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Vacuum 75 (2004) 57–69

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Electro-Mechani

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0042-207X/$ - see

doi:10.1016/j.vac

Analytical and finite element method design of quartz tuningfork resonators and experimental test of samples manufactured

using photolithography 1—significant design parametersaffecting static capacitance C0

$

Sungkyu Leea,*, Yangho Moonb, Jeongho Yoonb, Hyungsik Chunga

aDepartment of Molecular Science and Technology, Ajou University, 5 Wonchon, Youngtong, Suwon, 443-749, South KoreabComputer-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 2003

Abstract

Resonance frequency of quartz tuning fork crystal for use in chips of code division multiple access, personal

communication system, and a global system for mobile communication was analyzed by an analytical method, Sezawa’s

theory and the finite element method (FEM). From the FEM analysis results, actual tuning fork crystals were

fabricated using photolithography and oblique evaporation by a stencil mask. A resonance frequency close to

31.964 kHz was aimed at following FEM analysis results and a general scheme of commercially available 32.768 kHz

tuning 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 discrepancy

explained and discussed. The average resonance frequency of the fabricated tuning fork samples at a vacuum level of

3� 10�2 Torr was 31.228–31.462 kHz. The difference between modeling and experimentally measured resonance

frequency is attributed to the error in exactly manufacturing tuning fork tine width by photolithography. The

dependence of sensitivities for other quartz tuning fork crystal parameter C0 on various design parameters was also

comprehensively analyzed using FEM and Taguchi’s design of experiment method. However, the tuning fork design

using FEM modeling must be modified comprehensively to optimize various design parameters affecting both the

resonance 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’s

theory; Crystal impedance; Photolithography; Oblique evaporation; Side-wall electrode; Static capacitance

g to this manuscript was conducted at Samsung

cs Co. Ltd. (SEMCO), Korea and all of the

the research belong to the SEMCO.

g author.

ss: sklee@ajou.ac.kr (S. Lee).

front matter r 2004 Elsevier Ltd. All rights reserv

uum.2003.12.156

1. Introduction

Tuning fork-type quartz crystals (32.768 kHz)are widely used as stable frequency sources oftiming pulse generator with very low powerconsumption and very small size not only in the

ed.

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S. Lee et al. / Vacuum 75 (2004) 57–6958

quartz-driven wristwatch but also in the portableand personal communication equipments. Tuningfork-type quartz crystals (32.768 kHz) are ofspecial interest here because they are widely usedas sleep-mode timing pulse generator of Qual-comms mobile station modem-3000t series cen-tral processing unit chips. These chips are essentialparts of mobile, personal telecommunication unitssuch as code division multiple access (CDMA),personal communication system (PCS), and globalsystem for mobile communication (GSM). Thetuning fork-type quartz crystals are favoredbecause the following user-specified requirementsare satisfied [1–4]: (1) low frequency for lowbattery power consumption and (2) minimalfrequency change with temperature and time afterthermal or mechanical shock.

Resonance frequency, crystal impedance, staticand motional capacitances are important crystalparameters of tuning fork-type crystals. Thesecrystal parameters depend on various designparameters, for example, shape and thickness oftuning fork blanks and electrodes [5], manufactur-ing considerations such as etching anisotropy atthe biforkation point, and other factors [6].Although significant design parameters contribut-ing to the resonance frequency and the crystalimpedance were already statistically analyzedusing finite element method (FEM) analysis [7],further literature search [5–12] revealed thatsimilar FEM analysis of device characteristicshas not been comprehensively made of staticcapacitance C0 of quartz tuning fork resonatorsand the individual contribution of design para-meters to C0 is to be detailed. It was also revealedfrom the extensive literature search [5–12] that acomparison has to be made in a more compre-hensive manner among tuning fork resonancefrequencies calculated by analytic cantilever beammodel, FEM analysis, and Sezawa’s approxima-tion where the effect of clamped position of tuningfork base is taken into account.

