Design of Active Filters
To design filters, the specifications that need to be established include:
- The range of desired frequencies (the passband) together with the shape of the frequency response. This indicates the variety of filter (see above) and the center or corner frequencies.
- Input and output impedance requirements. These limit the circuit topologies available; for example, most, but not all active filter topologies provide a buffered (low impedance) output. However, remember that the internal output impedance of operational amplifiers, if used, may rise markedly at high frequencies and reduce the attenuation from that expected. Be aware that some high-pass filter topologies present the input with almost a short circuit to high frequencies.
- Dynamic range of the active elements. The amplifier should not saturate (run into the power supply rails) at expected input signals, nor should it be operated at such low amplitudes that noise dominates.
- The degree to which unwanted signals should be rejected.
- In the case of narrow-band bandpass filters, the Q determines the -3dB bandwidth but also the degree of rejection of frequencies far removed from the center frequency; if these two requirements are in conflict then a staggered-tuning bandpass filter may be needed.
- For notch filters, the degree to which unwanted signals at the notch frequency must be rejected determines the accuracy of the components, but not the Q, which is governed by desired steepness of the notch, i.e. the bandwidth around the notch before attenuation becomes small.
- For high-pass and low-pass (as well as band-pass filters far from the center frequency), the required rejection may determine the slope of attenuation needed, and thus the "order" of the filter. A second-order all-pole filter gives an ultimate slope of about 12 dB per octave (40dB/decade), but the slope close to the corner frequency is much less, sometimes necessitating a notch be added to the filter.
- The allowable "ripple" (variation from a flat response, in decibels) within the passband of high-pass and low-pass filters, along with the shape of the frequency response curve near the corner frequency, determine the damping factor (reciprocal of Q). This also affects the phase response, and the time response to a square-wave input. Several important response shapes (damping factors) have well-known names:
- Chebyshev filter – slight peaking/ripple in the passband before the corner; Q>0.7071 for 2nd-order filters
- Butterworth filter – flattest amplitude response; Q=0.7071 for 2nd-order filters
- Linkwitz–Riley filter – desirable properties for audio crossover applications; Q = 0.5 (critically damped)
- Paynter or transitional Thompson-Butterworth or "compromise" filter – faster fall-off than Bessel; Q=0.639 for 2nd-order filters
- Bessel filter – best time-delay, best overshoot response; Q=0.577 for 2nd-order filters
- Elliptic filter or Cauer filter – add a notch (or "zero") just outside the passband, to give a much greater slope in this region than the combination of order and damping factor without the notch.
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