Quartz crystals are piezoelectric,and are broadly used to make acoustic wave crystal resonators and filters for frequency generation and operation. They have broad applications in time-keeping,telecommunication,and sensing. Most quartz resonators and filters are made from rectangular or circular plates with various electrode patterns. For rectangular plate devices,when the structural parameters vary along one of the two in-plane directions ($x$ or $z$) only,they are called the $x$-or $z$-strips (see Fig. 1 for the latter). This paper is concerned with $z$-strip resonators with one pair of electrodes (see Fig. 1(a)) and filters with two pairs of electrodes (see Fig. 1(b)).

Fig. 1 $z$-strip AT-cut quartz plates, where $x_1 =x$ and $x_3 =z$

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Quartz resonators and filters usually operate with thickness-shear (TS) modes of plates^[1]. Strictly speaking,pure TS modes independent of the in-plane coordinates $x$ and $z$ can only exist in infinite plates. For real devices of finite sizes,their operating modes have slow in-plane variations,and are called transversely varying TS modes^[2]. Because of the material anisotropy of quartz crystals and their electromechanical coupling effects,it is tremendously difficult to theoretically analyze quartz resonators and filters by three-dimensional (3D) linear piezoelectric equations. Tiersten^{[3, 4]} and Tiersten and Smythe^{[5, 6]} derived scalar differential equations for TS modes in AT-cut quartz plates. The scalar equations are comparatively simple and accurate,and have been broadly used to study quartz resonators^{[7, 8, 9, 10, 11, 12]}. For crystal filters, however,there is much less reported work. A filter is structurally more complicated than a resonator,and is equivalent to two elastically connected resonators with two pairs of electrodes covering the plate surfaces partially. When using the scalar equations to study a plate with partial electrodes,the electroded and unelectroded regions need to be treated separately by slightly different equations,considering the interfacial bonding conditions. This is very difficult for filters. Theoretical results on crystal filters are few and scattered^{[13, 14, 15, 16, 17]},and they are either for the modes varying in one of the two in-plane directions of a plate only or with various approximations. Chen et al.^[18] obtained the simple and exact solutions of the scalar equation proposed by Tiersten and Smythe for an $x$-strip rectangular resonator with partial electrodes. Zhao et al.^[19] obtained the simple and exact solutions of the scalar equation proposed by Tiersten and Smythe for an $x$-strip rectangular filter with two pairs of electrodes. The results given by Chen et al.^[18] and Zhao et al.^[19] are of fundamental importance for the understanding and design of quartz resonators and filters. Because of the in-plane material anisotropy of quartz crystals,$x$-and $z$-strips have different vibration characteristics. Therefore,in this paper,we theoretically analyze $z$-strip resonators and filters. A series of basic results on the resonant frequencies and vibration modes are obtained, providing a reference to the design and optimization of these devices.

Consider the AT-cut quartz plates constrained by two parallel planes at $x_2 =\pm b$ (see Fig. 1). $x_{2}$ is vertical to the plate. $x_{1}$ and $x_{3}$ are in the middle plane of the quartz plate. The structure is symmetric about all the three coordinate planes with $x_1 =0$,$x_2 =0$,and $x_3 =0$. For the TS motion,the dominating displacement component is $u_1 (x_1 ,x_2 ,x_3 ,t)$. For the operating modes of quartz resonators and filters,$u_1 (x_1 ,x_2 ,x_3 ,t)$ is antisymmetric about the middle plane of the quartz plate at $x_2 =0$. These modes can be excited in a simple manner by an electric field along the plate thickness through the piezoelectric constant $e_{26}$. In time-harmonic vibrations,the $n$th-order TS displacement component $u_1^n (x_1 ,x_3 )$ given by Tiersten^{[3, 4]} and Tiersten and Smythe^{[5, 6]} can be written as follows:

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$n=1$ represents the fundamental family of the modes. $n>1$ denotes the overtone families. For electroded and unelectroded areas,the scalar differential equations governing $u_1^n (x_1 ,x_3 )$ are slightly different,and are^{[3, 4, 5, 6]}

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for electroded areas and

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for unelectroded areas. In Eqs. (2) and (3),$M_{n}$ depends on the usual elastic constants $c_{pq} $,the piezoelectric constants $e_{ip}$,and the dielectric constants $\varepsilon_{ij} $ of the AT-cut quartz presented by Tiersten^{[3, 4]} and Tiersten and Smythe^{[5, 6]}.

