Design and fabrication of a length-extensional mode rectangular X-cut quartz resonator with zero temperature coefficient.

Rectangular X-cut quartz crystal resonators with cut angles theta > 5.0 degrees and aspect ratios Rzy (= width 2z0/length 2y0) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle theta and Rzy diagram. We proved that a fabricated resonator with Rzy = 0.400 and theta = 16.0 degrees, whose design parameter is located in the area of the contour, had a zero temperature coefficient.     

Design and fabrication of a mode rectangular X-cut quartz resonator with zero temperature coefficient

Rectangular X-cut quartz crystal resonators with cut angles /spl theta/ > 5.0/spl deg/ and aspect ratios R/sub zy/ (= width 2z/sub 0//length 2y/sub 0/) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle /spl theta/ and R/sub zy/ diagram. We proved that a fabricated resonator with R/sub zy/ = 0.400 and /spl theta/ = 16.0/spl deg/, whose design parameter is located in the area of the contour, had a zero temperature coefficient.     

First-order sensitivity analysis of quartz crystal resonator model parameters

A first-order measurement sensitivity analysis of a single mode, quartz resonator device model was conducted. This analysis predicts the variations in the motional arm resistance versus the resonance S-parameter as a function of measurement errors. The accuracy of this first-order sensitivity analysis is verified by computer simulation and experimentally. The model parameters were extracted following the recommended EIA-512 standard. The theoretical first-order sensitivity analysis, the computer simulation results, and the experimental measurements are found to be in good agreement.     

Force-frequency coefficient of symmetrical incomplete circular quartz crystal resonator

The changes in the resonance frequencies of the thickness-shear vibration of symmetrical incomplete circular AT-cut quartz crystal resonators (QXRs), which are used as sensing elements in digital force sensors or pressure transducers, by the application of diametrical forces are discussed by considering the piezoelectric effect and the anistropic characteristics of crystal plates. Two-dimensional motion equations for predicting the frequency changes are derived from three-dimensional piezoelectricity equations, and equations deduced by Janiaud for solving the stress distribution in crystal plates are used to calculate the force-frequency coefficients of incomplete circular resonators. The results show that the piezoelectric effect of crystal plates decreases the force-frequency coefficient of the azimuth angle 0 degrees by 7% as compared with no piezoelectricity assumption. The incomplete circular shape can be applied to a larger load than the circular disc because the flat regions of the incomplete shape provide a distributed load application without the problems of stress concentration of a point force inherent to a circular disc. The incomplete shape gives an increased force sensitivity when compared with the complete disc near azimuth angle 0 degrees. The theoretical curves agree well with Ratajski's experimental results.     

Double-resonance quartz crystal oscillator and excitation of a resonator immersed in liquid media

In this work we propose a novel circuit design: a double-resonance oscillator. Its oscillation shows two oscillation modes: frequency locking to the quartz crystal resonance and LC resonance oscillation. Transition of the oscillation mode and the strength of oscillation are analyzed and reviewed for the fundamental mode in comparison with a Colpitts oscillator. The experimental results support the estimates of negative resistance for the double-resonance oscillator compared with the LC oscillator.     

Determination of boron by using a quartz crystal resonator coated with N-methyl-D-glucamine-modified poly(epichlorohydrin)

A nongravimetric quartz crystal resonator for determination of boron was proposed. The key step is the preparation of a polymer that forms a complex with boron (from borate ion). The polymeric film is deposited on one face of an electrode-separated quartz crystal. The backbone of the polymer is poly(epichlorohydrin), which is modified to anchor N-methyl-D-glucamine. After reticulation and reduction, the film presents high stability and sensitivity to boron at pH 8.5. A carrier solution containing 50 mM EDTA ensures high conductivity and the elimination of several interfering metal ions. Boron is strongly retained by the film, and a positive shift of the oscillating frequency is proportional to its concentration. Boron is eluted with 1 mL of a 1 M mannitol solution. For a 0.160-mL sample loop and concentration up to 600 microM, the calibration sensitivity was 1.67 Hz/microM and the LOD was 2 microM. This limit could be lowered to 0.3 microM by using a 1.00-mL sample loop. In both cases, it was possible to detect 3 ng of boron. It was estimated that the nongravimetric sensor is at least 10 times more sensitive that a hypothetical gravimetric sensor.     

Thickness-shear mode shapes and mass-frequency influence surface of a circular and electroded AT-cut quartz resonator

Finite-element solutions for the fundamental thickness shear mode and the second-anharmonic overtone of a circular, 1.87-MHz AT-cut quartz plate with no electrodes are presented and compared with previously obtained results for a rectangular plate of similar properties. The edge flexural mode in circular plates, a vibration mode not seen in the rectangular plate is also presented. A 5-MHz circular and electroded AT-cut quartz plate is studied. A portion of the frequency spectrum is constructed in the neighborhood of the fundamental thickness-shear mode. A convergence study is also presented for the electroded 5-MHz plate. A new two-dimensional (2-D) technique for visualizing the vibration mode solutions is presented. This method departs substantially from the three-dimensional (3-D) ;wire-frame' plots presented in the previous analysis. The 2-D images can be manipulated to produce nodal line diagrams and can be color coded to illustrate mode shapes and energy trapping phenomenon. A contour plot of the mass-frequency influence surface for the plated 5-MHz resonator is presented. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied to the resonator surface.     

