Effects of mode coupling on the admittance of an AT-cut quartz thickness-shear resonator

We study the effects of couplings to flexure and face-shear modes on the admittance of an AT-cut quartz plate thickness-shear mode resonator. Mindlin’s two-dimensional equations for piezoelectric plates are employed. Electrically forced vibration solutions are obtained for three cases： pure thickness-shear mode alone; two coupled modes of thickness shear and flexure; and three coupled modes of thickness shear, flexure, and face shear. Admittance is calculated and its dependence on the driving frequency and the length/thickness ratio of the resonator is examined. Results show that near the thickness-shear resonance, admittance assumes maxima, and that for certain values of the length/thickness ratio, the coupling to flexure causes severe admittance drops, while the coupling to the face-shear mode causes additional admittance changes that were previously unknown and hence are not considered in current resonator design practice.

ENERGY TRAPPING OF THICKNESS-SHEAR AND THICKNESS-TWIST MODES IN A PARTIALLY ELECTRODED AT-CUT QUARTZ RESONATOR

The thickness-shear and thickness-twist vibrations of a finite and partially elec- troded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.

The Characteristics of the Shear-Vibrating Quartz Crystal Resonator

<正>According to the piezoelectric equation and the vibration theory of the quartz crystal,the relations between the vibrating frequency and structural parameters under the thickness-shear-vibration of AT-cut quartz crystal have been studied. The frequency conditions under which quartz crystal resonator formed stationary wave inside the electrode district and the transmission characteristics of wave outside the electrode district have also been discussed.A quartz crystal resonator was developed based on this analysis.The experiment showed that the force-sensing characteristics were independent of the fixation of the crystal edge.The detecting distinguish ability was up to 0.001°,and the short-term frequency stability was up to 1.38×10~ (-10)/min.

PARALLEL FINITE ELEMENT ANALYSIS OF HIGH FREQUENCY VIBRATIONS OF QUARTZ CRYSTAL RESONATORS ON LINUX CLUSTER

Quartz crystal resonators are typical piezoelectric acoustic wave devices for frequency control applications with mechanical vibration frequency at the radio-frequency （RF） range. Precise analyses of the vibration and deformation are generally required in the resonator design and improvement process. The considerations include the presence of electrodes, mountings, bias fields such as temperature, initial stresses, and acceleration. Naturally, the finite element method is the only effective tool for such a coupled problem with multi-physics nature. The main challenge is the extremely large size of resulted linear equations. For this reason, we have been employing the Mindlin plate equations to reduce the computational difficulty. In addition, we have to utilize the parallel computing techniques on Linux clusters, which are widely available for academic and industrial applications nowadays, to improve the computing efficiency. The general principle of our research is to use open source software components and public domain technology to reduce cost for developers and users on a Linux cluster. We start with a mesh generator specifically for quartz crystal resonators of rectangular and circular types, and the Mindlin plate equations are implemented for the finite element analysis. Computing techniques like parallel processing, sparse matrix handling, and the latest eigenvalue extraction package are integrated into the program. It is clear from our computation that the combination of these algorithms and methods on a cluster can meet the memory requirement and reduce computing time significantly.

Coupled vibrations and frequency shift of compound system consisting of quartz crystal resonator in thickness-shear motions and micro-beam array immersed in liquid

The dynamic characteristics of a quartz crystal resonator（QCR） in thicknessshear modes（TSM） with the upper surface covered by an array of micro-beams immersed in liquid are studied. The liquid is assumed to be inviscid and incompressible for simplicity. Dynamic equations of the coupled system are established. The added mass effect of liquid on micro-beams is discussed in detail. Characteristics of frequency shift are clarified for different liquid depths. Modal analysis shows that a drag effect of liquid has resulted in the change of phase of interaction（surface shear force）, thus changing the system resonant frequency. The obtained results are useful in resonator design and applications.

Effect of strain-gradients of surface micro-beams on frequency-shift of a quartz crystal resonator under thickness-shear vibrations

With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator（QCR） system,consisting of a QCR and surface micro-beam arrays.The frequency shift that was induced by micro-beams with consideration of strain-gradients is discussed in detail and some useful results are obtained,which have important significance in resonator design and applications.