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where $\rho $ is the mass density of the quartz. $2{b}'$ and ${\rho }'$ are the thickness and the density of the electrodes, respectively. $\overline {\omega }_\infty$ and $\omega _\infty $ denote the TS resonant frequencies for the unbounded electroded and unelectroded plates,respectively. The electrode inertia decreases the resonant frequencies. Therefore,we have $\omega _\infty>\overline {\omega }_\infty$. At the left and right edges of the plate where $x_1 $ is constant,the main stress component is ${T_{12}}$^[6]. Consequently,the main traction-free boundary condition where $T_{12}=0$ can be translated to $u_1^n = 0$^[6]. At the front and back edges where $x_3 $ is constant,the main stress component is ${T_{31}}$^[6]. The traction-free boundary condition where $T_{31}=0$ is equivalent to $u_{1,3}^n = 0$^[6]. At a junction between the electroded and unelectroded areas of the plate where $x_3 $ is constant,we have the continuity of $u_1^n $ and $ T_{31}$,or equivalently,$u_1^n $ and $u_{1,3}^n $.

In this section,we start our analysis with the resonator in Fig. 1(a). The boundary conditions at the four edges are

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At the joints between the electroded and unelectroded areas,the continuity conditions are

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where the pair of square brackets enclosing a physical quantity denotes the jump of the physical quantity across a joint. General solutions need to be found for the electroded and unelectroded areas separately and to be matched at the joint. For resonator operations, the interested modes are symmetric about both the $x_{1}$-axis and the $x_{3}$-axis. Since the structure in Fig. 1(a) is also symmetric,it is only necessary to analyze the front part of the plate with $x_{3}>0$.

For the electroded area $0

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where $\overline {A}$ is an undetermined coefficient. Equation (10) satisfies the boundary conditions at $x_1 =\pm a$ and Eq. (2) if

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From Eq. (11),$\overline {\nu }$ can be written in terms of the resonant frequency $\omega $. For the unelectroded area $c

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where $A$ and $B$ are undetermined coefficients. Equation (12) satisfies the boundary conditions at $x_1 =\pm a$ and Eq. (3) when

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From Eq. (13),we can see that $\nu $ can also be written in terms of $\omega $.

Substituting Eqs. (10) and (12) into the rest boundary condition at $x_3 =c+d$ and the continuity conditions at $x_3 =c$,we can obtain the set of equations for $\overline {A}$,$A$,and $B$ as follows:

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To obtain the nontrivial solutions of $\overline {A}$,$A$,and $B$, we equate the determinant of the coefficient matrix of Eq. (14) to be zero,which along with Eqs. (11) and (13) leads to the transcendental equation determing $\omega$. The nontrivial solutions of $\overline {A}$,$A$,and $B$ determine the corresponding vibration mode,which can be done by MATLAB.

The material constants of the AT-cut quartz are taken from Ref. ^[15]. The structural parameters of the resonator are $b=0.317 27 {\rm {mm}}$,$a=15b,$ $c=12b,$ and $d=10b.$ These are typical for a quartz resonator. When $n=1$ and the mass ratio of the electrode to the plate $R$ is 0.05,we have $\overline {\omega }_\infty=15.549 3 {\rm {MHz}}$ and $\omega _\infty =16.447 9 {\rm {MHz}}.$ For the determinant of Eq. (14),which is a complicated transcendental equation,we adopt a bisection method for the numerical computations. For the interested frequency range,i.e.,$\overline {\omega }_\infty <\omega <\omega _\infty $,we first plot the transcendental equation as a frequency function so as to determine the relatively narrow frequency range where the resonant frequency locates. After that,we use the bisection method to find the resonant frequency precisely.

When $n=1$ and $m=1$,two resonant frequencies are found,satisfying $\nu >0$ and $\overline {\nu }>0$. The corresponding mode shapes are plotted in Figs. 2(a) and 2(b),respectively. Similarly,the case of $n=1$ and $m=3$ also has two modes with positive $\nu $ and $\overline {\nu }$ (see Figs. 2(c) and 2(d)). These modes have slowly increasing frequencies. The mode in Fig. 1(a) is the operating mode of the resonator. Everywhere in the electroded area, the vibration is in phase,and the TS strain has the same sign. Then,the charges on an electrode produced by the TS strain through the piezoelectric coupling have the same sign,resulting in a large capacitance.

