Tong University, Shanghai , China Published online: 27 May 2014.

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1 This article was downloaded by: [Shanghai Jiaotong University] On: 29 July 2014, At: 01:51 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Ferroelectrics Letters Section Publication details, including instructions for authors and subscription information: Effects of Electrode Configuration on Vibration Characteristics of Quartz Thickness-Shear Mode Trapped-Energy Resonators Yan Zhang a & Tao Han a a Department of Instrument Science and Engineering, Shanghai Jiao Tong University, Shanghai , China Published online: 27 May To cite this article: Yan Zhang & Tao Han (2014) Effects of Electrode Configuration on Vibration Characteristics of Quartz Thickness-Shear Mode Trapped-Energy Resonators, Ferroelectrics Letters Section, 41:1-3, 44-50, DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Ferroelectrics Letters, 41:44 50, 2014 Copyright Taylor & Francis Group, LLC ISSN: print / online DOI: / Effects of Electrode Configuration on Vibration Characteristics of Quartz Thickness-Shear Mode Trapped-Energy Resonators YAN ZHANG AND TAO HAN Department of Instrument Science and Engineering, Shanghai Jiao Tong University, Shanghai , China 1. Introduction Communicated by Dr. George W. Taylor (Received in final form December 1, 2013) We study the effects of electrode configuration of quartz thickness-shear mode trappedenergy resonators on their basic vibration characteristics. Free and electrically forced vibration analyses are performed on a rectangular trapped-energy resonator by COM- SOL, focusing on the effects of the extended parts of the electrodes leading from the main rectangular parts of the crystal plate. Results show that the effect of the extended parts on resonant frequencies is of the order of 100 ppm. They also displace the mode center which has further implications in resonator frequency stability. These effects are nonnegligible, and should be taken into consideration in resonator design. Keywords Quartz; thickness-shear mode; trapped-energy; electrode configuration; extended parts; COMSOL; FEM Electrodes are necessary parts of crystal resonators. Their most basic function is for electrically exciting mechanical vibrations. In addition, the motional capacitance or admittance of a resonator is an important design consideration and is calculated from the charge (or current) and voltage on the electrodes in an electrically forced vibration. In plate thicknessshear (TSh) mode trapped-energy resonators [1], the electrodes cover the central portion of a crystal plate only. In these resonators, the electrode mass is responsible for what is called energy trapping [2] through which the vibration is confined mainly under the electrodes and decays rapidly outside them, a behavior crucial to resonator mounting. Recently, due to resonator miniaturization, the effects of electrode configuration have become more important in resonator design than before and have been studied from different angles. Electrodes of varying thickness were shown to be effective in producing strong energy trapping [3 7], which provides a possible future alternative to contoured resonators [8] with variable thickness that are known to be difficult to make. Electrodes of unequal thickness on the top and bottom of a plate resonator were studied in [9 12]. Effects of Corresponding author. than@sjtu.edu.cn Color versions of one or more of the figures in the article can be found online at 44

3 Electrode on Quartz Trapped-Energy Resonators 45 M N 2w 2b x 1 c 2a x 3 2l Figure 1. A rectangular trapped-energy resonator. misaligned electrodes were treated in [13]. In [14], it was shown that electrodes with corners cause field concentration and should be avoided in general. Electrode shape is another important factor. At present, circular [15, 16] and rectangular electrodes [17, 18] are routinely used in resonator manufacturing. Elliptical electrodes [19 22] are known to be optimal in certain ways but they are relatively more complicated and are not used often. This paper is concerned with the effects of the details of electrode configuration in trapped-energy resonators (see Fig. 1). The electrodes of these resonators are somewhat irregular. The main parts of them are rectangular or circular, covering the central region of a crystal plate only. They are identical on the plate top and bottom. The regular central part of each electrode also has an extension (see Fig. 1) leading from the central part to the edge of the plate, and thus making the shape of the entire electrodes somewhat irregular. In Figure 1, the central part of the top electrode and its extension to the right edge are filled with dark grey. The extended part of the bottom electrode leading to the left edge is bounded by dotted lines and filled with light grey. In theoretical analyses, the extended parts of the electrodes have to be neglected to simplify the problem. While the extended parts of the electrodes can be included in a finite element numerical analysis [23], they did not seem to have been the focus of existing finite analyses. Therefore, in this paper, we perform a finite element analysis to examine the effects of the fine details of real electrodes with both the regular central parts and the extended parts. Since the TSh vibration is strong under the electrodes and near the electrode edges where the extended parts of the electrodes begin, conceivably the extended parts may potentially affect the basic vibration characteristics of the resonator including resonant frequencies and mode shapes which are related to energy trapping, as well as the admittance of the resonator. 2. Finite Element Model Consider an AT-cut quartz resonator whose length, width and thickness are 2l = 25 mm, 2w = 20 mm, and 2h = 1 mm. The electrode is gold and its dimensions are 2a = 15 mm, 2b = 10 mm, and 2h = mm. The width of the extended parts c will be varied in the calculation. The mode we are interested in is the trapped fundamental TSh mode which has half a wavelength along the plate thickness and no more the one wavelength in the the x 3 -x 1 plane. It may seem that one does not need many elements to describe such a mode accurately.

