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ARTICLE IN PRESS Vacuum 75 (2004) 57–69 Analytical and ﬁnite element method design of quartz tuningfork resonators and experimental test of samples manufactured using photolithography 1—signiﬁcant design parameters affecting static capacitance C0$ Sungkyu Leea,*, Yangho Moonb, Jeongho Yoonb, Hyungsik Chunga a Department of Molecular Science and Technology, Ajou University, 5 Wonchon, Youngtong, Suwon, 443-749, South Koreab Computer-Aided Engineering (CAE) Team, R&D Support Division, Central R&D Center, Samsung Electro-Mechanics Co., Ltd., 314, Maetan 3-Dong, Youngtong, Suwon, 443-743, South Korea Received 23 December 2002; received in revised form 5 December 2003; accepted 29 December 2003Abstract Resonance frequency of quartz tuning fork crystal for use in chips of code division multiple access, personalcommunication system, and a global system for mobile communication was analyzed by an analytical method, Sezawa’stheory and the ﬁnite element method (FEM). From the FEM analysis results, actual tuning fork crystals werefabricated using photolithography and oblique evaporation by a stencil mask. A resonance frequency close to31.964 kHz was aimed at following FEM analysis results and a general scheme of commercially available 32.768 kHztuning fork resonators was followed in designing tuning fork geometry, tine electrode pattern and thickness.Comparison was made among the modeled and experimentally measured resonance frequencies and the discrepancyexplained and discussed. The average resonance frequency of the fabricated tuning fork samples at a vacuum level of3 Â 10À2 Torr was 31.228–31.462 kHz. The difference between modeling and experimentally measured resonancefrequency is attributed to the error in exactly manufacturing tuning fork tine width by photolithography. Thedependence of sensitivities for other quartz tuning fork crystal parameter C0 on various design parameters was alsocomprehensively analyzed using FEM and Taguchi’s design of experiment method. However, the tuning fork designusing FEM modeling must be modiﬁed comprehensively to optimize various design parameters affecting both theresonance frequency and other crystal parameters, most importantly crystal impedance.r 2004 Elsevier Ltd. All rights reserved.Keywords: Quartz; Surface mount device; Tuning fork; Resonance frequency; Finite element method; Analytical method; Sezawa’stheory; Crystal impedance; Photolithography; Oblique evaporation; Side-wall electrode; Static capacitance 1. Introduction $ Work leading to this manuscript was conducted at Samsung Tuning fork-type quartz crystals (32.768 kHz)Electro-Mechanics Co. Ltd. (SEMCO), Korea and all of thelegal claims for the research belong to the SEMCO. are widely used as stable frequency sources of *Corresponding author. timing pulse generator with very low power E-mail address: sklee@ajou.ac.kr (S. Lee). consumption and very small size not only in the0042-207X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.vacuum.2003.12.156
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ARTICLE IN PRESS58 S. Lee et al. / Vacuum 75 (2004) 57–69quartz-driven wristwatch but also in the portable following sections with primary focus on properand personal communication equipments. Tuning design to obtain a desirable resonance frequencyfork-type quartz crystals (32.768 kHz) are of of 31.964 kHz. 31.964 kHz was chosen as the ﬁrstspecial interest here because they are widely used target frequency considering a frequency increaseas sleep-mode timing pulse generator of Qual- of about 25,000 ppm during the subsequent lasercomms mobile station modem-3000t series cen- trimming of the tine tip electrodes to the exactlytral processing unit chips. These chips are essential desired resonance frequency of 32.768 kHz. Also,parts of mobile, personal telecommunication units tuning fork test samples were fabricated usingsuch as code division multiple access (CDMA), photolithography with side-wall electrodes andpersonal communication system (PCS), and global interconnections deﬁned by a stencil mask and thesystem for mobile communication (GSM). The assembled tuning forks evaluated to comparetuning fork-type quartz crystals are favored modeled resonance frequencies with experimentalbecause the following user-speciﬁed requirements ones. These results are reﬂected in further optimi-are satisﬁed [1–4]: (1) low frequency for low zation of tuning fork design to obtain pre-laserbattery power consumption and (2) minimal trimming resonance