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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