Couplings of asynchronously tuned coupled microwave resonators

The paper presents a comprehensive treatment of the couplings of asynchrono usly tuned coupled microwave resonators. The derived formulation allows one to extract coupling coefficients in terms of four characteristic frequenci es that can be easily determined numerically or experimentally. To demonstr ate the applications, numerical examples are given for asynchronously tuned microstrip coupled open-loop and dual-mode resonators.