9.13 : Active Filters

Active filters are electronic circuits that use operational amplifiers (op-amps), resistors, and capacitors to filter out unwanted frequency components from a signal. A first-order low-pass active filter is designed to pass signals with a frequency lower than a certain cutoff frequency and attenuate frequencies higher than that cutoff frequency. The transfer function for a first-order low-pass active filter is:

Where A_f is the gain at low frequencies also known as the dc gain, s is the complex frequency, and ω_c is the corner or cutoff frequency. A first-order high-pass active filter allows signals above a certain cutoff frequency to pass while attenuating signals below this frequency. Its transfer function can be expressed as:

Where A_f represents the gain at high frequencies, and ω_c is the cutoff frequency below which the output is significantly reduced. A band-pass filter has combined properties of low-pass and high-pass filters. It permits only a certain range of frequencies to pass through while attenuating frequencies outside this range. The transfer function of a band-pass filter has the form:

Here ω₀ is the center frequency, and Q is the quality factor. A band-reject filter, also known as a notch filter, does the opposite by allowing most frequencies to pass, except for a specific range that it attenuates. The standard transfer function for a band-reject filter is:

In active filters, the gain within the passband can be set independently of the cutoff frequency, the gain is determined by the ratio of the feedback resistor to the input resistor, and the design parameters, such as resistor and capacitor values are chosen to set the cutoff frequencies and to shape the overall frequency response of the filter to meet specific requirements.