A time domain based method for the accurate measurement of Q-factor and resonance frequency of microwave resonators

Abstract

We present a novel method to determine the resonant frequency and quality factor of microwave resonators which is faster, more stable, and conceptually simpler than the yet existing techniques. The microwave resonator is irradiated at a frequency away from its resonance. It then emits an exponentially decaying radiation at its eigen-frequency when the excitation is rapidly switched off. The emission is down-converted with a microwave mixer, digitized and its Fourier transformation (FT) directly yields the resonance curve in a single shot. Being an FT based method, this technique possesses the Fellgett (multiplex) and Connes (accuracy) advantages and it conceptually mimics that of pulsed nuclear magnetic resonance. We also establish a novel benchmark to compare accuracy of the different approaches of microwave resonator measurements. This shows that the present method have similar accuracy to the existing ones.

A highly accurate measurement of resonator Q-factor and resonance frequency

“…We note at this point that the use of FBO differs conceptually from our previous time domain method in Ref. 25: the present method acts as if two independent frequency sources were present with adjustable frequencies that allows the use of a high IF. In the previous work, the same LO was used to irradiate and downconvert the transient signal.…”

“…Standard deviations, σ (Q) and σ (f 0 ), of the relevant data are determined from the conventional definitions and the result for the present method is also given in Table II. Table II shows that all methods, including our previous transient work 25 , provide a δ (Q) ≈ δ (f 0 ) ≈ 10 −3 . In contrast, when normalized to a 1 second measurement time, the present FBO transient based method yields an order of magnitude improvement of δ (Q) = δ (f 0 ) ≈ 6 × 10 −5 .…”

“…In the following, we discuss the practical use of the FBO transients for the measurement of Q and f 0 and also evaluate the measurement performance. We believe that the resonator measurement methods can be classified into 4 main groups as shown in Table I: i) frequency swept methods 11,12,15 , ii) automatic frequency control (AFC) based methods 4,17,37 , iii) PLL stabilized frequency stepped methods 13 , and iv) time domain methods [18][19][20][21]25 such as the present work.…”

“…We would like to point out a compelling analogy between the error benchmark defined in Ref. 25 and the so-called Gabor uncertainty [41][42][43][44][45][46][47] . The latter states that for a a signal with a given time, σ t , and frequency uncertainty, σ ω it holds:…”

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“…where Q and ∆f are the mean values of Q and the resonator bandwidth ∆f , respectively. Q = f 0 /∆f , where f 0 is the resonator frequency 30 . We note that the error of f 0 is not σ(f 0 )/f 0 as it would be intuitive at first sight.…”