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Analogue Electronic 2
EMT 212
Chapter 5
Active Filters
By
En. Tulus Ikhsan Nasution
Filter is a circuit used for signal processing due to its
capability of passing signals with certain selected frequencies
and rejecting or attenuating signals with other frequencies.
This property is called 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 opamps in the design an addition to resistors
and capacitors.
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 electronic, etc.
Filters are usually categorized by the manner in which the
output voltage varies with the frequency of the input voltage.
In terms of general response, the four basic categories of
active filters are: low-pass, high-pass, band-pass, and bandstop.
A low-pass filter is a filter that passes frequencies from 0Hz to critical
frequency, fc and significantly attenuates all other frequencies.
Vo
Fig. 5-1: Low-pass filter responses
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.
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.
Vin
Vo
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.
The bandwidth of an ideal low-pass filter is equal to fc:
BW  f c
(5-1)
When XC = R, the critical frequency of a low-pass RC filter
can be calculated using the formula below:
1
fc 
2 RC
(5-2)
A high-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
Fig. 5-2: High-pass filter responses
The critical frequency for the high pass-filter also occurs when
XC = R, where
1
fc 
2 RC
(5-3)
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.
Fig. 5-3: General band-pass response curve.
The bandwidth (BW) is
defined as the difference
between the upper critical
frequency (fc2) and the
lower critical frequency
(fc1).
BW  f c 2  f c1
(5-4)
The frequency about which the passband is centered is called the
center frequency, fo, defined as the geometric mean of the
critical frequencies.
fo 
f c1 f c 2
(5-5)
The quality factor (Q) of a band-pass filter is the ratio of
the center frequency to the bandwidth.
fo
Q
BW
(5-6)
The quality factor (Q) can also be expressed in terms of the
damping factor (DF) of the filter as
1
Q
DF
(5-7)
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 outside
bandwidth are passed.
Fig. 5-4: General band-stop
filter 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.
The characteristics of filter response can be Butterworth, Chebyshev,
or Bessel characteristic.
Butterworth characteristic
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
Fig. 5-5: Comparative plots of
must have the same gain.
three types of filter response
characteristics.
Chebyshev characteristic
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.
Bessel characteristic
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.
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
Fig. 5-6: Diagram of an active filter.
R1
DF  2 
R2
(5-8)
The critical frequency, fc is determined by the values of R and C
in the frequency-selective RC circuit.
For a single-pole (first-order)
filter, the critical frequency is
1
fc 
2 RC
Fig. 5-7: One-pole (first-order)
low-pass filter.
(5-9)
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 -20 dB/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;
and so on.
The number of filter poles can be increased by cascading. To
obtain a filter with three poles, cascade a two-pole and onepole filters.
Fig. 5-8: 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.
Fig. 5-9: 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 close-loop voltage gain is set by the values of R1 and R2,
so that
Acl ( NI )
R1

1
R2
(5-10)
Sallen-Key is one of the most common configurations for a twopole 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.
Fig. 5-10: Basic Sallen-Key low-pass filter.
The critical frequency for the Sallen-Key filter is
1
fc 
2 RA RBC ACB
(5-11)
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 lowpass filter and a single-pole low-pass filter.
Fig. 5-11: Cascaded low-pass filter: third-order configuration.
Four-pole filter is obtained by cascading Sallen-Key (2-pole)
filters.
Fig. 5-12: Cascaded low-pass filter: fourth-order configuration.
In high-pass filters, the roles of the capacitor and resistor are
reversed in the RC circuits. The negative feedback circuit is the
same as for the low-pass filters.
Fig. 5-13: Single-pole active high-pass filter and response curve.
Components RA, CA, RB, and CB form the two-pole frequencyselective circuit.
The position of the resistors and capacitors in the frequencyselective circuit.
The response
characteristics can be
optimized by proper
selection of the feedback
resistors, R1 and R2.
Fig. 5-14: Basic Sallen-Key
high-pass filter.
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.
