This document discusses different types of filters including low-pass, high-pass, band-pass and band-stop filters. It describes how active filters using op-amps can overcome limitations of passive filters, providing advantages such as reduced size and cost. Single-pole active low-pass and high-pass filters are presented, which buffer the RC circuit to provide a zero output impedance and roll-off rate of -20dB per decade above the critical frequency.

1. Active Filters

2. IntroductionIntroduction
 Filters are circuits that are capable of passing signals
within a band of frequencies while rejecting or blocking
signals of frequencies outside this band.
This property of filters is also called “frequency
selectivity”.

3. Types of Filters
There are two broad categories of filters:
 An analog filter processes continuous-time signals
 A digital filter processes discrete-time signals.
The analog or digital filters can be subdivided into
four categories:
 Lowpass Filters
 Highpass Filters
 Bandstop Filters
 Bandpass Filters

4. Ideal Filters
Passband Stopband
Stopband Passband
Passband PassbandStopband
Lowpass Filter Highpass Filter
Bandstop Filter
PassbandStopband Stopband
Bandpass Filter
M(ω)
M(ω)
ω ω
ω ω
ω
c
ω
c
ω
c
1
ω
c
1
ω
c
2
ω
c
2

5. Analog Filter Responses
H(f)
ffc
0
H(f)
ffc
0
Ideal “brick wall” filter Practical filter

6. Filter can be also be categorized as passive or active..
Passive filtersPassive filters: The circuits built using RC, RL, or RLC
circuits.
Active filtersActive filters : The circuits that employ one or more
op-amps in the design an addition to
resistors and capacitors

7. Passive filters
Passive filters use resistors, capacitors, and inductors
(RLC networks).
To minimize distortion in the filter characteristic, it is
desirable to use inductors with high quality factors
practical inductors includes a series resistance.
 They are particularly non-ideal
 They are bulky and expensive

8. Active filters overcome these drawbacks and are
realized using resistors, capacitors, and active
devices (usually op-amps) which can all be
integrated:
 Active filters replace inductors using op-amp based
equivalent circuits.

9. Advantages
Advantages of active RC filters include:
 reduced size and weight
 increased reliability and improved performance
 simpler design than for passive filters and can realize a wider
range of functions as well as providing voltage gain
 in large quantities, the cost of an IC is less than its passive
counterpart

10. Disadvantages
Active RC filters also have some disadvantages:
 limited bandwidth of active devices limits the highest
attainable frequency (passive RLC filters can be used up
to 500 MHz)
 require power supplies (unlike passive filters)
 increased sensitivity to variations in circuit parameters
caused by environmental changes compared to passive
filters
For many applications, particularly in voice and data
communications, the economic and performance
advantages of active RC filters far outweigh their
disadvantages.

11. Bode Plots
Bode plots are important when considering the
frequency response characteristics of amplifiers.
 They plot the magnitude or phase of a transfer
function in dB versus frequency.

12. Bode plots use a logarithmic scale for
frequency.
where a decade is defined as a range of
frequencies where the highest and lowest
frequencies differ by a factor of 10.
10 20 30 40 50 60 70 80 90 100 200
One decade

13. The decibel (dB)
Two levels of power can be compared using a
unit of measure called the bel.
The decibel is defined as:
1 bel = 10 decibels (dB)
1
2
10log
P
P
B =

14. A common dB term is the half power point
which is the dB value when the P2 is one-
half P1.
1
2
10log10
P
P
dB =
dBdB 301.3
2
1
log10 10 −≈−=

15. 15
Decibel (dB) By Definition: 





=
1
2
10log10
P
P
dB
(1) Power Gain in dB :






=
in
o
p
P
P
dBA 10
log10)(






=
in
in
P
P
dB 10
log100










=−
in
in
P
P
dB 2
1
log103 10






=+
in
in
P
P
dB
2
log103 10
Pin Pout
(2) Voltage Gain in dB: (P=V2
/R)
vin vout






=
in
o
v
v
v
dBA 10
log20)(






=
in
in
v
v
dB 10
log200










=−
in
in
v
v
dB 2
1
log206 10






=+
in
in
v
v
dB
2
log206 10

16. 16
Cascaded System
Av1 Av2 Av3
x10 x10x10
vin
vout
20dB 20dB 20dB
321 vvvv
AAAA ××=
3
10101010 =××=v
A
( )32110
log20)( vvvv
AAAdBA ××=
( ) ( ) ( )310210110
log20log20log20)( vvvv
AAAdBA ++=
( ) ( ) ( )dBAdBAdBAdBA vvvv 321
)( ++=
dBdBdBdBAv
202020)( ++=
dBdBAv
60)( =
( ) dB2010log20 10
=
( ) dB6010log20 3
10
=

17. Poles & Zeros of the transfer
functionpole—value of s where the denominator goes to
zero.
zero—value of s where the numerator goes to zero.

