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Fou, Important Passband Shapes 317
the product of the resonator efficiencies, (8.37). In this case,
(
QL)N
L(wo) = - 10 10gIO 1- Q, dB. (8.101)
However, 10gIOx = (loglOe)(ln x), and Dwight (1961, p. 137) give the series
x 2 x 3
In(l-x)= -x- '2 - 3 - ... -. (8.102)
In this case, x = QdQ, is assumed to be small, so that
(8.103)
This valuable relationship is simple to compute and shows that the decibel
midband loss is inversely proportional to tbe resonator unloaded Q.
Furthermore, (8.99) may be solved for N,
N= 6+L,
2010g(QLF,) ,
(8.104)
and (8.103) may be substituted to obtain
L = 4.34(L, + 6)(QLF,)
o cc(Q""'L'"""F'"",)""'20""'I--'-og-'-c(Q';;'L'""F'"'-,)'
(8.105)
Differentiating (8.105) with respect to QLF, and setting this to zero gives a
minimum midband-loss condition of
10glO(QLF,) =0.434.
(8.106)
Putting this back into (8.104) yields
L,+6
No = 8.686 .
(8.107)
This is the optimal number of resonators for a minimum midband loss when
the stopband attenuation (LJ is specified. The nearest integer value would be
used, of course.
The singly terminated response function can be obtained using (8.14); it
applies to Figure 8.1 and includes the load conductance. There is no source
conductance in the singly terminated case. The overall ABCD matrix consists
of (8.94) for the resonators and inverters, but it must be postmultiplied by the
ABCD matrix for the unity-load conductance (see Section 4.2.1). The resulting
C element of the overall ABCD matrix for this case is
(8.108)