Robust Algorithms for Fitting Q-Factor in the Complex Domain

This paper describes robust vector-fitting algorithms for determining the Q-factor and resonant frequency of spectrally-isolated resonances from frequency-swept S-parameter measurements for both one-port (reflection) and two-port (transmission) systems. It also provides guidance on measurement techniques, and gives measurement examples from the electromagnetic and acoustic domains. These include measurements on a LC resonator (unloaded Q-factor $Q_{{ o}}\approx 57$ ), a photonic-crystal resonator ( $Q_{{ o}}\approx 123\,000$ ) and a superconducting notch resonator ( $Q_{{ o}}\approx 1.5\times 10^{6}$ ). The vector techniques advocated are often advantageous compared to scalar techniques because they are more informative, and in many cases more precise. Among the most common applications is the measurement of dielectric permittivity and loss by resonance at RF and microwave frequencies by using Vector Network Analysers. The algorithms described, however, are applicable more generally to sensing and imaging applications that use vector instrumentation. This is demonstrated by one of the measurement examples, which shows that acoustic Q-factor can be fitted to vibrational data obtained by Resonant Ultrasound Spectroscopy. Open-source software implementations (Python and Matlab) of the algorithms have been made available.