Doubly rotated contoured quartz resonators

Doubly rotated contoured quartz resonators are used in the design of temperature-compensated stable clocks and dual-mode sensors for simultaneous measurements of pressure and temperature. The design of these devices is facilitated by models that can predict frequency spectra associated with the three thickness modes and temperature and stress-induced frequency changes as a function of crystalline orientation. The Stevens-Tiersten technique for the analysis of the C-mode of a doubly rotated contoured quartz resonator is extended to include the other two thickness modes. Computational results for harmonic and anharmonic overtones of all three thickness modes of such resonators help in optimizing the radius of curvature of the contour and electrode shape for suppression of unwanted modes and prevention of activity dips. The temperature and stress-induced changes in thickness-mode resonator frequencies are calculated from a perturbation technique for small dynamic fields superposed on a static bias. The static bias refers to either a temperature or stress-induced static deformation of the resonator plate. Phenomenological models are also used for calculating the temperature and stress-induced changes in resonant frequencies as a function of crystalline orientation. Results for the SBTC-cut quartz plate with a spherical convex contour of 260 mm indicate that normal trapping occurs for the third (n=3) and fifth (n=5) harmonic of the A-mode, the fundamental (n=1) and third (n=3) harmonic of the B-mode, and the fundamental (n=1) and fifth (n=5) harmonic of the C-mode     

Thickness-shear and thickness-twist vibrations of an AT-cut quartz mesa resonator

We study thickness-shear and thickness-twist vibrations of an AT-cut quartz plate mesa resonator with stepped thickness. The equations of anisotropic elasticity are used with the omission of the small elastic constant c(56). An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and energy trapping are calculated and examined. The solution shows that a mesa resonator exhibits strong energy trapping of thickness-shear and thickness-twist modes, and that the trapping is sensitive to some of the structural parameters of the resonator.     

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.     

Acoustic Wave Flow Sensor Using Quartz Thickness Shear Mode Resonator

A quartz thickness shear mode (TSM) bulk acoustic wave resonator was used for in situ and real-time detection of liquid flow rate in this study. A special flow chamber made of 2 parallel acrylic plates was designed for flow measurement. The flow chamber has a rectangular flow channel, 2 flow reservoirs for stabilizing the fluid flow, a sensor mounting port for resonator holding, one inlet port, and one outlet port for pipe connection. A 5-MHz TSM quartz resonator was edge-bonded to the sensor mounting port with one side exposed to the flowing liquid and other side exposed to air. The electrical impedance spectra of the quartz resonator at different volumetric flow rate conditions were measured by an impedance analyzer for the extraction of the resonant frequency through a data-fitting method. The fundamental, 3rd, 5th, 7th, and 9th resonant frequency shifts were found to be around 920, 3572, 5947, 8228, and 10 300 Hz for flow rate variation from 0 to 3000 mL/min, which had a corresponding Reynolds number change from 0 to 822. The resonant frequency shifts of different modes are found to be quadratic with flow rate, which is attributed to the nonlinear effect of quartz resonator due to the effective normal pressure imposing on the resonator sensor by the flowing fluid. The results indicate that quartz TSM resonators can be used for flow sensors with characteristics of simplicity, fast response, and good repeatability.     

Theory of the rooftop resonator: resonant frequencies and eigenpolarizations

The eigenpolarizations and resonant frequencies are derived for a circular dielectric waveguide resonator (EH11 mode) terminated by perfectly reflecting metallic rooftop mirrors. It is demonstrated that, in the far-IR region, circularly polarized helical standing waves result from an arbitrary angular misalignment of the rooftop creases. Further, the right and left circularly polarized modes have different resonant frequencies.     

A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case

A modal representation of the thickness vibrations of rotated Y-cut quartz plates, which was used in the treatment of driven transversely varying thickness modes, is shown to be defective in certain respects. The differential equations and edge conditions for transversely varying thickness modes have been used in the accurate treatment of trapped energy resonators, monolithic crystal filters, and contoured quartz resonators, even though those defects were present. In this work those defects in the thickness solution are corrected along with the influence on the differential equations and edge conditions in the transversely varying case. The corrected modal representation shows that, because in practical applications to the above mentioned devices, the driving frequency is always near a thickness resonant frequency, essentially the same results will be obtained with the corrected representation as were obtained with the defective one, which explains why the results obtained with the defective equations were so accurate.     

