1. Lecture 11
CHAPTER 4: Active Filter
1

2. Lecture 11
2. High Pass Filter
2
a) 1st
Order
AV (dB)
AV(max)
AV(max) - 3
fc
0.1fc
0.01fc
AV(max) - 20
AV(max) - 40
+20dB/dec
f (Hz)
 A high-pass filter is a filter that passes high frequencies well, but
attenuates (or reduces) frequencies lower than the cutoff frequency
 The actual amount of attenuation for each frequency varies from filter to
filter
 A high-pass filter is the opposite of a low-pass filter
 It is useful as a filter to block any unwanted low frequency components of
a complex signal while passing the higher frequencies
R1
R2
_
+
Vin
+V
-V
RA
CA
Vo

3. Lecture 11
Analysis
 It is high pass filter if
The numerator has s
V+
= VR
3
Vo = 1+ V+
V+
= Vi
RA
1
sCA
+RA
R1
R2
Vo = 1+
R1
R2
Vi
V+
= Vi
sRACA
sRACA+1
V+
= VRA
AV = = 1+
R1
R2Vi
Vo
R1
R2
_
+
Vin
+V
-V
RA
CA
Vo
sRACA
sRACA+1
sRACA
sRACA+1

4. Lecture 11
Analysis
 At 0 < f < fc, high pass filter will attenuate the
frequencies at roll-off of +20dB/decade because
XCA is decreasing starts from ∞. When XCA=∞,
V+
=VRA = 0
AV = 0
4
AV(max) = 1+
R1
R2
 At f = fc, the gain is
0.707AV(max) or in dB, AVdB(max)-3
where the magnitude of the
capacitive reactance, XCA
equals the resistance of the
resistor, RA
 At f > fc, high pass filter will
pass the frequencies
because XCA is decreasing
to 0Ω. When XCA= 0,
V+
=VRA=Vi
1
2πfcCA
= RA
fc = 1
2πRACA
AV (dB)
AV(max)
AV(max) - 3
fc
0.1fc
0.01fc
AV(max) - 20
AV(max) - 40
+20dB/dec
f (Hz)

5. Lecture 11
21
2
1
1 1 1 2 2 2
22
1 2 1 2 1 2 1 2
1
1 1
1
o
in
R
s
RV
V R
R C R C R C
R
s s
R R C C R R C C
 
+ ÷
 =
   
+ + − + ÷ ÷ ÷ ÷   + +
5
b) 2nd
Order (Sallen-Key)
Vo
R1
R2
_
+
Vin
+V
-V
RA
CA
CB
RB
AV (dB)
AV(max)
AV(max) - 3
fc
0.1fc
0.01fc
AV(max) - 40
AV(max) - 80
+40dB/dec
f (Hz)
AV(max) = 1+
R1
R2
1
2π RARBCACB
fc = fAfB
1
2πRACA
1
2πRBCB
•=
=

6. Lecture 11
6
R1=2.55kΩ, R2=4.3k Ω, RA=RB=1.6kΩ, CA=CB=0.01µF

7. Lecture 11
Higher Order High Pass Filter
7
Table for Butterworth response
3rd
order system
4th
order system
R1
R2
_
+
Vin
+V
-V
RA
CA
CB
RB
R1
R2
_
+
Vin
+V
-V
RA
CA
CB
RB
Vo
R3
R4
_
+
+V
-V
RC
CC
CD
RD
R3
R4
_
+
+V
-V
RC
CC
Vo

8. Lecture 11
3. Band Pass Filter
8
fo = fC1fC2
BW = fC2 – fC1
 The band-pass filter allows frequencies between fC1 and fC2 (fC1< f < fC2)
to pass while effectively blocking all others.
AV (dB)
AV(max)
AV(max) - 3
fo 10fC20.01fC1
AV(max) - 20
+20dB/dec
f (Hz)
fC1 fC2
-20dB/dec
fC1 < fC2
BW
(HP) (LP)
Cascaded Low Pass
and High Pass Filter

9. Lecture 11
9
HP
R1
R2
A
_
+
Vin
+V
-V
RA
CA
R3
R4
B
_
+
+V
-V
RB CB
Vo
R1
R2
A
_
+Vi
+V
-V
RA CA
R3
R4
B
_
+
+V
-V
RB
CB
Vo
LP
fcA < fcB
HP
LP
fcA > fcB

10. Lecture 11
10
RA=1kΩ, RB=10kΩ, R1=R2=R3=R4= 1kΩ, CA=CB=0.01µF
LP HPVi Vo

11. Lecture 11
11
Multiple-Feedback
Band-Pass Filter
Vo
+
-
Vin
+V
-V
RA
CA
Rf
CB
RB
fo =
1
2π (RA||RB)RfCACB
Q = πfoRf CACB
BW =
Q
fo
2
2
1
2
A B
o A
A Bin
f A B f C C
s
V R C
R RV s s
R C R R R C
=
= −
+
+ +

12. Lecture 11
12
RA=3.18kΩ, RB=352 Ω, Rf=31.8kΩ, CA=CB=0.01µF

13. Lecture 11
4. Band Stop Filter
13
Vo
+
-
Vin
+V
-V
R1
C2
Rf
C1
R3
R2
fo =
1
2π R1RfC1C2
Multiple-Feedback
Band-Stop Filter

14. Lecture 11
14
State-Variable
Band Stop Filter
_
+
Vin
Rf
R1 +V
-V
VoR2
LP
fC1
HP
fC2
fC1< fC2