A new technique is presented which makes it possible to measure the inherent short-term stability of quartz crystal resonators in a passive circuit. Comparisons with stability measurements made on crystal controlled oscillators indicate that noise in the electronics of the oscillators very seriously degrades the inherent stability of the quartz resonators for times less than 1 s. A simple model appears to describe the noise mechanism in crystal controlled oscillators and points the way to design changes which should improve their short-term stability by two orders of magnitude. Calculations are outlined which show that with this improved short-term stability it should be feasible to multiply a crystal controlled source to 1 THz and obtain a linewidth of less than 1 Hz. In many cases, this improved short-term stability should also permit a factor of 100 reduction in the length of time necessary to achieve a given level of accuracy in frequency measurements.

Measurement of the Short-Term Stability of Quartz Crystal Resonators and the Implications for Crystal Oscillator Design and Applications

This paper is devoted to one aspect of the study of phase noise in oscillators: how an oscillator reacts to internal noise that occurs in the active element. The following theoretical analysis will lead us to express the Leeson's model in a more general form.

Phase Noise in Oscillators: A Mathematical Analysis of Leeson's Model

Expressions for the averaged two-sample variance as well as expressions for the uncertainty on its estimation are given, in cases where dead time is present between measurements. The effect of this dead time on the precision of the time domain measure of frequency stability, in the presence of a limited number of data, is specified. The possibility to distinguish between white and flicker noises is pointed out.

Effect of Dead-Time on the Estimation of the Two-Sample Variance

This paper presents a study of the transfer of frequency stability from an atomic reference to a quartz-crystal oscillator. The study is done for the cases of active and passive atomic frequency standards in which one makes use of a phase-lock loop and of a frequency-lock loop, respectively. The analysis is made in both the frequency and time domains and covers the cases of the hydrogen and rubidium masers and the passive cesium-and rubidium-frequency standards. The results obtained from numerical calculations are presented under the form of graphs. These results include the fractional frequency spectral density Sy(f) as a function of the Fourier frequency f, and the two sample variance ?2(?) as a function of ?, the sampling time.

Transfer of Frequency Stability from an Atomic Frequency Reference to a Quartz-Crystal Oscillator

We considered the general sampling form for the estimate of the Allan variance which is the proposed measure of frequency stability in the time domain, and we defined a variable proportional to the difference between the average fractional frequency fluctuations over the time interval ? to derive the autocorrelation coefficient of the process to which the variable belongs. Calculations of the variance of the estimated Allan variance proved that it may be convergent to its true value with infinite sample number for considered spectral densities of frequency noise. We also applied the results to estimations of frequency measurements to know the influence of the autocorrelation of the process considered. In order to obtain some direct estimates of the confidence of the estimate, distributions of the estimate were plotted by means of computer simulations, and were compared with the chi-square distribution. Those results suggested that for white-and flicker-phase noises (and white-frequency noise) we have to take into account the autocorrelation of the process, while for flicker-or random-walk-frequency noise we may regard the process as a nearly independent (and Gaussian) one.

R. Winkler

Papers

Characterization ofFrequency Stability