Dissipation Loss in Multiple-Coupled-Resonator Filters

This paper examines the effect of dissipation on the response of multiple-coupled-resonator filters designed from lossless prototype low-pass filters. A simple approximate formula makes possible the computation of center-frequency loss in terms of bandwidth, unloaded Q of the resonators, and the parameters of the prototype filter. It is shown how the insertion loss elsewhere in the pass band and stop band may be computed from the prototype filter after modifying its circuit to take account of dissipation in the resonators. The computational techniques are used to compare several symmetrical designs with each other and with unsymmetrical designs having exact maximally-flat and equal-ripple response in the presence of dissipation. It is found that the symmetrical designs offer lower pass-band loss than the unsymmetrical designs. It is proven that a symmetrical filter based on an equal-element prototype has minimum-possible center-frequency loss for given unloaded Q and stop-band bandwidth, subject to assumptions that the loss per resonator is small and that the stop-band insertion loss may be determined accurately from the highest-power term of the insertion-loss polynomial.