LICA-

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

Passive elements : Inductor
BIG PROBLEM!
 High accuracy (1% or 2%), small physical size, or large
inductance values are required ??
 Standard values of inductors are not very closely spaced
 Difficult to find an off-the-shelf inductor within 10 percent
of any arbitrary value .
 Adjustable inductors are used.
 Tuning such inductors to the required values is time-
consuming and expensive for larger quantities of filters.
 Inductors are often prohibitively expensive.

Active Filter
 No inductors
 Made up of op-amps, resistors and capacitors
 Provides virtually any arbitrary gain.
 Generally easier to design.
 High input impedance prevents excessive loading of the
driving source .
 Low output impedance prevents the filter from being
affected by the load
 At high frequencies is limited by the gain-bandwidth of
the op-amps
 Easy to adjust over a wide frequency range without
altering the desired response

Advantages of Active Filters over Passive Filters
 Active filters can be designed to provide required gain,
and hence no attenuation as in the case of passive filters
 No loading problem, because of high input resistance
and low output resistance of op-amp.
 Active Filters are cost effective as a wide variety of
economical op-amps are available.

Applications
 Active filters are mainly used in communication and signal
processing circuits.
 They are also employed in a wide range of applications
such as entertainment, medical electronics, etc.

Active Filters
1. Low-pass filters
2. High-pass filters
3. Band-pass filters
4. Band-reject filters
5. All-pass filters
 Each of these filters can be built by using op-amp as the
active element combined with RC, RL or RLC circuit as the
passive elements.
 There are 4 basic categories of active filters:

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

Stopband is the range of frequencies that have the most attenuation.
Critical frequency, fc, (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.
Passband 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 region shows the
area where the fall-off occurs.
roll-off rate

 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 gain falls and drops off gradually
as the frequency is increased.
Vo

 The bandwidth of an ideal low-pass filter is equal to fc:
cfBW 
The critical frequency of a low-pass RC filter occurs when
XC = R and can be calculated using the formula below:
RC
fc
2
1


 A high-pass filter is a filter that significantly attenuates or rejects
all frequencies below fL and passes all frequencies above fL.
 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

 The critical frequency of a high-pass RC filter occurs when
XC = R and can be calculated using the formula below:
RC
fc
2
1


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

 The bandwidth (BW) is defined as the difference between
the upper critical frequency (fc2) and the lower critical
frequency (fc1).
12 cc ffBW 

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

 The quality factor (Q) of a band-pass filter is the ratio of
the center frequency to the bandwidth.
BW
f
Q o

 The quality factor (Q) can also be expressed in terms of the
damping factor (DF) of the filter as :
DF
Q
1

 The higher value of Q, the narrower the bandwidth and the
better the selectivity for a given value of fo.
 (Q>10) as a narrow-band or (Q<10) as a wide-band

 Band-stop filter is a filter
which its operation is opposite to
that of the band-pass filter
because the frequencies within
the bandwidth are rejected, and
the frequencies above fc1 and fc2
are passed.
Actual response  For the band-stop filter,
the bandwidth is a band of
frequencies between the 3
dB points, just as in the
case of the band-pass filter
response.
Ideal response

 There are 3 characteristics of filter response :
i) Butterworth characteristic
ii) Chebyshev characteristic
iii) Bessel characteristic.
 Each of the characteristics is identified by the shape of the response
curve
Comparative plots of three types
of filter response characteristics.

 The damping factor (DF) of an active filter determines which
response characteristic the filter exhibits.
 This active filter consists of
an amplifier, a negative
feedback circuit and RC circuit.
 The amplifier and feedback
are connected in a non-inverting
configuration.
DF is determined by the
negative feedback and defined
as :
2
1
2
R
R
DF  General diagram of active filter

 The value of DF required to produce a desired response
characteristics depends on order (number of poles) of the filter.
 A pole (single pole) is simply one resistor and one capacitor.
 The more poles filter has, the faster its roll-off rate

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

 For a single-pole (first-order) filter, the critical frequency is :
RC
fc
2
1

 The above formula can be used for both low-pass and high-
pass filters.

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

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

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 attenuation
as the signal passes through the filter.
2. No 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 range without
altering the desired response.

