Minimum Q-factor Elliptic Filters
See Lutovac & et al. (2001, § 12.11, 13.14).
Elliptic filters are generally specified by requiring a particular value for the passband ripple, stopband ripple and the sharpness of the cutoff. This will generally specify a minimum value of the filter order which must be used. Another design consideration is the sensitivity of the gain function to the values of the electronic components used to build the filter. This sensitivity is inversely proportional to the quality factor (Q-factor) of the poles of the transfer function of the filter. The Q-factor of a pole is defined as:
and is a measure of the influence of the pole on the gain function. For an elliptic filter, it happens that, for a given order, there exists a relationship between the ripple factor and selectivity factor which simultaneously minimizes the Q-factor of all poles in the transfer function:
This results in a filter which is maximally insensitive to component variations, but the ability to independently specify the passband and stopband ripples will be lost. For such filters, as the order increases, the ripple in both bands will decrease and the rate of cutoff will increase. If one decides to use a minimum-Q elliptic filter in order to achieve a particular minimum ripple in the filter bands along with a particular rate of cutoff, the order needed will generally be greater than the order one would otherwise need without the minimum-Q restriction. An image of the absolute value of the gain will look very much like the image in the previous section, except that the poles are arranged in a circle rather than an ellipse. They will not be evenly spaced and there will be zeroes on the ω axis, unlike the Butterworth filter, whose poles are also arranged in a circle.
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