A Primer on the Design of Active Filters - Part I
Actual Versus Ideal Filters
An ideal filter would provide 100% transmission within the specified frequency passband and 100% attenuation outside of that passband.
In practise, an analog filter can only approximate this performance (hence ‘approximation’), and will have some attenuation in the passband, a transition region between passband and stopband, and less than 100% attenuation over the stopband frequency range. Two examples are shown below:
Cascading Simple filter stages
The most common way to construct an active filter, and the technique used in Filter Wiz PRO, is to cascade a series of simpler 1st or 2nd order filter stages. In the example below, the overall filter will be of 7th order:
1st Order Stages
There are only 2 types of 1st order stages, lowpass and highpass. Each has a single pole.
2nd Order Stages
There are 6 types of 2nd order stages. Each has a pair of poles. The notch stages also have a pair of zeros that cause the gain to dip to zero at a particular frequency.
Note that all notch stages force the gain to zero at a particular frequency. Mathematically, they have a zero pair as well as the poles common to all stages.
If a filter does not have a lowpass notch, highpass notch or notch(bandstop) stage (ie has no zero pairs) then it is called an All-Pole filter.
Notch stages have the advantage of steepening the filter sides (transition region) and thus perhaps requiring a lower filter order. On the other hand, notch stages tend to be more complex when implemented by electronic circuits.
Consider, for instance, the simplest 2nd order low pass topology (Sallen-Key), and the simplest 2nd order low pass notch topology (Boctor):
Shown below are two Approximations for a 4th order lowpass filter. The Butterworth Approximation is all-pole, while the Elliptic Approximation is not, having two zero pairs that force the gain to zero in the stopband, thus steepening the filter edge.
Stage Gains are Multiplicative
The gain of a filter is the product of the gain of each stage. Shown below are two 8th order filter, one lowpass and the other bandpass. The overall filter response is shown in blue or red, while the individual stage gain responses are drawn in black.
The Quality Factor - Q
2nd order filter stages can vary in the degree to which the gain-frequency response peaks. The frequency responses of both a lowpass and a bandpass filter stage are shown below. A higher Quality Factor (Q) dictates a more peaked response.
Filter with narrower transition zones (steeper sides) require filter stages with higher Q values. But higher Q values are harder to implement electronically, given that the frequency and magnitude of stage response "peak" will be dependent on resistor and capacitor values, with their tolerance variablity and sensitivity to age and temperature.
Generally speaking, Q values should be kept as low as possible (under 10 is optimum). For that reason, Filter Wiz PRO offers several "Low-Q" Approximations that will provide more stable filters for a modest increase in filter size.
An Example Bandpass Filter
Using Filter Wiz PRO a bandpass filter is specified as follows:
- Passband ripple 1dB or less
- Passband bandwidth of 2khz at 1dB gain attenuation
- Stopband bandwidth of 4khz at 40dB gain attenuation
- Center frequency of 20khz
- Gain of 1
Charts of Approximation type versus Filter Order and Maximum Quality Factor (Q) are shown below:
A plot of Filter Order versus Maximum Stage Q of the filter demonstrates that while the Elliptic Approximation will yield a filter of the lowest order, for critical designs it may be wise to choose another Approximation of higher order but of significantly reduced stage Quality Factors.
For instance increasing the filter order from 8 to 10 will cut the maximum stage Q by more than half (Inverse Chebyshev or Low-Q Elliptic I), while an order 12 filter will reduce the maximum stage Q by a factor of 5 (Low-Q Elliptic II).