7.Active Filters using Opamp

09/12/15 Active Filters by Prof.Satheesh MB, INA
Active Filters
Prof.Satheesh Monikandan B
HOD-ECE
sathy24@gmail.com

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”.
 Filter can be passive or active filter.
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

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:
– Low pass Filters
– High pass Filters
– Band stop Filters
– Band pass Filters

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

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

Passive Filters
• Made up of passive components - resistors, capacitors and
inductors
• No amplifying elements (transistors, op-amps, etc)
• No signal gain
• 1st
order - design is simple (just use standard equations to
find resonant frequency of the circuit)
• 2nd
order - complex equations
• Require no power supplies
• Buffer amplifiers might be required
●
Desirable to use inductors with high quality factors

Inductor - BIG PROBLEM!
• Physical size, and large inductance values are required.
• Tuning inductors to the required values is time-consuming
and expensive for larger quantities of filters.
• Often prohibitively expensive.
●
Difficult to implement at frequencies below 1 kHz.
●
Lossy

Active Filter
• No inductors
• Made up of op-amps, resistors and capacitors
• Provides 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
• Easy to adjust over a wide frequency range without altering the
desired response

ApplicationsApplications
 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.

Op Amp 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

Op Amp Disadvantages
• Active RC filters also have some disadvantages:
– limited bandwidth of active devices limits the highest
attainable pole frequency and therefore applications nearby
100 kHz (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.

Categories of Filters
-3dB {
f 2
f
A v(dB)
-3dB {
f 1
f
A v(dB)
Low-pass response High-pass response
Low Pass Filters:
Pass all frequencies from dc
up to the upper cutoff
frequency.
High Pass Filters:
Pass all frequencies that are
above its lower cutoff
frequency

Categories of Filters
-3dB {
f 2
f
A v(dB)
f 1
-3dB {
f
f 2f 1
A v(dB)
Band Pass Response Band Stop Response
Band Pass Filters:
Pass only the frequencies
that fall between its values
of the lower and upper
cutoff frequencies.
Band Stop (Notch) Filters:
Eliminate all signals within
the stop band while passing
all frequencies outside this
band.

Filter Response Characteristics
Av
Butterworth
Bessel
Chebyshev
f

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

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

 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 bandwidthbandwidth of an idealideal low-pass filter is equal to
ffcc:
BW= f c
The critical frequency of a low-pass RC filter occurs
when
XXCC = R= R and can be calculated using the formula
below:
f c=
1
2π RC

Passive single pole low pass filter
R
C VoVi
V o=
XC
X C + R
V i
V o=
1
jωC
1
jωC
+R
V i=
1
1+ jωCR
Vi
H ( jω)=
1
1+ j
ω
ω0
ωo=
1
RC
where
or
H ( s)=
ω0
s+ω0
s= jω
where
φ(ω)=−tan
−1
(ω
ω0
)

H ( jω)=
1
1+ jωCR
V o=
1
1+ jωCR
Vi
ω → 0 ⇒ |Vo
| = |Vi
| ← max.
value
ω → ∞ ⇒ |Vo
| = 0 ← min. value
⇒ |Vo
| = ??ω=
1
RC
V o=
1
1+ j
Vi
∣V o∣=
1
√12
+12
∣V i∣=
1
√2
∣V i∣
ωc=ωo=
1
RC
(cut-off frequency)
ω
c
ω
o
v
maxo
v
2
maxo
v
ω
c
ω
)( ωjH
2
1
1

Bode Plot (single pole)
H ( jω)=
1
1+ jωCR
=
1
1+ j
(ω
ωo)
∣H ( jω)∣=
1
√1+
(ω
ωo)
2
∣H ( jω)∣dB=20 log10∣H ( jω)∣=20 log10(1/
√1+
(ω
ωo
)
2
)
⇒
∣H ( jω)∣dB≈−20 log10(ω
ωo
)
For ω>>ωo
R
C VoVi
Single pole low-pass filter

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 vout
vin
=
ZC
R+ZC
=
1/ sC
R+1/sC
¿
1
sCR+1
=
1/ RC
s+1/ RC

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

 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.

 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)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

 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-pass filter.

 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

Advantages of active filters over passive filters
1. By containing the op-amp, active filters can be
designed to provide required gain, and hence nono
signal attenuationsignal 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.

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

 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 :
Acl (NI )=
R1
R2
+1
 The critical frequency of the single-pole filter is :
f c=
1
2π RC

 Sallen-KeySallen-Key is one of the most common configurations for
a second ordersecond order (two-pole) filter.
Basic Sallen-Key low-pass filter.
 There are two low-pass
RC circuits that provide
a roll-off of -40roll-off of -40
dB/decade above fdB/decade above fcc
(assuming a Butterworth
characteristics).
 One RC circuit consists
of RRAA and CCAA, and the
second circuit consists of
RRBB and CCBB.

