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Designing with L and C Inverters 289
Note the difference in L, and L\. The nomenclature is significant; henceforth,
only prototype (synchronous resonator) reactances will have roman
subscripts. Table 8.1 shows the final element values for the band-center
frequency of 50 MHz. It is in the format for running analysis Program B4-1
(Section 4.1.4). Several results are provided in Table 8.1. The 90-degree nodal
phase relationship shown in Figure 8.12c was also confirmed.
8.2.5. Summary of De.,igning With Simple Lande Inverters. The ABCD
chain matrix for a pi arrangement of L's or C's was obtained for the case
where both shunt branches were negative elements. Comparison with the
ABCD matrix of a lossless, 90-degree transmission line revealed that these pi
inverters are 90 degrees long at all frequencies and have characteristic impedances
(Zo) that are proportional or inversely proportional to frequency for
Land C inverters, respectively. It was also shown that the magnetic transformer
equivalent circuit incorporates an inductive inverter with shunt inductances
left over on each side. When these shunt inductances constitute the
total synchronous nodal inductances, then the coupling coefficient K'j (flux
linkage) between windings is (QLiQLj)-1/2 between the coupled nodes.
The prototype asymptotic selectivity estimate that was connected with a
breakpoint graph in Section 8.1.4 was modified to account for inverter
frequency dependence. It turned out that each inductive inverter added 6
dB/octave to the upper stopband slope and subtracted that amount in the
lower stop band. Also, capacitive inverters affected the slope in the opposite
manner. This comprehensive estimate of stopband selectivity is valid for
frequencies greater than 1.2f o (geometrically symmetric about the tune frequency)
or for losses greater than about 20 dB. Program A8-l was furnished
to make the selectivity estimate from any set of dependent variables in order
to find the remaining variable. An extended example of program utilization
was included.
Finally, a complete design example for a three-resonator filter was furnished.
It contained all the major steps in direct-coupled filter design; subsequent
refinements will not alter this fundamental procedure. Design success
was confirmed by analysis using Program B4-1; the desired selectivity and
impedance match were obtained. The choice of loaded-Q distribution in the
ratios 1,2, 1 anticipated the maximally flat response described in Section 8.5.
However, any arbitrary loaded-Q distribution could have been used without
affecting the selectivity and impedance-matching outcome. Similarly, an arbitrary
choice of resistance level within the filter provided acceptable element
values. It was shown in Section 8.1.3 that such choices of resistance levels (and
related inverter impedances) do not affect the selectivity or response shape.
A rule was provided for checking any basic direct-coupled design: all the
L's and C's touching a node should resonate. This check and the use of an
analysis program justify carrying about five significant figures in calculations,
even thOUgh such accuracy has little meaning in the real world of physical
components. Otherwise, numerical or procedural mistakes are very easy to
overlook.