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.

Second order Low-pass Frequency Response

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.

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.

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

Sallen-Key Low-pass Filter (1955)

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

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 !!

Accuracy of integrated capacitors

Frequency denormalization

Impedance denormalization

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

Sallen-Key High-pass Filter

Enhanced positive feedback general structure

Multiple Feedback SAB One virtual ground node with no voltage swing

BP Multiple Feedback SAB

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

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