Cut Quartz Resonator Research Articles

SC-Cut Quartz Resonators for Dynamic Liquid Viscosity Measurements.

This article proposes an innovative viscosity sensor based on the thickness-shear vibration of stress compensated (SC)-cut quartz resonator. The thickness-shear mode is first analyzed and further studied with fluid-structure interaction between the resonator and the viscous fluid loading. The characteristic equation is derived based on the 3-D linear piezoelectric equations and solved for sensitivity analysis. Then laboratory experiment is carried out to validate the theory. To conduct the viscosity measurement, the SC-cut quartz resonator is integrated with a U-tube test fixture, which is designed and fabricated for sensor housing to avoid the influence of the mass of the fluid. The resonator is tested with various viscosities by tuning the ratio of glycerol/water mixture. Experiment results show consistency with the analytical solution, which together present an improved sensitivity of viscosity measurement by using SC-cut quartz resonator comparing to other resonator-based viscosity sensors. The proposed viscosity sensor is sensitive, accurate, and portable, and therefore can be applied to real-time, on-site measurement or sampling of fluidic samples.

Simultaneous Multimode Oscillation of Stress-Compensated Cut Quartz Resonator with Narrow-Band Wide Variable-Range Quartz Crystal Oscillator

A multimode-quartz-crystal oscillator was developed to excite stable dual-mode resonance at different frequencies: The oscillation of the 3rd harmonic resonance of the principle C-mode and an additional resonance B-mode of SC-cut crystal. Harmonic combinations of the 3rd and fundamental mode of B-mode with the 3rd harmonics of C-mode are demonstrated. The measurement of the temperature dependence of the oscillation frequency is demonstrated along with the stability determined by root Allan variance. Dependence on the open conductance of the active circuit and the dependence on the coupling capacitors are discussed.

Equivalent-Circuit Model for Quartz Resonators Effects of Finite Element Analysis, Acceleration, and Mass Loading

In radar equipments and communication systems, quartz resonators are key components. So as to enhance the resonator performances, the development of accurate modelings taking into account the environment interaction (temperature, acceleration, magnetism, …) is one of the major economic and strategic stakes. The main idea is to produce a realistic modeling. In literature, a lot of modelings are based on finite element analysis. But, for industrial applications, it is essential to link these accurate modelings to electric simulation softwares, such as Spice. Moreover, it is necessary to develop a simple computation method in order to answer the industrial time constraints. The idea is to use a Butterworth-Van Dyke (BVD) model based on an electrical equivalent circuit with, as input data, the studied natural frequency and the antiresonance frequency of an accurate finite element modeling. For the numerical example, a SC-cut (Stress-Compensated cut) quartz resonator is studied in this article for its fundamental thickness-shear resonance, the third and the fifth overtone. The influence of the model mesh quality, the electrode mass-loading, and the acceleration sensitivity on the motional parameters are analyzed. First, the choice of the finite element number along the thickness axis is crucial for the computation accuracy of the motional parameters. Then, the more the electrode material has a high density, the more the mass loading has a strong influence of the motional parameter values. Finally, the motional parameters are not really influenced by the acceleration field.

Drive level dependency in quartz resonators

Common piezoelectric resonators such as quartz resonators have a very high Q and ultra stable resonant frequency. However, due to small material nonlinearities in the quartz crystal, the resonator is drive level dependent, that is, the resonator level of activity and its frequency are dependent on the driving, or excitation, voltage. The size of these resonators will be reduced to one fourth of their current sizes in the next few years, but the electrical power which is applied will not be reduced as much. Hence, the applied power to resonator size ratio will be larger, and the drive level dependency may play a role in the resonator designs. We study this phenomenon using the Lagrangian nonlinear stress equations of motion and Piola–Kirchhoff stress tensor of the second kind. Solutions are obtained using COMSOL for the AT-cut, BT-cut, SC-cut and other doubly rotated cut quartz resonators and the results compared well with experimental data. The phenomenon of the drive level dependence is discussed in terms of the voltage drive, electric field, power density and current density. It is found that the drive level dependency is best described in terms of the power density. Experimental results for the AT-, BT- and SC-cut resonators in comparison with our model results are presented. Results for new doubly rotated cuts are presented. The effects of spurious modes, quality factor and air damping on DLD are presented.

Acceleration insensitive encapsulated silicon microresonator

This paper presents the acceleration sensitivity and the resulting vibration-induced phase noise of an electrostatically coupled encapsulated silicon microresonator. External vibrations can produce phase noise in microresonators by generating time-varying stress in the resonant beams. A spring mounted resonator design that reduces the induced axial stress is presented here. Measurements and simulations show that the acceleration sensitivity and the vibration-induced phase noise of this device can be reduced 1000 times smaller than that of the previously published silicon microresonator and 10 times smaller than the commercially used high end SC (stress compensated)-cut quartz resonator.

Isochronism defect for various doubly rotated cut quartz resonators.

