The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation

Recently, as the dissipation of quartz crystal through material viscosity is being considered in vibrations of piezoelectric plates, we have the opportunity to obtain electrical parameters from vibration solutions of a crystal plate representing an ideal resonator. Since the solutions are readily available with complex elastic constants from Mindlin plate equations for thickness-shear vibrations, the calculation of resistance and other parameters related to both mechanical deformation and electrical potential is straightforward. We start with the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model. The electrical parameters are derived with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All the electrical parameters are frequency dependent, enabling the study of the frequency behavior of crystal resonators with a direct formulation. Through the full consideration of complications like partial electrodes and supporting structures, we should be able obtain electrical parameters for practical applications in resonator design.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Effects of mass layer imperfect bonding on the electrical impedance of a quartz crystal microbalance

We study electrically forced vibrations of a crystal plate of AT-cut quartz carrying a thin mass layer operating as a quartz crystal microbalance for mass sensing. The mass layer is imperfectly bonded to the crystal plate with their interface described by the so-called interface-slip model which allows a discontinuity of the tangential interface displacement. Mindlin’s equations for piezoelectric plates are used. An analytical solution is obtained. The electrical impedance is calculated. The effects of an elastic interface and a viscoelastic interface are examined.     

Effects of radially applied angular sector stresses on the frequency of doubly rotated quartz crystal

Summary A theoretical study is presented to determine the sensitivity of a doubly rotated crystal resonator, subject to radial compression along two diametrically opposed zones. The variation of frequency is calculated as a function of the stresses applied to the crystal, then a coefficient of sensitivity is deduced. Some solutions are put forward for obtaining the stability of frequency of a resonator in the particular case where a quartz crystal, stress compensated (SC), is used. Moreover, the particular conditions are described for limiting the influence of possible faults in the construction of such quartz resonators, which must be employed when a very high stability of frequency is needed.     

Investigation of ring resonators in photonic crystal circuits

In this paper, we propose the implementation of waveguide-coupled ring resonators in photonic crystal integrated circuits. Using two-dimensional finite difference time domain (2D FDTD) method, we study the spectral characteristics of a waveguide-coupled ring carved in two-dimensional photonic crystal of square lattice (2D SLPC) and based on the results, we suitably modify the structure geometry to establish its performance as a ring resonator. We further investigate the effects of ring dimension and crystal parameters on the resonance properties of the ring resonator.     

Crystal quartz optical whispering-gallery resonators

A quality factor exceeding 5x10(9) is obtained in whispering-gallery mode (WGM) resonators fabricated of crystalline quartz. We observe significant electrical tunability of WGMs in x-cut resonators and demonstrate an electro-optic modulator with a submegahertz passband at 12 GHz. We discuss other photonics applications of the crystal quartz WGM resonators in narrowband agile tunable filters, compact narrow linewidth lasers, and microwave and millimeter wave oscillators.     

超材料谐振子间的电耦合谐振理论与实验研究

通过将两个金属开口环谐振器口对口地放置, 实现了超材料谐振子间的电耦合谐振. 对电耦合谐振的微波等效电路进行了理论分析和数值计算, 结果表明耦合后的超材料谐振子能产生两个谐振频率, 其中一个随耦合强度的增加逐渐向低频方向移动, 而另一个固定在单谐振子的谐振频率处不变. 微波透射谱的实验测试和电磁仿真结果表明, 两个谐振峰随耦合强度的增加分别向低频和高频方向移动. 分析表明: 低频谐振峰的位置主要是由超材料谐振子间的电耦合强度决定的; 高频谐振偏离单谐振子的谐振频率主要是由不可避免的磁耦合引起的, 而且在耦合间距越小时磁耦合影响越大. 提出的基于超材料谐振子间的电磁耦合实现的双频谐振及其可调性极大地增加了超材料的设计与应用空间.     

On optimal design of half-wave resonators for acoustic damping in an enclosure

Acoustic design parameters of a half-wave resonator are studied experimentally for purely acoustic tuning of the resonator. According to the standard acoustic-test procedures, acoustic-pressure signals in the model enclosure with the resonators are measured. Based on the signals, quantitative acoustic properties of damping factor and sound absorption coefficient are evaluated and thereby, the acoustic-damping capacity of the resonator is characterized. Sound absorption coefficient has the advantages of the damping factor in various aspects. The coefficient indicates clearly the tuning frequency of the resonator, absorption effectiveness as a function of frequency, and overall damping capacity. The diameter and the number of a half-wave resonator, its distribution, and the blockage ratio at its inlet are selected as design parameters for optimal tuning of the resonator in the model enclosure. The resonators with larger diameter have the advantage of those with smaller one with respect to purely acoustic damping at the tuning frequency. The optimum number of resonators or the optimum open-area ratio decreases as boundary absorption decreases. When the open-area ratio exceeds the optimum value, over-damping appears, leading to a decrease in peak absorption coefficient and a broadening of absorption bandwidth. Blockage at the resonator inlet controls both peak absorption coefficient and its absorption bandwidth and it can be considered one of design factors for acoustic tuning.     

