Coupling Matrix Representation of Nonreciprocal Filters Based on Time-Modulated Resonators

This article addresses the analysis and design of nonreciprocal filters based on time-modulated resonators. We analytically show that time modulating a resonator leads to a set of harmonic resonators composed of the unmodulated lumped elements plus a frequency-invariant element that accounts for differences in the resonant frequencies. We then demonstrate that harmonic resonators of different orders are coupled through nonreciprocal admittance inverters, whereas harmonic resonators of the same order couple with the admittance inverter coming from the unmodulated filter network. This coupling topology provides useful insights to understand and quickly design nonreciprocal filters and permits their characterization using an asynchronously tuned coupled resonators network together with the coupling matrix formalism. Two designed filters, of orders three and four, are experimentally demonstrated using quarter wavelength resonators implemented in microstrip technology and terminated by a varactor on one side. The varactors are biased using coplanar waveguides integrated into the ground plane of the device. Measured results are found to be in good agreement with numerical results, validating the proposed theory.