frequency of 31.964 kHz usingfrequency change with temperature and time after theoretical modeling and actual fabrication of testthermal or mechanical shock. samples using photolithography. It is sincerely Resonance frequency, crystal impedance, static hoped that ordinary readers understand thisand motional capacitances are important crystal unique piezoelectric device that recently emergesparameters of tuning fork-type crystals. These as a key electronic part for use in mobile andcrystal parameters depend on various design personal telecommunication units.parameters, for example, shape and thickness oftuning fork blanks and electrodes [5], manufactur-ing considerations such as etching anisotropy at 2. Modeling of tuning fork crystalsthe biforkation point, and other factors [6].Although signiﬁcant design parameters contribut- 2.1. Analytical solution of a cantilever beaming to the resonance frequency and the crystalimpedance were already statistically analyzed Tuning fork crystals have been mathematicallyusing ﬁnite element method (FEM) analysis [7], analyzed as a cantilever beam vibrating in afurther literature search [5–12] revealed that ﬂexural mode [9,10,12–14] and an analyticalsimilar FEM analysis of device characteristics solution of the equation of motion for tuninghas not been comprehensively made of static forks has been obtained with pertinent boundarycapacitance C0 of quartz tuning fork resonators conditions. The ﬂexural mode vibration of aand the individual contribution of design para- tuning fork crystal is modeled by a cantilevermeters to C0 is to be detailed. It was also revealed beam with one end clamped and the other end freefrom the extensive literature search [5–12] that a as shown in Fig. 1. A vibrating beam of uniformcomparison has to be made in a more compre- cross-section and stiffness with this boundaryhensive manner among tuning fork resonance condition is rather easily dealt with analyticallyfrequencies calculated by analytic cantilever beam [9–10,12–14] and resonance frequency is obtainedmodel, FEM analysis, and Sezawa’s approxima- from analytic solution as follows:tion where the effect of clamped position of tuningfork base is taken into account. sﬃﬃﬃﬃﬃﬃﬃﬃﬃ To this effect, research began with Samsung m2 2x0 1 f ¼ pﬃﬃﬃ 2 ð1ÞElectro-Mechanics Co. Ltd. (SEMCO) and those 2p2 3 ð2y0 Þ rs22aforementioned modeling methods of resonancefrequency and FEM analysis on sensitivity of Resonance frequencies and other importantstatic capacitance C0 for various tuning fork functional relationships can thus been calculateddesign parameters were to be presented in the for various tine-width to tine-length ratios. For
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ARTICLE IN PRESS S. Lee et al. / Vacuum 75 (2004) 57–69 59Fig. 1. Coordinate system of cantilever beam in ﬂexural modevibration.mathematical details leading to Eq. (1), also seeRefs. [9,10,12–14].2.2. Sezawa’s theory Fig. 2. (a) Overall conﬁguration and (b) right half section of the fork. In the analytic solution of a quartz crystaltuning fork cantilever beam vibrating in a ﬂexuralmode, the tuning fork base has been assumed to be Also see Ref. [11] for relevant mathematicalnon-vibrating and neglected in the analysis of procedure leading to Eq. (2).Section 2.1. In the present paper, in order to clarify The equations of motion of the beams A1, A2both the vibration mode of the base of tuning fork and A3 in ﬂexural vibration as shown in Fig. 2 areand the inﬂuence of clamped position of the base expressed byon resonance frequency from different analyticalviewpoints, the right half section of a quartz q2 y1 q4 y1crystal tuning fork has been approximated to an rA1 þ E1 I1 4 ¼ 0; qt2 qx1L-shaped bar, in which the right half section oftuning fork, as shown in Fig. 2(b), can be q2 y2 q4 y2 rA2 þ E2 I2 4 ¼ 0;represented by a series of two bars corresponding qt2 qx2to the base (designated by the beams A1 and A2) q2 y3 q4 y3and the bar corresponding to the arm (designated rA3 þ E3 I3 4 ¼ 0: qt2 qx3by the beam A3). The beam A1 is joined to thebeam A2 and the beams A1, A2, and A3 are If we write y1=u1 cos pt, y2=u2 cos pt, andconsidered to be in bending vibration as illustrated y3=u3 cos pt, thenin Fig. 2. The conﬁguration of Fig. 2 was chosen tosimulate actual mounting structure of the quartz d 4 u1 ¼ l4 u1 ; 1 ð3Þtuning fork resonators as explained in detail in dx4 1Section 3. The resonance frequency of the vibrat-ing tuning fork system depicted in Fig. 2 was d 4 u2obtained from Sezawa’s theory of Ref. [11] as ¼ l4 u2 ; 2 ð4Þfollows: dx4 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃ g2 E3 I3 d 4 u3f ¼ 2 ð2Þ ¼ l4 u3 ; 3 ð5Þ 2pL3 rA3 dx4 3
