Active filters are filters which contain an active element like an opamp in addition to the passive components (resistors, capacitors and inductors).
Order of an active filter is determined by number of R-C sections (or poles) used in the filter, which for a few exceptions equals the number of capacitors.
Figure below shows the circuit of a first-order active low pass filter. As we can see from the figure, it comprises of a lag type R-C section connected to the non-inverting input of the opamp wired as a voltage follower.
First-order active low pass filter
At low frequencies, reactance offered by the capacitor is much larger than the resistance value. Therefore, the applied input signal appears at the output mostly unattenuated. At high frequencies, the capacitive reactance becomes much smaller than the resistance value thus forcing the output to be near zero. The cut-off frequency is given by
At the cut-off frequency, the capacitive reactance equals the resistance value and the output is 0.707 times the input. The gain rolls off at a rate of 6 dB per octave or 20 dB per decade beyond the cut-off frequency.
The circuit of a first-order active low pass filter with gain is given in figure below. The voltage gain is given by
First-order active low pass filter with gain
Figure below shows the circuit of a first-order active high pass filter. As we can see from the figure, it comprises of a lead type R-C section connected to the non-inverting input of the opamp wired as a voltage follower.
First-order active high pass filter
At low frequencies, reactance offered by the capacitor is much larger than the resistance value and therefore applied input signal does not appear at the output. At high frequencies, the capacitive reactance becomes much smaller than the resistance value, therefore almost the whole input appears at the output. The cut-off frequency is given by
At the cut-off frequency, the capacitive reactance equals the resistance value and the output is 0.707 times the input. The gain increases at a rate of 6 dB per octave or 20 dB per decade before the cut-off frequency.
The circuit of a first-order active high pass filter with gain is given in figure below. he voltage gain is given by
First-order active high pass filter with gain
Roll-off rate for an n-order filter is 6n dB per octave or 20n dB per decade.
Figure below shows the circuit of a single-order low pass active filter which employs the opamp in inverting amplifier configuration.
Single-order low pass active filter employing opamp in inverting amplifier configuration
Cut-off frequency is given by
Mid-band gain is given by
Figure below shows the circuit of a single-order high pass active filter which employs the opamp in inverting amplifier configuration.
First-order active high pass amplifier employing opamp in inverting amplifier configuration
Commonly used opamp filters include the Butterworth filter (or maximally flat filter), Chebychev, Bessel and Cauer filters.
Butterworth filter offers a relatively flat pass and stop band response but has the disadvantage of relatively sluggish roll-off.
Chebychev filters offer much faster roll-off but their pass band has ripple.
Cauer filters have fastest transition from pass-band to stop-band but has rippled pass and stop bands.
Figure below shows the circuit of a generalized form of a second-order Butterworth active filter
Generalized form of second-order Butterworth filter
If Z1 = Z2 = R and Z3 = Z4 = C, we get a second-order low-pass filter. The cut-off frequency is given by
Pass band gain (AF) is given by
If Z1 = Z2 = C and Z3 = Z4 = R, we get a second-order high-pass filter. The cut-off frequency is given by
Pass band gain (AF) is given by
Wide band-pass filters can be formed by cascading the high-pass and the low-pass filter sections in series. These filters are simple to design and offer large bandwidth.
Figure below shows the circuit diagram of a second order narrow band-pass filter.
Narrow band-pass filter
At very low frequencies, C1 and C2 offer very high reactance. As a result, the input signal is prevented from reaching the output. Therefore, the output is zero. At very high frequencies, the output is shorted to the inverting input, which converts the circuit to an inverting amplifier with zero gain. Again, the output is zero. Thus, at both very low and very high frequencies, the output is zero. At some intermediate band of frequencies, the gain provided by the circuit offsets the loss due to the potential divider R1–R3. Important mathematical expressions governing the design of the filter circuit are given below. Resonant frequency is given by
Where, Q is the quality factor. For C1 = C2 = C, the quality factor is given by
The voltage gain is
Figure below shows the circuit diagram of second-order band reject filter. As we can see from the figure, it employs a twin-T network that is connected in series with the non-inverting input of the opamp. A twin-T network offers very high reactance at the resonance frequency and very low reactance at frequencies off-resonance. This phenomenon explains the behavior of the circuit.
Second-order band reject filter
commercially available.