Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour

A theoretical analysis is performed on thicknessshear vibrations of a contoured AT-cut quartz resonator with a hyperbolic thickness variation using the Legendre equation and hypergeometric function. Based on the solution, resonant frequencies and modes are calculated. Strong energy trapping of the modes is observed. The effects of the parameters of the hyperbolic contour on resonant frequencies and modes are examined. A comparison with the conventional contoured resonator in the literature with a quadratic thickness variation is made. The behaviors of the two types of resonators are qualitatively similar, with small but significant quantitative differences.     

Effects of electromagnetic radiation on the Q of quartz resonators

The quartz resonator Q with aluminum electrodes was studied with respect to its fundamental thickness shear mode frequency and its viscoelastic, viscopiezoelectric, and viscopiezoelectromagnetic behaviors. The governing equations for viscoelasticity, viscopiezoelectricity, and viscopiezoelectromagnetism were implemented for an AT-cut quartz resonator. To simulate the radiation conditions at infinity for the viscopiezoelectromagnetic model, perfectly matched layers over a surface enclosing the resonator were implemented to absorb all incident electromagnetic radiation. The shape of the radiation spectrum of a 5.6 MHz AT-cut quartz resonator was found to compare relatively well the measured results by Campbell and Weber. The mesa-plate resonator was studied for a frequency range of 1.4 GHz to 3.4 GHz. The resonator Q was determined to be influenced predominantly by the quartz viscoelasticity; however at frequencies greater than 2.3 GHz, the quartz electromagnetic radiation had an increasingly significant effect on the resonator Q. At 3.4 GHz, the electromagnetic radiation accounted for about 14% of the loss in resonator Q. At frequencies less than 2 GHz, the calculated resonator Q compared well with the intrinsic Q_x provided by the formula Q_x = 16 times 10⁶/f where f was in MHz. At frequencies higher than 2.3 GHz, the aluminum electrodes had significant effects on the resonator Q. At 3.4 GHz, the electromagnetic radiation loss in the electrodes was an order of magnitude greater than their viscoelastic loss; hence, the vibrating aluminum electrodes became an efficient emitter of electromagnetic waves. The effects of electrical resistance in both the electrodes and quartz were determined to be negligible.     

Estimation of quartz resonator Q and other figures of merit by an energy sink method

An important determinant of the quality factor Q of a quartz resonator is the loss of energy from the electrode area to the base via the mountings. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base substrate. The base substrate affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a base. Hence, the base is an energy sink and the Q will be affected by the shape and size of this base. A lower bound Q will be obtained if the base is a semi-infinite base since it will absorb all acoustical energies radiated from the resonator. A scaled boundary finite element method is employed to model a semi-infinite base. The frequency spectra of the quartz resonator with and without the base are presented. In addition to the loss of energy via the base, there are other factors which affect the resonator Q, such as, for example, material dissipation, and damping at the interfaces of quartz and electrodes. The energy dissipation due to material damping increases with the resonant frequency and the reduction of resonator size; hence material damping becomes important in the current and future miniaturized resonators operating at very high frequencies. An energy sink model along with material dissipation would provide realistic Q, motional capacitance, motional resistance, and other figures of merit useful for designing resonators. The model could be used for evaluating resonator and mountings designs of microelectromechanical systems and miniaturized devices. The effect of the mountings, and plate and electrode geometries on the resonator Q and other electrical parameters are presented for AT-cut quartz resonators. Model results from the energy sink method were compared with experimental results and were found to be good.     

Doubly rotated resonators for sensing the properties of liquids

When a doubly rotated resonator is operated in a liquid, the displacement of the surface is partly out of the plane of the plate of the resonator. The out-of-plane component of the displacement propagates a damped compressional wave into the liquid, and the in-plane component propagates a damped shear wave. In this paper, we report the measurements of the series resonant frequency and the motional arm resistance of doubly rotated quartz resonators (theta approximately 35 degrees and phi = 7 degrees) in liquids to compare with singly rotated AT-cut resonators (theta approximately 35 degrees and phi = 0 degrees). A modified Butterworth-Van Dyke (BVD) equivalent circuit model is suggested to analyze doubly rotated cut resonators under liquid loading.     

Thickness-shear vibrations of rotated Y-cut quartz plates with imperfectly bonded surface mass layers

A solution is obtained from the three-dimensional equations of linear piezoelectricity for the pure thickness-shear vibration of rotated Y-cut quartz or langasite plates with imperfectly bonded surface mass layers. The solution includes a few results in the literature as special cases. It is shown that the mass layers lower the resonant frequencies when they are relatively perfectly bonded to the crystal surface, and that loosely bonded mass layers may raise the frequencies. The results are useful in the analysis of frequency stability of quartz resonators and acoustic wave sensors.     

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.     