4 46 Y. Zhang and T. Han Figure 2. Finite element mesh (c=0). However, since this is treated as a three-dimensional problem and the plate has a large length/thickness ratio, a considerable amount of elements are in fact needed. Tetrahedron elements are used in the analysis. The finite element mesh is determined by specifying the number of element layers along the plate thickness and the element size. We varied these two numbers to test for convergence and accuracy, and finally chose twelve layers of elements with a maximal size of 0.35 mm. This givesa total number of 123,432 elements when c = 0. When this mesh is varied a little, the change of the frequency of the fundamental TSh mode, the one that we are most interested in, is only a few Hz among about a couple of MHz. The mesh is shown in Fig. 2 in the case of c = 0. In finite element analysis, the ratios among the sides of the elements should be neither too large nor too small. The elements in Fig. 2 look reasonable. The electrodes have only one layer of elements and they are very thin. This is not a concern because during TSh motions the electrodes deform very little and essentially only their inertia matters. 3. Numerical Results and Discussion When c = 0, we denote the frequency of the fundamental TSh mode of interest by ω 0 which is used as a reference for later comparisons. Through a free vibration analysis it is found that ω 0 = MHz. The distribution of the corresponding major displacement component u 1 is shown in Figure 3. When the plate is vibrating at this mode, u 1 at the top

5 Electrode on Quartz Trapped-Energy Resonators 47 Figure 3. Distribution of u1 of the fundamental TSh mode (c=0). and bottom of the plate at y =±h are out of phase. u 1 is large in the electrode region and decays rapidly outside the electrodes. This is the so-called energy trapping [2]. To examine the effects of the extended parts of the electrodes described by c, weplot all three displacement components along MN in Figure. 1 on the top surface of the plate at y = h for c = 0, 2, 3, and 4 mm, respectively, in Figures 4(a d). Clearly, u 1 is much larger than the other two components, indicating that the mode is the TSh dominated mode of interest. In Fig. 4(a) when the electrodes are without extended parts, the u 1 displacement looks symmetric about x = 0, which agrees with that in theoretical analyses u 1 is usually described by a cosine function of x. When c is not zero, as c increases in Figures 4 (b d), u 1 gradually loses its symmetry. Its amplitude at the left edge with x = 12.5 mm becomes clearly larger than that at the right edge, indicating that energy trapping has been affected and the left edge can no longer be considered as well trapped in Fig. 4(d). It is also visible, although barely, in Fig. 4(d) that the peak of u 1 in fact is no long at x = 0 and has displaced a little to the left. This seemingly small displacement of the peak or mode center has important implications in resonator frequency stability [24]. A perfectly symmetric resonator with a symmetric mode has zero normal acceleration sensitivity. Any displacement of the mode center destroys the symmetry and results in a nonzero normal acceleration sensitivity. This causes concern as the U.S. military is trying to improve the relative acceleration sensitivity of the current technology from /g to /g in the near future. The resonant frequencies of the fundamental TSh mode for different values of c are summarized in Table 1, along with the maximal values of the real and imaginary parts of the admittance (conductance and susceptance). As c increases, the extended parts of the electrodes become wider. This results in more electrode inertia and lowers the resonant frequencies as expected. This effect if of the order of 100 ppm and is relevant in resonator performance. As the electrode width increases from 0 to 3 mm, the admittance increases too as expected because bigger electrodes collect more charges. However, this trend stops

6 48 Y. Zhang and T. Han Figure 4. Effect of c on the displacement distributions along MN on the top of the plate (a) c=0, (b) c=2 mm, (c) c=3 mm, (d) c=4 mm. when c goes from 3 to 4 mm, which may be due to that the mode center displacement is in the opposite direction of the electrode extension. The maximal values of the conductance and susceptance occur at slowlychanging frequencies as c varies. This can be seen in Fig Conclusions COMSOL can calculate quartz resonator frequencies as accurate as a few ppm. This is sufficient for analyzing electrode configuration details. Resonator electrodes are irregular, Table 1 Effect of c on the frequency and admittance of the fundamental TSh mode c (mm) Frequency ω (MHz) ω 0 = Frequency shift ω ω 0 (Hz) ω ω 0 /ω 0 (ppm) Max. conductance (S) Max. susceptance (S)

7 Electrode on Quartz Trapped-Energy Resonators 49 Figure 5. Effect of c on the conductance and susceptance of the resonator (a) c=0, (b) c=2 mm, (c) c=3 mm, (d) c=4 mm. with extended parts which usually destroy the symmetry of the resonator structures. The extended parts of the electrodes have a frequency effect of the order of 100 ppm, and cause mode center displacement which has implications in resonator frequency stability. These effects indicate that in resonator design attention should also be paid to the extended parts of the electrodes. An electrode configuration that preserves the symmetry of the structure will be ideal. The admittance is not very sensitive to the extended parts of the electrodes. Funding This work was supported by NSFC with Grant No and Program for New Century Excellent Talents in University (NCET ). References 1. H. F. Tiersten, Analysis of trapped-energy resonators operating in overtones of coupled thickness shear and thickness twist. J. Acoust. Soc. Am. 59(4), (1976). 2. V. E. Bottom, Introduction to Quartz Crystal Unit Design. New York: Van Nostrand Reinhold, (1982). 3. S. Y. Pao, R. Huang, and C. S. Lam, Energy trapping effect of stepped electrode in miniature ATcut quartz resonator. in Piezoelectricity, Acoustics Waves and Device Applications. Singapore: World Scientific (2007).

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