easy to obtain a similar expression for a series LC network
and then to recognize the significance of the lowpass-to-bandpass transformation
considered in (6.83). The importance of the bandpass frequency variable
F in (8.2) cannot be overemphasized; it will occur in nearly every selectivity
expression for direct-coupled filters.
The main parameter is the loaded Q of the Kth resonator:
R KK
QLK = X
K
' (8.3)
where R KK
= I/G KK
. For I volt across the resonator in Figure 8.4, it is easy to
see that Q is the reactive power divided by the real power. The reactive power
is stored in the resonator, and the real power is that which proceeds toward
the load as delivered to G KK
. The nodal parallel reactance X K in (8.3) is
determined at the midband geometric mean frequency wo:
X K
= _1_ =wOL K
• (8.4)
wOCK
These definitions are consistent with those in Section 6.1.3. This is singly
terminated loaded Q; it does not consider any resistive loading-real or
through the intervening circuit-that occurs on the source side of the resonator.
Finally, (8.1) may be put in terms of the loaded Q:
(8.5)
The ABCD chain parameters were defined in Section 4.2.1. The ABCD
parameters for the Klh resonator in Figure 8.4 are
(8.6)
These will be used in conjunction with those of the inverters to obtain
expressions for the overall ABCD matrix of the prototype network in Figure
8.1.
8,1.2. Ideal Inverters. The hypothetical lossless transmission line segments
in Figure 8.1 are defined to have frequency-independent characteristic impedance
Zoo and a constant quarter-wave length, where the ij subscripts denote
the adjacent nodes that they connect. These are variously called impedance or
admittance inverters because they invert impedances according to (6.38),
which is repeated:
This is shown in Figure 8.5.
(8.7)