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

Coupling Coefficients of Resonators in Microwave Filter Theory

“…Exact values of coupling coefficients |k i,i+1 | were obtained in [23] for a bandpass network comprising a ladder chain of alternate series and parallel nondissipative resonant circuits and featuring Chebyshev frequency response. In order to calculate the coupling coefficient for a certain asymmetrical pair of adjacent resonant circuits, formula (9) was rewritten in the form…”

“…The isolation of a pair was achieved by disconnecting the external port of the parallel resonant circuit and closing the external port of the series resonant circuit. The coupling coefficients acquired the following values [23]:…”

“…The values of |k i,i+1 | obtained from (58) for a bandpass network of alternate series and parallel resonant circuits were compared in [23] with the corresponding values of |k i,i+1 | in an optimized microwave bandpass filter of cascaded alternate low-impedance and high-impedance half-wavelength transmission line sections having the same passband. A pair of cascaded sections in microwave filters may be treated as coupled series and parallel half-wavelength resonators.…”

“…A pair of cascaded sections in microwave filters may be treated as coupled series and parallel half-wavelength resonators. Resonant frequencies ω ± of their coupled oscillations are determined from [23]:…”

Analysis of the quality factor of the optical half‐wavelength resonator

“…In narrowband filters and some other resonant devices, all the adjacent resonators must have weak couplings, whereas the input and output resonators must have high external quality factors [6]. But external Q factors of the input and output resonators in the PC as well as the coupling coefficients of the adjacent resonators depend on the difference of RIs of the contacting mediums—the larger the difference, the smaller the coupling coefficient and the higher the Q factor of the resonator [7, 8]. In view of the fact that RIs of materials in optics are not so high, the external Q factor of a half‐wavelength (HWL) dielectric layer being in free space is not high too.…”