Modeling of a short-term stability measuring system of quartz crystal resonators

A new numerical model of a short-term stability measuring system of quartz crystal resonators is presented. It is based on the phase bridge method using a pair of resonators driven by a low-noise source. The output signal, obtained with a phase detector, is proportional to the phase difference introduced by the resonators. The numerical transfer function of each bridge path is given by the model. The output spectral density of the phase fluctuations is computed from these transfer functions and the numerical approximation of the low-noise source. The model was applied to third overtone, SC-cut, 10 MHz BVA quartz crystal resonators. It enables the rejection of the source noise versus the resonant frequency of quartz crystal resonators to be quantified.     

Phase noise measurements of 10-MHz BVA quartz crystal resonators

In this paper, we review a new piece of equipment that allows one to characterize the phase noise of crystal resonators using a phase bridge system with carrier suppression. This equipment allows one to measure the inherent phase stability of quartz crystal resonators in a passive circuit without the noise usually associated with an active oscillator. We achieved a system noise floor of approximately -150 dBc/Hz at 1 Hz and -160 dBc/Hz, at 10 Hz. A SPICE characterization of the carrier suppression system is given. An investigation of the phase modulation (PM) noise in 10 MHz BVA, SC-cut quartz crystal resonator pairs is presented.     

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

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.     

1/f noise in etched groove surface acoustic wave (SAW) resonators

Measurements of 1/f (or flicker) frequency fluctuations in SAW resonators fabricated with etched groove reflectors on single crystal quartz have shown that the observed noise levels vary inversely with device size. These measurements were made on sixteen 450 MHz resonators of four different sizes. The 1/f noise levels were also evaluated on twenty-eight other SAW resonators ranging in frequency from 401 to 915 MHz. This additional data provides valuable information on the dependence of the flicker noise levels on resonator frequency. A model based an localized, independent velocity fluctuations in the quartz is proposed which correctly fits the observed size and frequency dependence of the measured 1/f noise levels. This model suggests that the velocity fluctuations originate in small regions (much less than ~5 mum in diameter) randomly distributed throughout the quartz with an average separation of about 5 mum between independent (incoherent) sources. The magnitude of the localized fractional velocity fluctuations, Deltav/v, averaged over a 5 micron cube is on the order of 1x10 (-9).     

Reduction of quartz crystal oscillator flicker-of-frequency and white phase noise (floor) levels and acceleration sensitivity via use of multiple resonators

Through the use of N series-connected quartz crystal resonators in an oscillator circuit, a 10 log N reduction in both flicker-of-frequency noise and white phase-noise (floor) levels has been demonstrated. The reduction in flicker noise occurs as a result of the uncorrelated short-term frequency instability in each of the resonators, and the reduction in noise floor level is a simple result of the increase in net, allowable crystal drive level. This technique has been used in 40-, 80-, and 100-MHz AT-, BT-, and SC-cut crystal oscillators using low flicker-of-phase noise modular amplifier sustaining stages, and four series connected crystals. Total (four crystal) power dissipations of up to 30 mW have been utilized. State-of-the-art, flicker-of-frequency noise levels have been obtained with noise-floor levels (80 MHz) as low as -180 dBc/Hz. Four- to five-fold reduction in acceleration sensitivities has been determined     

An X-ray double crystal topographic assessment of defects in quartz resonators

This paper describes an X-ray double crystal topographic study of defects in eighteen quartz resonators designed to operate at 1.4 MHz. The types of defects found in quartz are described, together with their reported effects on resonator performance. The mode of operation of the bulk resonator and the technique of X-ray double crystal reflection topography are outlined. Topographs reveal the electrode structures and surface features of the resonators together with the presence of growth defects such as dislocation cells, sub-boundaries, growth striations and growth sector boundaries. Spurious flexure modes in two resonators are also shown. It is demonstrated that a correlation exists between the presence of growth striations (and probably sub-boundaries) and a higher equivalent series resistance of the resonators. It is shown that such defects change the contributions to the losses, possibly by changing the nature of the vibration pattern of the resonator.     

