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Second order Low-pass Frequency Response

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 << w o , filter has unity gain.

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Second order Low-pass Frequency Response

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  1. 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>>woresponse exhibits two-pole roll-off at -40dB/decade. At w=wo, gain of filter =Q.

  2. Second order Low-pass Frequency Response

  3. 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.

  4. 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.

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

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

  7. 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

  8. 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 !!

  9. Accuracy of integrated capacitors

  10. Frequency denormalization

  11. Impedance denormalization

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

  13. Sallen-Key High-pass Filter

  14. Enhanced positive feedback general structure

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

  16. BP Multiple Feedback SAB

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

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

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