由非諧波泛音引起的寄生模式是AT切割石英晶體諧振器最常見的問題之一.Mindlin和Lee在1960年代在理論上詳細解釋了這一現象。他們的計算結果也與Curran和Koneval的實驗數據吻合良好，這些實驗數據基于長度，寬度和厚度相對較大的石英板。今天，由于連續小型化，石英板的尺寸逐漸減小。
Mindlin和Lee的理論計算是否仍然與小型化AT切割石英板的測量數據有關，這是我們必須解決的問題。本文計算了不同尺寸的部分電極石英板的頻譜，并與微型板的實驗數據進行了對比。還研究了石英晶體板和電極的長度和寬度以及安裝面積的大小的影響。結果表明，該理論仍然可以很好地估計出諧波的頻率，板的尺寸對非諧波泛音的影響很小。這些結果在晶振的設計改進過程中是有用的。
由于石英的高度各向異性特性，石英板的振動非常復雜。Mindlin在水晶板振動方面做了系列工作，作為AT切割石英板振動分析的基礎。Mindlin等人。介紹了自由，受力和輪廓形狀條件下晶體板的厚度-剪切和彎曲振動;然后考慮了壓電效應和不完全電極來解決矩形板問題。
Mindlin和Lee研究了部分鍍晶體板的厚度剪切，非諧波泛音和彎曲振動，沿對角線軸是無限的，并且很好地解釋了Bechmann數和電極尺寸對Q的影響。Wang和Zhao提出了厚度-剪切，彎曲和伸展振動的分散關系以及模式圖，以確定有限尺寸但沒有電極的晶體坯料的最佳長度。
今天，最小商業化AT切割石英晶振芯片的尺寸約為1.0mm×0.8mm。電極面積不應太小，因為電阻太高而無法工作。對于小型AT切割石英芯片，晶體板和電極的真實布局與無限維度假設完全不同。同時，電極效應非常重要，不容忽視。然而，如果晶體工程師僅通過商業有限元分析工具設計諧振器，則效率低且成本高。
在本文中，Mindlin和Lee的理論考慮了三種振動模式，厚度剪切，非諧波泛音和彎曲。計算了帶狀電極的有限維晶體板的模式圖和位移，并測量了相應的實驗數據。結果表明，Mindlin和Lee的理論開發的仿真工具可以為晶體工程師提供有用的小尺寸石英水晶振子設計信息。
通過檢查部分電鍍晶體板的Mindlin一階方程，我們得到了頻譜和位移。比較頻譜和測量數據，我們發現理論計算仍然可以很好地匹配小型化石英晶體諧振器。得到了電極長度與第一次非諧音的關系。測量數據和理論計算都證明了石英板的長度和寬度以及電極的寬度對非諧波泛音的影響很小。轉而設計石英晶體的改進，得出的結論是如果虛假問題被非諧波泛音控制，改變電極長度的尺寸將是一種有效的方法。
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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