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

## TL;DR

This paper introduces a matrix-based approach to analyze and design non-reciprocal filters using time modulated resonators, providing a new framework for understanding their coupling and behavior.

## Contribution

It presents a novel coupling matrix formalism for non-reciprocal filters based on time modulated resonators, with experimental validation in microstrip technology.

## Key findings

- Validated the coupling matrix model with experimental results
- Demonstrated non-reciprocal filter designs of orders three and four
- Achieved good agreement between measured and numerical results

## Abstract

This paper addresses the analysis and design of non-reciprocal 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 order are coupled through non-reciprocal 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 non-reciprocal 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 in the ground plane of the device. Measured results are found to be in good agreement with numerical results, validating the proposed theory.