Op Amps for Everyone - The Repeater Builder's Technical ...

Fundamentals of Low-Pass Filters
40
30
20
|A| — Gain — dB
Overall Filter Q 5
10
0
1st Stage
–10 2nd Stage
3rd Stage
–20 4th Stage
5th Stage
–30
0.01 0.1 1 10
Frequency — Ω
Figure 16–10.
Graphical Presentation of Quality Factor Q on a Tenth-Order
Tschebyscheff Low-Pass Filter with 3-dB Passband Ripple
The gain response of the fifth filter stage peaks at 31 dB, which is the logarithmic value
of Q 5 :
Q 5
[dB] 20·logQ 5
Solving for the numerical value of Q5 yields:
Q 5
10 31
20 35.48
which is within 1% of the theoretical value of Q = 35.85 given in Section 16.9, Table 16–9,
last row.
The graphical approximation is good for Q > 3. For lower Qs, the graphical values differ
from the theoretical value significantly. However, only higher Qs are of concern, since the
higher the Q is, the more a filter inclines to instability.
16.2.5 Summary
The general transfer function of a low-pass filter is :
A(s)
A 0
i 1 ai s b i
s 2
(16–1)
The filter coefficients a i and b i distinguish between Butterworth, Tschebyscheff, and Bessel
filters. The coefficients for all three types of filters are tabulated down to the tenth order
in Section 16.9, Tables 16–4 through 16–10.
16-10