Application of the photo-elastic method to measurement of dynamicstress distribution for NS-GT cut quartz crystal resonators

NS-GT cut quartz crystal resonators are widely used as a frequency standard element in consumer products and communication equipment. The vibration mode of the resonators was analyzed by the finite element method (FEM) because they have a complicated shape. As a result, an asymmetrical vibration mode at the main resonant frequency has been obtained by the FEM simulation. But, it is necessary to confirm the asymmetrical vibration mode experimentally because it is just a simulation. In this paper, stress distributions of the NS-GT cut quartz crystal resonators are measured experimentally by using a dynamic photo-elastic method when the resonators are vibrating in the resonant frequency; thereafter, vibration modes of the NS-GT cut resonators are estimated with the experimental data of the stress distributions. This experiment for the NS-GT cut quartz crystal resonators exposes the existence of a twisted asymmetrical vibration mode at the main resonant frequency, with the magnitude of the twisted vibration in proportion to thickness of the resonators     

Resonant frequency function of thickness-shear vibrations of rectangular crystal plates

The resonant frequencies of thickness-shear vibrations of quartz crystal plates in rectangular and circular shapes are always required in the design and manufacturing of quartz crystal resonators. As the size of quartz crystal resonators shrinks, for rectangular plates we must consider effects of both length and width for the precise calculation of resonant frequency. Starting from the three-dimensional equations of wave propagation in finite crystal plates and the general expression of vibration modes, we obtained the relations between frequency and wavenumbers. By satisfying the major boundary conditions of the dominant thickness-shear mode, three wavenumber solutions are obtained and the frequency equation is constructed. It is shown the resonant frequency of thickness-shear mode is a second-order polynomial of aspect ratios. This conforms to known results in the simplest form and is applicable to further analytical and experimental studies of the frequency equation of quartz crystal resonators.     

Strained surface layers of quartz plates produced by lapping and polishing and their influence on quartz resonator performance

The strained surface layers of AT-cut quartz plates produced by lapping and polishing have been studied using the X-ray double crystal method. In addition, the influence of surface strain on the characteristics of quartz resonators has been investigated. Experimental results from X-ray double crystal measurements indicate that a residual stress layer in the lapped and polished surface is created and that the quartz resonator performance is also affected.     

A new method for continuous monitoring of series resonance frequency and simple determination of motional impedance parameters for loaded quartz-crystal resonators

A new electronic design for continuous motional series resonant frequency monitoring of loaded quartz crystal resonators is presented. Using this circuit, a low-cost method for a simple determination of equivalent circuit parameters of quartz crystal resonators is described. Measurements made with the proposed system on typical AT cut quartz crystals are in good agreement with those of an Impedance Analyzer.     

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.     

Predicting phase noise in crystal oscillators

In order to predict the phase noise in crystal oscillators an enhanced phase-noise model has been built. With this model, the power spectral densities of phase fluctuations can be computed in different points of the oscillator loop. They are calculated from their correlation functions. The resonator-caused noise as well as the amplifier-caused noise are taken into account and distinguished. To validate this enhanced model, the behavior of a batch of 10 MHz quartz crystal oscillators is observed and analyzed. The tested batch has been chosen in a facility production. Their associated resonators have been selected according to the value of their resonant frequency and their motional resistance. Open-loop and closed-loop measurements are given. The phase noise of the overall oscillator working in closed loop is provided by the usual active method. Theoretical and experimental results are compared and discussed.     

HF and VHF Source Stabilization Technique Incorporating Q-Multiplied Quartz Crystal Resonator

A new approach is described for the desiga of HF/VHF crystal-controlled frequency sources exhibiting theoretical short-term stability unattainable through the use of conventional quartz oscillator design. The signal generator design uses the concept of AFC stabilization of a conventional quartz oscillator (VCXO) by means of a crystal-controlled highly selective active frequency reference. The AFC reference is a phase-shift type frequency discriminator that employs a product detector and an active Q-multiplied quartz crystal resonator. The extremely selective transmission response, large group delay, and power gain exhibited by the resonator, together with resonator phase noise levels comparable to that exhibited by the oscillator-maintaining circuit, provide the principal means for prediction of superior output signal spectral purity. Models of the resonators have been designed and constructed at 30 and 80 MHz, exhibiting 3-dB bandwidths of 30 and 160 Hz, respectively. Based on actual measurement of VHF Q-multiplied crystal resonator performance characteristics, approximately 16 dB improvement in VHF crystal-controlled frequency source spectral purity at low and moderate modulation rates is possible, compared to that attainable using the best available VHF quartz oscillator circuit designs.     

Multichannel quartz crystal microbalance

Arrays of quartz crystal resonators are fabricated on a single quartz wafer as a multichannel quartz crystal microbalance (MQCM). Three types of four-channel array of 10-MHz resonators are prepared and tested. Mechanical oscillation of each channel is entrapped within the channel almost completely, so that the interference between the channels via the quartz crystal plate is almost negligible. A mass change on each channel is quantitatively evaluated on the basis of Sauerbrey's law. Thus, each channel of a MQCM device can be used as an independent QCM. Influence from a longitudinal wave generated from another channel is also negligible compared to the influence from the wave from the monitored channel itself. The simultaneous oscillation of channels is also possible. The potential applicability of MQCM to the two-dimensional mapping of mass changes is demonstrated.     