Fig. 5-15: A six-pole high-pass filter consisting of three Sallen-Key
two-pole stages with the roll-off rate of -120 dB/decade.
Filters that build up an active band-pass filter consist of a SallenKey High-Pass filter and a Sallen-Key Low-Pass filter.
Fig. 5-16: Band-pass filter formed by cascading a two-pole highpass and a two-pole low-pass filters.
Both filters provide the roll-off rates of –40 dB/decade, indicated
in Fig. 5-17.
The critical frequency of the high-pass filter, fC1 must be lower
than that of the low-pass filter, fC2 to make the center frequency
overlaps.
Fig. 5-17: The composite response curve of a high-pass filter and a
low-pass filter.
The lower frequency, fc1 of the pass-band is calculated as follows:
1
f C1 
2 RA1 RB1C A1CB1
(5-12)
The upper frequency, fc2 of the pass-band is determined as follows:
fC 2
1

2 RA2 RB 2C A2CB 2
(5-13)
The center frequency, fo of the pass-band is calculated as follows:
fo 
f C1 f C 2
(5-14)
Multiple-feedback band-pass
filter is another type of filter
configuration.
The feedback paths of the
filter are through R1 and C1.
R1 and C1 provide the lowpass filter, and R2 and C2
provide the high-pass filter.
The center frequency is given
as
1
fo 
2 ( R1 // R3 ) R2C1C2
Fig. 5-18: Multiple-feedback
band-pass filter.
(5-15)
For C1 = C2 = C, the resistor values can be obtained using the
following formulas:
Q
R1 
2f oCAo
(5-16)
Q
R2 
f oC
(5-17)
Q
R3 
2f oC (2Q 2  Ao )
(5-18)
The maximum gain, Ao occurs at the center frequency.
State-variable filter contains a summing amplifier and two opamp integrators that are combined in a cascaded arrangement to
form a second-order filter.
Besides the band-pass (BP) output, it also provides low-pass (LP)
and high-pass (HP) outputs.
Fig. 5-19: State-variable filter.
Fig. 5-20: General state-variable response curve.
In the state-variable filter, the bandwidth is dependent on
the critical frequency and the quality factor, Q is independent
on the critical frequency.
The Q is set by the feedback resistors R5 and R6 as follows:
1  R5 
Q    1
3  R6 
(5-19)
Biquad filter contains an integrator, followed by an inverting
amplifier, and then an integrator.
In a biquad filter, the bandwidth is independent and the Q is
dependent on the critical frequency.
Fig. 5-21: A biquad filter.
Band-stop filters reject a specified band of frequencies and
pass all others.
The response are opposite to that of a band-pass filter.
Band-stop filters are sometimes referred to as notch filters.
Fig. 5-22: Multiple-feedback
band-stop filter.
This filter is similar to the
band-pass filter in Fig. 5-18
except that R3 has been
moved and R4 has been
added.
Summing the low-pass and the high-pass responses of the
state-variable filter with a summing amplifier creates a state
variable band-stop filter.
Fig. 5-23: State-variable band-stop filter.
Example-1:
Determine the cutoff frequency, the pass-band gain in dB,
and the gain at the cutoff frequency for the active filter of
Fig. 5-7 with C = 0.022 μF, R = 3.3 kΩ, R1 = 24 kΩ, and R2
= 2.2 kΩ
Fig. 5-7: One-pole (first-order) low-pass filter.
Example-2:
Determine the cutoff frequency, the pass-band gain in dB,
and the gain at the cutoff frequency for the active filter of
Fig. 5-7 with C = 0.02 μF, R = 5.1 kΩ, R1 = 36 kΩ, and R2 =
3.3 kΩ
Fig. 5-7: One-pole (first-order) high-pass filter.
Example-2:
Determine the critical frequency, the pass-band gain, and damping
factor for the second-order low-pass active filter with the following
circuit components:
R1 = R2 = 3 kΩ,
C1 = C2 = 0.05 μF, R1 = 10 kΩ, R2 = 15 kΩ
Fig. 5-10: Basic Sallen-Key/second-order low-pass filter.