Influence of surface micro-beams with large deflection on the resonance frequency of a quartz crystal resonator in thickness-shear mode vibrations

We study the dynamic behavior of a quartz crystal resonator （QCR） in thickness-shear vibrations with the upper surface covered by an array of micro-beams （MBs） under large deflection. Through taking into account the continuous conditions of shear force and bending moment at the interface of MBs/resonator, dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations. It is found that the frequency shift produces a little right （left） translation for increasing elastic modulus （length/radius ratio） of MBs. Moreover, the frequency right （left） translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one,

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.

RESEARCH ON A SCANNING NEAR-FIELD MICROSCOPY BASED ON A QUARTZ RESONATOR

A scanning near field acoustic microscopy(SNAM) used in imaging the topography of precise surfaces is presented. As a micro force sensor, a non encapsulated 1 MHz quartz resonator is damped by hydrodynamic forces when approaching an object. Thus the oscillating characteristics of the leg of the quartz resonator change. While measuring the decrease of the amplitude of vibrating, the shape of the object can be obtained. Based on the analyses of the principle and key parameters of SNAM, a SNAM is exploited. A lateral resolution of 0.5 μm and vertical resolution of 2 nm has been achieved in the experiments.

Thickness shear mode quartz crystal resonators with optimized elliptical electrodes

Quartz crystal resonators （QCRs） with circular electrodes have been widely used for various liquid and gas sensing applications. In this work, quartz crystal resonators with elliptical electrodes were studied and tested for liquid property measurement. Mindlin＇s theory was used to optimize the dimension and geometry of the electrodes and a 5-MHz QCR with minimum series resistance and without any spurious modes was obtained. A series of AT-cut QCRs with elliptical electrodes of different sizes were fabricated and their sensing performances were compared to devices with circular electrodes. The experimental result shows that the device with elliptical electrodes can obtain lower resonance impedance and a higher Q factor, which results in a better loading capability. Even though the sensitivities of devices with elliptical and circular electrodes are found to be similar, the sensor with elliptical electrodes has much higher resolution due to a better frequency stability. The study indicates that the performance of QCRs with elliptical electrodes is superior to that of traditional QCRs with circular electrodes.

Electric admittance analysis of quartz crystal resonator in thickness-shear mode induced by array of surface viscoelastic micro-beams

The electric admittance of a compound system composed of a thickness-shear mode （TSM） quartz crystal resonator （QCR） and an array of surface viscoelastic micro-beams （MBs） is studied. The governing equations of the MBs are derived from the Timoshenko-beam theory in consideration of shear deformation. The electrical admittance is described directly in terms of the physical properties of the surface epoxy resin （SU-8） MBs from an electrically forced vibration analysis. It is found that both the inertia effect and the constraint effect of the MBs produce competitive influence on the resonant frequency and admittance of the compound QCR system. By further comparing the numerical results calculated from the Timoshenko-beam model with those from the Euler-beam model, the shear deformation is found to lead to some deviation of an admittance spectrum. The deviations are revealed to be evident around the admittance peak（s） and reach the maximum when a natural frequency of the MBs is identical to the fundamental frequency of the QCR. Besides, a higher order vibration mode of the MBs corresponds to a larger deviation at the resonance.

Coupled extension and thickness-twist vibrations of lateral field excited AT-cut quartz plates

In this paper, the coupled extension and thickness- twist vibrations are studied for AT-cut quartz plates under Lateral Field Excitation （LFE） with variations along the x1- direction. Mindlin＇s two-dimensional equations are used for anisotropic crystal plates. Both free and electrically forced vibrations are considered. Important vibration characteristics are obtained, including dispersion relations, frequency spectra, and motional capacitances. It is shown that, to avoid the effects of the couplings between extension and thickness-twist vibrations, a series of discrete values of the length/thickness ratio of the crystal plate need to be excluded. The results are of fundamental significance for the design of LFE resonators and sensors.