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ARTICLE IN PRESS60 S. Lee et al. / Vacuum 75 (2004) 57–69where expressed in terms of g: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 rA1 p2 4 A1 E3 I3 L1 A 1 E 3 I3l4 ¼ ; a¼ g ; l1 ¼ 4 g; 1 E1 I1 A3 E1 I1 L4 3 A 3 E 1 I1 L 4 3 rA2 p2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃl4 ¼ 2 ; 4 A2 E3 I3 L2 4 A 2 E 3 I3 E2 I2 b¼ 4 g ; l2 ¼ 4 g; A3 E2 I2 L3 A 3 E 2 I2 L 4 3 rA3 p2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃl4 ¼ 3 ; 3 E3 I3 4 A3 E3 I3 1 x¼ 3 g4 ; l3 ¼ g; A2 E2 I2 L3 The solutions of Eqs. (3)–(5) are generally rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃsimpliﬁed by the use of the following boundary A3 E3 I3 Z¼ :condition notations following the procedure of A2 E2 I2Ref. [11]: Substituting these notations a; b; x; Z; l1 ; l2 ; andl1 L1 ¼ a; l2 L2 ¼ b; l3 L3 ¼ g; l3 for Eq. (7), g is evaluated as described in Section 4.2. Therefore, the resonance frequency of the vibrating system depicted in Fig. 2 is thusrA3 L3 p2 E 3 I3 l 2 3 obtained from Sezawa’s theory using Eq. (2): ¼ x; ¼ Z: ð6Þ From both g and Eq. (2), the resonance fre- E2 I2 l3 2 E 2 I2 l 2 2 quency of a quartz tuning fork crystal is obtained Then the eigenvalue equation (7) can be with the effect of the clamped position of the baseobtained: taken into account as depicted in Fig. 2. In the present research, Bechmann’s constants [11,15,16]½X l2 fcos bðcosh b þ e sinh bÞ were used as material constants and density and À cosh bðe sin b À cos bÞg elastic compliance constants [15,16] were inserted to calculate Young’s modulus in Eq. (2). The þ l2 cos aðsin b þ sinh bÞ 1 tuning fork base vibration is thus taken into Â ðcosh b þ e sinh b À e sin b þ cos bÞ consideration in calculation of the resonance þ Y l2 ðcosh b À cos bÞ frequencies. 2 If we let the length of the base w1 (=w2 ) be Â 2Zl5 ðcos g À sinh g À sin g cosh gÞ 3 equal to inﬁnity, both x and g become inﬁnite 2 l l3 because I2 in Eq. (6) becomes inﬁnite. From these À 1 cos aðcos b þ cosh bÞfxðsin b À sinh bÞ conditions and Eq. (7), the well-known cantilever l2 beam’s eigenvalue equation is expressed by Àcosh b À cos bg þ l3 X fsin bðcosh b þ x sinh bÞ þsinh bðx sin b À cos bÞg À l2 l3 Y ðsin b þ sinh bÞ 1 þ cosh g cos g ¼ 0: ð8Þ Â l5 fðcos g þ cosh gÞ2 3 þ ðsin g À sinh gÞðsinh g þ sin gÞg ¼ 0; ð7Þ 2.3. FEM analysiswhere In actual design of tuning fork crystals, how- ever, other important design parameters must also cos a be considered such as geometry of tuning forkX ¼ l1 sin a þ l1 sinh a; cosh a blanks and electrodes [5] and other manufacturing l2 1 requirements [6,17]. The dependence of the in-Y ¼ cos aðsin b À sinh b þ x cos b À x cosh bÞ: l2 2 dividual crystal parameter sensitivity on various design parameters can be comprehensively ana- In order to describe Eq. (7) only in terms of the lyzed by using FEM and detailed informationeigenvalue g; the signs a; b; x; Z; l1 ; l2 ; and l3 are on geometry of tuning fork blanks and electrodes
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ARTICLE IN PRESS S. Lee et al. / Vacuum 75 (2004) 57–69 61[5–10,12]. In FEM analysis, the resonance fre-quency and vibration mode analysis are carriedout by harmonic analysis [6,7,12,18]. Consideringsolid material with losses, stress, electric potentialdistribution and equivalent circuit parameters ofFig. 3 could also be obtained by FEM analysis asdescribed elsewhere [6,7,18]. Therefore, in thepresent research, various tuning fork designparameters and their levels have been laid out bywell-known Taguchi’s design of the experimentmethod [19]. Design parameters for FEM analysisare schematically illustrated in Fig. 4 and are listedalong with levels according to L27 (313) matrix ofTaguchi’s method [19] in Table 1. In the presentpaper, FEM modeling was used for both reso-nance frequency and static capacitance C0 : sensi-tivity analysis was subsequently carried out onlyfor C0 using statistical F-test method [20,21]to distinguish relevant design parameters andtheir individual contribution to the static capaci-tance C0 : Tuning fork crystals were theoretically modeledin essentially the same way as previously described[6,7,12] by using commercially available FEM !software (Atila code of Institut Superieur d’Elec-tronique du Nord, Acoustics Laboratory, France)and the resonance frequency and the sensitivity ofthe static capacitance C0 for various