18. Actual response
Vo
 A low-pass filterlow-pass filter is a filter that passes frequencies from 0Hz to
critical frequency, fc and significantly attenuates all other frequencies.
Ideal response
 Ideally, the response drops abruptly at the critical frequency, fH
roll-off rateroll-off rate

19. StopbandStopband is the range of frequencies that have the most attenuation.
Critical frequencyCritical frequency, ffcc, (also called the cutoff frequency) defines the
end of the passband and normally specified at the point where the
response drops – 3 dB (70.7%) from the passband response.
PassbandPassband of a filter is the
range of frequencies that are
allowed to pass through the
filter with minimum attenuation
(usually defined as less than
-3 dB of attenuation).
Transition regionTransition region shows
the area where the fall-off
occurs.
roll-off rateroll-off rate

20.  At low frequencies, XC is very high and the capacitor circuit can be
considered as open circuit. Under this condition, Vo = Vin or AV = 1
(unity).
 At very high frequencies, XC is very low and the Vo is small as
compared with Vin. Hence the gainfalls and drops off gradually as the
frequency is increased.
Vo

21.  The bandwidthbandwidth of an idealideal low-pass filter is equal to ffcc:
cfBW =
The critical frequency of a low-pass RC filter occurs when
XXCC = R= R and can be calculated using the formula below:
RC
fc
π2
1
=

22.  A high-pass filterhigh-pass filter is a filter that significantly attenuates or rejects
all frequencies below fc and passes all frequencies above fc.
 The passband of a high-pass filter is all frequencies above the
critical frequency..
Vo
Actual response Ideal response
 Ideally, the response rises abruptly at the critical frequency, fL

23.  The critical frequency of a high-pass RC filter occurs when
XXCC = R= R and can be calculated using the formula below:
RC
fc
π2
1
=

24.  A band-pass filterband-pass filter passes all signals lying within a band
between a lower-frequency limitlower-frequency limit and upper-frequency limitupper-frequency limit
and essentially rejects all other frequencies that are outside
this specified band.
Actual response Ideal response

25.  The bandwidth (BW)bandwidth (BW) is defined as the differencedifference between
the upper critical frequency (fupper critical frequency (fc2c2)) and the lower criticallower critical
frequency (ffrequency (fc1c1)).
12 cc ffBW −=

26. 21 cco fff =
 The frequency about which the pass band is centered is called
the center frequencycenter frequency, ffoo and defined as the geometric mean of
the critical frequencies.

27.  Band-stop filterBand-stop filter is a filter which
its operation is oppositeopposite to that of
the band-pass filter because the
frequencies withinwithin the bandwidth
are rejectedrejected, and the frequencies
above ffc1c1 and ffc2c2 are passedpassed.
Actual response
For the band-stop filter, the
bandwidthbandwidth is a band of
frequencies between the 3 dB
points, just as in the case of the
band-pass filter response.
Ideal response

28. RC
fc
π2
1
=
cXR =
 Figure below shows the basic Low-Pass filter circuit
Cf
R
cπ2
1
=
C
R
cω
1
=
At critical frequency,
Resistance = Capacitance
So, critical frequency ;

29. RC
fc
π2
1
=
cXR =
 Figure below shows the basic High-Pass filter circuit :
Cf
R
cπ2
1
=
C
R
cω
1
=
At critical frequency,
Resistance = Capacitance
So, critical frequency ;

30. Single-Pole Passive Filter
First order low pass filter
Cut-off frequency = 1/RC rad/s
Problem : Any load (or source) impedance will
change frequency response.
vin vout
C
R
RCs
RC
sCR
sCR
sC
ZR
Z
v
v
C
C
in
out
/1
/1
1
1
/1
/1
+
=
+
=
+
=
+
=

31. Ref:080222HKN EE3110 Active Filter (Part 1)31
Bode Plot (single pole)






+
=
+
=
o
j
CRj
jH
ω
ωω
ω
1
1
1
1
)(
2
1
1
)(






+
=
o
jH
ω
ω
ω














+==
2
1010
11log20)(log20)(
o
dB
jHjH
ω
ω
ωω
⇒






−≈
o
dB
jH
ω
ω
ω 10
log20)(
For ω>>ωo
R
C VoVi
Single pole low-pass filter

32. 32
dB
jH )( ω
(log)ω
x
ω x
ω2 x
ω10
6dB20dB
slope
-6dB/octave
-20dB/decade






−≈
o
jH
ω
ω
ω 10log20)(
For octave apart,
1
2
=
o
ω
ω
dBjH 6)( −≈ω
For decade apart,
1
10
=
oω
ω dBjH 20)( −≈ω

33. Single-Pole Active Low-Pass Filter
Same frequency response as passive filter.
Buffer amplifier does not load RC network.
Output impedance is now zero.
vin
vout
C
R

34. Single-pole active low-pass filter and response curve.
 This filter provides a roll-off rate of -20 dB/decade above the
critical frequency.