Vibration analysis of a new polymer quartz piezoelectric crystal sensor for detecting characteristic materials of volatility liquid

We present a new polymer quartz piezoelectric crystal sensor that takes a quartz piezoelectric crystal as the basal material and a nanometer nonmetallic polymer thin film as the surface coating based on the principle of quartz crystal microbalance (QCM). The new sensor can be used to detect the characteristic materials of a volatile liquid. A mechanical model of the new sensor was built, whose structure was a thin circle plate composing of polytef/quartz piezoelectric/polytef. The mechanical model had a diameter of 8 mm and a thickness of 170 μ. The vibration state of the model was simulated by software ANSYS after the physical parameters and the boundary condition of the new sensor were set. According to the results of experiments, we set up a frequency range from 9.995850 MHz to 9.997225 MHz, 17 kinds of frequencies and modes of vibration were obtained within this range. We found a special frequency f_sp of 9.996358 MHz. When the resonant frequency of the new sensor’s mechanical model reached the special frequency, a special phenomenon occurred. In this case, the amplitude of the center point O on the mechanical model reached the maximum value. At the same time, the minimum absolute difference between the simulated frequency based on the ANSYS software and the experimental measured stable frequency was reached. The research showed that the design of the new polymer quartz piezoelectric crystal sensor perfectly conforms to the principle of QCM. A special frequency value f_sp was found and subsequently became one of the most important parameters in the new sensor design.     

Open Resonators Excited by Wave Beams

We study open resonators on the basis of the scattering theory. In particular, we develop the theory of an open resonator with a dielectric plate. This theory makes it possible to find the eigenfrequencies of such a system and their dependences on various parameters. Dispersion due to the dielectric plate is determined. The theory developed can be used as a basis for the method of measuring extremely small dielectric losses in the short-wavelength part of the millimeter-wave band. We also analyze the properties of a two-mirror resonator as a reflective filter and formulate the quality requirements for the exciting beam.     

Natural frequency for exponential shaped thermoacoustic resonator

The two-dimensional nonlinear acoustic field of eleven exponential shaped resonators was simulated with a computational fluid dynamics software Fluent.The influence of driving frequency and driving intensity on pressure in resonator as well as its natural frequency was investigated.The relationship between natural frequency and theoretical calculation resonance frequency was also explored.It is found that beating phenomena can be observed in the resonator when the driving frequency deviates from the natural frequency.Moreover,the natural frequency of resonator increases with the increasing of driving intensity,which shows a hard spring effect.However,the driving intensity plays little effect on natural frequency and the natural frequencies are smaller than the theoretical calculation values in any driving intensity.Meanwhile,a formula between the natural frequency and its first-order resonance frequency from theoretical calculation was obtained by linear fitting for all these exponential shaped resonators under consideration.It is also found that the highest pressure amplitude and highest pressure ratio can be obtained from the exponential shaped resonator of m=2.8 under the same driving intensity.Moreover,the relation between natural frequency and the theoretical resonance frequency for m=2.8-tube is slightly different from other tubes in consideration.     

指数型热声谐振管的固有频率

采用计算流体力学软件Fluent模拟研究了11种不同形状参数的指数型热声谐振管内二维非线性声场特性,分析了驱动频率和驱动强度对管内声压演化过程及固有频率的影响,并探索了指数管的固有频率与理论计算谐频之间的关系.研究发现:当驱动频率偏离谐振管固有频率时,管内将出现明显的"拍"现象;指数管的固有频率随驱动强度的增加而增加,呈现硬弹簧效应,但驱动强度对固有频率的影响较小并且在任何驱动下指数管的固有频率均小于理论计算谐频.针对所研究的指数型管,获得了其固有频率与理论计算谐频之间的关系式.结果表明,相同驱动下,形状参数m值约等于2.8的指数管所能获得的压力幅值及压比最大,且m=2.8指数管的固有频率与理论计算谐频之间的关系式与其他管型略有不同.      

Highly selective surface-wave resonators for terahertz frequency range formed by metallic Bragg gratings

In the frame of the quasi-optical approach we solve the diffraction problem and describe surface modes confined at a metallic plate with a shallow grating of finite length. We prove that such planar grating can form a highly selective surface-wave Bragg resonator. For a given material conductivity and grating length, we find the optimum corrugation depth that provides the maximum value of Q factor. These results are applicable for developing resonators for terahertz frequency bands.     