designparameters were calculated. For the analysis bythe FEM, the tuning fork half blank was dividedinto 438 rectangular elements, 262 elements in Fig. 4. Design of a tuning fork with (a) blank (2x0 ; 0.26 mm; 2y1 ; 2.43 mm; Rarc —radius of arc, 0.04 mm); (b) electrodethe bare quartz portion and 176 elements in the dimensions and (c) cross-section of tuning fork tines acrosselectrode portion. Of 262 elements in the bare A–A0 .quartz portion, 168 elements are in the armportion and 94 elements are in the base portionas shown in Fig. 5. Due to symmetry of the tuning fork blank, the number of rectangular elements has only to be doubled to account for the entire blank area. The piezoelectricity of the specimen was taken into account and relevant elastic and piezoelectric constants were used [15,16]. 3. Fabrication of the tuning fork crystals Based on the analytical modeling, Sezawa’s theory, FEM analysis and F-test results forFig. 3. Electrical equivalent circuit for tuning fork quartz resonance frequency and static capacitance C0 ;crystal. which are depicted in Figs. 6 and 7, but otherwise
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ARTICLE IN PRESS62 S. Lee et al. / Vacuum 75 (2004) 57–69Table 1Thirteen design parameters and three levels according to L27(313) matrix of Taguchi’s method [19]: see Fig. 4 for schematics.Part No Design parameter Symbol Levels 1 2 3Crystal bank 1 Length y 2.358 mm 2.368 mm 2.378 mm 2 Width x 0.208 mm 0.218 mm 0.228 mm 3 Thickness t 0.12 mm 0.13 mm 0.14 mm 4 Side notch radius Rsn 0(none) 33 mm 66 mm 5 Radius of arc Rarc 0.055 mm 0.065 mm 0.075 mm 6 Misalignment Gb 0 10 mm 20 mm 7 Cutting angle Y 0.5 1 1.5Face electrode 8 Thickness te 2000 A( 3000 A( 4000 A( 9 Width We 0.158 mm 0.168 mm 0.178 mm 10 Error — — — — 11 Window win none 1/2 FullSide electrode 12 Thickness ts 1000 A( 2000 A( 3000 A(Tine tip Electrode 13 Thickness tt 5000 A( 7500 A( 10000 A ( Fig. 5. Rectangular elements partition of tuning fork half blank.following a general scheme of commercially and important design parameters. Tine face andavailable 32.768 kHz tuning fork resonators, irre- side electrode thicknesses (te and ts ; respectively)levant design parameters and levels were elimi- were assumed to be the same and commonlynated so that tuning fork samples could be more designated as electrode thickness (tall ). Designeffectively fabricated using photolithography: the parameters of tuning forks were thus ﬁnally laidthickness of tuning fork blank, tuning fork side out in an L12 ð211 Þ matrix following Taguchi’snotch radius, misalignment of face top and bottom method [19] as shown in Table 2 to fabricateelectrodes and cutting angle were excluded as tuning fork samples. Using Table 2 as a designsigniﬁcant design parameters affecting resonance basis, L12 design of experiment table was alsofrequency. Instead, tine width asymmetry (asw), prepared as shown in Table 3 according totine electrode length (dl), and chromium adhesion Taguchi’s method [19] and 12 different tuninglayer thickness (tcr ) were added as other relevant fork samples were fabricated at SEMCO using
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ARTICLE IN PRESS S. Lee et al. / Vacuum 75 (2004) 57–69 63 Table 2 39 Eleven design parameters and two levels according to L12 (211) 37 matrix of Taguchi’s method [19]: see Table 1 and Fig. 4 forResonance Frequency (kHz) symbols of design parameters and schematics, respectively. 35 No. Parameters Levels 33 1 2 31 Analytic 1 y 2.368 mm 2.373 mm Atila 2 x 0.228 mm 0.232 mm 29 Sezawa 3 Rarc 0.065 mm 0.075 mm Experiment 4 asw 0 2 mm 27 5 dl 1466 mm 1536mm 6 We 0.148 mm 0.158 mm 25 7 tcr ( 60 A ( 100 A 0.1 0.105 0.11 0.115 0.12 Rxy (=2x0/2y0) 8 Window — — 9 Hair line — — Rxy (=2x0/2y0) Analytic Atila Sezawa Experiment 10 tall ( 1000 A ( 2000 A 0.1015 32. 734 31.470 34. 797 11 tt ( 1000 A ( 5000 A 0.1052 30. 335 29.187 32. 395 0.107 36. 105 34.730 38. 620 31.528 0.1154 36. 063 34.711 38. 925Fig. 6. Resonance frequency values for various tuning forkdimensions obtained by analytical cantilever beam model, FEM subsequently assembled and evaluated in the same(atila solution) analysis, and Sezawa’s approximation. way as that described in a previous research paper Sensitivity of parameters [17]. The SEMCO tuning fork resonators with a 50% 44% 47% complete surface, side-wall electrodes and inter- 45% 40% connections are shown in Fig. 8 and a tuning fork 35% sample complete with packaging is shown in Fig. 9. 