A new method of determining the equivalent circuit parameters of piezoelectric resonators and analysis of the piezoelectric loading effect

An inductive method of piezoelectric resonance detection is applied to the determination of equivalent circuit parameters of piezoelectric resonators. Using this method one can measure the resonance frequency and mechanical Q-factor of a resonator directly as well as their dependences on the electrical impedance which is connected to the resonator. From the equivalent circuit analysis the changes in resonance frequency and Q-factor due to the piezoelectric loading effects are determined. Measurements on two typical commercial piezoelectric resonators, an AT-cut quartz crystal and a PZT ceramic resonator, are in good agreement with the analysis.     

Pressure Dependence of Bis(2-ethylhexyl) Sebacate and VG32 Hydraulic Oil Viscosities Using a Quartz Crystal Resonator

Conductances versus sweep frequencies for AT-cut quartz crystal resonators have been measured for bis(2-ethylhexyl) sebacate and several machine oils under high pressure. The frequencies versus viscosities of these liquids appeared on the correlation line between frequency shift and viscosity. The correlation line agreement is essentially a criterion for when quartz resonators may be used because non-Newtonian silicone oils deviated from the correlation line. The responses of resonators immersed in bis(2-ethylhexyl) sebacate were obtained at pressures up to 350 MPa. Viscosities were calculated from the resonance frequencies. The pressure dependence of the bis(2-ethylhexyl) sebacate viscosity agreed well with that obtained with the rolling ball method. By using reference liquids, it was found that resonator responses were independent of temperature over the range 295–372 K. Viscosity measurements for VG32 hydraulic oil were taken at pressures up to 250 MPa and at the temperature at which the commercial viscosity grade was defined [313 K (40 °C)].     

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.     

Strained surface layers of quartz plates produced by lapping and polishing and their influence on quartz resonator performance

The strained surface layers of AT-cut quartz plates produced by lapping and polishing have been studied using the X-ray double crystal method. In addition, the influence of surface strain on the characteristics of quartz resonators has been investigated. Experimental results from X-ray double crystal measurements indicate that a residual stress layer in the lapped and polished surface is created and that the quartz resonator performance is also affected.     

Bulk acoustic wave sensors for sensing measurand-induced electrical property changes in solutions

A variety of quartz thickness shear mode (TSM) resonant sensors with different electrode configurations have been designed, fabricated, and tested in liquids for probing liquid electrical property changes. The resonant frequency of the sensors was found to have more than an order of magnitude increase in sensitivity over the standard AT-cut quartz resonant sensor. The increase in sensitivity is due to the ability of the sensors to detect changes in the electrical properties in the liquid     

Analysis of piezoelectric bulk-acoustic-wave resonators as detectors in viscous conductive liquids

An analytical solution for the resonance condition of a piezoelectric quartz resonator with one surface in contact with a viscous conductive liquid is presented. The characteristic equation that describes the resonance condition and accounts for all interactions including acoustoelectric interactions with ions and dipoles in the solution is obtained in terms of the crystal and liquid parameters. A simple expression for the change in the resonance frequency is obtained. For viscous nonconductive solutions, the frequency change is reduced to a relationship in terms of the liquid density and viscosity. For dilute conductive liquid, the change in frequency is derived in terms of the solution conductivity and dielectric constant. The boundary conditions for the problem are defined with and without the electrical effects of electrodes. Experiments were conducted with various viscous and conductive chemical liquids using a fabricated miniature liquid flow cell containing an AT-cut quartz crystal resonator. The results, which show good agreement with the theory, on the use of quartz crystal resonators as conductivity and/or viscosity sensors are reported.     

Chemical sensor based on surface acoustic wave resonator using Langmuir-Blodgett film

A one-port surface acoustic wave (SAW) resonators incorporating Langmuir-Blodgett (LB) films has been investigated. SAW sensors are one potential applications of SAW devices. Most of the work reported on SAW sensor concerns delay lines. In this paper we characterize the mass loading effects of one-port resonators by depositing successive monolayers of LB films onto the surface. A 90 MHz SAW gas-phase sensor has been fabricated on an ST cut quartz substrate, and one-port resonator configurations have been used as the sensing element. Ultra thin monolayers of arachidic acid and arachidic acid ethyl ester have been deposited using the LB method. The resonant frequencies and the Q values have been measured as sensor response. Experimental results show that the Q values and the resonant frequencies of the one-port SAW resonator vary with film mass loading on the SAW device surface.     

