CHAPTER 4:
ACTIVE FILTERS
OBJECTIVES:


Describe three types of filter response characteristics
and other parameters.
Describe and analyze the gain-versus-frequency
responses of basic types of filters.

Identify and analyze active low-pass filters.

Identify and analyze active high-pass filters.

Analyze basic types of active band-pass filters.

Describe basic types of active band-stop filters.
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.
Filter is a circuit that passes certain frequencies and
rejects all others. The passband is the range of frequencies
allowed through the filter. The critical frequency defines
the end (or ends) of the passband.
A low-pass filter is one that passes frequency from dc to fc and
significantly attenuates all other frequencies. The simplest low-pass
filter is a passive RC circuit with the output taken across C.
Vo
Fig. 1-1: 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
(4-1)
When XC = R, the critical frequency of a low-pass RC filter
can be calculated using the formula below:
1
fc 
2 RC
(4-2)
A high-pass filter is one that significantly attenuates or rejects all
frequencies below fc and passes all frequencies above fc. The simplest
low-pass filter is a passive RC circuit with the output taken across R.
Vo
Fig. 1-2: High-pass filter responses
The critical frequency for the high pass-filter also occurs
when XC = R, where
1
fc 
2 RC
(4-3)
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. The simplest
band-pass filter is an RLC circuit.
Fig. 1-3: 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
(4-4)
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
(4-5)
The quality factor (Q) of a band-pass filter is the ratio of
the center frequency to the bandwidth.
fo
Q
BW
(4-6)
The quality factor (Q) can also be expressed in terms of the
damping factor (DF) of the filter as
1
Q
DF
(4-7)
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. Its also
known as notch, band-reject or band-elimination filter
Fig. 1-4: General band-stop filter response.
EXAMPLE 1
A certain band-pass filter has a center frequency
of 15 kHz and a bandwidth of I kHz. Determine
the Q and classify the filter as narrow-band or
wide-band.
Answer: Q = 15
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. 1-5: Comparative plots of three
types of filter response characteristics.
must have the same gain.
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) primarily determines if the filter will
have a Butterworth, Chebyshev, or Bessel response.
This active filter consists of an
amplifier, a negative feedback
circuit and RC circuit.
The amplifier and feedback
are connected in a noninverting configuration.
DF is determined by the
negative feedback and defined
as
Fig. 1-6: Diagram of an active filter.
DF  2 
R1
R2
(4-8)
Parameter for Butterworth filters up to four poles are given in the
following table.
Notice that the gain is 1 more than this resistor ratio. For example, the gain
implied by the the ratio is 1.586 (4.0dB).
VALUES FOR THE BUTTERWORTH
RESPONSE
EXAMPLE 2
If resistor R2 in the feedback circuit of an active
single-pole filter of the type in figure below is
10kΩ, what value must R1 be to obtain a
maximally flat Butterworth response?
Answer: R1 = 5.86kΩ
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. 1-7: One-pole (first-order)
low-pass filter.
The above formula can be
used for both low-pass and
high-pass filters.
(4-9)
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
one-pole filters.
Fig. 1-8: 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. 1-9: 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
(4-10)
Sallen-Key is one of the most common configurations for a two-pole filter.
It is also known as a VCVS (voltage-controlled voltage source) filter.
There are two low-pass RC
circuits that provide a rolloff 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. 1-10: Basic Sallen-Key low-pass filter.
The critical frequency for the Sallen-Key filter is
1
fc 
2 RA RB C AC B
If RA = RB = R and CA = CB = C, the critical frequency can be
expressed as:
1
fc 
2RC
(4-11)
EXAMPLE 3
Determine the critical frequency of the Sallen-Key lowpass filter in Figure below, and set the value of R1 for
an appropriate Butterworth response.
Answer: fc = 7.23 kHz, R1 = 586Ω
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.
Fig. 1-11: Cascaded low-pass filter: third-order configuration.
Four-pole filter is obtained by cascading Sallen-Key (2-pole) filters.
Fig. 1-12: Cascaded low-pass filter: fourth-order configuration.
Example 4
Determine the cutoff frequency, the pass-band gain in dB, and the
gain at the cutoff frequency for the active filter of Fig. 1-7 with C =
0.022 μF, R = 3.3 kΩ, R1 = 24 kΩ, and R2 = 2.2 kΩ
Fig. 1-7: One-pole (first-order) low-pass filter.
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 lowpass filters.
Fig. 1-13: 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. 1-14: 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. 1-15: A six-pole high-pass filter consisting of three Sallen-Key two-pole
stages with the roll-off rate of -120 dB/decade.
Example 5
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. 1-7: One-pole (first-order) high-pass filter.
Filters that build up an active band-pass filter consist of a Sallen-Key HighPass filter and a Sallen-Key Low-Pass filter.
Fig. 1-16: Band-pass filter formed by cascading a two-pole high-pass 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. 1-17: The composite response curve of a high-pass filter and a lowpass filter.
The lower frequency, fc1 of the pass-band is calculated as follows:
1
f C1 
2 RA1 RB1C A1CB1
(4-12)
The upper frequency, fc2 of the pass-band is determined as follows:
fC 2
1

2 RA2 RB 2C A2CB 2
(4-13)
The center frequency, fo of the pass-band is calculated as follows:
fo 
f C1 f C 2
(4-14)
Multiple-feedback band-pass
filter is another type of filter
configuration.
The feedback paths of the
filter are through R2 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. 1-18: 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 o CAo
(4-16)
Q
R2 
f oC
(4-17)
Q
R3 
2f oC (2Q 2  Ao )
(4-18)
R2
Ao 
2R1
EXAMPLE 6
Determine the center frequency, maximum gain
and bandwidth for the filter in figure below.
Answer: fo = 736 Hz, Ao = 1.32, BW = 177 Hz
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. 1-19: State-variable filter.
Fig. 1-20: 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 
(4-19)
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. 1-21: 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. 1-22: Multiple-feedback
band-stop filter.
This filter is similar to the
band-pass filter in Fig. 1-18
except that R3 has been
moved and R4 has been
added.
Summing the low-pass and the high-pass responses of the statevariable filter with a summing amplifier creates a state variable
band-stop filter.
Fig. 1-23: State-variable band-stop filter.
~ END OF CHAPTER 4 ~