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

Four Important Passband Shapes 313
Table 8.6.
Fano ~injmum Lowpass·Overshoot Droop Values
N 2 3 4 5 6 7 8
L d (dB) 15.40 19.49 27.11 34.76 42.41 50.Q7
MaxL d 9.54 16.99 24.61 32.26 39.91 47.57 55.22
A typical set of values for the undercoupled, lossy-source case is shown in
Table 8.5. Note the available arbitrary choices that do not affect the response
shape: not only resistance levels, but also distributions for a given decibel
droop. However, two of each set in Table 8.5 simply turn the network end for
end. An undercoupled example for N=4 and a 6=dB droop is worked in
complete detail in Section 8.6.
Fano examined the step response of his lowpass filters and recommended
certain decibel droop values versus N for minimum overshoot. Although the
frequency mapping in (6.83) causes some distortion of group delay, and the
inverters will cause more, Fano's criterion for good transient response is a
useful guide. His equation for the recommended decibel droop applies for
N>2:
I + (sinh' 0.8814N)
L d = 1010g,o 3.28 dB. (8.86)
Some of these values are tabulated in Table 8.6 along with the maximum
possible droop values from (8.80).
8.4.4. Comparison ofElliptic Family Responses. The defining response equations
for the overcoupled, maximally flat, and undercoupled response shapes
may be compared to the ideal selectivity asymptote equation, represented in
(8.18), to identify the loaded-Q product (semilogarithmic breakpoint) in each
case. Ignoring the - 6-dB term, the loaded-Q product for overcoupled filters is
20 log IIQ LK = - N2010g F p
+ 2010ge+6(N - I) dB. (8.87)
The loaded-Q product for maximally flat filters is
2010gIIQLK = - N20 log F p
+ 2010ge dB. (8.88)
The loaded-Q product for undercoupled filters is
2010gTIQLK = - N20 log F d +2010gq +6(N-I) -2010g(sinh O.8814N) dB.
(8.89)
Table 8.7 compares the overcoupled and undercoupled cases to the Butterworth
case on the basis of (I) relative decibel selectivity for constant
passband width and (2) percentage of Butterworth passband width for a fixed
stopband selectivity (L,). It is noted that the defined Fano undercoupled
bandwidth is the decibel droop in Table 8.7. However, an arbitrary basis for
defining the undercoupled bandwidth is available using (8.82).