Rectangular AT-cut Quartz Plates Research Articles

Thickness-shear vibration characteristics of an AT-cut quartz resonator with rectangular ring electrodes

We studied thickness-shear vibration frequencies and modes in a rectangular AT-cut quartz plate resonator with ring electrodes. The electrodes are also rectangular, with their central parts missing and thus form rectangular ring electrodes. It was found that the vibration tends to be trapped in the electrode area and decays away from the electrode edges. By properly designing the electrode dimensions, the vibration can be made nearly uniform in the central part of the plate. This is particularly suitable for resonator-based sensor applications where the sensitivity is determined theoretically from the uniform vibration distribution of pure thickness-shear modes in an unbounded plate.

Effects of Unequal Electrode Thickness, Location, and Size on Thickness-Shear Mode Quartz Piezoelectric Resonators

We studied thickness-shear vibrations of a rectangular AT-cut quartz plate piezoelectric resonator. The finite element method was used with COMSOL, a commercial software. The plate analyzed had two rectangular electrodes at its top and bottom surfaces near the center, forming a trapped energy resonator. The two electrodes had slightly different thickness, location, or size. The effects of electrode differences on the basic vibration characteristics of the resonator including resonant frequencies and mode shapes were calculated and examined. Results showed that the effects of electrode variations were small but were not negligible, and should be taken into consideration in resonator design.

On the acceleration sensitivity and its active reduction by edge electrodes in AT-cut quartz resonators.

Incremental piezoelectric equations for small vibrations superposed on initial deformations are presented. The equations are implemented in COMSOL finite element models (FEA). Equations are validated by comparing the results for the force sensitivity coefficient Kf of a circular quartz plate subjected to a pair of diametrical forces with measured data. The model results show a consistent trend with the experimental results, and the relative difference between our FEA results and Ballato's measured result is about 13%. A detailed study of the acceleration sensitivity of a rectangular AT-cut quartz plate is presented. The plate resonator is fixed along one edge as a cantilever. For AT-cut quartz resonators with the crystal digonal X-axis perpendicular to plate X-axis, the in-plane acceleration sensitivity is found to be negligible compared with the out-of-plane (Y-axis) acceleration sensitivity. For AT-cut quartz resonators with the crystal digonal X-axis parallel to plate X-axis, the Y-axis acceleration sensitivity is found to be rectified, that is the fractional change in frequency is positive with respect to both positive and negative Y-axis accelerations. The Y-axis acceleration sensitivity is small in comparison with the in-plane acceleration sensitivity for small body forces. However, for large body forces, the Y-axis acceleration sensitivity dominates because it increases nonlinearly with the Y-axis acceleration. The resonator rectified acceleration sensitivity is confirmed by phase noise measurements. For reduced acceleration sensitivity, two pairs of electrodes along the plate edges reduce the bending of the plate resonator and subsequently reduce acceleration sensitivity. We present a new method using these edge electrodes in which a dc bias field is employed to control the resonant frequency of resonator subjected to g body forces. A dc bias field with an appropriate dc bias voltage could potentially yield a reduction of acceleration sensitivity in Y-axis direction of about two orders of magnitude.

Three-Dimensional Analysis of Coupled Vibrations of Rectangular AT-Cut Quartz Plates with Tab Electrodes

We numerically analyzed the influence of strip tab electrodes on the coupled vibrations of the fundamental thickness-shear (TS) and flexural modes in a rectangular AT-cut quartz plate with rectangular central electrodes. It was assumed that the tabs were extended along the x1 (X) axis and softly supported at two points on a single x1-edge of the plate. We calculated the frequency spectra near the main TS response for three types of tab configurations that possessed a twofold rotational symmetry about the crystal X-axis of quartz as well as the rectangular plate and central electrodes. These results verified that the tabs affect the characteristics of coupling between the TS and spurious modes, and demonstrated that the coupling strength depended on both tab width and tab arrangement. We also clarified each property for the three types of tab configurations.

Numerical Analysis of Three-Dimensional Coupled Vibrations of Rectangular AT-Cut Quartz Plates with X-Edge Regions of Heavy Viscosity

Coupled vibrations in AT-cut quartz plates were analyzed. In this analysis, the viscosity constants of edge regions were multiplied by the weighting coefficient Wv. The viscosity data of quartz measured by Lamb and Richter were introduced into the analysis. The admittance charts and complex resonant frequencies near the main thickness-shear (TS) response were calculated as a function of increasing loss coefficient Wv. The calculation results revealed that for two strongly coupled vibrations, a single resonance was observed on the real frequency axis and the resonant resistance decreased as Wv increased. Moreover, there was an optimal Wv to obtain the highest Q of the main TS resonance and to control the spurious pole. It was also shown that even at optimum, the suppression of spurious resonance was incomplete, and an undesirable jump of resistance occurred in the temperature behavior of the main resonance.