30% Tine length and tine width of 2.43 and 0.26 mm 25% 20% were ﬁnally selected for actual fabrication of the 15% tuning fork as illustrated in Fig. 4, simultaneously 10% 5% 5% 3% taking the major design parameters affecting the 0% crystal impedance into account as discussed in x t Gb we (a) Factors affecting static capacitance Section 4.4. Static Capacitance(C0) according to each factor 4E−13 3.5E−13 4. Results and discussion 3E−13 2.5E−13 4.1. Analytical modeling F 2E−13 x 1.5E−13 t 1E−13 Gb The results of the analytic solution of the 5E−14 we equation of motion for the deﬂection of a 0 x t Gb we cantilever beam are shown in Fig. 6. The (b) Effects of various design parameters resonance frequency modeled by this method isFig. 7. Design parameters affecting static capacitance: each generally lower than that expected by Sezawa’sfactor is deﬁned in Fig. 4 and Table 1. (a) Factors affecting approximation. This is probably attributed to thestatic capacitance and (b) effects of various design parameters. boundary condition of the cantilever beam model where vibration of the tuning fork base is notphotolithography with side-wall electrodes and taken into account. More speciﬁcally, the analy-interconnections deﬁned by a stencil mask as tical modeling corresponds to L1 ¼ 0 in Fig. 2 and,outlined in a previous research paper [17] and according to Sezawa’s theory, the resonance
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ARTICLE IN PRESS64 S. Lee et al. / Vacuum 75 (2004) 57–69Table 3Design of experiment table following Taguchi’s method [19]: see Tables 1 and 2 and Fig. 4 for symbols of design parameters andschematics, respectively.Case Parameter y x Rarc asw dl We te win hl ts tt 1 2 3 4 5 6 7 8 9 10 111 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 2 2 1 1 2 23 1 1 2 2 2 1 1 1 1 2 24 1 2 1 2 2 1 2 1 1 1 25 1 2 2 1 2 2 1 1 1 2 16 1 2 2 2 1 2 2 1 1 1 17 2 1 2 2 1 1 2 1 1 2 18 2 1 2 1 2 2 2 1 1 1 29 2 1 1 2 2 2 1 1 1 1 110 2 2 2 1 1 1 1 1 1 1 211 2 2 1 2 1 2 1 1 1 2 212 2 2 1 1 2 1 2 1 1 2 1 frequency increases with L1 [11]. In view of this, the resonance frequency modeled by the analytical method is expected to be lower than that calculated by Sezawa’s approximation except at L1 ¼ 0 where the resonance frequencies calculated by this theory and Sezawa’s approximation are identical [11]. The analytical method can only be used on a limited number of geometries [10] and the resonance frequency is calculated as a function of tine width and tine length in Fig. 6. Therefore, the analytical method is simpler than the FEM to model tuning fork crystals. However, approxima- tions are often needed for the analytical modeling approach to be manageable and the analytic expression should be reﬁned by using FEM analysis to properly simulate part of the geometry,Fig. 8. Sample tuning fork resonator chips demonstrating the electro-mechanical and other relevant physicalgeometries and electrodes under consideration: etching aniso-tropy is shown in circled region. effects of the piezoelectric quartz crystals [10]. Besides, some difﬁculties arise in the calculation of the temperature vs. frequency behavior using the analytical method: (1) The tuning fork dimension and quartz density depend on temperature, which is written mathematically as follows: Df Tf ¼ ¼ F ðTcð¼ F ðy; cÞÞ; Tr; TlÞ; f0 T; f ; f0 are the temperature, frequency, resonance frequency, respectively. Subscripts of T mean Fig. 9. Mounting structure of tuning fork resonator. causes of temperature deviation for quartz:
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ARTICLE IN PRESS S. Lee et al. / Vacuum 75 (2004) 57–69 65c; y; r; and l are stiffness coefﬁcient, cutting angle, Table 4density, and length, respectively. These tempera- Solving for g value satisfying Eq. (8) by a trial and error method.ture effects cannot be properly modeled analyti-cally to reﬁne frequency–temperature curves. (2) Gamma cosh r cos r 1 þ cosh r cos rThe effects of overtone mode and details of tuning 0.1 1.005004 0.995004 1.999983fork geometry cannot be properly modeled using 0.2 1.020067 0.980067 1.999733the analytical method, either. 0.3 1.045339 0.955336 1.99865 0.4 1.081072 0.921061 1.9957344.2. Sezawa’s theory 0.5 1.127626 0.877583 1.989585 0.6 1.185465 0.825336 1.978407 0.7 1.255169 0.764842 1.960006 The results of Sezawa’s theory of the equation 0.8 1.337435 0.696707 1.9318of motion for the vibrating tuning fork system 0.9 1.433086 0.62161 1.890821depicted in Fig. 2 are also incorporated in Fig. 6. 