"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

Fundamentals of Low-Pass Filters
16-10
|A| — Gain — dB
40
30
20
10
0
–10
–20
1st Stage
2nd Stage
3rd Stage
4th Stage
Overall Filter
5th Stage
–30
0.01 0.1
Frequency — Ω
1 10
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 Q5: Q5 [dB] 20·logQ5 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 a i s b i s 2
Q5
(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.