Influence of a magnetic field on quartz crystal resonators

The magnetic sensitivity of quartz crystal resonators is a consequence of the ferromagnetic properties of the metal used as support for the vibrating plate. Various magneto-mechanic interactions can contribute to the overall sensitivity, the most important of which is shown to be the change in Young's modulus of the spring material submitted to a magnetic field, which in turn modifies the stress in the quartz plate and then induces a change in the crystal resonant frequency. The experimental setup and procedure are described and a large number of experimental results obtained with resonators of different technologies are presented and discussed. A comparison between the magnetic behavior of identical resonators mounted with different materials definitely proves the responsibility of the supports in the magnetic sensitivity of resonators and gives interesting information on its reduction     

Analysis of aging of piezoelectric crystal resonators

Aging of piezoelectric (quartz crystal) resonator has been identified as one of the most important quality control problems of quartz crystal products. Aging is defined as frequency change with time. Aging in quartz resonators can be due to several sources: mass transfer due to contamination inside the resonator enclosure, stress-strain in the resonator blank, quartz defect, etc. In this study, the stress-strain effect, which has been believed as a dominant factor contributing to aging, is studied. The stress-strain effect is caused mainly by the long-term viscoelastic properties of bonding adhesive that attach quartz crystal plate to the ceramic base package. With the available accelerating testing method under elevated temperatures, the stress-strain induced aging in the quartz crystal resonators can be investigated. Because of the miniaturized size of the resonator, a digital image analysis method called image intensity matching technique (IIMT) is applied to obtain deformation patterns in the quartz blank due to thermal load. Our preliminary results showed that the unsymmetric thermal deformations may be a dominant contributing factor to aging. For simulation purposes, finite-element analysis is used to investigate the deformation patterns (i.e., stress-strain distributions) and corresponding natural frequency shift in the piezoelectric resonators. The viscoelastic behavior of mounting adhesives is incorporated into the analysis to show the dominant effect of long-term behavior of stress-strain developed in the crystal resonators. Also, some geometrical aspects-such as uneven mounting supports due to distances, volumes and heights of the adhesives-are simulated in the model.     

Doubly rotated quartz resonators with a low amplitude-frequencyeffect: the LD-cut

This article deals with a crystal quartz cut with virtually no amplitude-frequency effect (also called "isochronism defect" or "anisochronism"). The cut, called LD-cut (Low isochronism Defect), is set in a BVA-type structure for later use in an Ultra-Stable Oscillator (USO). Various design parameters are presented, along with the properties of the resonator. The resonance frequencies of various modes, the temperature dependence, the motional parameters and the phase noise in relation to the power supplied to the resonator are mainly studied. These properties are compared to those of the SC-cut resonators     

Development of a 10 MHz oscillator working with an LGT crystal resonator: preliminary results

Presently, to our knowledge, measurement of the noise of langatate (LGT) crystal oscillators has not previously been reported. First results of such a measurement are given in this paper. They have been obtained from 10 MHz resonator prototypes tested with a dedicated electronics. The main steps of the resonator manufacturing are described in this paper. Good quality factors, close to 1.4 10(6), have already been achieved on the 5th overtone of the thickness shear mode of LGT Y cuts, even if the energy trapping should still be optimized. The motional parameters of these resonator prototypes are quite different from those of usual quartz crystal resonators. As a consequence, dedicated sustaining electronics have been designed. The explored options are reported to justify the implemented one. Moreover, the high thermal sensitivity of LGT crystal resonators (parabolic f-T curve) requires that particular attention be paid to the oven thermal stability. This important feature is also pointed out in the paper. The preliminary version of the resulting system exhibits a relative frequency stability of 6 10(-12).     

Quartz crystal oscillator at very low temperature

The most significant results concerning quartz crystal resonators and semiconductors properties at liquid helium temperature are presented. A cryogenic quartz oscillator is realized with these elements and some values of its frequency stability are given.     

Fast three-step method for shear moduli calculation from quartz crystal resonator measurements

Quartz crystal resonator measurements can be used for polymer material characterization. The non-gravimetric regime of these resonators is exploited: the electrical response of polymer-coated quartz resonators depends on the polymer shear modulus. Previously reported methods employ an electrical admittance analysis together with difficult and time-consuming data fitting procedures to calculate the film shear modulus. This contribution presents a fast and accurate three-step method for the calculation of complex shear moduli of polymer films from quartz crystal resonator measurements. In the first step, the acoustic load impedance is calculated from the electrical admittance of the quartz crystal. The key point of this method is the application of a family of approximations for the calculation of the shear modulus from the acoustic load impedance in the second step. In the third step, the best approximation is improved further in an iterative procedure.