Quantum 1/f effect in resonant biochemical piezoelectric and MEMS sensors

Piezoelectric sensors used for the detection of chemical agents and as electronic nose instruments are based on bulk and surface acoustic wave resonators. Adsorption of gas molecules on the surface of the polymer coating is detected by a reduction of the resonance frequency of the quartz disk, subject also to fundamental quantum 1/f frequency fluctuations. The quantum 1/f limit of detection is given by the quantum 1/f formula for quartz resonators. Therefore, for quantum 1/f optimization and for calculation and improvement of the fundamental sensitivity limits, we must avoid closeness of the crystal size to the phonon coherence length, which corresponds to the maximum error and minimal sensitivity situation, as shown here. Adsorbed masses below the pg range can be detected. Microelectromechanical system (MEMS) resonators have provided a possibility for the nanominiaturization of these sensors. Essential for integrated nanotechnology, these resonant silicon bars (fingers) are excited magnetically or electrically through external applied forces, since they are not piezoelectric or magnetostrictive. The application of the quantum 1/f theory to these systems is published here for the first time. It provides simple formulas that yield much lower quantum 1/f frequency fluctuations for magnetic excitation, in comparison with electrostatically driven MEMS resonators.     

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     

Thermal transient model of a crystal resonator employing thickness-shear vibrations

This paper presents an improved model of thermally induced frequency transients in vacuum-enclosed thickness-shear mode quartz crystal resonators. The response times to temperature changes for different parts of the resonator and resulting thermal dynamic coefficients are examined and are related to Ballato's coefficient through a function defined by the resonator design, dependent on thermal response times only. A method is worked out for response time calculations for the different contributions to the static and dynamic temperature behavior of general and anharmonic modes. The model has been used to examine thermally induced frequency transients of the AT-cut resonator h(513) anharmonic mode excited by the modulational method within an ovenized Colpitts oscillator. A good agreement is shown between the predicted curves and experimental data over a variety of temperature ranges.     

An interrogation unit for passive wireless SAW sensors based on fourier transform

The application of surface acoustic wave (SAW) resonators as sensor elements for different physical parameters such as temperature, pressure, and force has been well-known for several years. The energy storage in the SAW and the direct conversion from physical parameter to a parameter of the wave, such as frequency or phase, enables the construction of a passive sensor that can be interrogated wireless. This paper presents a temperature-measurement system based on passive wireless SAW sensors. The principle of SAW sensors and SAW sensor interrogation is discussed briefly. A new measurement device developed for analyzing the sensor signals is introduced. Compared to former interrogation units that detect resonance frequency of the SAW resonator by comparing amplitudes of sensor response signals related to different stimulating frequencies, the new equipment is able to measure the resonance frequency directly by calculating a Fourier transformation of the resonator response signal. Measurement results of an experimental setup and field tests are presented and discussed.     

Effect of metal coating and residual stress on the resonant frequency of MEMS resonators

MEMS resonators are designed for a fixed resonant frequency. Therefore, any shift in the resonant frequency of the final fabricated structure can be a denting factor for its suitability towards a desired application. There are numerous factors which alter the designed resonant frequency of the fabricated resonator such as the metal layer deposited on top of the beam and the residual stresses present in the fabricated structure. While the metal coating, which acts as electrode, increases the stiffness and the effective mass of the composite structure, the residual stress increases or decreases the net stiffness if it is a tensile or compressive type respectively. In this paper, we investigate both these cases by taking two different structures, namely, the micro cantilever beam with gold layer deposited on its top surface and the MEMS gyroscope with residual stresses. First, we carry out experiments to characterize both these structures to find their resonant frequencies. Later, we analytically model those effects and compare them with the experimentally obtained values. Finally, it is found that the analytical models give an error of less than 10% with respect to the experimental results in both the cases.     

Quartz crystal resonator g sensitivity measurement methods and recent results

A technique for accurate measurements of quartz crystal resonator vibration sensitivity is described. The technique utilizes a crystal oscillator circuit in which a prescribed length of coaxial cable is used to connect the resonator to the oscillator sustaining stage. A method is provided for determination and removal of measurement errors normally introduced as a result of cable vibration. In addition to oscillator-type measurements, it is also possible to perform similar vibration sensitivity measurements using a synthesized signal generator with the resonator installed in a passive phase bridge. Test results are reported for 40 and 50 MHz, fifth overtone AT-cut, and third overtone SC-cut crystals. Acceleration sensitivity (gamma vector) values for the SC-cut resonators were typically four times smaller (5x10(-10) per g) than for the AT-cut units. However, smaller unit-to-unit gamma vector magnitude variation was exhibited by the AT-cut resonators. Oscillator sustaining stage vibration sensitivity was characterized by an equivalent open-loop phase modulation of 10(-6) rad/g.     

Self-temperature-testing of the quartz resonant force sensor

Temperature characteristics of a quartz resonant force sensor are important features, which should be seriously considered in the sensor's practical application. This paper analyzes the temperature characteristics of a quartz resonant force sensor and presents a self-temperature-testing method for the sensor by analyzing the different temperature characteristics when the quartz resonator vibrates in its fundamental mode and in its third overtone mode. A beat frequency results from the resonator's fundamental and third overtone frequencies. Experimental result show that the sensor's operating temperature can be measured by making use of this beat frequency rather than applying a temperature sensor