https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1223&context=mechengfacpub

Thickness-shear and thickness-twist vibrations of circular AT-cut quartz resonators

The determination of the precise thickness-shear frequency of electroded crystal plates has practical importance in quartz crystal resonator design and fabrication, especially when the high fundamental thickness-shear frequency has reduced the crystal plate thickness to such a degree that proper consideration of the effect of electrodes is very important The electrodes effect as mass loading in the estimation of the resonance frequency has to be modified to consider the stiffness of electrodes, as the relative strength is increasingly noticeable By following a known procedure in the determination of the thickness-shear frequency of an infinite AT-cut crystal plate, frequency equations of crystal plate without and with piezoelectric effect are obtained in terms of elastic constants and the electrode material density After solving these equations for the usual design parameters of crystal resonators, the design process can be optimized to pinpoint the precise configuration to avoid time-consuming trial and reduction steps Since these equations and solutions are presented for widely used materials and parameters, they can be easily integrated into the existing crystal resonator design and manufacturing processes

Exact thickness-shear resonance frequency of electroded piezoelectric crystal plates

We studied mode couplings in thin film bulk acoustic wave resonators of a piezoelectric film on a dielectric layer operating with the fundamental thickness-extensional mode A system of plate equations derived in our previous paper was used which includes the couplings to the unwanted in-plane extension, flexure, fundamental and second-order thickness shear modes It was shown that the couplings depend strongly on the plate length/thickness ratio For a relatively clean operating mode with weak couplings to unwanted modes, a series of discrete values of the plate length/thickness ratio should be avoided and these values were determined in the present paper The results can be of great significance to the design and optimization of film bulk acoustic wave resonators

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Effects of aspect ratio on the mode couplings of thin-film bulk acoustic wave resonators

The Mindlin plate equations have been widely used in the analysis of high-frequency vibrations of quartz crystal resonators with accurate solutions, as demonstrated by the design procedure based on analytical results in terms of frequency, mode shapes, and optimal parameters for the ATcut quartz crystal plate, which is the core element in a resonator structure. Earlier studies have been focused on the AT-cut (which is one type of rotated Y-cut) quartz crystal plates because it is widely produced and has relatively simple couplings of vibration modes at thickness-shear frequencies of the fundamental and overtone modes. The simplified equations through the truncation, correction, and modification of the Mindlin plate equations have been widely accepted for practical applications, and further efforts to expand their applications to similar problems of other material types, such as doubly-rotated quartz crystals, with the SC-cut being a typical and popular one, are also naturally expected. We have found out that the Mindlin plate theory can be truncated and corrected for the SC-cut quartz crystal plates in a manner similar to the AT-cut plates. The analytical results show that the corrected Mindlin plate equations are equally accurate and convenient for obtaining essential design parameters of resonators for the thickness-shear vibrations of SC-cut quartz crystal plates.

An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

In our previous work, a system of two-dimensional equations for piezoelectric thin-film acoustic wave resonators (FBAR) has been derived and then successfully applied on the free vibration analysis of a finite FBAR plate. In this paper, we analyzed the forced vibrations of FBAR with alternating voltage applied on its electrodes. The derived two dimensional equations were simplified based on straight-crested wave approximations and were rewritten into matrix formalism by state-vector approach. With the knowledge of matrix function, the state-vector equation was successfully solved. Results of admittance were obtained with the dependence on the driving frequency and the aspect ratio. The obtained figure was compared with the frequency spectra obtained in free vibration analysis. Different aspect ratios and quality factors were taken to study the effects of them on the admittance. It is shown that admittance is highly influenced by mode couplings. Thus the work in this paper will be important for the thorough understanding of the coupling vibration of FBAR and its optimization design.In our previous work, a system of two-dimensional equations for piezoelectric thin-film acoustic wave resonators (FBAR) has been derived and then successfully applied on the free vibration analysis of a finite FBAR plate. In this paper, we analyzed the forced vibrations of FBAR with alternating voltage applied on its electrodes. The derived two dimensional equations were simplified based on straight-crested wave approximations and were rewritten into matrix formalism by state-vector approach. With the knowledge of matrix function, the state-vector equation was successfully solved. Results of admittance were obtained with the dependence on the driving frequency and the aspect ratio. The obtained figure was compared with the frequency spectra obtained in free vibration analysis. Different aspect ratios and quality factors were taken to study the effects of them on the admittance. It is shown that admittance is highly influenced by mode couplings. Thus the work in this paper will be important for the thoroug...

Forced coupling vibration analysis of FBAR based on two-dimensional equations associated with state-vector approach

We study straight-crested waves and vibration modes with variations along the x3 direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.

Five-mode frequency spectra of x 3 -dependent modes in AT-cut quartz resonators

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations.

The Mindlin plate theory plays a vital role in the analysis of high-frequency vibrations of quartz crystal resonators in the thickness-shear mode. The coupled equations with an infinite number of vibration modes have been truncated to retain essential modes with strong couplings for simplified analysis, and correction factors have been introduced to ensure exact solutions at cut-off frequencies as part of the validation procedure. Starting from plates without complications, correction factors have also been modified to include electrodes for their mass effect. Such results have been important for higher-order correction factors to ensure accurate consideration of electrodes in AT-cut quartz crystal plates. The findings in this systematic study of correction factors are readily available for the analytical equations with selected modes with strong couplings, and equally convenient for the implementation of the finite element analysis with the Mindlin plate equations.

Guijia Chen

Papers

Five-mode frequency spectra of x 3 -dependent modes in AT-cut quartz resonators

An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations.

Correction factors of the mindlin plate equations with the consideration of electrodes [Correspondence]

Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations.