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338 Other Direct Filter Design Methods
where P is a polynomial in variable q':
q=cosO. (9.3)
Angle 0 is the filter component electrical length at any frequency f:
0=I fa . (9.4)
It is easy to show that such functions have arithmetic symmetry according to
f, f,
-=2-- (9.5)
f o f o
between any two frequencies (f, and f,) having the same selectivity. The
polynomial P must be obtained for each particular case; it will be obtained
here for the network in Figure 9.1 c.
Storch's matrix result was applied to replicated filter subsections in Section
8.4.5. In the case shown in Figure 9.lc, the typical section to be replicated is
shown in Figure 9.3. To find the ABCD matrix for the typical subnetwork,
consider the two cascade lines on both sides of the shunt resonator as three
cascaded sections. Because of their symmetry, the ABCD matrix of the
subsection in Figure 9.3 is:
(9.6)
where
and
o
A=cos2" '
B .. 0
=JSI0
2 ,
Y= -jKcotO.
(9.7)
(9.8)
(9.9)
K is the characteristic admittance of the shorted-stub resonator. Therefore, the
ABCD matrix of the ith subsection is
T= [(ABY+A'+B')
, (A'Y+ 2AB)
(B'Y +2AB) ]
(ABY+A'+B') .
(9.10)
Storch's matrix result in (8.90) (Section 8.4.5) can be applied to obtain the
elements of the overall ABCD matrix TN' For lossless, reciprocal, symmetric
networks, (8.96) reduces to
(B C)'
IS'II'=I- N~ N (9.11)