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An iterative curve fitting method for accurate calculation of quality factors in resonators
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Image of FIG. 1.
FIG. 1.

Schematic of the measurement setup, including the resonator and network analyzer, with the corresponding signal and noise inputs marked. At node 1: is the input signal from the network analyzer, and are the noises associated with the input signal and the ambience (e.g., thermomechanical noise), respectively. At node 2: is the resonator output signal and is the intermediate stage noise. The measured output signal is .

Image of FIG. 2.
FIG. 2.

(a) The magnitude of the measured signal amplitude compared to the contributing SHO transfer function , the initial least-square curve fit , and root-mean-square noise , (b) illustration of the proposed iterative procedure for step-by-step elimination of the noise effect, with intermediate signals and of the iteration step are marked. In both graphs the signal amplitude is shown in linear scale. In the presented data, the SNR at the resonance frequency, i.e., , is 3.0; also, the extracted -factor based on is 1074, while the actual resonance -factor, which is extracted based on , is 1280.

Image of FIG. 3.
FIG. 3.

Flowchart of proposed iterative curve fitting method.

Image of FIG. 4.
FIG. 4.

Measured resonance characteristics (magnitude) for different signal powers. The corresponding measurements are performed with minimum sequential time delay to prevent possible variations in the noise power density. (The presented magnitudes are in decibel, with the noise power set as reference.)

Image of FIG. 5.
FIG. 5.

Implementation of the proposed iterative fitting method on an actual measured signal . Because of magnitude fluctuations, the partial noise power is calculated by considering the average of squares of the first 10 data points of . In this graph, the SNR is 3.5, and the magnitude is in linear scale.

Image of FIG. 6.
FIG. 6.

-factor as a function of the SNR, extracted from the measurement data by using the two conventional methods, i.e., single least-square curve fit (square) and the 3 dB bandwidth methods (circle), and also the proposed iterative fitting method (diamond). No meaningful -factor data were obtained with the 3 dB bandwidth method for SNR smaller than 2, and with the single fit method for SNR smaller than 1.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: An iterative curve fitting method for accurate calculation of quality factors in resonators