A coupled-mode theory for periodic piezoelectric composites

A coupled oscillator model to calculate the resonance spectrum of a one-dimensional piezoelectric composite plate, used in ultrasonic transducers, is proposed. Two resonant modes, one produced by the elastic wave reflection on the plate boundaries (thickness resonance) and the other by the reflection on the periodic discontinuities (lateral resonance) are considered. A Kronig-Penney model is used to calculate the lateral resonances. The thickness resonance is obtained with an effective medium model. The coupling of these two modes is described by a biquadratic equation whose solutions are the resonant frequencies of the piezoelectric composite plate. A criterion for a distribution of phases to keep the spurious lateral resonances away from the thickness resonance vicinity is obtained.     

Equivalent circuit for the microstrip ring resonator suitable for broadband materials characterisation

A simple equivalent circuit of the edge coupled microstrip ring resonator is developed based on both circuit and electromagnetic theory. The new model extends the work done by previous authors by including the effects of radiation loss, the coupling gap and the feed network as well as extending the frequency range by including higher-order modes. The model accurately predicts the resonant frequencies of the ring including the effects of dispersion and thick conductors. The equivalent circuit allows measurement of the resonance frequencies and the Q factors to be made independently of the coupling gap dimensions, which traditionally have been difficult to accurately model. A method of determining radiation loss is also presented, which has often been incorrectly neglected in the past. Experimental results support the accuracy of the equations and measurements on alumina rings that demonstrate a frequency accuracy of better than 1% over 12 resonant modes in the frequency range 3-33 GHz.     

Analysis of whispering-gallery superconducting dielectric resonator modes

The whispering-gallery (WG) modes of a superconducting dielectric resonator (SDR) based on a sapphire cylindrical dielectric resonator and a YBa₂Cu₃O_{7 –} shielding cylinder have been studied. A method for the determination of the resonant frequencies and the maximum quality factor of such modes is presented. Calculations have shown that most of the mode energy could be confined between the caustic surface of the WG modes provided the dimensions of the SDR are properly selected, and a magnitude of 10⁹ forQ of the SDR could be estimated. A phenomenal explanation is given to account for such outstanding microwave behavior.     

A thickness mode acoustic wave sensor for measuring interface stiffness between two elastic materials

We studied thickness vibration of 2 elastic layers with an elastic interface mounted on a plate piezoelectric resonator. The effect of the interface elasticity on resonant frequencies was examined. The result obtained suggests an acoustic wave sensor for measuring the elastic property of an interface between 2 materials.     

Analysis of the Internal Noise of Quartz-Crystal Oscillators

This paper describes the influence of the parallel capacitance of a quartz-crystal resonator on the amplitude-frequency coupling and particularly on the internal noise spectra of the oscillator working at the series resonance. A theoretical analysis which is a first order perturbation method is used. It is shown that the parallel capacitance of the quartz-crystal resonator increases the amplitude-frequency coupling and drastically modifies both amplitude and phase spectra of the internal noise. The 1/f2 phase spectrum of the internal thermal noise is transformed into a white phase spectrum for noise component frequencies greater than f0 + f? or less than f0 - f?, where f0 is the resonator series resonant frequency and f?, the difference between antiresonant and resonant frequencies of the quartz crystal. A "noise quieting" phenomenon appears when the noise component frequencies are in the vicinity of the antiresonant frequency fp. A good agreement between theoretical and experimental results for different values of the parallel-capacitance proves the validity of the mathematical model.     

圆柱共鸣腔力学特性模拟分析

采用ABAQUS有限元软件模拟了圆柱共鸣腔在不同螺栓预紧力下的变形以及共鸣腔的固有频率,实验测量了圆柱共鸣腔的实际固有频率,模拟仿真结果与实验测量结果具有良好的一致性,为下一步壳体振动修正理论模型的建立以及共鸣腔谐振模式的选取提供了基础。     

Analysis of a monolithic, two-dimensional array of quartz crystal microbalances loaded by mass layers with nonuniform thickness

We study free thickness-shear vibrations of a monolithic, two-dimensional, and periodic array of quartz crystal microbalances loaded by mass layers with gradually varying thickness. A theoretical analysis is performed using Mindlin's two-dimensional plate equation. It is shown that the problem is mathematically governed by Mathieu's equation with a spatially varying coefficient. A periodic solution for resonant frequencies and modes is obtained and used to examine the effects of the mass layers. Results show that the vibration may be trapped or untrapped under the mass layers. The trapped modes decay differently in the two in-plane directions of the plate. The mode shapes and the decay rate of the trapped modes are sensitive to the mass layer thickness.     

