A finite element analysis of frequency–temperature relations of AT-cut quartz crystal resonators with higher-order Mindlin plate theory

SummaryThe frequency–temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes are strongly coupled, has been an important requirement in most applications. The analytical work on frequency–temperature relations has been done over the last decades in many aspects, ranging from the fundamental theory for the thermal effect on vibrations of elastic solids to simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to complications of the resonator structures such as the presence of a mounting structure and electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects. In this paper, we incorporate the frequency–temperature theory for crystal plates based on the incremental thermal field formulation by Lee and Yong into our finite element analysis implementation, which is then used to analyze the free vibrations of crystal plates with the higher-order Mindlin plate theory. The effect of electrodes on the frequency–temperature relation is also considered. The computational results are compared with experimental ones from actual products. The satisfactory agreement demonstrates the precise prediction of the frequency–temperature behavior and practical applications of the finite element analysis in product modeling and development.

[1]  Honggang Zhou,et al.  Thickness-shear vibration of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness. , 2005, IEEE transactions on ultrasonics, ferroelectrics, and frequency control.

[2]  Y. Yong,et al.  Characteristics of a Lagrangian, high-frequency plate element for the static temperature behavior of low-frequency quartz resonators , 1988, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[3]  Raymond D. Mindlin,et al.  High frequency vibrations of piezoelectric crystal plates , 1972 .

[4]  Pcy Lee,et al.  Frequency‐temperature behavior of thickness vibrations of doubly rotated quartz plates affected by plate dimensions and orientations , 1986 .

[5]  T. Imai,et al.  On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates , 1999, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[6]  H. F. Tiersten,et al.  Linear Piezoelectric Plate Vibrations , 1969 .

[7]  S. Goka,et al.  Relationship between Mass Loading and Frequency Temperature Characteristics of AT-Cut Quartz Resonators , 2003 .

[8]  Ji Wang,et al.  On the correction of the higher-order Mindlin plate theory , 2005 .

[9]  Y K Yong.  Three-Dimensional Finite-Element Solution of the Lagrangean Equations for the Frequency-Temperature Behavior of Y-Cut and NT-Cut Bars. , 1987, IEEE transactions on ultrasonics, ferroelectrics, and frequency control.

[10]  T. Imai,et al.  A new theory for electroded piezoelectric plates and its finite element application for the forced vibrations of quartz crystal resonators , 2000 .

[11]  Raymond D. Mindlin,et al.  An Introduction to the Mathematical Theory of Vibrations of Elastic Plates , 2006 .

[12]  Yook-Kong Yong,et al.  Third-order Mindlin plate theory predictions for the frequency-temperature behavior of straight crested wave modes in AT- and SC-cut quartz plates , 1996, Proceedings of 1996 IEEE International Frequency Control Symposium.

[13]  J. Wang.  The frequency-temperature analysis equations of piezoelectric plates with Lee plate theory. , 1999, IEEE transactions on ultrasonics, ferroelectrics, and frequency control.

[14]  Stavros Syngellakis,et al.  A two‐dimensional theory for high‐frequency vibrations of piezoelectric crystal plates with or without electrodes , 1987 .

[15]  Ji Wang,et al.  The Thermal Effect of Metal Electrodes on Thickness-Shear Vibrations of Crystal Plates , 2007, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[16]  A Lagrangean, High Frequency Plate Element for the Static Temperature Behavior of Low Frequency Quartz Resonators , 1987, 41st Annual Symposium on Frequency Control.

[17]  Ji Wang,et al.  Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory , 1999 .

[18]  Pcy Lee,et al.  Frequency–temperature relations of thickness‐shear and flexural vibrations of contoured quartz resonators , 1996 .

[19]  Shen Li-jun,et al.  Exact thickness-shear resonance frequency of electroded piezoelectric crystal plates , 2005 .

[20]  Raymond D. Mindlin,et al.  Frequencies of piezoelectrically forced vibrations of electroded, doubly rotated, quartz plates , 1984 .

[21]  Yook-Kong Yong,et al.  Effects of Thermal Stresses on the Frequency-Temperature Behavior of Piezoelectric Resonators , 2007 .

[22]  J. Wang,et al.  A set of hierarchical finite elements for quartz plate resonators , 1996, 1996 IEEE Ultrasonics Symposium. Proceedings.

[23]  J. Zelenka.  The influence of the dimensions of electrodes on the frequency-temperature characteristics of at and BT-cut quartz resonators , 1996 .