Download Active Filters (Chp 4)

Electronic Circuit
DKT 214
Chapter 4
Active Filters
By
Pn. Hazila Othman
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 lowerfrequency 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 bandpass 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.
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
must have the same gain.
Fig. 1-5: Comparative plots of 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) 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 non-inverting
configuration.
DF is determined by the negative
feedback and defined as
Fig. 1-6: Diagram of an active filter.
R1
DF  2 
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).
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 opamp 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 RBC ACB
(4-11)
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 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.
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.
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 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. 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 oCAo
(4-16)
Q
R2 
f oC
(4-17)
Q
R3 
2f oC (2Q 2  Ao )
(4-18)
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. 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.
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. 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.
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. 1-7: One-pole (first-order) high-pass filter.
Example-3:
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. 1-10: Basic Sallen-Key/second-order low-pass filter.
Example-4:
Determine the capacitance value required to produce a critical
frequency of 2.68k Hz if all the resistors in the RC low-pass circuits
are 1.8kΩ. Also select values for the feedback resistors to get a
Butterworth response.