Active filter

An example of high-pass active filter of the Sallen–Key topology. The operational amplifier is used as a buffer amplifier.

An active filter is a type of analog electronic filter that uses active components such as an amplifier. Amplifiers included in a filter design can be used to improve the performance and predictability of a filter,^[1] while avoiding the need for inductors (which are typically expensive compared to other components). An amplifier prevents the load impedance of the following stage from affecting the characteristics of the filter. An active filter can have complex poles and zeros without using a bulky or expensive inductor. The shape of the response, the Q (quality factor), and the tuned frequency can often be set with inexpensive variable resistors. In some active filter circuits, one parameter can be adjusted without affecting the others. ^[1]

Using active elements has some limitations. Basic filter design equations neglect the finite bandwidth of amplifiers. Available active devices have limited bandwidth, so they are often impractical at high frequencies. Amplifiers consume power and inject noise into a system. Certain circuit topologies may be impractical if no DC path is provided for bias current to the amplifier elements. Power handling capability is limited by the amplifier stages.^[2]

Active filter circuit configurations (electronic filter topology) include:

• Sallen-Key, and VCVS filters (low sensitivity to component tolerance)
• State variable filters and biquadratic or biquad filters
• Dual Amplifier Bandpass (DABP)
• Wien notch
• Multiple Feedback Filters
• Fliege (lowest component count for 2 opamp but with good controllability over frequency and type)
• Akerberg Mossberg (one of the topologies that offer complete and independent control over gain, frequency, and type)

Active filters can implement the same transfer functions as passive filters. Common transfer functions are:

• High-pass filter – attenuation of frequencies below their cut-off points.
• Low-pass filter – attenuation of frequencies above their cut-off points.
• Band-pass filter – attenuation of frequencies both above and below those they allow to pass.
• Notch filter – attenuation of certain frequencies while allowing all others to pass.

Combinations are possible, such as notch and high-pass (in a rumble filter where most of the offending rumble comes from a particular frequency). Another example is an elliptic filter.

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 ratio or damping factor (= 1/(2Q)). This also affects the phase response, and the time response to a square-wave input. Several important response shapes (damping ratios) have well-known names:
  • Chebyshev filter – peaking/ripple in the passband before the corner; Q>0.7071 for 2nd-order filters
  • Butterworth filter – maximally flat amplitude response; Q=0.7071 for 2nd-order filters
  • Linkwitz–Riley filter – desirable properties for audio crossover applications, fastest rise time with no overshoot; 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 – maximally flat group delay; 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 ratio without the notch.

References

1. Don Lancaster, Active-Filter Cookbook, Howard W. Sams and Co., 1975 ISBN 0-672-21168-8 pages 8-10
2. Muhammad H. Rashid, Microelectronic Circuits: Analysis and Design, Cengage Learning, 2010 ISBN 0-495-66772-2, page 804

External links

• Split-Supply Analog Filter Expert
• Single-Supply Analog Filter Expert
• Introduction to active filters
• Lacanette, Kerry (April 21, 2010), A Basic Introduction to Filters–Active, Passive, and Switched-Capacitor (PDF), Application Note, National Semiconductor, AN-779
• Active filter design - related articles
• Analog Filter Wizard: Design tool for active filters