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324 Direct-Coupled Filters
dependent values to see if any are out of bounds. A convenient and wellconditioned
measure of violations is the sum of squared differences between
squares of top-coupling reactance (in kilohms). The constraint on the variables,
namely the shunt inductors, is on their net values after combination of
adjacent shunt inverter inductances and possibly a transformer or L-section
inductance.
It may be possible to improve the design by adjustment of the shunt-L
values if there are violations of element bounds. Step 10 in Figure 8.28
represents this process, which involves N variables in the usual situation.
Adjustment of a particular shunt L, therefore that node's parallel resistance,
sometimes succeeds on a cut-and-try basis because of the designer's insight
into the seesaw impedance reaction of ideal (quarter-wave) inverters. Sometimes,
it is useful to construct a constrained optimization (nonlinearprogramming)
problem. This procedure can minimize the sum of squared
differences over the prototype shunt-L space with constraints on the final
shunt-L reactance values after combining adjacent inverters and, perhaps, an
L-section inductor. This may be accomplished by a conjugate gradient algorithm
incorporating the nonlinear constraints by penalty functions, as described
in Chapter Five. A simple enumeration search scheme may suffice
when only discrete-valued sets of inductors are available.
8.5.4. Sensitivities. Design step II in Figure 8.28 requires design adjustment
for dissipative effects, as described in Section 8.3.3. The physical suitability of
the resulting filter should then be determined by computing performance and
sensitivities according to step 12. This will determine if the component and
load tolerances are compatible with the performance expectations. Some
important sensitivities of direct-coupled filters are remarkably simple.
To derive the sensitivity of filter input impedance with respect to resonator
capacitance, consider the lossless prototype network at the tune frequency.
For N=3,
Similarly, for N = 4,
Z
- Z512 -Z2 y2 Z
in-R- 012 023 L"
22
(8.113)
(8.114)
Now consider a filter that has more than four resonators, but the interest is in
the fourth resonator (K= 4). Suppose that there is a slight surplus of node
capacity, say 8CK . Looking at the (8.114) result,
y L=GKK +j"'o8CK= GKK(I +j8QLK)' (8.115)
Then the perturbed input impedance is
8Zin=jZ5GKK8QLK' (8.116)
Therefore, (4.82) yields the direct-coupled-filter sensitivity of input impedance