Direct derivation and refinement of generalized Cohn-Matthaei formulas for resonator coupling coefficients in a microwave filter

Generalized formulas for direct synthesis of bandpass filters are presented. The formulas are derived on the basis of a lowpass prototype filter formed from lumped reactive elements and on the definitions of the coupling coefficient and the quality factor of the filter resonators. In deriving the formulas, neither the equivalent circuits with impedance inverters nor linear approximations of the frequency dependences of the immitances of reactive elements are used. The frequency characteristics of a simple filter that is formed by a cascade connection of alternating low-impedance and high-impedance half-wavelength sections and is synthesized using both the approximate and refined formulas are compared.