To this effect, research began with SamsungElectro-Mechanics Co. Ltd. (SEMCO) and thoseaforementioned modeling methods of resonancefrequency and FEM analysis on sensitivity ofstatic capacitance C0 for various tuning forkdesign parameters were to be presented in the

following sections with primary focus on properdesign to obtain a desirable resonance frequencyof 31.964 kHz. 31.964 kHz was chosen as the firsttarget frequency considering a frequency increaseof about 25,000 ppm during the subsequent lasertrimming of the tine tip electrodes to the exactlydesired resonance frequency of 32.768 kHz. Also,tuning fork test samples were fabricated usingphotolithography with side-wall electrodes andinterconnections defined by a stencil mask and theassembled tuning forks evaluated to comparemodeled resonance frequencies with experimentalones. These results are reflected in further optimi-zation of tuning fork design to obtain pre-lasertrimming resonance frequency of 31.964 kHz usingtheoretical modeling and actual fabrication of testsamples using photolithography. It is sincerelyhoped that ordinary readers understand thisunique piezoelectric device that recently emergesas a key electronic part for use in mobile andpersonal telecommunication units.

2. Modeling of tuning fork crystals

2.1. Analytical solution of a cantilever beam

Tuning fork crystals have been mathematicallyanalyzed as a cantilever beam vibrating in aflexural mode [9,10,12–14] and an analyticalsolution of the equation of motion for tuningforks has been obtained with pertinent boundaryconditions. The flexural mode vibration of atuning fork crystal is modeled by a cantileverbeam with one end clamped and the other end freeas shown in Fig. 1. A vibrating beam of uniformcross-section and stiffness with this boundarycondition is rather easily dealt with analytically[9–10,12–14] and resonance frequency is obtainedfrom analytic solution as follows:

f ¼m2

2p2ffiffiffi3

p 2x0

ð2y0Þ2

ffiffiffiffiffiffiffiffiffi1

rs22

sð1Þ

Resonance frequencies and other importantfunctional relationships can thus been calculatedfor various tine-width to tine-length ratios. For

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Fig. 1. Coordinate system of cantilever beam in flexural mode

vibration.

Fig. 2. (a) Overall configuration and (b) right half section of the

fork.

S. Lee et al. / Vacuum 75 (2004) 57–69 59

mathematical details leading to Eq. (1), also seeRefs. [9,10,12–14].

2.2. Sezawa’s theory

In the analytic solution of a quartz crystaltuning fork cantilever beam vibrating in a flexuralmode, the tuning fork base has been assumed to benon-vibrating and neglected in the analysis ofSection 2.1. In the present paper, in order to clarifyboth the vibration mode of the base of tuning forkand the influence of clamped position of the baseon resonance frequency from different analyticalviewpoints, the right half section of a quartzcrystal tuning fork has been approximated to anL-shaped bar, in which the right half section oftuning fork, as shown in Fig. 2(b), can berepresented by a series of two bars correspondingto the base (designated by the beams A1 and A2)and the bar corresponding to the arm (designatedby the beam A3). The beam A1 is joined to thebeam A2 and the beams A1, A2, and A3 areconsidered to be in bending vibration as illustratedin Fig. 2. The configuration of Fig. 2 was chosen tosimulate actual mounting structure of the quartztuning fork resonators as explained in detail inSection 3. The resonance frequency of the vibrat-ing tuning fork system depicted in Fig. 2 wasobtained from Sezawa’s theory of Ref. [11] asfollows:

f ¼g2

2pL23

ffiffiffiffiffiffiffiffiffiE3I3

rA3

sð2Þ

Also see Ref. [11] for relevant mathematicalprocedure leading to Eq. (2).