RC
fc
2
1

cXR 
 Figure below shows the basic Low-Pass filter circuit
Cf
R
c2
1

C
R
c
1

At critical frequency,
Resistance = Capacitance
So, critical frequency ;

Low-Pass Filter
O
VF
f
i
O
j
A
cRj
R
R
jV
jV
jA









11
1
)(
)(
)( 1
First-order (one-pole)
Filter
VA
-20dB/decade
1
1
R
Rf

RC
f
2
1
0 
0 

0
1 1
1
1 1
1
i
f f
V
R R j c
V V
R R R
j c




   
       
    

 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
2
1


Ch7 Operational Amplifiers and Op Amp Circuits
7.3 Active Filter
Low-Pass Filter
2
)(31
1
)(
)(
)(
RCjRCj
A
jV
jV
jA VF
i
O





Second-order (two-pole) Filter
VA
-20dB/decade
1
1
R
Rf

RC
f
2
1
0 
0 

-40dB/decade

0
1
1 fR
V V
R

 
   
 
1
1
1
1 1
/ /
i
f
V
R j c
R
R R
j c j c

 

 
  
         
   

fRR
R
VV

 
3
3
7.3 Active Filter
Low-Pass Filter
Voltage-controlled voltage
source
(VCVS) filter A
For simplicity, RRR  21 CCC  21
0
2
0
2
1
)(1
)()3(1
f
f
Q
j
f
f
A
RCjRCjA
A
V
V
A
f
f
f
i
o



 



3
1
3
1
R
R
A
A
Q
f
f
f




f
f
o AV
R
R
VV 





 

3
1
fRR
R
VV

 
3
3
7.3 Active Filter
Low-Pass Filter
Voltage-controlled voltage
source
(VCVS) filter A
For simplicity, RRR  21
CCC  21
     fo AVVV  

21
 
 































cj
R
cj
R
cj
VV
cj
R
cj
cj
RR
cj
cj
RR
VV






1
//
1
1
1
1
1
//
1
1
//
12
01
f
f
o AV
R
R
VV 





 

3
1
Using super position:

Active high-pass filter
SS
S
F
S
S
F
S
SS
S
F
S
out
CRj
R
R
Cj
R
R
jA
Cj
RZ
Z
Z
V
V
jA





1
1
1
1
)(
1
)(






e.g. RF/RS=10; 1/RFCF=1
Max Amplification: RF/RS
High pass factor: 1/(1+ 1/jωRSCS)
Cut-off frequency: ωRSCS=1

 Sallen-Key is one of the most common configurations for a
second order (two-pole) filter.
Basic Sallen-Key low-pass filter.
 There are two low-pass
RC circuits that provide a
roll-off of -40 dB/decade
above fc (assuming a
Butterworth characteristics).
 One RC circuit consists of
RA and CA, and the second
circuit consists of RB and CB.

 The critical frequency for the Sallen-Key filter is :
BABA
c
CCRR
f
2
1

For RA = RB = R and CA = CB = C, thus the critical frequency :
RC
fc
2
1


21 586.0 RR 
 kR 5861
• Determine critical frequency
• Set the value of R1 for Butterworth response by giving
that Butterworth response for second order is 0.586
kHz
RC
fc 23.7
2
1


• Critical frequency
• Butterworth response given
R1/R2 = 0.586

 A three-pole filter is required to provide a roll-off rate of -60
dB/decade. This is done by cascading a two-pole Sallen-Key low-
pass filter and a single-pole low-pass filter.
Cascaded low-pass filter: third-order configuration.

Cascaded low-pass filter: fourth-order configuration.
 A four-pole filter is required to provide a roll-off rate of -80
dB/decade. This is done by cascading a two-pole Sallen-Key low-
pass filter and a two-pole Sallen-Key low-pass filter.

RC
fc
2
1

• Both stages must have the same fc. Assume equal-value of capacitor
F
Rf
C
c


033.0
2
1

CA1=CB1=CA2=CB2=0.033µf
• Determine the capacitance values required to produce a critical
frequency of 2680 Hz if all resistors in RC low pass circuit is
1.8k

RC
fc
2
1

cXR 
 Figure below shows the basic High-Pass filter circuit :
Cf
R
c2
1

C
R
c
1

At critical frequency,
Resistance = Capacitance
So, critical frequency ;

 In high-pass filters, the roles of the capacitor and resistor are
reversed 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.