 The critical frequency for the Sallen-Key filter is :
f c=
1
2π√RA RB CACB
For RA = RB = R and CA = CB = C, thus the critical
frequency :
f c=
1
2π RC

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

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

f c=
1
2π RC
• Both stages must have the same fc. Assume equal-value of
capacitor
C=
1
2πfc R
=0.033 μF
CA1=CB1=CA2=CB2=0.033
µf
• Determine the capacitance values required to produce a
critical frequency of 2680 Hz if all resistors in RC low pass circuit
is 1.8kΩ

Low-Pass and High-Pass
Designs
High Pass Low Pass
vout
vin
=
1
1
sCR
+1
=
1
1+sRC
sCR
¿
sRC
RC ( s+1/ RC )
=
s
( s+1/ RC )
vout
vin
=
1/ RC
s+1/ RC

 The critical frequency of a high-pass RC filter occurs
when
XXCC = R= R and can be calculated using the formula
below:
f c=
1
2π RC

f c=
1
2π RC
R=X c
 Figure below shows the basic High-Pass filter circuit :
R=
1
2πfc C
R=
1
ωcC
At critical frequency,
Resistance = Capacitance
So, critical frequency ;

 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.

 Components RRAA, CCAA, RRBB, and CCBB form the second ordersecond order
(two-pole) frequency-selective circuit.
 The position of the resistors and capacitors in the
frequency-selective circuit are oppositeopposite in low pass
configuration.
 The responseresponse
characteristicscharacteristics can be
optimized by proper
selection of the
feedback resistorsfeedback resistors, R1
and R2. Basic Sallen-Key high-pass filter.
 There are two high-
pass RC circuits that
provide a roll-off of -40roll-off of -40
dB/decade above fcdB/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 filtersix-pole high-pass filter consisting of three Sallen-Keythree Sallen-Key
two-poletwo-pole stages with the roll-off rate of -120 dB/decade-120 dB/decade.
Sixth-order high-pass filter

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

 The bandwidth (BW)bandwidth (BW) is defined as the differencedifference
between the upper critical frequency (fupper critical frequency (fc2c2)) and the
lower critical frequency (flower critical frequency (fc1c1)).
BW= f c2− f c1

f o=√f c1 f c2
 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.

 The quality factor (Q)quality factor (Q) of a band-pass filter is the
ratio of the center frequency to the bandwidth.
Q=
f o
BW
 The quality factor (Q) can also be expressed in
terms of the damping factor (DF) of the filter as :
Q=
1
DF
 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

OP AMP BANDPASS FILTERS
A bandpass filter consists of three separate components
1. A unity-gain low-pass filter whose cutoff frequency is wc2,
the larger of the two cutoff frequencies
2. A unity-gain high-pass filter whose cutoff frequency is wc1,
the smaller of the two cutoff frequencies
3. A gain component to provide the desired level of gain in
the passband.
These three components are cascaded in series. The resulting
filter is called a broadband bandpass filter, because the
band of frequencies passed is wide.

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.

f 0=√f c1 f c2f c1=
1
2π√RA1 RB1C A1CB1
f c2=
1
2π √RA2 RB2C A2CB2
 The following formulas express the three frequencies of
the band-pass filter.
 If equal-value components are used in implementing
each filter,
f c=
1
2π RC

 It has outputs for low-pass, high-pass, and band-
pass.
 The center frequency is set by the integrator RC
circuits.
 The critical frequency of the integrators usually
made equal
 The band-pass output peaks sharply the center
frequency giving it a high Q.

 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

OP AMP BANDREJECT FILTERS
Like the bandpass filters, the bandreject filter consists of
three separate components
• The unity-gain low-pass filter has a cutoff frequency of wc1,
which is the smaller of the two cutoff frequencies.
• The unity-gain high-pass filter has a cutoff frequency of wc2,
which is the larger of the two cutoff frequencies.
• The gain component provides the desired level of gain in the
passbands.
The most important difference is that these components are
connected in parallel and using a summing amplifier.

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

+
+vi
RL
RL
+
RH
RH
CH
+
vo
+
Ri
Ri
Rf
CL

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

Filter response is characterized
by flat amplitude responseflat 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 gainsame gain.

Filter response is characterized
by overshootovershoot or ripplesripples in the
passband.
 Provides a roll-off rate
greater than -20dB/decade/pole.
 Filters with the Chebyshev
response can be
implemented with fewerfewer
polespoles and less complexless complex
circuitrycircuitry for a given roll-off
Rate.

Filter response is characterized by
a linear characteristiclinear 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.

 The damping factor (DF)damping factor (DF) of an active filter determines which
response characteristic the filter exhibits.
 This active filter consists
of an amplifieran amplifier, a negativea negative
feedback circuitfeedback circuit and RCRC
circuitcircuit.
 The amplifier and
feedback are connected in a
non-inverting configurationnon-inverting configuration.
General diagram of active
filter

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

Pole
• A pole is nothing more than an RC circuit
• n-pole filter → contains n-RC circuit.

Frequency transfer function of filter
H(jω)
( I ) Low-Pass Filter
∣H ( jω)∣=1 f < f o
∣H ( jω)∣=0 f > f o
(II) High-Pass Filter
∣H ( jω)∣=0 f < f o
∣H ( jω)∣=1 f > f o
(III) Band-Pass Filter
∣H ( jω)∣=1 f L< f < f H
∣H ( jω)∣=0 f < f L and f > f H
(IV) Band-Stop (Notch) Filter
∣H ( jω)∣=0 f L< f < f H
∣H ( jω)∣=1 f < f L and f > f H
(V ) All-Pass (or phase-shift) Filter
∣H ( jω)∣=1 for all f
has a specific phase response

 Higher order active filters have multiple poles in
their transfer functions, resulting in a sharper
transition from the passband to the stopband and
thus a more nearly ideal frequency response.
Higher Order Op Amp Filters

 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 dB
 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.

7.Active Filters using Opamp

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