It has been shown in earlier works that the amplitude-frequency effect [also called isochronism defect (ID) or anisochronism] could be a limitation factor on ultrastable oscillators. Theoretical studies based on the nonlinear theory of piezoelectricity have already been developed to explain the amplitude-frequency effect. So, it is possible to estimate the dependence of the ID versus various parameters of the resonator design (overtone rank, radius of curvature, electrodes diameter, etc.). However, because of the lack of available fourth-order elastic coefficients, it is not possible to predict the ID of any resonant frequency of a given trapped energy resonator. To tentatively find orientations of plates exhibiting a quasi-null ID, we have realized electroded resonators with different orientations and curvatures. We present results that verify, particularly, the R(-1/2) dependence of the amplitude-frequency effect versus radius of curvature. Moreover, we show that the ID can be positive or negative, that it can vary from one orientation to other one of about one order of magnitude, and that there exists a thermal compensated mode for which the amplitude-frequency effect is null.

Determination of the vibration amplitude of thickness shear modes using white beam stroboscopic topography at E.S.R.F.

Abstract A new topographic method allows a direct mapping of the amplitude of the vibration modes of piezoelectric devices. With this method it is possible to measure the amplitude of the vibration of a thickness shear mode at each point of an AT cut quartz resonator. The principle of this method is based on the use of stroboscopic section topography obtained from the white beam delivered by a third generation synchrotron soure at ESRF.

Experimental study on the characteristic of the NS-GT cut quartz crystal resonator oscillating in the sub-resonant frequency

We previously reported that the dynamic photo-elastic method was a very effective measuring technique for the stress distribution of vibrating quartz crystal resonators. The existence of a twisted asymmetrical vibration mode has been verified experimentally when the NS-GT cut quartz crystal resonator was vibrating in the main resonant frequency (MRF). A MRF and a sub-resonant frequency (SRF) of the NS-GT cut quartz resonator were defined as follows. If a mechanical standing wave was in the x' or y' direction of the resonator, the former was MRF vibration and the latter was SRF vibration, respectively. In this paper, stress distributions of two samples of the NS-GT cut quartz crystal resonator, one of which had a thickness of 80 mum and the other 150 mum, were measured by the dynamic photo-elastic method when the resonators were vibrating in each SRF. Thereafter, vibration modes of those resonators were estimated by the experimental data of stress distributions. We find that the vibration mode of the 80-mum resonator had a simple mechanical standing wave on the y' direction and the vibration mode of the 150-mum resonator was combined with a shearing mode in the SRF vibration. From the experiment, we decided that vibration modes of the NS-GT cut quartz crystal resonator were composed of the longitudinal stress T(3)' belonging to the z' direction of the plate and of the shearing stress T(5)' when the plate thickness was thickened and the resonator was oscillating in the SRF.

Thermosensitive quartz resonators at cryogenic temperatures

The temperature-frequency characteristics (TFCs) of thermosensitive yxbl/10°54′/11°06′-cut quartz resonators are investigated in the temperature range 300-4.2 K. The linear character of the TFC is kept within the range 300–170 K but after that a non-linearity is observed, which can be expressed approximately by a polynomial of third order. The resonators have high temperature sensitivity, 80 mK at 4.2 K and 1 mK at temperatures over 30 K, and are insensitive towards magnetic fields. The investigations show that the temperature-sensitive sensor could be used as a temperature sensor for cryogenic temperatures.

Voltage controlled self‐temperature compensated crystal oscillators with NS‐GT cut quartz resonators

AbstractIn this paper, voltage controlled self‐temperature compensated oscillators made with NS‐GT cut quartz resonators (VC‐S‐TCXOs in what follows) are introduced and their frequency stability (obtained by controlling input voltage to VC‐S‐TCXO) is discussed. An oscillator circuit consists of a CMOS inverter, a variable capacitor diode, resistors, and capacitors. The primary oscillator frequency of the coupled quartz resonator is first derived from the electric equivalent circuit of the oscillator. The oscillator frequency is given by a function of the coupling coefficient, which in turn is given as a function of the load capacitance CL. Since frequency stability by controlling the input voltage to VC‐S‐TCXO is dictated by the VC circuit, the VC circuit was investigated in detail in order for the circuit independent of temperature to be obtained. Using the VC circuit, the oscillator frequency was controlled by the input voltage and the temperature stability of the oscillator frequency was less than 4 × 10−9 at a temperature range of ‐30 to 85 °C. Furthermore, it has been demonstrated experimentally that the frequency stability of the VC‐S‐TCXO directly after voltage control does not depend on the frequency temperature characteristics of the quartz resonator, time interval between the voltage control, or rate of temperature change.