Influence of Viscosity Loss on Three-Dimensional Vibrations of Rectangular AT-Cut Quartz Plates

We introduced the viscosity loss of quartz, and analyzed three-dimensional (3-D) coupled vibrations of a very high frequency (VHF) rectangular AT-cut quartz plate with partial electrodes. Classical mode matching was utilized to solve the 3-D problem for forced vibration. From the admittance chart near the main thickness-shear (TS) response, we extracted the variation of resonant frequencies and resistances with the magnitudes of mode coupling. The results revealed that we could estimate the lower bound of resonant resistances or the upper bound of Qs for VHF rectangular AT-cut plates by introducing the viscosity constants of quartz that were measured by Lamb and Richter. We also found that two series resonances on the real frequency axis could occur only within a limited range of the length-to-thickness ratio, and although each of two resonant frequencies was almost the same as that evaluated with an assumption of no losses, the corresponding resistances increased with the magnitudes of mode coupling.

P1-14 Influence of Viscosity Loss on 3-D Vibrations of VHF Rectangular AT-cut Quartz Plates(Poster session 1)

P1-14 Influence of Viscosity Loss on 3-D Vibrations of VHF Rectangular AT-cut Quartz Plates(Poster session 1)

Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates.

Thickness-shear vibrations of a rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2-D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1-D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerting on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin.

Analysis of 3-D vibrations of rectangular AT-cut quartz plates with a bi-mesa structure.

A method to analyze 3-D vibrations of rectangular AT-cut quartz bi-mesa-shaped plates is developed. The method is based on a classical approach. As in 2-D analysis, the half structure of a plate is separated into a thick bi-mesa and a thin-side portion, and the displacement field of each region is represented by a linear combination of guided waves. In the 3-D analysis, we apply the 2-D finite element method (FEM) to obtain the waves guided by two pairs of parallel surfaces. The orthogonal property of guided modes is incorporated to approximately fulfill the continuity conditions at the interface between the thick and thin portions. The stress-free conditions on the plate edges are satisfied by employing the method of weighted residuals (MWR). The computational advantage of this method is that it can greatly reduce the matrix size compared with the 3-D FEM. As a numerical example, the frequency spectra are calculated for X-elongated plates of bi-mesa shape, and the strong energy-trapping effect on the fundamental thickness-shear (TS) resonance is verified.

Frequency-temperature behavior of spurious vibrations of rectangular AT-cut quartz plates

We evaluate the frequency-temperature behavior of spurious modes in a rectangular AT-cut quartz plate resonator based on three-dimensional linear equations. The elastic constants and three geometrical dimensions of the resonator are defined in terms of cubic polynomials of a temperature change. Assuming that the resonator holds its rectangular plate shape irrespective of temperature, we can determine the relationship between frequency and the dimensions of the resonator for a given temperature using the previous technique. We compare the calculated results with our own experimental data, and show that agreement between the calculated and observed data is excellent.

Three-Dimensional Analysis of Spurious Vibrations of Rectangular AT-cut Quartz Plates

We have developed a technique for the three-dimensional (3-D) analysis of all the spurious vibrations of rectangular AT-cut quartz plates. The 3-D displacements are described with a linear combination of modes guided along the X crystal axis of an AT-cut quartz strip. We accurately calculate each guided mode by applying the one-dimensional finite element method to the two-dimensional strip waveguide problem. To uniquely determine the amplitudes of guided modes in the 3-D solutions we employ the variational expression for the stress-free conditions on the X-length edges. The present technique makes it possible to perform 3-D analysis of spurious vibrations within a reasonable computational time. Frequency spectra for rectangular AT-cut quartz plates are computed as a numerical example and are compared with the numerical results obtained using an extended version of the previous method and with Koga's experimental data. Vibrational patterns of spurious modes are shown, and a convergence study is carried out.

Application of Lee's Plate Equations to Analysis of Spurious Vibrations of Rectangular AT-Cut Quartz Plates

Strip AT-cut quartz resonators have recently been developed and are widely used in frequency applications because of their small size. Many spurious resonances can be supported in a rectangular AT-cut plate. Therefore, analysis of the influence of both the X and Z′ dimensions of the plate upon the spurious characteristics is necessary to design the strip resonators. However, it is very difficult to exactly solve three-dimensional equations of elasticity. The practical analysis of spurious characteristics has been carried out on the basis of approximate two-dimensional equations. In this paper, the applicability of Lee's two-dimensional equations to AT-cut plates is examined. The resonant frequencies are calculated as a function of X-length, and are compared to Koga's experimental data. The result shows that new correction factors should be introduced in the equations in order to adjust the approximate values of wave numbers for the thickness-flexure waves propagating in X and Z′ directions.