The equations of motion of the beams A1, A2

and A3 in flexural vibration as shown in Fig. 2 areexpressed by

rA1q2y1

qt2þ E1I1

q4y1

qx41

¼ 0;

rA2q2y2

qt2þ E2I2

q4y2

qx42

¼ 0;

rA3q2y3

qt2þ E3I3

q4y3

qx43

¼ 0:

If we write y1=u1 cos pt, y2=u2 cos pt, andy3=u3 cos pt, then

d4u1

dx41

¼ l41u1; ð3Þ

d4u2

dx42

¼ l42u2; ð4Þ

d4u3

dx43

¼ l43u3; ð5Þ

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S. Lee et al. / Vacuum 75 (2004) 57–6960

where

l41 ¼rA1p2

E1I1;

l42 ¼rA2p2

E2I2;

l43 ¼rA3p2

E3I3;

The solutions of Eqs. (3)–(5) are generallysimplified by the use of the following boundarycondition notations following the procedure ofRef. [11]:

l1L1 ¼ a; l2L2 ¼ b; l3L3 ¼ g;

rA3L3p2

E2I2l32

¼ x;E3I3l

23

E2I2l22

¼ Z: ð6Þ

Then the eigenvalue equation (7) can beobtained:

½Xl2fcos bðcosh bþ e sinh bÞ

� cosh bðe sin b� cos bÞg

þ l21cos aðsin bþ sinh bÞ

� ðcosh bþ e sinh b� e sin bþ cos bÞ

þ Yl22ðcosh b� cos bÞ

� 2Zl53ðcos g� sinh g� sin g cosh gÞ

�l21l3l2

cos aðcos bþ cosh bÞfxðsin b� sinh bÞ�

�cosh b� cos bg þ l3Xfsin bðcosh bþ x sinh bÞ

þsinh bðx sin b� cos bÞg � l2l3Y ðsin bþ sinh bÞ

� l53fðcos gþ cosh gÞ2

þ ðsin g� sinh gÞðsinh gþ sin gÞg ¼ 0; ð7Þ

where

X ¼ l1 sin aþcos acosh a

l1 sinh a;

Y ¼l21l22

cos aðsin b� sinh bþ x cos b� x cosh bÞ:

In order to describe Eq. (7) only in terms of theeigenvalue g; the signs a; b; x; Z; l1; l2; and l3 are

expressed in terms of g:

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1E3I3L4

1

A3E1I1L43

g4

s; l1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1E3I3

A3E1I1L43

g4

s;

b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2E3I3L4

2

A3E2I2L43

g4

s; l2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2E3I3

A3E2I2L43

g4

s;

x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA3

3E3I3

A32E2I2

g44

s; l3 ¼

1

L3g;

Z ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA3E3I3

A2E2I2

r:

Substituting these notations a; b; x; Z; l1; l2; andl3 for Eq. (7), g is evaluated as described inSection 4.2. Therefore, the resonance frequency ofthe vibrating system depicted in Fig. 2 is thusobtained from Sezawa’s theory using Eq. (2):

From both g and Eq. (2), the resonance fre-quency of a quartz tuning fork crystal is obtainedwith the effect of the clamped position of the basetaken into account as depicted in Fig. 2. In thepresent research, Bechmann’s constants [11,15,16]were used as material constants and density andelastic compliance constants [15,16] were insertedto calculate Young’s modulus in Eq. (2). Thetuning fork base vibration is thus taken intoconsideration in calculation of the resonancefrequencies.

If we let the length of the base w1 (=w2) beequal to infinity, both x and g become infinitebecause I2 in Eq. (6) becomes infinite. From theseconditions and Eq. (7), the well-known cantileverbeam’s eigenvalue equation is expressed by

1þ cosh g cos g ¼ 0: ð8Þ

2.3. FEM analysis

In actual design of tuning fork crystals, how-ever, other important design parameters must alsobe considered such as geometry of tuning forkblanks and electrodes [5] and other manufacturingrequirements [6,17]. The dependence of the in-dividual crystal parameter sensitivity on variousdesign parameters can be comprehensively ana-lyzed by using FEM and detailed informationon geometry of tuning fork blanks and electrodes

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Fig. 4. Design of a tuning fork with (a) blank (2x0; 0.26mm;

2y1; 2.43mm; Rarc—radius of arc, 0.04mm); (b) electrode

dimensions and (c) cross-section of tuning fork tines across

A–A0.