Start-Up Acceleration of Quartz Crystal Oscillator Using Active Inductance Double Resonance and Embedded Triggering Circuit

Low-frequency double-resonance quartz crystal oscillator was developed with active inductance circuit aiming the start-up of stable oscillation of tuning fork-type quartz crystal resonator at 32.768 kHz within 0.37 ms. The initial oscillation is triggered by a part of crystal oscillator forming a CR oscillator. The negative resistance ranges to 4 MΩ at gmf of 4.1 μA/V. In a limited frequency range, the circuit shows negative reactance Ccci = -3.4 pF equivalent to inductance Lcc = 9.8 H. The Allan standard deviation indicated 10-11 to 10-10, showing high stability comparable to general quartz crystal oscillator.

The Multi-field Coupled Vibration Analysis of AT-Cut Quartz Crystal Resonators with Parallelism Error

During the fabrication of quartz crystal resonators(QCRs),parallelism error is inevitably generated,which is rarely investigated.In order to reveal the influence of parallelism error on the working performance of QCRs,the coupled vibration of a non-parallel AT-cut quartz crystal plate with electrodes is systematically studied from the views of theoretical analysis and numerical simulations.The two-dimensional thermal incremental field equations are solved for the free vibration analysis via the coefficient-formed partial differential equation module of the COMSOL Multiphysics software,from which the frequency spectra,frequency–temperature curves,and mode shapes are discussed in detail.Additionally,the piezoelectric module is utilized to obtain the admittance response under different conditions.It is demonstrated that the parallelism error reduces the resonant frequency.Additionally,symmetry broken by the non-parallelism increases the probability of activity dip and is harmful to QCR’s thermal stability.However,if the top and bottom surfaces incline synchronously in the same direction,the influence of parallelism error is tiny.The conclusions achieved are helpful for the QCR design,and the methodology presented can also be applied to other wave devices.

三种石英晶体谐振器频率测试方法对比分析研究

测试石英晶体谐振器频率的传统方法有间接计算法、信号源起振法和网络分析仪法,三种方法各有大的缺陷。根据石英晶体谐振器频率的特点,提出三种新的测试方法:示波器测试法、频率计测试法和频谱仪测试法。这三种方法在基本测试原理、设备配置要求、测试精度和适用环境等方面存在差异。对三种方法的优缺点进行了详细比较,并针对不同应用条件提出了选用建议:如果只单纯测量石英晶体谐振器频率,选用频谱仪测试法;如果需要测量频率的同时还需要知道频率的特性等内容,宜选用频率计测试法;示波器测试法则可用在对石英晶体谐振器频率定性的粗测。此外,还根据各自的优势探讨了仪器协作测试的可能性。

压实石英颗粒中孔隙水微观动力学性质研究

土水作用使土体中孔隙水具有与自由水完全不同的物理状态,从而影响了土体的物理特性。作为地球表面含量最多的矿物之一,石英广泛存在于各种岩土体中,石英颗粒与水的相互作用极大的影响各种岩土体的工程特性。为了研究石英颗粒与水的相互作用,利用核磁共振技术探测了非饱和压实石英颗粒中不同温度条件下的纵向弛豫时间T_(1)分布曲线和常温下T_(1)-横向弛豫时间T_(2)相关谱图。通过分析孔隙水T_(1)/T_(2)分布曲线,发现峰点处T_(1)/T_(2)约为6.31。基于BPP(Bloembergen-Purcell-Pound)模型,计算得到非饱和石英土样中孔隙水分子的转动特征时间远大于自由水的转动特征时间,说明压实石英颗粒中的孔隙水活动度受到较大的抑制。利用Korb模型,计算得到了吸附水在石英颗粒表面的特征停留时间和亲和性。结果表明:水分子在离开石英颗粒表面时在颗粒表面移动了4.2×10^(9)步,说明石英颗粒对水分子具有较大的亲水性或束缚性。基于Arrhenius定理分析T_(1)分布曲线随温度的变化得到石英颗粒中表面吸附水的吸附活化能为26.8 kJ/mol,远大于黏土颗粒表面吸附水的吸附活化能。这表明石英颗粒表面对水分子的束缚作用大于黏土颗粒表面。经分析认为这主要是因为石英颗粒表面的硅醇基与表面吸附水形成大量的氢键使得石英颗粒对表面吸附水具有较大的束缚作用。

