The paper discusses approximate statistical estimates of limiting errors associated with single differential phase
measurement of a time delay (phase difference) between two reflectors of the passive surface acoustic wave (SAW)
sensor.
The paper discusses a probabilistic model of a crystal oscillator at low drive levels where the noise intensity
is comparable with the oscillation amplitude. The stationary probability density of the oscillations envelope is
derived and investigated for the nonlinear resonator loses. A stochastic explanation is given for the well-known
phenomenon termed sleeping sickness associated with losing a facility of self-excitation by a crystal oscillator
after a long storage without a power supply. It is shown that, with low drive levels leading to an insufficient
feedback, a crystal oscillator generates the noise-induced oscillations rather than it absolutely "falls in sleep".
Dynamics of the mean amplitude of oscillations of a crystal oscillator with a linear feedback is outlined for low drives when the losses (friction) of a resonator become large and nonlinear after a long storage. The drive-level-dependence (DLD) of the crystal resonator losses is assumed to change inversely to the piezoelectric current. A stochastic differential equation for the mean amplitude is derived and solved in a sense of Ito. The development and attenuation processes are learned and it is shown that attenuation finishes at some non-zero level associated with the effect termed "sleeping sickness." The critical value of the friction is calculated and the conditions are discussed to avoid attenuation. Based upon, we show in that (1) if the value of the DLD coefficient of the resonator losses ranges below the critical point, the effect occurs primarilly in a delay of self-excitation; (2) contrary, noise drives the crystal oscillator.
We present the generalized asymptotical model of a single-mode crystal oscillator with a nonlinear resonator. The motional and static equivalent parameters associated with the fundamental mode of a crystal resonator are assumed to be drive level dependent (DLD) and flicker noisy. A thermal noise of the resonator losses is also taken into account. A concept of the feedback amplifier is discussed resulting in a series branch of the equivalent resistance and inductance (both flicker-noisy and negative) along with the thermal additive noise source. The generic differential equation of the second order is derived, in which the auxiliary sub-functions are introduced being responsible for the equilibrium, anisochronisms, thermal noise (additive and multiplicative) effect, and colored (flicker) noise effect. We show that such a model covers all three special ranges of the piezoelectric current, namely those with low, normal, and high drives.
This paper discusses the noise performance of a crystal oscillator. It is assumed that the crystal resonator motional losses, inductance, and capacity as well as its static capacity are flicker noisy. The Johnson thermal additive noise of the motional losses is also taken into account. To convert the intrinsic noise sources to the amplitude and phase fluctuations, the crystal resonator and the feedback amplifier are performed by noisy impedances. The crystal oscillator is reduced to the closed loop of the resonator impedance and the feedback amplifier impedance. The transformation coefficients are derived to convert the resonator and amplifier amplitude and phase power spectral density functions to those of the oscillator amplitude and phase. Noise performance of a crystal oscillator with different excitation angles of a resonator is studied in detail.
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