Quartz Strip Research Articles

Quartz strip resonators as a temperature sensor

The miniature frequency–temperature sensor in the form of a small quartz strip vibrating in the thickness-shear mode with the resonant frequency in the range from 4 to 8 MHz is considered. Y-cut strips rotated in the range from 0° to 27° are the subject of the study. The temperature coefficients of the resonance frequency for the different rotation angles are given. The different temperature dependence of the resonant frequency of the quartz temperature sensors vibrating on the fundamental and harmonic frequency of the thickness-shear modes is pointed out.

Forced vibrations of the fundamental family of modes in AT-cut quartz strip resonators

Mindlin’s equations for the vibrations of elastic crystal plates are employed in the description of AT-cut quartz strip resonators. The electrically driven three-dimensional piezoelectric pure thickness solution is incorporated in the treatment. The driving voltage appearing in this thickness solution is included in the variational principle from which the plate equations are obtained. In this way the resulting Mindlin equations contain the driving voltage and hold for plates with small piezoelectric coupling. The equations are applied in the analysis of strip resonators. The eigensolutions are obtained by solving a sequence of one-dimensional problems that are defined by utilizing the results from the previous problem variationally. The driven solution is obtained by means of an expansion in the eigensolutions and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. Calculated results are presented for a range of geometries and the influence of the couplings is exhibited.

Thickness-shear and flexural vibrations of linearly contoured crystal strips with multiprecision computation

The effect of linear contouring of crystal strips on the resonance frequencies, the coupling among vibration modes, and the spatial variation of mode shapes have attracted analytical and experimental studies for many decades for its application in crystal resonator design. In the present paper, exact solutions are obtained for the coupled thickness–shear and flexural vibrations of quartz strips with linearly varying thickness by the Frobenius method. Four special functions in terms of infinite power series are obtained from which displacements and stresses can be derived. The efficient and precise evaluation of these functions is aided by a multiprecision program. Frequency spectra and mode shapes are computed for various parameters of crystal strip resonators with beveled edges. Some important effects of contouring are discussed from these results.

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.

The piezoelectrically forced vibrations of AT-cut quartz strip resonators

The one-dimensional vibration equations of crystal strips of AT-cut quartz with narrow width and finite length are given by Lee and Wang [J. Appl. Phys. 75, 7681 (1994)]. The mechanical vibrations of such strips have been studied by neglecting the coupling due to piezoelectric effect. In the present study, the one-dimensional equations are further expanded to include the piezoelectric effect. New sets of coupled equations which are much larger than the ones for the mechanical vibrations are obtained. By neglecting the components of high-order displacements and electric potentials, particular attention is given to the thickness-shear vibrations of the strips. Analytical solutions of the free vibrations, including components of thickness-shear, flexural, width-shear, width-stretch, width-flexural, and the zeroth- and first-order electric potentials, are obtained. The dispersion relations, frequency spectra, and mode shapes of the free vibrations are given as functions of the ratios of length to thickness and width to thickness. The forced vibrations are studied with these solutions, and forced response and capacitance ratios are obtained. Comparisons are made with known experimental results. A quartz strip resonator is studied with this seven modes model. The effect of the length of the electrodes, the width of the crystal, and the forcing frequency on the resonance frequency, vibration mode shapes, and capacitance ratios are obtained.

Thickness-shear and flexural vibrations of contoured crystal strip resonators

A system of two-dimensional equations of motion of successively higher-order approximations for contoured crystal plates are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions of the thickness coordinate of the plates. By removing the first-order thickness-stretch mode and the first-order x2x3 (or fast) thickness-shear mode and all the higher ones, a set of first-order equations of motion is obtained for contoured crystal plates and for frequencies up to and including those of the fundamental x1x2 (or slow) thickness-shear mode. The coupled thickness-shear and flexural vibrations are studied for contoured quartz strip resonators having the shape of (a) a double wedge and (b) a plate with beveled edges. Exact solutions in terms of infinite power series are obtained by Frobenius method for the plates with linearly varying thickness in the x1 direction. Frequency spectra and mode shapes are computed for both types of resonators. The effects of the contour on the frequencies and modes are examined.

Vibrations of AT-cut quartz strips of narrow width and finite length

A system of one-dimensional equations of motion for AT-cut quartz strip resonators with narrow width and for frequencies up to and including the fundamental thickness shear is deduced from the two-dimensional, first-order equations for piezoelectric crystal plates by Lee, Syngellakis, and Hou [J. Appl. Phys. 61, 1249 (1987)] by expanding the mechanical displacements and electric potentials in a series of trigonometric functions of the width coordinate. By neglecting the piezoelectric coupling and the weak mechanical coupling through c56, four groups of coupled equations of motion are obtained. For the equations of each group, closed form solutions are obtained and the traction-free conditions at four edges are accommodated. Dispersion curves and frequency spectrum are computed for quartz strips. The predicted frequency as a function of the length-to-thickness ratio and as a function of the width-to-thickness ratio of the quartz strips is compared with experimental data with good agreement.

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.

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.

Third overtone quartz resonator

The Lee-Nikodem equations of motion of elastic plates are solved for the case of vibrations of an AT-cut quartz strip, with free faces and edges, at frequencies up to and including the third harmonic thickness-shear overtone.

Thickness-shear vibrations of a beveled AT-cut quartz plate

Thickness-shear vibration modes of a beveled AT-cut quartz plate have been analyzed from an equivalent three-dimensional equation of motion. This equation has recently been applied in the analysis of plano-convex contoured quartz crystal resonators by Tiersten and Smythe, and has been derived from the dispersion relation obtained in the course of an investigation on overtone modes of coupled thickness–shear and thickness–twist vibrations of trapped energy resonators with rectangular electrodes. Calculations based on the analysis have been shown to have excellent agreement with experimental results. The analysis accounts for the anisotropy, piezoelectricity, and mass loading effects due to the electrodes. The solution for a beveled quartz strip is expressed in terms of Bessel functions of the order ±1/3, and the eigenfrequencies are obtained for harmonic thickness and anharmonic lateral overtones for a given bevel angle and thickness-to-length ratio of the linearly tapered crystal strip. Both the clamped and free-edge boundary conditions are considered.

On a Piezoelectric Flapper Type Servovalve Operated by a Pulse-Width-Modulated-Signal

As a first step to developing a servovalve with minimum moving parts, a new servovalve is devised in which a piezoelectric flapper (a quartz strip) is used instead of a conventional flapper actuated through a torque motor. A servovalve of this type is desirable from the viewpoint of fast response, however it has not been in common use because of the strong hysteresis of piezoelectric material. In order to linearize the nonlinear characteristics of the flapper, a pulse width modulated wave is used as the input signal to the flapper. The frequency characteristics of the pulse width modulator is theoretically analyzed in detail to establish the theoretical basis of the pulse width modulation control system. As well-known, this basis has been ambiguous so far. The experimental frequency responses show that the bandwidth of the servovalve using this new technique is comparable to that of conventional valves having the same power output in spite of its simple construction in comparison with conventional ones.

Thickness-twist vibrations of a quartz strip

An exact solution of the three-dimensional equations of elasticity has been found for the thicknesstwist modes of vibration of a rotated-Y-cut quartz plate with a pair of parallel, free edges. The solution has a simple form: made possible by relaxing the condition that the edge-planes be perpendicular to the main faces of the plate. For the AT-cut, the edges are off perpendicular by about five degrees.