Design of Lossy Filters

In the design of frequency selective filters, it is often necessary to take into account the parasitic dissipation of the reactive components that will be used to realize the filter. This report presents design parameters for filters having Butterworth, Chebyshev, and Bessel transfer functions for a specific but quite useful distribution of parasitic dissipation. The particular case considered is that in which all circuit elements are assumed to have the same value of unloaded Q (such a circuit is said to be uniformly lossy). The resulting designs are most useful for the designs are most useful for the design of narrow bandpass filters realized as a cascade of coupled resonant circuits; here the uniformly lossy assumption reduces to that of equal Q for each of the coupled resonant circuits. However, the design parameters apply equally well to the realization of lowpass and wide bandpass filters. In all cases the design is based in a normalized, equivalent, lowpass filter.

For each of the filter characteristics, design parameters are given for filters having two through five poles. In the case of Chebyshev filters, pass-band ripples of 0.001, 0.01, 0.03, 0.10, 0.30, and 1.0 dB are included.

In addition to the design data, normalized gain curves are given from which one can calculate the magnitude of the transfer function of each filter at the center frequency. Useful characteristics of the normalized, lowpass filters are also presented, including plots of unit impulse and unit step response and attenuation, phase, and a group delay. From the impulse and step responses the envelope of the corresponding responses of a narrow bandpass filter can be determined.

The author presents a complete theoretical development of the design, and the solution for the five-pole case is apparently new. Multiple solutions for the design parameters are discussed in the light of modern network synthesis, and the correspondence between the two is established.