1 1.543081 0.540302 1.83373In Fig. 6, the resonance frequency is calculated as 1.1 1.668519 0.453596 1.756834 1.2 1.810656 0.362358 1.656105a function of tine width and tine length based on 1.3 1.970914 0.267499 1.527217Sezawa’s approximation of Ref. [11]. The observed 1.4 2.150898 0.169967 1.365582discrepancy between Sezawa’s approximation and 1.5 2.35241 0.070737 1.166403the experimentally measured resonance frequen- 1.6 2.577464 À0.0292 0.924739cies indicates that the base of a quartz crystal 1.7 2.828315 À0.12884 0.635587 1.8 3.107473 À0.2272 0.293976tuning fork behaves more rigidly than the ﬂexural 1.81 3.137051 À0.23693 0.256742bar model of Fig. 2 [22]. As discussed in the next 1.82 3.166942 À0.24663 0.21893section, the conﬁguration of Fig. 2 was modeled to 1.83 3.19715 À0.25631 0.180536simulate actual mounting of the quartz tuning fork 1.84 3.227678 À0.26596 0.141554resonators. The tuning fork resonator base part is 1.85 3.258528 À0.27559 0.101981 1.86 3.289705 À0.28519 0.061812placed onto the two adhesive-dabbed shelves of 1.87 3.32121 À0.29476 0.021042the ceramic package base as shown in Fig. 9. In 1.871 3.324379 À0.29571 0.016932this case, it is strongly inferred that the nodal 1.872 3.327551 À0.29667 0.012816points (x1 ¼ L1 in Fig. 2(b)) of the tuning fork 1.873 3.330726 À0.29762 0.008693resonator are ﬁxed to the ceramic package base. 1.874 3.333905 À0.29858 0.004565 1.875 3.337087 À0.29953 0.000431There are neither displacements nor vibrations at 1.8751 3.337405 À0.29963 1.68E-05the nodal points of the quartz crystal resonators 1.8752 3.337724 À0.29972 À0.0004[23] and the quartz tuning forks are mounted to 1.8753 3.338042 À0.29982 À0.00081the ceramic package base at their nodal points. 1.8754 3.338361 À0.29992 À0.00122However, the ﬂexural vibration of the base as per 1.8755 3.338679 À0.30001 À0.00164 1.8756 3.338998 À0.30011 À0.00205Fig. 2(b) is not signiﬁcantly contributing to the 1.8757 3.339316 À0.3002 À0.00247resonance frequency as shown by the disparity 1.8758 3.339635 À0.3003 À0.00288between the resonance frequencies calculated by 1.8759 3.339954 À0.30039 À0.0033Sezawa’s theory and the experiment (y bar ofFig. 6) following previous arguments. Because ofthe ﬁnite length of L1 taken into account in The solution of g ¼ 1:875 thus obtained wasSezawa’s approximation, the resonance frequency inserted into Eq. (2) along with other tuning forkmodeled by Sezawa’s theory is always higher than design parameters, Young’s modulus, moment ofthat calculated by the cantilever beam model inertia of the A3 beam, and other relevant materialfollowing the arguments of Ref. [11]. constants of the a-quartz and the resonance To calculate the resonance frequency of Eq. (2) frequency was subsequently calculated (Table 5).as a function of tine width and tine length using For the purpose of comparison, frequenciesSezawa’s theory [11], g value satisfying Eq. (8) was calculated by analytical cantilever beam modeling,obtained by a trial and error method (Table 4). FEM analysis, and Sezawa’s approximation are
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ARTICLE IN PRESS66 S. Lee et al. / Vacuum 75 (2004) 57–69Table 5Calculation of the resonance frequency (2).f Gamma 2x0 (m) 2y0 (m) t (m) E3 I3 Density (kg/m3) A334797.17 1.8751 0.00026 0.00256 0.00013 7.81E+10 1.9EÀ16 2650 3.38EÀ0832395.31 1.8751 0.0003 0.00285 0.00013 7.81E+10 2.93EÀ16 2650 3.9EÀ0838619.92 1.8751 0.00026 0.00243 0.00013 7.81E+10 1.9EÀ16 2650 3.38EÀ0838924.68 1.8751 0.0003 0.0026 0.00013 7.81E+10 2.93EÀ16 2650 3.9EÀ08also comprehensively tabulated in Fig. 6 alongwith the experimentally measured ones.4.3. FEM analysis FEM modeling results for resonance frequencyare also incorporated into Fig. 6 and sensitivityanalysis results of static capacitance C0 for varioustuning fork design parameters are illustrated inFig. 7 which shows that the most signiﬁcantfactors affecting static capacitance C0 are tinesurface electrode width and tine width. Thestatistical F-test procedure leading to the sensitiv-ity analysis of static capacitance C0 for varioustuning fork design parameters is essentially similar Fig. 10. Electric potential distribution across a beam cross-to that described in a previous research paper [7]. section.4.3.1. Resonance frequency The electric potential distribution across a tinecross-section and vibration mode of a tuning forkblank are obtained following the methods outlinedin the literature [6,7,10,12,16,17] and illustrated in Fig. 11. Vibration mode of a tuning fork blank.Figs. 10 and 11. Fig. 10 clearly illustrates that amechanical deformation can create large voltageswhen the applied harmonic voltage