Fig. 2 Effects of $R$ on trapped modes in resonator when $n=1$, $a=15b$, $c=12b$, $d=10b$, and $R=5$%

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It is important to notice that the plate vibration is mainly confined under the electrode. Outside the electrodes,near the front and back edges $x_3 =\pm (c+d)$,there is essentially no vibration left,where mounting may be designed without affecting the vibration of the plate. This is the so-called energy trapping of the TS modes. The modes in Figs. 2(b) and 2(d) are not trapped as well as the ones in Figs. 2(a) and 2(c),with some vibration remaining at the front and back edges. The mode in Fig. 2(b) has two nodal lines (zeros) parallel to the $x$-axis. Across these nodal lines,the TS displacements are out of phase,and the TS strain has a sign reversal. This mode is called a thickness-twist (TT) mode. The locations of these nodal lines are within the electroded area. Therefore,there is charge cancelation on the electrodes due to the sign change of the TS strain. This causes a capacitance drop,and is undesirable in the resonator performance. The mode in Fig. 2(c) has two nodal lines parallel to the $z$-axis,and is still called a TS mode. It also has the charge cancelation problem. The mode in Fig. 2(d) has nodal lines in both directions,and is more complicated and not very useful. In the resonator operation,the driving frequency is at the resonant frequency of the first mode in Fig. 2(a). The modes in Figs. 2(c) and 2(d) are at different frequencies,and are only weakly excited. The resonator operation is dominated by the first mode. We note that a $z$-strip resonator is fundamentally different from an $x$-strip resonator^[18],since the former does not have cylindrical or straight-crested modes and depends on one of the two in-plane coordinates only,and the latter has the modes depending on the $x$-axis only. They are only electrically excitable modes in a symmetric $x$-strip resonator.

In Fig. 3,all structural parameters are kept the same as those used in Fig. 2 except that the electrode/plate mass ratio is increased to 13%. This is larger than what happens in real applications,but needs to be exaggerated to show a phenomenon, i.e.,in this case,the number of the modes satisfying $\nu >0$ and $\overline {\nu }>0$ has increased to three for each $m$ although only two are shown in the figure for each $m$. In addition to the increase in the number of the trapped modes,a comparison with Fig. 2 shows that,when $R$ increases,the modes trap better under the electrodes. At the same time,the resonant frequencies become lower as expected because of the larger electrode inertia.

Fig. 3 Effects of $R$ on trapped modes in resonator when $n=1$, $a=15b$, $c=12b$, $d=10b$, and $R=13$%

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When plotting Fig. 4,we keep all the structural parameters the same as those used for Fig. 2 except that the electrode dimension 2$c$ along the $z$-direction increases while $d$ is reduced accordingly so that the total length of the plate in the $z$-direction remains the same. In this case,the two trapped modes with positive $\nu $ and $\overline {\nu }$ are found for each $m$. Following the electrodes,the distribution of the vibration moves toward the front and back edges. The resonant frequencies are slightly lower than those in Fig. 2 as expected when the electrodes become larger.

Fig. 4 Effects of $c$ on trapped modes in resonator when $n=1$, $a=15b$, $c=16b$, $d=6b$, and $R=5$%

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In Fig. 5,the structural parameters are kept completely the same as those in Fig. 2,but $n=3$ is used for the overtone modes. In this case,four trapped modes with positive $\nu$ and $\overline {\nu }$ are found for each $m$,but only two are shown for each $m$. The frequencies of these modes are about three times of those in Fig. 2,and the modes are more confined under the electrodes.

Fig. 5 Trapped overtone modes in resonator when $n=3$, $a=15b$, $c=12b$, $d=10b$, and $R=5$%

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For the filter shown in Fig. 1(b),the boundary and continuity conditions are

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Because of the symmetry of the structure,we only analyze the front part of the plate $x_{3}>0$. For filter operations,the interested modes are symmetric about $x_1 =0$,but may be either symmetric or antisymmetric about $x_3 =0$. We treat the symmetric and antisymmetric modes separately below.

For symmetric modes about $x_3 =0$,in the unelectroded region where $0

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where $A$ is an undetermined coefficient. Equation (17) satisfies the boundary conditions at $x_1 =\pm a$. It also satisfies Eq. (3) under Eq. (13).

Similarly,in the electroded region where $g

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where $\overline {A}$ and $\overline {B}$ are undetermined coefficients. Equation (18) satisfies the boundary conditions at $x_1 =\pm a$ and also Eq. (2) under Eq. (11).

In the unelectroded region near the front edge where $g+c

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where $B$ and $D$ are undetermined coefficients. Equation (19) satisfies the boundary conditions at $x_1 =\pm a$ and also Eq. (3) under Eq. (13).