 Components RA, CA, RB, and CB form the second order (two-
pole) frequency-selective circuit.
 The position of the resistors and capacitors in the frequency-
selective circuit are opposite in low pass configuration.
 The response
characteristics can be
optimized by proper
selection of the feedback
resistors, R1 and R2.
Basic Sallen-Key high-pass filter.
 There are two high-pass
RC circuits that provide a
roll-off of -40 dB/decade
above fc

 As with the low-pass filter, first- and second-order high-pass
filters can be cascaded to provide three or more poles and thereby
create faster roll-off rates.
A six-pole high-pass filter consisting of three Sallen-Key two-pole
stages with the roll-off rate of -120 dB/decade.
Sixth-order high-pass filter

R A 1
R B 1
R A 2 R B 2
C A 1
C B 1
C A 2
C B 2
R 1
R 2
R 3
R 4
V in
V out
Two-pole high-pass Two-pole low-pass
 Band-pass filter is formed by cascading a two-pole high-pass and
two pole low-pass filter.
 Each of the filters shown is Sallen-Key Butterworth configuration,
so that the roll-off rate are -40dB/decade.

A v (dB)
0
3
Low-pass response High-pass response
f c 1 f c 2f o
f
 The lower frequency fc1 of the passband is the critical frequency
of the high-pass filter.
 The upper frequency fc2 of the passband is the critical frequency
of the low-pass filter.

210 cc fff 
1111
1
2
1
BABA
c
CCRR
f


2222
2
2
1
BABA
c
CCRR
f


 The following formulas express the three frequencies of the
band-pass filter.
 If equal-value components are used in implementing each filter,
RC
fc
2
1


R1
R2
R3
C1
C2
Vin
Vout
 The low-pass circuit consists of
R1 and C1.
 The high-pass circuit consists of
R2 and C2.
 The feedback paths are through
C1 and R2.
 Center frequency;
  21231
0
//2
1
CCRRR
f



 The maximum gain, Ao occurs at the center frequency.
1
2
2R
R
Ao 
321
31
0
2
1
RRR
RR
C
f



ooCAf
Q
R
2
1 
 By making C1 = C2 =C, yields
 The resistor values can be found by using following formula
Cf
Q
R
o
2
)2(2 23
oo AQCf
Q
R




Ref:080222HKN EE3110 Active Filter (Part 1) 51
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.
f1
f2
vin vout
Low pass
filter
High pass
filter
Summing
amplifier

-3dB{
f
f2
f1
Av(dB)
low-pass high-pass
Block diagram Frequency response

A parallel op amp
bandreject filter
(a) The block diagram
(b) The circuit
Band reject Filters

R 1
R 2
R 3
R 4
C 1
C 2
Vin
Vout
 The configuration is similar to the band-pass version BUT R3
has been moved and R4 has been added.
The BSF is opposite of BPF in that it blocks a specific band of
frequencies

Measuring frequency response can be performed
with typical bench-type equipment.
 It is a process of setting and measuring frequencies
both outside and inside the known cutoff points in
predetermined steps.
 Use the output measurements to plot a graph.
 More accurate measurements can be performed with
sweep generators along with an oscilloscope, a
spectrum analyzer, or a scalar analyzer.

 The bandwidth of a low-pass filter is the same
as the upper critical frequency.
 The bandwidth of a high-pass filter extends
from the lower critical frequency up to the
inherent limits of the circuit.
 The band-pass passes frequencies between
the lower critical frequency and the upper
critical frequency.

 A band-stop filter rejects frequencies within
the upper critical frequency and upper critical
frequency.
 The Butterworth filter response is very flat
and has a roll-off rate of –20 B
 The Chebyshev filter response has ripples and
overshoot in the passband but can have roll-
off rates greater than –20 dB

 The Bessel response exhibits a linear phase
characteristic, and filters with the Bessel
response are better for filtering pulse
waveforms.
 A filter pole consists of one RC circuit. Each
pole doubles the roll-off rate.
The Q of a filter indicates a band-pass filter’s
selectivity. The higher the Q the narrower the
bandwidth.
 The damping factor determines the filter
response characteristic.

n Denominator of H(s) for Butterworth Filter
1 s + 1
2 s2 + 1.414s + 1
3 (s2 + s + 1)(s + 1)
4 (s2 + 0.765 + 1)(s2 + 1.848s + 1)
5 (s + 1) (s2 + 0.618s + 1)(s2 + 1.618s + 1)
6 (s2 + 0.517s + 1)(s2 + 1.414s + 1 )(s2 + 1.932s + 1)
7 (s + 1)(s2 + 0.445s + 1)(s2 + 1.247s + 1 )(s2 + 1.802s + 1)
8 (s2 + 0.390s + 1)(s2 + 1.111s + 1 )(s2 + 1.663s + 1 )(s2 + 1.962s + 1)

