Wang J
Epson Palo Alto Lab., CA.
IEEE Trans Ultrason Ferroelectr Freq Control. 1999;46(4):1042-6. doi: 10.1109/58.775672.
Frequency-temperature analysis theory of crystal plates based on the incremental field theory (IFT) and the Mindlin plate theory have been widely used in the vibration analysis of crystal resonators subject to temperature change. As one of the two major plate theories for the resonator analysis, the Lee plate theory based on the trigonometric series expansion of displacements has been extensively used for both analytical and numerical analyses of quartz resonators, and efforts have also been made in correcting and perfecting the theory for much broader applications. In this paper, the earlier frequency-temperature analysis equations based on the IFT is further extended to the trigonometric series expansion in a systematic manner. By incorporating the frequency-temperature analysis into the Lee plate theory, the complete analysis of crystal resonators can be made in a consistent way. The thickness-shear and flexural vibration equations have been compared with the Mindlin plate theory to demonstrate the similarity and consistency.
基于增量场理论（IFT）和明德林板理论的晶体板频率-温度分析理论已广泛应用于温度变化下晶体谐振器的振动分析。作为谐振器分析的两种主要板理论之一，基于位移三角级数展开的李板理论已广泛用于石英谐振器的解析分析和数值分析，并且还在对该理论进行修正和完善以实现更广泛的应用方面做出了努力。本文将基于IFT的早期频率-温度分析方程系统地进一步扩展到三角级数展开。通过将频率-温度分析纳入李板理论，可以以一致的方式对晶体谐振器进行完整分析。已将厚度剪切和弯曲振动方程与明德林板理论进行比较，以证明其相似性和一致性。
