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Designing with L and C Inverters 285
reveal the inductive inverter equivalent to that in Figure 8.IOb. The inverter
Zo=wL12 ·
Figure 8.lOd is slightly less general. It assumes that no other inductance is
connected to nodes I and 2 and that each node is parallel resonated by
capacitors. In this special case, (8.9) can be used to show that
But the coefficient of flux-linkage coupling is defined to be
_ M
K I2----·
~LpL,
Therefore, when Figure 8.lOd applies,
K 12
=
I
VQL1QL2
( W O L I2)2L,L p
(w o Ld 2M2 (8.24)
(8.2S)
(8.26)
There are many filters involving more than two resonators where (8.26) is
valid; otherwise, the .coupling coefficient may still be calculated using Figure
8.10c and (8.2S).
8.2.3. All Accurate StopbamJ Selectivity Estimate. The asymptotic behavior
of loss function (8.14) is easily modified to account for the Zo frequency
dependence. Consider the role of the Z's in (8.11) and (8.12) for N =3 and
N = 4, respectively. In all cases, the Zo terms are factors in the coefficient of
the Nth-degreejF frequency variable. The necessary modification to (8.18) for
the upper stop band is
L = - 6 + 20 log IIQLK + N20 10giFI + (NMI - NCI)20 log fa dB, (8.27)
where L;;. 20 dB or f/f o ;;' 1.2 ensure the validity of the estimate. NMI stands
for the number of magnetic inverters, and NCI stands for the number of
capacitive inverters. Figure 8.11 illustrates how the prototype selectivity in
Figure 8.8 is affected when there are two more inductive than capacitive
inverters present. According to (8.27), it is possible to make NMI equal to
NCI, so that there is no inverter selectivity tilt in the stop band. However, a
certain amount of passband distortion will remain, even in that case. It will be
more severe for wider passband widths and for response shapes having ripple.
Program A8-1 in Appendix A computes (8.27) in a very flexible fashion. If
all but one variable in (8.27) are given, the remaining variable will be
calculated. Frequencies fifo and F are found implicitly by secant search; all
others can be solved explicitly. As indicated in the program documentation,
the loaded-Q product may be input directly or calcUlated from the sequence
of individual resonator loaded-Q values input one at a time.