35. The op-amp in single-pole filter is connected as a
noninverting amplifier with the closed-loop voltage gain in the
passband is set by the values of R1 and R2 :
1
2
1
)( +=
R
R
A NIcl
 The critical frequency of the single-pole filter is :
RC
fc
π2
1
=

36.  The critical frequencycritical frequency, ffcc is determined by the values of R
and C in the frequency-selective RC circuit.
 Each RCRC set of filter components represents a polepole.
 Greater roll-off ratesGreater roll-off rates can be achieved with more polesmore poles.
 Each pole represents a -20dB/decade-20dB/decade increase in roll-off.
One-pole (first-order) low-pass filter.

37.  In high-pass filters, the roles of the capacitorcapacitor and resistorresistor are
reversedreversed in the RC circuits as shown from Figure (a). The negative
feedback circuit is the same as for the low-pass filters.
 Figure (b) shows a high-pass active filter with a -20dB/decade roll-off
Single-pole active high-pass filter and response curve.

38. The op-amp in single-pole filter is connected as a
noninverting amplifier with the closed-loop voltage gain in the
passband is set by the values of R1 and R2 :
1
2
1
)( +=
R
R
A NIcl
The critical frequency of the single-pole filter is :
RC
fc
π2
1
=

39. The number of poles determines the roll-off rate of the filter.
 A Butterworth response produces -20dB/decade/pole
This means that:
 One-pole (first-order)One-pole (first-order) filter has a roll-off of -20 dB/decade
 Two-pole (second-order)Two-pole (second-order) filter has a roll-off of -40
dB/decade
 Three-pole (third-order)Three-pole (third-order) filter has a roll-off of -60
dB/decade

40.  The number of filter poles can be increased by cascadingcascading.
To obtain a filter with three poles, cascade a two-pole with
one-pole filters.
Three-pole (third-order) low/high pass filter.

41. 41
Two-Stage Band-Pass Filter
R2 R1
vin
C1
C2
Rf1
Rf2
C4 C3
R3
R4
+V
-V
vout
Rf3
Rf4
+
-
+
-
+V
-V
Stage 1
Two-pole low-pass
Stage 2
Two-pole high-pass
BW
f1
f2
f
Av
Stage 2
response
Stage 1
response
fo
BW = f2
– f1
Q = f0
/ BW

42. 42
Band-Stop (Notch) Filter
The notch filter is designed to block all frequencies that fall
within its bandwidth. The circuit is made up of a high pass
filter, a low-pass filter and a summing amplifier. The
summing amplifier will have an output that is equal to the
sum of the filter output voltages.
f 1
f 2
v in v out
Lowpass
filter
Highpass
filter
Summing
amplifier
Σ
-3dB{
f
f2
f1
Av(dB)
low-pass high-pass
Block diagram Frequency response

43. 43
Notch filter

44. 44
Transfer function H(jω)
Transfer
Function
)( ωjH
VoVi
)(
)(
)(
ω
ω
ω
jV
jV
jH
i
o=
)Im()Re( HjHH +=
22
)Im()Re( HHH +=

45. 45
Frequency transfer function of filter
H(jω)
HL
HL
o
o
o
o
ffffjH
fffjH
ffjH
ffjH
ffjH
ffjH
><=
<<=
>=
<=
>=
<=
and0)(
1)(
FilterPass-Band(III)
1)(
0)(
FilterPass-High(II)
0)(
1)(
FilterPass-Low(I)
ω
ω
ω
ω
ω
ω
responsephasespecificahas
allfor1)(
Filtershift)-phase(orPass-All(V)
and1)(
0)(
Filter(Notch)Stop-Band(IV)
fjH
ffffjH
fffjH
HL
HL
=
><=
<<=
ω
ω
ω

46. Advantages of active filters over passive filters (R, L, and C
elements only):
1. By containing the op-amp, active filters can be designed to
provide required gain, and hence no signal attenuationno signal attenuation
as the signal passes through the filter.
2. No loading problemNo loading problem, due to the high input impedance of
the op-amp prevents excessive loading of the driving
source, and the low output impedance of the op-amp
prevents the filter from being affected by the load that it is
driving.
3. Easy to adjust over a wide frequency rangeEasy to adjust over a wide frequency range without
altering the desired response.