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 <<o, filter has unity gain.At >>o response exhibits two-pole roll-off at -40dB/decade.At =o, 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 >o, the lower cutoff frequency of the filter.

222

)(oQ

ossss

HPA

Second-order Band-pass Frequency Response

Response peaks approximately at o.

At <<o or >>o, filter response corresponds to single-pole high-pass or low-pass filter changing at a rate of 20dB/decade.

22)(

oQossoss

BPA

12 QoBW

o21.2

Single amplifier Biquad (SAB)

Enhanced Positive Feedback (EPF) Enhanced Negative Feedback (ENF)

Sallen-Key Low-pass Filter (1955)

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

The transfer function is:

21

21

1

212

21

21

)(sV)(oV)(

CC

GG

C

GGss

CC

GG

ss

sLPA

Sallen-Key using voltage follower

In standard form,

22

2)(

oQossos

LPA

2121

1CCRRo

21

21

2

1RR

RR

C

CQ

For equal R design

Equal-R & Equal-C design (K≠1)

2121

1CCRRo RCo

1

)3(1K

Q

RCKBW 3

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

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

One virtual ground node with no voltage swing

Multiple Feedback SAB

21

1RRCo

1

221R

RQ

BP Multiple Feedback SAB

Deliyannis SAB (1968)

Enhanced negative feedback ENF

Equal-C & a=1

Q enhancement

Friend SAB (1970)

Need to have Feedforward paths to implement notch filters