Measurement of the Resonance Frequency by the Amplitude and Phase Methods with the Use of Digital Frequency Scanning

We perform theoretical and experimental investigations of the methods of measurement of resonance frequencies according to the amplitude-frequency and phase-frequency characteristics of resonators (amplitude and phase methods, respectively). For the evaluation of the resonance frequency, we used the procedure of digital frequency scanning. On the basis of the probability theory, we deduce analytic expressions that describe the dependences of the systematic and random errors on the location of the resonance frequency in the interval between the closest discrete frequencies and on the level of noise. The reliability of these expressions is confirmed in the course of virtual experiments performed with a computer model of the resonator. It is also shown that the errors of the amplitude and phase methods practically coincide for the noise level at which we record at most two discrete frequencies but are noticeably different if the number of recorded discrete frequencies is greater than two. In this case, the errors of the resonance frequency measured by the phase method are practically independent of the steps of frequency tuning, as these step decrease and linearly depend on the level of phase noise. In measuring the resonance frequency by the amplitude method, the measurement errors decrease as the step of frequency tuning becomes smaller. For this case, we propose an empirical formula that describes the dependence of the systematic and random errors on the step of frequency tuning and the level of amplitude noise. The accumulated results can be used in the construction of digital resonance gauges.