Capillary aging of the contacts between glass spheres and a quartz resonator surface

The strength of the contacts between small glass spheres and the surface of a quartz crystal resonator has been probed based on the increase of resonance frequency induced upon sphere contact. The acoustic interaction between the sphere and the plate is modeled as a low-frequency coupled resonance; the dependence of the resonant parameters on overtone order lends support to this model. After exposing the sample to humid air and drying it again, the contact strength increases at least tenfold due to capillary forces--we observe a hysteretic form of the sand-castle effect. Repeated wet-dry cycles reveal logarithmic capillary aging with time. The experiments suggest that the drying of the liquid bridges leads to a contraction of small voids in the contact zone, subsequently increasing cohesion.     

Low-frequency shear parameters of liquid viscoelastic materials

An experimental study of the shear parameters of viscoelastic liquids is carried out by the acoustic resonance method based on the changes in the natural frequency and Q factor of a piezoelectric quartz resonator. The liquid to be studied is placed between a stationary quartz strap and the piezoelectric quartz crystal vibrating at the resonance frequency. For a set of drilling muds, the values of the real and imaginary shear moduli are obtained at a frequency of 74 kHz. The measurements are performed with a liquid layer thickness much smaller than the shear wavelength. It is shown that the shear modulus decreases with increasing strain amplitude. A cluster model based on the Isakovich-Chaban nonlocal diffusion theory is proposed for explaining the low-frequency viscoelastic relaxation process.     

Analysis of the electrically forced vibrations of piezoelectric mesa resonators

We study the electrically forced thickness-shear and thickness-twist vibrations of stepped thickness piezoelectric plate mesa resonators made of polarized ceramics or 6-ram class crystals. A theoretical analysis based on the theory of piezoelectricity is performed, and an analytical solution is obtained using the trigonometric series. The electrical admittance, resonant frequencies, and mode shapes are calculated, and strong energy trapping of the modes is observed. Their dependence on the geometric parameters of the resonator is also examined.     

Energy trapping in high-frequency vibrations of piezoelectric plates with partial mass layers under lateral electric field excitation

We study coupled face-shear (FS) and thickness-twist (TT) motions of a piezoelectric plate of monoclinic crystals with mass layers on the central parts of the plate surfaces. The plate is driven by a lateral electric field. Mindlin’s first-order theory of piezoelectric plates is used. An analytical solution is obtained. Numerical results are presented for an AT-cut quartz plate, including the motional capacitance of the plate as a resonator and the vibration modes trapped under the mass layers in the central portion of the plate. The relationship between the dimension of the mass layers and the number of trapped modes is examined.     

Effect of mass layer stiffness on propagation of thickness-twist waves in rotated Y-cut quartz crystal plates

We analyze the effect of mass layer stiffness on thickness-twist waves propagating in a rotated Y-cut quartz crystal plate with thin mass layers on its surfaces. An equation that determines the dispersion relation of the waves is given and is solved numerically. Quantitative results of the effect of the mass layer stiffness on wave frequencies are presented. These results are important to the understanding and design of resonators and mass sensors.     

Nonlinear frequency shift in surface acoustic wave resonator operating as a gas sensor

The shift in the resonance frequency of a two-port quartz surface acoustic wave (SAW) resonator operating as a gas sensor without a selective layer is studied versus the power of an SAW excited in the resonator. At working frequencies of the resonator (≈389 MHz) placed in the flow of moisture-containing nitrogen gas, an anomalously large positive shift of the resonance frequency is observed as the SAW power exceeds 1 mW. This shift is one order of magnitude larger than that due to the nonlinear amplitude-frequency effect, which is known for quartz SAW resonators. Possible physical mechanisms underlying this phenomenon are analyzed. Experimental data indicate that such a shift is associated with the influence of a powerful SAW on sorption processes taking place on the active surface of the resonator rather than being a direct consequence of heating of the SAW substrate by the powerful SAW.     

Prediction of the frequency spectrum of AT-cut contoured quartz resonators by means of X-ray diffraction topography

This paper presents some important relationships relating to frequencies in contoured AT-cut quartz resonators. It is shown that frequency interval relations are not affected by the piezoelement geometry but are functionally related solely to the indices of the Hermite functions. According to the analysis of trapped-energy resonators, an X-ray technique for predicting the frequency spectrum can be derived. It is based on the use of X-ray patterns of two wave motions and enables one to determine the whole frequency spectrum in the vicinity of any odd harmonic overtone of vibration. Two contoured resonator analyses show that the X-ray topography predictions are in good agreement with experimental data.