9.13:
Active Filters
Certain passive filters incorporate expensive inductors designed not to exceed unity gain. These filters may exhibit reduced attenuation efficacy at frequencies below the audio range.
Active filters consist of operational amplifiers, resistors, and capacitors.
Employed in audio processing, telecommunications, and power supply units, these filters are categorized as first-order low-pass, first-order high-pass, band-pass, and band-reject filters.
Consider the transfer functions and frequency responses of low-pass and high-pass active filters.
Unlike passive filters, the finite low-frequency gain in low-pass active filters and the high-frequency gain in high-pass active filters depend on the feedback-to-input resistor ratio.
Low-pass filters attenuate frequencies above the corner frequency, while high-pass filters suppress those below the corner frequency.
The combination of low-pass and high-pass filters results in a band-pass filter, while a band-reject filter combines both types with a summing amplifier.
Band-pass filters transmit frequencies between the corner frequencies and reject those outside; band-reject filters block frequencies within the range and allow external frequencies.
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 Af is the gain at low frequencies also known as the dc gain, s is the complex frequency, and ωc 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 Af represents the gain at high frequencies, and ωc 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.