Electroelastic effect of thickness mode langasite resonators

Langasite is a very promising material for resonators due to its good temperature behavior and high piezoelectric coupling, low acoustic loss, and high Q factor. The biasing effect for langasite resonators is crucial for resonator design. In this article, the resonant frequency shift of a thickness-mode langasite resonator is analyzed with respect to a direct current (DC) electric field applied in the thickness direction. The vibration modes of a thin langasite plate fully coated with an electrode are analyzed. The analysis is based on the theory for small fields superposed on a bias in electroelastic bodies and the first-order perturbation integral theory. The electroelastic effect of the resonator is analyzed by both analytical and finite-element methods. The complete set of nonlinear elastic, piezoelectric, dielectric permeability, and electrostrictive constants of langasite is used in the theoretical and numerical analysis. The sensitivity of electroelastic effect to nonlinear material constants is analyzed.     

Tunable ultracompact electro-optical photonic crystal ring resonator

A tunable ultracompact electro-optical photonic crystal ring resonator with high transmission is reported. The photonic crystal ring resonator is obtained by removing a ring shape of cylinders from a square lattice of dielectric cylinders in air. The transmission spectra of this ring resonator have been investigated by using the finite-difference time-domain technique. The general characteristics of the ring elements to achieve resonant tunneling are determined. By modulating the conductibility of the inner cylinders in the ring resonator, the electrical tunability of the resonant modes is observed in the transmission spectrum. The research results should open opportunities for this ring resonator as ultracompact filters, optical add-drop multiplexers, electro-optical N × N switches, and modulators.     

Analysis of microwave ovens loaded with lossy process materials using the transmission-line-matrix method

The transmission-line-matrix (t.l.m.) numerical analysis technique is used in this paper to evaluate resonant frequencies and field parameters of the generated modes in microwave ovens loaded with lossy process materials. Analysis is made for the variation of resonant frequencies with the loss factor of the loading material. Temperature rise profile through the load is predicted from the calculated results of field parameters of sustained modes. The demonstrated data are for the dimensional parameters of a rectangular cavity of a commercial microwave oven operating at the standard heating frequency 915 MHz.     

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.     

Analysis of acoustic resonator with shape deformation using finite element method

An acoustic resonator with shape deformation has been analysed using the finite element method. The shape deformation is such that the volume of the resonator remains constant. The effect of deformation on the resonant frequencies is studied. Deformation splits the degenerate frequencies.     

Theoretical analysis of a ceramic plate thickness-shear mode piezoelectric transformer

We perform a theoretical analysis on a ceramic plate piezoelectric transformer operating with thickness-shear modes. Mindlin's first-order theory of piezoelectric plates is employed, and a forced vibration solution is obtained. Transforming ratio, resonant frequencies, and vibration mode shapes are calculated, and the effects of plate thickness and electrode dimension are examined.     

Ultra-wideband bandpass filters with improved out-of-band behaviour via embedded electromagnetic-bandgap

A class of ultra-wideband microstrip-line bandpass filters with improved out-of-band behaviours is presented by using an electromagnetic (EM) bandgap embedded multimode resonator. At first, the two EM bandgap structures are constructed by periodically loading uniform or non-uniform open-circuited stubs. Their performances are numerically characterised, aiming at exhibiting the existence of a bandstop in a frequency range of 15.0?30.0 GHz. These bandgap structures are then internally embedded in the middle part of an open-ended microstrip line resonator, constituting a modified type of multimode resonator. In the design, the dimensions of this resonator need to be properly adjusted towards allocating the first three resonant modes in the 3.1?10.6 GHz ultra-wide passband. To achieve a preferably wide upper stopband, the transmission zero of parallel-coupled lines is used to cancel the spurious fourth-order resonant mode. Also, higher-order modes in the upper band are inhibited by the employed bandgap structures. Two bandpass filters using the nonuniform stub-loaded resonator are finally designed, fabricated and tested. Predicted insertion/return losses and group delays are experimentally confirmed in a wide frequency range of 1.0?30.0 GHz.