$Q$ -Factor Estimation From the Return Loss of Low- $Q$ Microwave Resonators

We propose a novel approach for evaluating the resonance curves of low-Q microwave resonators, which are associated with the in situ monitoring of large and lossy material samples in cavities. It is assumed that the process observed is stationary and thus admits measurements over long enough time intervals to produce statistical data on selected features of the resonance curve. These features, such as resonance frequency and half-power bandwidth, are random variables owing to process and measurement noise. We show by numerical experiments and by measurements that the Q-factor of the resonator can be extracted from the probability density function (pdf) of the random variable “resonance frequency” or even from only a single moment of the pdf. This is demonstrated by way of the standard deviation (the square root of the second central moment), which, in practice, is estimated from the empirical standard deviation or measurement uncertainty. The method can be applied to lower Q-factors than any other method to date.