Mass-frequency influence surface, mode shapes, and frequency spectrum of a rectangular AT-cut quartz plate

The mass-frequency influence surface and frequency spectrum of a rectangular AT-cut quartz plate are studied. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied on the plate surface. Finite-element solutions of R.D. Mindlin's (1963) two-dimensional plate equations for thickness-shear, thickness-twist, and flexural vibrations are given. Spectrum splicing, and an efficient eigenvalue solver using the C. Lanczos (1950) algorithm are incorporated into the finite-element program. A convergence study of the fundamental thickness-shear mode and its first symmetric, anharmonic overtone is performed for finite-element meshes of increasing fineness. As a general rule, more than two elements must span any half-wave in the plate or spurious mode shapes will be obtained. Two-dimensional (2D) mode shapes and frequency spectrum of a rectangular AT-cut plate in the region of the fundamental thickness-shear frequency are presented. The mass-frequency influence surface for a 5-MHz rectangular, AT-cut plate with patch electrodes is obtained by calculating the frequency change due to a small mass layer moving over the plate surface. The frequency change is proportional to the ratio of mass loading to mass of plate per unit area and is confined mostly within the electrode area, where the magnitude is on the order 10(8) Hz/g.

Coupled and energy-trapped thickness vibrations in piezoelectric crystal plates

Simple form solutions are obtained for approximate two-dimensional equations of motion and dominant conditions for free edges of energy-trapped thickness vibrations of piezoelectric crystal plates. The resulting algebraic frequency equations are employed to calculate frequencies of fundamental and anharmonic overtones of thickness-shear vibrations in infinite quartz strips of SC cut with a pair of free edges and in rectangular AT-cut quartz plates with all four edges free. For circular quartz disks with free edge, frequencies of fundamental and harmonic overtones of thickness shear in AT cut, and frequencies of fundamental and anharmonic overtones of thickness shear in BT cut are computed. Presently predicted results are compared with either previously calculated results or existing experimental data. It is found that comparison is very close for all these cases.

Thickness shear, thickness twist, and flexural vibrations of rectangular AT-cut quartz plates with patch electrodes

The influence of the size and the mass of patch (rectangular electrodes on the vibrations and energy trapping in a plate rectangular crystal plate is studied. Mindlin’s two-dimensional approximate equations of vibrations of crystal plates are employed for which the coupled modes of thickness shear, thickness twist, and flexure are retained. Numerical solutions that satisfy the free-edge conditions of the plate and the continuity conditions of stresses and displacements at the interface of the plated and unplated regions of plate are obtained by the finite element method using displacement formulation. Numerical computations are made for rectangular AT-cut quartz plates with symmetrically placd, patch electrodes. Resonance frequencies and their corresponding two-dimensional modes of vibrations are obtained for various dimensions of plate and electrodes, and for different masses of electrodes. Calculated results are compared with existing analytical results by Mindlin and Lee [Int. J. Solids Structures, 2, 125 (1966)], Lee and Spencer [J. Acoust. Soc. Am. 45, 637 (1969)], and Tiersten [J. Acoust. Soc. Am. 59, 879 (1976)], and experimental data by Curran and Koneval [J. Acoust. Soc. Am. 34, 981 (1967)]. A two-dimensional Bechmann’s number is obtrained which depends essentially on three parameters; the electrode length-to-plate thickness ratio, the electrode wide-to-plate thickness ratio, and the ratio of the mass of electrodes to the mass of the plate per unit area.

Temperature Characteristics of Rectangular AT-cut Quartz Plates

Temperature Characteristics of Rectangular AT-cut Quartz Plates

The influence of electrodes on the resonance frequency of AT-cut quartz plates

The shear-flexural-twist vibrations in rectangular AT-cut quartz plates with partial electrodes are considered and the influences of metalic electrodes on the resonance frequency and on the resonance frequency temperature dependence are calculated. Different elastic stiffnesses were supposed in the plated and unplated part of the plate and in this way the influence of piezoelectricity is considered. The influences of mass loading and the length of electrodes have been calculated and examples of computed results are given.

Thickness-Twist Vibrations in Beveled AT -Cut Quartz Plates

Frequencies and mode shapes of thickness-twist vibrations in beveled, rectangular AT-cut quartz plates have been determined from an infinite-strip model. In addition, the possible use of beveling for mode suppression has been studied. Experimental results from beveled plates verify the theoretical predictions. Descriptions of the calculations and comparisons with the experimental results are included. Thickness-shear frequencies and mode shapes were also calculated using a similar model and are compared with thickness-twist results.

X-Ray Diffraction from Vibrating Quartz Plates

X-ray diffraction topographs of modes in rectangular AT-cut quartz plates are shown. The change in the resulting patterns when different Bragg planes are used is demonstrated. Diffracted x -ray intensity was measured as a function of strain in a vibrating quartz plate . The Lame solutions for quartz plates are briefly described and some experiments suggested for x -ray topography. Finally modes in quartz plates with multiple electrodes are shown indicating the complexity of this type of vibration problem.

Parametric excitation of contour modes of vibration in AT-cut quartz plates

Parametrically excited contour modes of vibration were discovered in a rectangular AT-cut quartz plate. These modes of vibration were found to satisfy the relationship f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> = f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> + f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> where f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is the driving frequency at the principal thickness-shear mode, and f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> are the vibration frequencies of two contour modes.