Coupled-mode theory for RF and microwave resonators

Abstract

The case of two dielectric resonators inserted in a cavity is fully analyzed. It is shown that the three uncoupled modes interact to form three coupled ones. Accordingly, Energy Coupled-Mode Theory, a coupled mode formalism where energy is conserved, is developed to study the coupling of RF and microwave resonators. The governing equations are written in the form of an eigenvalue problem where the eigenvalues represent the square of the frequencies and the eigenvectors are the fields’ coefficients. Both external and internal boundary conditions are discussed. The coupled mode formalism is capable of analyzing a system consisting of an arbitrary number of resonators. Using the proposed equations, the physical origin of the coupling coefficient is found and interpreted based on the energy conservation principle. The interpretation is general and universal and is believed to encompass cases where dielectrics and conductors are present. An important electron paramagnetic resonance probe, namely a cavity with a tiny insert, is studied. It is shown that when the frequency of the cavity is equal to that of the insert, the resulting fields are complete mixes of the two uncoupled modes. This finding, together with others, finds applications in the magnetic resonance and dielectric measurements fields. Different practical scenarios (large/small cavities, high/moderate relative permittivity values) are discussed in detail. Expressions for field dependent parameters such as coupling coefficients, Q values, filling factors and resonator efficiency are derived. It is shown how these parameters contribute to the probe performance and how, in some circumstances, trade-offs need to be made. Design procedures for electron paramagnetic resonance probes, verified using finite-element simulations, are proposed. It is found that proper design can indeed enhance the electron paramagnetic resonance signal. Particularly, the shield can help in boosting the resonator efficiency of a lossy dielectric resonator. The method of images, area commonly used electromagnetic technique, is used together with the coupled mode theory to study cases where resonators are placed close to conducting planes. The aforementioned situations occur in the field of magnetic resonance spectroscopy when dielectric resonators are used as tuners. These situations also exist when conducting planes are used to enhance wireless power transfer using resonant inductive coupling.