Synthesis of RC Bandpass Filters

Abstract

For the first time a method of designing RC bandpass filters is presented. The method consists of two steps. The first step is a scheme to locate the necessary poles and zeros that are RC realizable to produce certain bandpass characteristics. The second step is the synthesis of RC networks to produce these poles and zeros. In the first step, the conformal transformati \begin{equation*}s(z) = {\left(\frac{sn^{2}(z,k) - sn^{2}(\alpha{K,k})}{sn^{2}(\alpha{K,k})[1 - k^{2}sn^{2}(\alpha{K,k})sn^{2}(z,k)]}\right)}^{1/2}\end{equation*} is used to map the complex frequency s plane into a rectangle in the z plane such that the passband becomes one side, and a part of the negative real axis becomes the opposite side of the rectangle. In the z plane, if poles are located along certain portions of the border and zeros in the interior of the rectangle, certain passband and stopband behavior can be achieved. Among the useful characteristics obtainable by this scheme, the following are three outstanding examples: 1) characteristics that are equal-ripple in the passband and monotonic in the stopband; 2) characteristics that are equal-ripple in the passband and have a number of transmission zeros in the stopband; and 3) characteristics with a maximum gain at the band center and monotonic elsewhere. The steepness of attenuation outside the passband can be altered by a change in the numbers of zeros at the origin and infinity.