力センサとしての矩形ATカット水晶振動子

力センサとしての矩形ATカット水晶振動子

Scanning shearing-stress microscope

A new scanning probe microscope based on a scanning tunneling microscope (STM) and a frequency shift of an AT cut quartz resonator has been developed. The quartz resonator coupled to a STM sample is oscillated at its resonance frequency. The shift of the resonance frequency corresponds to the strength of the shearing stress in the sample, and is caused by the shearing force interaction between the STM tip and the sample under the tip scanning. The preliminary images presented show simultaneously STM surface topographies and the changes of subsurface shearing stresses in gold thin films. The sensitivity of our microscope is 0.30 N/m for a 0.2-Hz frequency shift of the quartz crystal resonance frequency.

A new cut for torsional mode quartz crystal resonators of tuning fork type

AbstractThis paper proposes and describes a new cut torsional quartz resonator which is designated “TT‐Cut,” and clarifies its frequency characteristics, frequency temperature behavior, and electrical equivalent circuit parameters.First, in the analysis procedure, an equation of motion is derived from an energy method. Next, by solving the equation of motion under the boundary conditions of “free‐free bar” or “clamped‐free bar,” the frequency equations are derived as a function of thickness z0, width x0, and length y0.Furthermore, from the frequency equation a relationship of frequency constant (f. y0) versus thickness‐to‐width ratio Rzx (= z0/x0) and a relationship between thickness‐to‐width ratio Rzx and cut angles (ϕ, θ) where the first‐order temperature coefficient α reaches zero are derived theoretically. As a result, numerous relationships where α reaches zero are found to exist between thickness‐to‐width ratio Rzx and cut angles (ϕ, θ). Especially, the second‐order temperature coefficient β has a small value of ‐1.16 × 10−8/°C2 whose absolute value is approximately one‐third of the well‐known flexural mode quartz crystal resonator. The value of β then is‐compared with the measured data of ‐1.29 × 10−8/°C2, so that both results are found to agree sufficiently well.Finally, series resistance R1 and a quality factor Q of a tuning‐fork‐type resonator are examined. Consequently, it is shown that a tuning‐fork‐type torsional quartz crystal resonator is obtained with a small R1 of 2.2 to 14.4 k° and a large Q value of 276, 000 to 378, 000 in frequency range of 385 to 444 kHz.

Experimental verification of stress compensation in the SBTC-cut

The authors have demonstrated experimental verification of the stress compensation feature for the fast thickness shear mode of vibration of stress-compensated for B-mode and temperature-compensated for C-mode (SBTC)-cut quartz resonators. For the resonator design used in the cylindrical probe structure, the motional resistance for the B-mode of vibration was approximately 12% of that of the C-mode. The relatively large motional resistance for the C-mode of vibration of the SBTC-cut was found to be largely due to the lower piezoelectric coupling for the thickness excitation of this mode. In addition the proximity of the third overtone of the A-mode to the fifth overtone of the C-mode also contributed to the increase in the motional resistance. The authors have obtained experimental data on the temperature dependence of the planar stress coefficient and pressure dependence of the frequency-temperature characteristic for both the thickness-shear modes of the SBTC-cut. It is noted that such a doubly rotated cut can have applications in the design of either stable frequency sources or sensors for pressure and temperature measurements.

A K-cut quartz resonators

We calculated the turnover temperatures Tto of crystallographically doubly rotated quartz plates vibrating in the thickness mode. In addition to the known cuts situated along the loci of zero value of the first-order temperature coefficient of frequency, there is another family of orientations, with Tto≥75 °C, which, compared to other singly or doubly rotated cuts, are relatively insensitive to crystallographic orientations. We designate these orientations as the AK cut. We report experimental static frequency as a function of temperature characteristics f(T) for 14 AK-cut orientations, with angular combinations 30°≤φ≤46.1° and 21° ≤θ≤28.44°. Tto values are between 80 and 150 °C, in good agreement with computations. The f(T) curves have been fitted with third-order polynomials. Compared to the fundamental mode of vibrations, the 3rd overtone shows a larger number of modes, and resonance amplitudes are weaker. The frequencies of the harmonic and anharmonic modes are in fairly good agreement with calculations based on a trapped energy resonator model, including effects of coupled thickness shear and thickness twist.

Development of a Plano Convex AT- cut Quartz Resonator for wrist Watch

Development of a Plano Convex AT- cut Quartz Resonator for wrist Watch

Theoretical and Experimental Study of Modes of Vibration in Contoured, AT-Cut Quartz Resonator Plates

Thickness-shear and flexural modes of vibrations and their anharmonic overtones in rectangular AT-cut quartz plates of varying thickness are considered. Mathematical solutions are obtained analytically for plates with partial electrode platings and four free-edge conditions. The plated portion of the plate is of uniform thickness while in the unplated portion, the thickness varies as an exponential function of the length coordinate of the plate. A study is made of the effects on the resonant frequencies and mode shapes due to changes of plate parameters, such as the rate of decrease of the thickness of the plate and the ratio of the length of the plate to the length of the plating. The measured frequencies are compared with the computed values.