Accurate equation for determining resonator length in half-wavelength parallel-coupled bandpass filter

An accurate closed-form formula for calculating the resonator length of the half-wavelength parallel-coupled bandpass filter, taking into account the open-end effects, is presented for the first time. Using the resonator lengths determined from the derived equation results in filter responses which are very close to the design specifications without the need for iterating or foreshortening the resonators. Validation has been accomplished by comparing calculated results of various half-wavelength parallel-coupled bandpass filters, based on the new resonator lengths, with their design specifications.