Using the binomial transformer to approximate the Q distribution for maximally flat quarter-wavelength-coupled filters

With the binomial transformer as a basis, a closed-form expression is derived for the Q distribution used to design maximally flat quarter-wavelength-coupled transmission-line filters. The derived expression is shown to be valid for filters with small values of total Q. Also, an existing Q-distribution expression derived from the lumped-element prototype (LEP) circuit is discussed, and is shown to be valid for filters with large values of total Q. By combining the derived binomial transformer Q distribution and the LEP Q-distribution expressions, a piecewise closed-form Q-distribution expression is developed, which is applicable over a wide range of total Q values.