download searchable PDF of Circuit Design book - IEEE Global ...

356 Other Direct Filter Design Methods
which further reduces to
[(h 2 - hi) + (L2I - L,IlK II ]2
K 22 = 2
(h2-hl)/K21-(L2I-Lll) K II
Amstutz (1978) rearranges (9.49) into the last of four main expressions:
(9.49)
B 22 = V 2 B21 - (V+ 1)2B Il . (9.50)
The calculation illustrated in Figure 9.17 proceeds as follows: obtain L II
and L21 according to (9.35) and then compute U from (9.44). Compute B 'I
and B2I according to (9.37) and then compute V from (9.46). Thus L 22 may be
obtained from (9.45), and B 22 from (9.50). Repeating, the input trap-section
values in Figure 9.17 are determined from the known input impedances and
slopes at the trap frequencies. Then (9.44), (9.46), (9.45), and (9.50) are used,
in that order, to determine the three values of the second trap section in
Figure 9.17a.
Example 9.3. The element values of a type-c (antimetric) filter are shown in
Figure 9.18. A ladder network analysis program similar to Program B4-l was
used to compute the input impedance and its approximate frequency derivative
(group delay) using 0.01% finite differences, as described in Section 4.7.2.
Table 9.1 shows these values at the three trap frequencies. To find L 4 , M 4 , and
C4, begin by using (9.35)-(9.37) and the w, data in Table 9.1 to compute
L2= 1.0915, M2=0.11898I, and C2= I/B2= 1.377177. A similar calculation
using (9.35)-(9.37) and the W 4 data in Table 9.1 yields L41 = Z4/S4 =0.842907,
M41 ~(Zl- L 41)/2 = 0.455362, and B41 = M41W2 =0.990429. Therefore, with appropriate
changes in subscripts, (9.44) yields
Then (9.46) yields
U = L2 - L 41 = 0.248593. (9.51)
V-I = U 1=0.156978, (9.52)
(w 4
2_"'2 2)B 2
or V = 6.370313. Finally, (9.45) yields L 4 = UV = 1.5836, and (9.50) yields
0-
1.0915 W2
1.5849
w 1.3973 w 1.3500
4 , w,
I
L, • L, L, L,
0.11900 0.34829
. 0.23101 1.2034
_L1.3Z00 1.5233
f Z;70
;:: c, : :=: c,
I
c~ :: :=: 1
(.0)2 '" 2.4704 w 4 '" 1.4748 w3 '" 1.6858 w, ---> 00
Figure 9.18. A type-c antimetric elliptic filter appearing in Example 9.3.