Narrowband Electromagnetic Coupling Matrix in Coupled-Resonator Microwave Circuits
Valentin de la Rubia, David Young
TL;DR
This paper presents an enhanced electromagnetic coupling matrix (EMCM) framework that extracts all EM couplings among resonators and ports directly from a single full-wave Maxwell solve. By expressing the impedance matrix $\mathbf{Z}(k)$ as a pole-residue expansion (Kurokawa theory) and decomposing it into in-band and out-of-band parts, the authors derive a dynamical system whose in-band component yields a circuit-like representation; a narrowband approximation around a normalized frequency $\mathcal{K}$ then bridges EMCM with the classical circuit-theory coupling matrix while preserving out-of-band effects. A change of basis to local resonator fields reveals detailed EM couplings beyond the idealized topology, including leakage and higher-order-mode loading that classical models miss. Numerical examples across a dual-mode filter, inline dielectric filter, combline diplexer, and dual-passband combline filter demonstrate excellent agreement with full-wave simulations and illustrate how EM couplings inform design decisions, with out-of-band terms playing a critical role in accurate predictions. Overall, the work enables design insights from one full-wave simulation, unifying EM fidelity with circuit-language interpretability for complex microwave circuits.
Abstract
A novel methodology to unleash electromagnetic coupling matrix information in coupled-resonator microwave circuits has been recently proposed [1]. This information is derived from Maxwell's equations and the natural language of electromagnetics is employed. As a result, the coupling matrix coefficients stand only for electromagnetics. In this work, we enhance this approach to reveal valuable design information for microwave engineering, showing the electromagnetic (EM) coupling among all EM resonators and ports. By the same token, the similarities with the well-known classical coupling matrix theory are addressed. We bridge this gap since the classical theory is the preferential language among microwave engineers. Classical coupling matrix theory is a narrowband model for electromagnetics. Thus, we carry out a narrowband approximation in the electromagnetic coupling matrix. This makes it possible to describe the EM coupling coefficients in the same framework as classical circuits. As a result, proper comparison between both coupling matrices is allowed. Finally, both coupling matrix approaches have a common ground, namely, get physical insight and valuable information for design purposes in coupled-resonator microwave circuits. However, only the electromagnetic coupling matrix details all EM behavior, including parasitic and leakage couplings, and the higher-order mode influence in the microwave circuit. Several microwave circuits, such as filters and diplexers, will show the possibilities of this new technique and its relation to classical coupling matrix theory.