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380 Other Direct Filter Design Methods
marked accordingly. Of course, the lowpass network may be transformed into
bandpass networks, as discussed in Section 6.5.
The maximum possible input SWR, SK' may be defined with respect to
R,,= R, the value of the resistors in Figure 9.42. Then (9.86) yields the
maximum possible input SWR. This is plotted in Figure 9.44.
9.6.1. Three-Pole Invulnerable Filter. The invulnerable filter in Figure 9.45
is considered because it is easily compared to a classical design when R = O.
The parameters defined in (9.93)-(9.95) also apply for this network. The
development is accomplished using y parameters:
[
-, (I '-'d')]
I (;i'd' . d w - +w
Yll = R I +w'd' +JiJ I +w'd' '
I (. d)
Y21= R JiJ '
where y"=Yll by symmetry. Then the real-part parameter is
(9.100)
(9.101)
d -3
0l.- w
- ( I +w'd'
)'
(9.102)
Again, the minimum attenuation of this invulnerable filter, according to
(9.92), increases at a rate of only 6 dB/octave for large w. Proceeding as
before for the derivative of 0l. with respect to decrement d, yields the
condition for maximizing L min at a given w. Th.e result is d = l/w, a function
of frequency. According to Guillemin (1957, p. 196), the symmetric lattice for
the network in Figure 9.45 is not reciprocal under any conditions. Therefore,
this three-pole network cannot have the constant-resistance property: this will
complicate the calculation of L min
for cascaded sections.
The selectivity response according to (9.92) and (9.102) is shown in Figure
9.46. The da'shed line bounds the family of decrement curves according to the
requirement of maximum L min • namely d= l/w.
o
Figure 9.45.
R
R
l_ C C
__ l
o
A three-pole lowpass invulnerable filter.