Substituting Eqs. (17),(18),and (19) into the rest boundary condition at $x_3 =g+c+d$ and the continuity conditions in Eq. (16) gives the set of equations for $A$,$\overline {A}$,$\overline {B}$,$B$,and $D$,which determines the frequencies and modes as follows:

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Similarly,for the modes that are antisymmetric about $x_3 =0$, Eq. (17) becomes

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Equations (18) and (19) are valid. Substituting Eqs. (18),(19), and (21) into the boundary condition at $x_3 =g+c+d$ and the continuity conditions in Eq. (16),we can obtain the set of equations as follows:

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The structural parameters of the filter are $b=0.317 27 {\rm {mm}}$,$a=15b$,$c=24b$,$d=10b$,and $R=5\%.$ These are the same as those in the above. The distance between the two sets of electrodes is $g=2b$. The calculation approach selected here is the same as those in the previous section.

Figure 6 shows the result for the symmetric modes. For $n=1$ and $m=1$,three trapped modes are found,but only the two with relatively lower frequencies are shown in Figs. 6(a) and 6(b). The mode in Fig. 6(a) is suitable for filter applications. The two electroded areas vibrate in the phase. There are no nodal lines in the electroded regions. Near the front and back edges,there is good energy trapping. The mode in Fig. 6(b) has nodal lines parallel to the $x$-axis in the electroded areas. Therefore,the related charge cancellation problem is undesirable. When $n=1$ and $m=3$,there are three trapped modes. Only the first two trapped modes are shown in Figs. 6(c) and 6(d). These modes have more nodal lines parallel to both the axes.

Fig. 6 Symmetric trapped modes in filter when $n=1$, $a=15b$, $c=24b$, $d=10b$, $g=2b$, and $R=5$%

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Figure 7 shows the antisymmetric trapped modes in a filter when $n=1$,$a=15b$,$c=24b$,$d=10b$,$g=2b$,and $R=5$%. It shows that the antisymmetric modes and their behaviors are similar. In the first antisymmetric mode in Fig. 7(a),the two electroded areas vibrate out of the phase,and there are no nodal lines in the electroded areas. This mode is also useful for filter or other device applications.

Fig. 7 Antisymmetric trapped modes in filter when $n=1$, $a=15b$, $c=24b$, $d=10b$, $g=2b$, and $R=5$%

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In Fig. 8,the mass ratio $R$ is increased to 8%,and everything else is kept the same as those for Figs. 6 and 7. In this case, four trapped symmetric modes and three antisymmetric modes are found,but only the first ones are shown. Comparing Fig. 8(a) with Fig. 6(a),we can see clearly that,when $R$ becomes larger,the vibration is more trapped under the electrodes. The frequency also decreases a little when the electrodes become thicker. For the first antisymmetric modes,the difference is not pronounced between Fig. 8(b) and Fig. 7(a).

Fig. 8 Effects of $R$ on trapped modes when $n=1$, $m=1$, $a=15b$, $c=24b$, $d=10b$, $g=2b$, and $R=8$%

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Figure 9 shows the effects of 2$g$, i.e.,the distance between the two pairs of electrodes,on the first symmetric mode and the first antisymmetric mode along the $x_{3}$-axis when $n=1$ and $m=1$. When $g$ changes,the distance $d$ from the electrodes to the front and back edges changes accordingly,resulting in that the total length of the strips in the $z$-direction remains the same. $g$ is an important parameter in the filter design. A monolithic filter is equivalent to two elastically coupled resonators. If $g$ is too large,the vibration of one resonator may die out before reaching the other resonator, and the two resonators become disconnected. Figure 9 shows that the symmetric mode is more sensitive to $g$ than the antisymmetric mode.

Fig. 9 Effects of $g$ on trapped modes when $n=1$, $m=1$, $a=15b$, $c=24b$, and $R=5$%

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The exact solutions from the scalar equations by Tiersten and Smythe are obtained for $z$-strip AT-cut quartz resonators and filters. For symmetric resonators and filters,the modes can be separated into symmetric and antisymmetric ones. The results show the existence of a finite number of trapped modes,whose vibrations are mainly confined in the electroded areas. The number of the trapped modes depends on the electrode thickness and size. Thicker and larger electrodes lead to more trapped modes. There are more trapped overtone modes with $n=3$,and they are trapped better than the fundamental modes with $n=1$. For filters,the symmetric operating mode is more sensitive to the distance between the two pairs of electrodes than the antisymmetric mode. More complicated than $x$-strip resonators and filters,$z$-strips do not allow cylindrical or straight-crested modes,depending on only one of the two in-plane coordinates.