由非谐波泛音引起的寄生模式是AT切割石英晶体谐振器最常见的问题之一.Mindlin和Lee在1960年代在理论上详细解释了这一现象。他们的计算结果也与Curran和Koneval的实验数据吻合良好，这些实验数据基于长度，宽度和厚度相对较大的石英板。今天，由于连续小型化，石英板的尺寸逐渐减小。
Mindlin和Lee的理论计算是否仍然与小型化AT切割石英板的测量数据有关，这是我们必须解决的问题。本文计算了不同尺寸的部分电极石英板的频谱，并与微型板的实验数据进行了对比。还研究了石英晶体板和电极的长度和宽度以及安装面积的大小的影响。结果表明，该理论仍然可以很好地估计出谐波的频率，板的尺寸对非谐波泛音的影响很小。这些结果在晶振的设计改进过程中是有用的。
由于石英的高度各向异性特性，石英板的振动非常复杂。Mindlin在水晶板振动方面做了系列工作，作为AT切割石英板振动分析的基础。Mindlin等人。介绍了自由，受力和轮廓形状条件下晶体板的厚度-剪切和弯曲振动;然后考虑了压电效应和不完全电极来解决矩形板问题。
Mindlin和Lee研究了部分镀晶体板的厚度剪切，非谐波泛音和弯曲振动，沿对角线轴是无限的，并且很好地解释了Bechmann数和电极尺寸对Q的影响。Wang和Zhao提出了厚度-剪切，弯曲和伸展振动的分散关系以及模式图，以确定有限尺寸但没有电极的晶体坯料的最佳长度。
今天，最小商业化AT切割石英晶振芯片的尺寸约为1.0mm×0.8mm。电极面积不应太小，因为电阻太高而无法工作。对于小型AT切割石英芯片，晶体板和电极的真实布局与无限维度假设完全不同。同时，电极效应非常重要，不容忽视。然而，如果晶体工程师仅通过商业有限元分析工具设计谐振器，则效率低且成本高。
在本文中，Mindlin和Lee的理论考虑了三种振动模式，厚度剪切，非谐波泛音和弯曲。计算了带状电极的有限维晶体板的模式图和位移，并测量了相应的实验数据。结果表明，Mindlin和Lee的理论开发的仿真工具可以为晶体工程师提供有用的小尺寸石英水晶振子设计信息。
通过检查部分电镀晶体板的Mindlin一阶方程，我们得到了频谱和位移。比较频谱和测量数据，我们发现理论计算仍然可以很好地匹配小型化石英晶体谐振器。得到了电极长度与第一次非谐音的关系。测量数据和理论计算都证明了石英板的长度和宽度以及电极的宽度对非谐波泛音的影响很小。转而设计石英晶体的改进，得出的结论是如果虚假问题被非谐波泛音控制，改变电极长度的尺寸将是一种有效的方法。
Parasitic modes caused by non-harmonic overtones are one of the most common problems with AT-cut quartz resonators. Mindlin and Lee explained this phenomenon
in theory in the 1960s. Their calculations are also in good agreement with the experimental data of Curran and Koneval, which are based on quartz plates of relatively large length, width and thickness. Today, the size of quartz plates is gradually reduced due to continuous miniaturization.
The theoretical calculations of Mindlin and Lee are still related to the measurement data of miniaturized AT-cut quartz plates, which is a problem we must solve. In this paper, the spectrum of partial electrode quartz plates of different sizes is calculated and compared with experimental data of microplates. The effects of the length and width of the quartz crystal plates and electrodes and the size of the mounting area were also investigated. The results show that the theory can still estimate the frequency of harmonics well, and the size of the plate has little effect on non-harmonic overtones. These results are useful in the design improvement of quartz crystal resonators.
Due to the highly anisotropic nature of quartz, the vibration of quartz plates is very complex. Mindlin has done a series of work on the vibration of crystal plates as the basis for the vibration analysis of AT-cut quartz plates. Mindlin et al. The thickness-shear and bending vibration of the crystal plate under the conditions of freedom, force and contour shape are introduced; then the piezoelectric effect and incomplete electrode are considered to solve the rectangular plate problem.
Mindlin and Lee studied thickness shearing, non-harmonic overtones and bending vibrations of partially plated crystal plates, which are infinite along the diagonal axis and well explain the effect of Bechmann number and electrode size on Q. Wang and Zhao proposed the dispersion relationship of thickness-shear, bending and stretching vibrations and the pattern diagram to determine the optimum length of the crystal blank of finite size but without electrodes.
Today, the smallest commercial AT-cut quartz crystal chip has a size of about 1.0 mm x 0.8 mm. The electrode area should not be too small because the resistance is too high to work. For small AT-cut quartz chips, the true layout of the crystal plates and electrodes is completely different from the assumption of infinite dimensions. At the same time, the electrode effect is very important and cannot be ignored. However, if the crystal engineer designs the resonator only through commercial finite element analysis tools, it is inefficient and costly.
In this paper, Mindlin and Lee's theory considers three modes of vibration, thickness shearing, non-harmonic overtones and bending. The pattern and displacement of the finite-dimensional crystal plate of the strip electrode were calculated, and the corresponding experimental data were measured. The results show that Mindlin and Lee's theoretically developed simulation tools can provide crystal engineers with useful small-sized resonator design information.
By examining the Mindlin first-order equation of a partially plated crystal plate, we obtained the spectrum and displacement. Comparing the spectrum and the measured data, we found that the theoretical calculations still fit well with miniaturized quartz crystal resonators. The relationship between the electrode length and the first non-harmonic is obtained. Both the measured data and theoretical calculations demonstrate that the length and width of the quartz plate and the width of the electrode have little effect on the non-harmonic overtones. Turning to the improvement of crystal resonators, the conclusion is that if the false problem is controlled by non-harmonic overtones, changing the size of the electrode length will be an effective method.

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