Reconstructing design parameters of a rectangular resonator via peak signal-to-noise ratio and global optimization algorithms

This article proposes an approach for reconstructing physical parameters of a sample in a rectangular resonator during microwave radiation, knowing a priori, its electric field distribution. The inverse problem was solved using two global optimization algorithms and the peak signal-to-noise ratio (PSNR) criterion. First, the Self-regulated Fretwidth Harmony Search algorithm (SFHS) identified suitable resonant frequencies for a given configuration. Next, the unified Particle Swarm Optimization (UPSO) optimized said configuration. Together, they became a maximization strategy of the PSNR through a dual optimization process. Results showed the ability of the approach for estimating the height of each sample block and the resonating frequency of the cavity. This process takes longer to finish as a higher PSNR is demanded (mainly due to the aforementioned dual optimization). Even so, it allows for more similar electric field distributions between both, the direct and inverse problems.     

Thickness-shear and thickness-twist vibrations of rectangular quartz crystal plates with nonuniform thickness

An analysis was performed on the vibration of a rectangular contoured quartz plate resonator. The surface of the resonator has different curvatures in the two in-plane directions. The analysis was based on the equation by Tiersten and Smythe, which has variable coefficients. The power series method was used. Thickness-shear and thickness-twist vibration frequencies and modes were obtained. The effects of the curvatures of the resonator surface were examined. Two types of boundary conditions were considered from which the proper size of the resonator was determined. The results are useful in the understanding of these resonators and their design.     

Probing near-surface nanoscale mechanical properties of low modulus materials using a quartz crystal resonator atomic force microscope

We describe the development of a technique for making indentations on the top 5-20 nm of the surfaces of relatively low modulus materials using a high spatial and force sensitivity atomic force microscope (AFM) whose optical cantilever has been replaced by a quartz crystal resonator (QCR). Unlike conventional optical-cantilever-based AFMs, the accuracy of this technique is not compromised by the compliance of the loading system due to the high stiffness of the QCR. To obtain material modulus values from the indentation results, we find the commonly used Oliver-Pharr model to be unsuitable because of our use of a sharp tip and relatively deep indentation. Instead, we develop a new analysis that may be more appropriate for the geometry we use as well as the non-linear constitutive behavior exhibited by the materials we examined. We calculated values for the moduli of several different materials, which we find to be consistent with the range of published data.     

The shear coefficient for quartz crystal of rectangular crosssection in Timoshenko's beam theory

In this paper, we derive the shear coefficient K for quartz crystal of rectangular cross section in Timoshenko's beam theory of three-dimensional (3-D) elasticity. The theoretical result gives K as a function of the transformed elastic stiffness constants C₂₂ ^', cˆ₄₄, C₆₆'. For an isotropic material it agrees completely with that derived by G.R. Cowper (1966). In addition, a value of K and its temperature coefficient versus cut angle is calculated so that K has a value of 0.796-0.867 and its temperature coefficient of the first order varies from -15.3×10 ^-6/°C to +13.2×10^-6/°C, when a Z-plate of quartz crystal is rotated with a cut angle &thetas; of 0-180° about the x axis. The values of K are then compared with those for isotropic materials obtained by other authors     

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.     

Measured resonance characteristics of a 2-ghz-fundamental quartz resonator

I fabricated an inverted-mesa AT-cut quartz resonator in which the thickness of the vibrating area was adjusted to 0.6 /spl mu/m by wet etching, and I demonstrated the excitation of a fundamental thickness vibration of 2.074 GHz. This main mode was inductive, and Q was 1037. However, all inharmonic modes were capacitive.     

A simple single model for quartz crystal resonator low level drivesensitivity and monolithic filter intermodulation

The nature of the observed phenomenon of “starting resistance” is reviewed, along with the accompanying experimental evidence of high drive-level altering and sometime “curing” of the malady. A simple phenomenological model is proposed, and then it is shown that this model does indeed predict all of the observed properties of starting resistance with the exception of “high-drive curing”. This same model is then shown to predict the low level intermodulation effects seen in monolithic crystal filters. The ability to predict both starting resistance and intermodulation phenomena with the same model is taken as an indication that both phenomena have a common cause. Explanations of high-drive-level curing of the phenomena are contemplated     

Survey of quartz bulk resonator sensor technologies

The term ;bulk resonator' is used to include a variety of vibrational modes. The survey is broken down by type of resonant mode, namely thickness shear, single-ended flexural, double-ended flexural, and torsional. Where appropriate, the discussion of each type of resonant mode includes items related to the frequency-control applications of the particular mode to emphasize the cross fertilization occurring between frequency control and sensor work.     

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 fulfil 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     

Calculation of thermoelastic stresses in a rectangular heated ingot

Thermoelastic stresses in a heated ingot are determined by the finite-difference method. We present the principal results of the numerical analysis. Dneprodzerzhinsk State Technical University, Dneprodzerzhinsk, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 130–136, July–August, 1997.     

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.