FeGa膜/AT切石英晶片双面复合蝶形谐振磁场传感器

蝶形磁致伸缩膜由于磁汇聚,其中心区域的磁致伸缩应变增强,但是和矩形AT切石英晶片复合时,磁膜耦合至矩形晶片电极区的应变会向周围无磁膜区域扩散,降低应变对晶片厚度剪切振动的作用,降低磁场传感器灵敏度。提出AT切石英晶片和FeGa膜形状一样的蝶形谐振磁场传感器,蝶形晶片可以将蝶形磁膜增强的晶片应变汇聚至晶片电极区,以提高传感器灵敏度。仿真结果预测,在23.8~118.9 Oe的磁场范围内,蝶形结构的灵敏度为蝶形磁膜/矩形晶片结构的3.73倍。通过微机械加工,制作了蝶形器件,对实际器件测试的结果表明,在76.4~117 Oe的线性区间内,蝶形传感器的灵敏度可达-29.08 Hz/Oe。

基于Chip-on-TEC的高精度晶体振荡器研究

提高晶体振荡器频率稳定度的首要手段是保持温度的稳定。基于热电制冷器具有的恒温特性,采用Chip-on-TEC方法,给出了一种高精度的晶体振荡器的设计方案。利用多物理仿真软件,针对商用的音叉型石英晶体,首次建立了32.768 kHz石英晶片的完整的有限元多物理场仿真模型。仿真结果表明,晶体的谐振频率与商用产品完全相同;基于TEC的晶体振荡器设计方案可以使得晶片温度变化低至0.008℃,对应的频率稳定度5.4×10^(-9)。考虑到高精度切型晶片材料的非易得性,所给出的方案为采用低精度切型晶片设计高精度的晶体振荡器提供了思路。

基于核磁共振技术云母石英片岩干湿循环损伤特性研究

为研究片岩在干湿循环作用下的微观损伤特性,对不同干湿循环作用下云母石英片岩试样进行核磁共振测试,并定量分析其损伤变量变化规律。结果表明:不同干湿循环作用下云母石英片岩孔隙之间连通性较好,孔径较小孔隙集中发育,随着干湿循环次数的增加,孔径逐渐扩展增大;云母石英片岩孔径发育具有关联性,干湿循环5次前小孔、中孔数量增加较快,5~10次循环后小孔数量趋于饱和,部分中孔发育扩展为大孔,导致中孔数量减少,大孔数量从5次干湿循环后开始显著增加;云母石英片岩受干湿循环影响损伤明显,损伤变量曲线在前5次干湿循环作用下近似呈直线上升,试样内部孔隙、裂隙发育扩展较快,5次干湿循环后,孔隙、裂隙的大小和数量发育减缓,云母石英片岩损伤变量增速逐渐变缓。

Quartz-enhanced multiheterodyne resonant photoacoustic spectroscopy

The extension of dual-comb spectroscopy(DCS)to all wavelengths of light along with its ability to provide ultralarge dynamic range and ultra-high spectral resolution,renders it extremely useful for a diverse array of applications in physics,chemistry,atmospheric science,space science,as well as medical applications.In this work,we report on an innovative technique of quartz-enhanced multiheterodyne resonant photoacoustic spectroscopy(QEMR-PAS),in which the beat frequency response from a dual comb is frequency down-converted into the audio frequency domain.In this way,gas molecules act as an optical-acoustic converter through the photoacoustic effect,generating heterodyne sound waves.Unlike conventional DCS,where the light wave is detected by a wavelengthdependent photoreceiver,QEMR-PAS employs a quartz tuning fork(QTF)as a high-Q sound transducer and works in conjunction with a phase-sensitive detector to extract the resonant sound component from the multiple heterodyne acoustic tones,resulting in a straightforward and low-cost hardware configuration.This novel QEMRPAS technique enables wavelength-independent DCS detection for gas sensing,providing an unprecedented dynamic range of 63 dB,a remarkable spectral resolution of 43 MHz(or~0.3 pm),and a prominent noise equivalent absorption of 5.99×10^(-6)cm^(-1)·Hz-1/2.