Normalized Butterworth Polynomials
)1932.1)(12)(1518.0(s6
)1618.1)(1618.01)(s(s5
)1848.1)(1765.0(s4
)11)(s(s3
)12(s2
1)(s1
222
22
22
2
2






sssss
sss
sss
s
s

Standard Transfer Functions Butterworth
 Flat Pass-band.
 20n dB per decade roll-off.
 Chebyshev
 Pass-band ripple.
 Sharper cut-off than Butterworth.
 Elliptic
 Pass-band and stop-band ripple.
 Even sharper cut-off.
 Bessel
 Linear phase response – i.e. no signal distortion in pass-band.

7.3 Active Filter
Basic Filter Responses
)(
)(
)(
sv
sv
sA
i
O
voltage gain
)()(
)(
)(
)( 


 jjA
jV
jV
jA
i
o



jS 
Basic Filter Responses
bandwidth
cutoff frequency
Transition region
stopband region
Low-Pass Filter
Filter .. Vo(t)Vi(t)

7.3 Active Filter
Low-Pass Filter
2
0
0
0
1
1
1
1
1
/1












 










j
j
cj
R
cj
V
V
A
i
O
  
450
0
1 


 tg
)
1
(
RC
O 

7.3 Active Filter
High-Pass Filter



o
i
O
j
Rcjcj
R
R
V
V
A





 


1
1
1
1
1
1

7.3 Active Filter
Advantages of Filter
L
L
L
L
L
L
iL
Li
cRj
R
R
cRj
R
cj
R
R
cj
R
VR
cj
R
cj
RV
A


























 




1
1
1
1
)//
1
(
)
1
//(
1
1
)1)((
RR
cRjR
RR
R
L
L
L
L




'
1'1
)/(
O
v
L
LL
j
A
cRj
RRR

 





)( LLV RRRA 

CRL
O
'
1
'where
RL
1|| max

A

7.3 Active Filter
High-Pass Filter
H
SRC
SRC
L A
SRC
RCj
A 

 



1
1
1
1
1
1

• Transfer functions:
• Circuit: R↔C
• Frequency domain

7.3 Active Filter
Band-Pass Filter
Low-Pass High-
Pass
iV
oV
A
Af AfAf
1
ωL ωH ωωH ω
1
A
ωL ω
A
Lower-frequency Upper-frequency
fA
A
fA
A
fA
A

7.3 Active Filter
Band-Stop Filter
A A
Af
ωh ωL ω ωh ωL ω
AfAf
A
1
Low-Pass
High-
Pass
iV
oV
fA
A
fA
A
fA
A

7.3 Active Filter
Example 2 For the circuit shown, show that what it is filter?
(a)
1// R
v
ZR
v i
Cf
O

o
VF
f
f
f
fCf
i
o
j
A
cRjR
R
cj
R
cj
R
R
R
ZR
V
V
A











 


1
1
1
1
1
1
//
1
11
The Inverting First-order Low-Pass Filter.

7.3 Active Filter
(b)
C
i
f
o
ZR
v
R
v


1



o
VF
f
f
i
C
f
i
i
o
j
A
CRjR
R
cj
R
R
V
ZR
R
V
V
V
A








 


1
1
/11
1
1
1
11
1
1
The Inverting First-order High-Pass Filter.

7.3 Active Filter
(c)
The Non-Inverting Band-Stop Filter(Second-order).

7.3 Active Filter
The Inverting Band-Pass Filter.
(Second-order)
The Inverting High-Pass Filter.
(Second-order)

Filter response is characterized by
flat amplitude response in the
passband.
Provides a roll-off rate of -20
dB/decade/pole.
Filters with the Butterworth
response are normally used when
all frequencies in the passband
must have the same gain.

 Filter response is characterized
by overshoot or ripples in the
passband.
 Provides a roll-off rate greater
than -20 dB/decade/pole.
 Filters with the Chebyshev
response can be implemented
with fewer poles and less
complex circuitry for a given
roll-off rate

Filter response is characterized by a
linear characteristic, meaning that the
phase shift increases linearly with
frequency.
Filters with the Bessel response are
used for filtering pulse waveforms
without distorting the shape of
waveform.

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