reaches its the tine width and length via the modal analysismaximum. FEM can thus be used to study and this is very close, but not equal, to that deﬁnedphysical, piezoelectric and other electro-mechan- as the frequency at which the imaginary part of theical effects of the quartz tuning fork crystals that dynamic deﬂection has its maximum [10]. Besides,are difﬁcult and laborious to analyze and visualize the tuning fork shape and electrode conﬁgurationwith other methods [10]. FEM analysis of reso- are also considered in the FEM analysis and thenance frequency is subsequently made, the results resonance frequency calculated by FEM moreshown in Fig. 6 and compared with analytical accurately approximates experimental resultsmodeling and Sezawa’s theory results. Reasonable (marked by y bar) at Rxy = 0.107 as in Fig. 6.and consistent agreement showed the validity of From FEM analysis, it was shown that the tinethe FEM analysis results but the lower frequency width and the tine tip electrode thickness areof the FEM (Atila) results should be accounted major factors affecting the resonance frequency offor. In the analytical method, the resonance tuning fork crystals [6,7,12,17]: the resonancefrequency of the tuning forks was calculated from frequency is proportional to the tine width and
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ARTICLE IN PRESS S. Lee et al. / Vacuum 75 (2004) 57–69 67inversely proportional to tine tip electrode thick- levels according to L27 ð313 Þ matrix of Taguchi’sness. Increase of the tine width by 10 mm increased method as outlined in the previous research [7].the resonance frequency by about 1.265 kHz and Vibration mode analysis was carried out for each (1000 A increase of the tine tip electrode thickness case and the sensitivity analysis was subsequentlyreduced the resonance frequency by about 118 Hz. performed using the same statistical F-test methodAlso, the resonance frequency is inversely propor- [7,19–21] to distinguish relevant design parameterstional to the square of the tine length. Therefore, and their individual contribution to the staticthe precise control of the tine width is crucial to capacitance C0 : FEM modeling results of Fig. 7obtaining the desired resonance frequency of show that the most signiﬁcant factors affectingtuning fork crystals. static capacitance is tine face electrode width (we ) Although the FEM analysis of Section 2.3 is and tine width (x). Although quartz crystal blankcapable of specifying the direction and magnitude thickness (t) and misalignment of face top andof the tuning fork base displacement and, accord- bottom electrodes (Gb ) give minor contributions ofingly, of giving calculated resonance frequencies in 5 and 3%, respectively, to the static capacitancereasonably better agreement with experimentally C0 ; these were excluded as signiﬁcant designmeasured ones than those calculated by the parameters affecting static capacitance C0 :cantilever beam model as shown in Fig. 6, itcannot specify the vibration mode for the dis- 4.4. Fabrication and test of manufactured tuningplacement of the tuning fork base. Therefore, the fork samplesresonance frequency of a quartz tuning forkcrystal was further analyzed in Section 2.2 using Tine length and tine width of 2.43 and 0.26 mmSezawa’s theory [11], considering vibration of both were ﬁnally selected for actual fabrication of thetuning fork tine and base. However, resonance tuning fork as illustrated in Fig. 4. From previousfrequency calculated by Sezawa’s theory is also discussions, it is clear that a precise process controlhigher than the experimentally measured reso- and a reproducible tine width formation arenance frequency and it is also strongly inferred required for an additional ﬁne-tuning of thethat the base of a quartz crystal tuning fork resonance frequency by subsequently controllingbehaves more rigidly than the ﬂexural bar model the tine tip electrode thickness. Variations inof Fig. 2 [22]. frequency and crystal impedance are summarized in Table 6 along with vacuum levels of the4.3.2. Sensitivity analysis of static capacitance C0 packages. These experimentally measured reso-for various tuning fork design parameters nance frequency values are collectively depicted in In the FEM analysis of Ref. [7], resonance Fig. 6 as y bar which represents maximum 32.357frequency is modeled from detailed information on and minimum 30.759 kHz values listed in Table 6.the geometry of tuning fork blanks and electrodes. The resonance frequency values of SEMCOThe dependence