5. Second order Low-pass Frequency Response
For Q=0.707,magnitude response is maximally flat (Butterworth Filter: Maximum bandwidth without peaking)
For Q>0.707, response shows undesired peaking.
For Q<0.707: Filter’s bandwidth capability is wasted.
At w<<wo, filter has unity gain.
At w>>wo response exhibits two-pole roll-off at -40dB/decade.
At w=wo, gain of filter =Q.

6. Second order Low-pass Frequency Response

8. Second order High-pass Frequency Response
For Q=0.707,magnitude response is maximally flat (Butterworth Filter response).
Amplifier gain is constant at w>wo, the lower cutoff frequency of the filter.

9. Second-order Band-pass Frequency Response
Response peaks approximately at wo.
At w<<wo or w>>wo, filter response corresponds to single-pole high-pass or low-pass filter changing at a rate of 20dB/decade.

13. Single amplifier Biquad (SAB)
Enhanced Positive Feedback (EPF)
Enhanced Negative Feedback (ENF)

14. Sallen-Key Low-pass Filter (1955)

15. Sallen-Key using voltage follower
The transfer function is:
Op amp is voltage follower with unity gain over a wide range of frequencies.
Uses positive feedback through C1 at frequencies above dc to realize complex poles without inductors.
V1 and V2 are parasitic sensitive nodes with a large voltage swing
Large capacitor spread for large Q
In standard form,
For equal R design

16. Equal-R & Equal-C design (K≠1)
Can get high Q without high Capacitor spread
Condition: K < 3
Rarely used design due to high sensitivity !!

17. Accuracy of integrated capacitors

18. Frequency denormalization

19. Impedance denormalization

20. Equal-R & k=1 Sallen-Key LPF for Q=√2
Example:
Equal-R & k=1 Sallen-Key LPF for Q=√2
Frequency
Denormalization
1MHz
Impedance
Denormalization
10kΩ

21. Sallen-Key High-pass Filter

22. Enhanced positive feedback general structure

24. Multiple Feedback SAB
One virtual ground node with no voltage swing

25. BP Multiple Feedback SAB

26. Enhanced negative feedback ENF
Equal-C & a=1
Deliyannis SAB (1968)
Q enhancement

28. Need to have Feedforward paths to implement notch filters
Friend SAB (1970)
Need to have Feedforward paths to implement notch filters