Measurement of resonance frequency by amplitude and phase methods using digital frequency scanning

A theoretical and experimental study of methods for measuring the resonance frequency from the amplitude-frequency and phase-frequency characteristics of the resonator (amplitude and phase methods, respectively) has been carried out. In this case, digital frequency scanning was used to determine the resonant frequency. On the basis of the theory of probabilities, analytical expressions are derived that describe the dependences of systematic and random errors on the position of the resonance frequency in the interval between the nearest discrete frequencies, as well as on the noise level. The reliability of the derived expressions was confirmed in the course of a virtual experiment with a computer model of the resonator. It is also shown that the errors of the amplitude and phase methods for the noise level, at which no more than two discrete frequencies are recorded, practically coincide. However, if more than two discrete frequencies are recorded, then the indicated errors differ significantly, which is demonstrated using the experimental graphs. In this case, the errors in measuring the resonance frequency by the phase method practically do not depend on the frequency tuning step with a decrease in this step and linearly depend on the phase noise level. When measuring the resonance frequency by the amplitude method, the errors decrease with decreasing frequency tuning step, and for this case, an empirical formula is proposed for the dependence of systematic and random errors on the frequency tuning step and the amplitude noise level. The research results can be used in the construction of digital resonance sensors.