of sensitivities for other crystal samples in Table 6 are less than the targetparameter C0 on various design parameters can frequency value of 31.964 kHz by about 0.6 kHzthus be comprehensively analyzed in the same way at 3 Â 10À2 Torr. It is evident that the presentas that described in the previous research paper [7]. tuning fork sample design has to be modiﬁed andTherefore, FEM enables a more versatile analysis the tine width must be increased by 5–6 mm. Theas to the effects of tuning fork design parameters difference among tuning fork resonance frequen-on crystal performance. In the present research, cies calculated by analytic cantilever beam model,various tuning fork design parameters and their FEM analysis, Sezawa’s approximation and mea-levels have thus been laid out by the well-known sured by experiments is already accounted for inTaguchi’s design of experiment method [7,19]. the previous sections. However, the crystal im-Design parameters for FEM analysis of the static pedance is another important crystal parametercapacitance C0 are also schematically illustrated in and the major design parameters affecting theFig. 4 and are listed in Table 1 along with three crystal impedance have to be adjusted as well.
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ARTICLE IN PRESS68 S. Lee et al. / Vacuum 75 (2004) 57–69Table 6Variation of frequency and crystal impedance with increasing vacuum level. CI and fR represent crystal impedance and resonancefrequency, respectively. Vacuum level (Torr) SEMCO sample #1 SEMCO sample #2 SEMCO sample #3 2CI (kO) 7.6 Â 10 800 — — 1.0 104 154 — 3 Â 10À2 82 127 — 3 Â 10À5 74.4 125 80fR (kHz) 7.6 Â 102 — — — 1.0 31.384 32.357 — 3 Â 10À2 31.228 31.462 — 3 Â 10À5 30.759 31.500 30.700Since the resonance frequency and the crystal tuning fork shape and the electrode conﬁgurationimpedance are controlled rather independently of are also considered in the FEM analysis and theeach other by different design parameters, the resonance frequency is calculated more accuratelymost suitable combination of design parameters by FEM. The difference between modeling andmust be selected, following the arguments of Refs. experimentally measured resonance frequency is[6,7,10,12,17]. The tine length and tine width of attributed to the error in the exactly manufactur-2.43 and 0.26 mm were thus ﬁnally selected for ing tuning fork tine width by photolithography.actual fabrication of the SEMCO tuning fork The dependence of sensitivities for other crystalsamples. parameter C0 on various design parameters was also comprehensively analyzed using FEM and Taguchi’s design of experiment method. However,5. Summary the tuning fork design using FEM modeling must be modiﬁed comprehensively to optimize various The resonance frequency of tuning fork crystals design parameters affecting both the resonancewas obtained by the analytical solution of the frequency and other crystal parameters, mostequation of motion with pertinent boundary importantly crystal impedance.conditions, Sezawa’s theory and FEM analysis.Comparison was made among tuning fork reso-nance frequencies experimentally measured and Acknowledgementscalculated by analytic cantilever beam model,FEM analysis, and Sezawa’s approximation where Korean Ministry of Education and Humanthe effect of clamped position of tuning fork base Resources Development is gratefully acknowl-is taken into account. From the FEM analysis edged for support by Brain Korea (BK) 21 projectresults, actual tuning fork crystals were fabricated through Korea Research Foundation. This workusing photolithography and oblique evaporation was supported by the Multilayer and Thin Filmby a stencil mask. A resonance frequency close to Products Division of Samsung Electro-Mechanics31.964 kHz was aimed following the FEM results, Co. Ltd., Korea. The authors gratefully acknowl-but otherwise a general scheme of commercially edge the assistance of H.W. Kim and D.Y. Yangavailable 32.768 kHz tuning fork resonators was for modeling and analysis and of D.J. Na, C.H.followed. The difference among resonance fre- Jung, and J.P. Lee for fabrication of tuning forkquencies modeled by various methods and experi- samples. J.-H. Moon and S.-H. Yoo of Ajoumentally measured was discussed. The analytical University are also gratefully acknowledged forcantilever beam modeling is simpler than both artworks, preparation of the mathematical for-Sezawa’s theory and FEM analysis. However, the mulae, and helpful discussions.
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