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Four Important Passband Shapes 307
There is often a need to locate other points at F = F, on the curve in Figure
8.24 given the loss value L,. This may be accomplished by using
F = F,
p cosh[(l/N)cosh I(t/e)]'
(8.65)
where a new parameter, similar to e in (8.62), is
(8.66)
Example 8.5a. Suppose that the Chebyshev passband ripple is L p
= 0.1 dB
over bandwidth F p
=0.15, and the stopband loss is L,=30 dB at F,=0.195.
Find N and, using the next higher integer, find the frequencies where L=3
dB. Solving (8.64) yields N = 7.9668. Using N = 8 in (8.65) yields F, = 0.158.
Summarizing, when N = 8 there is a OJ-dB ripple over the 15% passband
width, 3 dB at the edges of a 15.8% bandwidth, and 30 dB at the edges of a
19.5% bandwidth.
The 2N zeros of the Chebyshev function in (8.61) are shown on the vertical
ellipse in Figure 8.23, where the dimensioning parameter is
The location r m
+ji m
and
I . h-I I
a= N sm eO
of the mth zero is
r m
=FpS N sin(2m-I)8,
im=Fp,jS~+I cos(2m-I)8,
(8.67)
(8.68)
(8.69)
(8.70)
In these equations, m= 1,2, ...,N/2 when N is even, and m= 1,2, ... ,
(N + 1)/2 when N is odd. Using N left-half-plane factors to create a polynomial
in (F/F p
) results in forms like (8.15). Green (1954) tabulated the
coefficient expressions through N = 5 and guessed the recursive relationships
for the general case. The coefficients of like powers of jF were compared, and
a general relationship for successive element values was obtained. The latter
result is equivalent to (6.72). In the present case, (8.16) and (8.17) showed that
loaded-Q parameters could be identified. The recursion for loaded-Q values
for the Chebyshev overcoupled case is given in Appendix G. A typical set of
values is shown in Table 8.3 for lossless and lossy sources (see Figure 8.1).