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(3

13) 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 FEMsoftware (Atila code of Institut Sup!erieur 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 inthe bare quartz portion and 176 elements in theelectrode portion. Of 262 elements in the barequartz 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

Fig. 3. Electrical equivalent circuit for tuning fork quartz

crystal.

fork blank, the number of rectangular elementshas only to be doubled to account for the entireblank area. The piezoelectricity of the specimenwas taken into account and relevant elastic andpiezoelectric constants were used [15,16].

3. Fabrication of the tuning fork crystals

Based on the analytical modeling, Sezawa’stheory, FEM analysis and F-test results forresonance frequency and static capacitance C0;which are depicted in Figs. 6 and 7, but otherwise

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Table 1

Thirteen 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 3

Crystal bank 1 Length y 2.358mm 2.368mm 2.378mm

2 Width x 0.208mm 0.218mm 0.228mm

3 Thickness t 0.12mm 0.13mm 0.14mm

4 Side notch radius Rsn 0(none) 33mm 66mm5 Radius of arc Rarc 0.055mm 0.065mm 0.075mm

6 Misalignment Gb 0 10mm 20mm7 Cutting angle Y 0.5� 1� 1.5�

Face electrode 8 Thickness te 2000 (A 3000 (A 4000 (A

9 Width We 0.158mm 0.168mm 0.178mm

10 Error — — — —

11 Window win none 1/2 Full

Side 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.

S. Lee et al. / Vacuum 75 (2004) 57–6962

following a general scheme of commerciallyavailable 32.768 kHz tuning fork resonators, irre-levant design parameters and levels were elimi-nated so that tuning fork samples could be moreeffectively fabricated using photolithography: thethickness of tuning fork blank, tuning fork sidenotch radius, misalignment of face top and bottomelectrodes and cutting angle were excluded assignificant design parameters affecting resonancefrequency. Instead, tine width asymmetry (asw),tine electrode length (dl), and chromium adhesionlayer thickness (tcr) were added as other relevant

and important design parameters. Tine face andside electrode thicknesses (te and ts; respectively)were assumed to be the same and commonlydesignated as electrode thickness (tall). Designparameters of tuning forks were thus finally laidout in an L12ð211Þ matrix following Taguchi’smethod [19] as shown in Table 2 to fabricatetuning fork samples. Using Table 2 as a designbasis, L12 design of experiment table was alsoprepared as shown in Table 3 according toTaguchi’s method [19] and 12 different tuningfork samples were fabricated at SEMCO using

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25

27

29

31

33

35

37

39

0.1 0.105 0.11 0.115 0.12

Rxy (=2x0/2y0)

Res

onan

ce F

requ

ency

(kH

z)

Analytic

Atila

Sezawa

Experiment

Rxy (=2x0/2y0) Analytic Atila Sezawa Experiment0.1015 32. 734 31.470 34.7970.1052 30. 335 29.187 32.3950.107 36. 105 34.730 38.620 31.5280.1154 36. 063 34.711 38.925

Fig. 6. Resonance frequency values for various tuning fork

dimensions obtained by analytical cantilever beam model, FEM

(atila solution) analysis, and Sezawa’s approximation.

Factors affecting static capacitance

Effects of various design parameters

44%

5% 3%

47%

0%5%

10%15%20%25%30%35%40%45%50%

x t Gb we

0

5E−14

1E−13

1.5E−13

2E−13

2.5E−13

3E−13

3.5E−13

4E−13

x t Gb we

xtGbwe

F

Sensitivity of parameters

Static Capacitance(C0) according to each factor

(a)

(b)

Fig. 7. Design parameters affecting static capacitance: each

factor is defined in Fig. 4 and Table 1. (a) Factors affecting

static capacitance and (b) effects of various design parameters.

Table 2

Eleven design parameters and two levels according to L12(211)

matrix of Taguchi’s method [19]: see Table 1 and Fig. 4 for

symbols of design parameters and schematics, respectively.

No. Parameters Levels

1 2

1 y 2.368mm 2.373mm

2 x 0.228mm 0.232mm

3 Rarc 0.065mm 0.075mm

4 asw 0 2mm5 dl 1466mm 1536mm6 We 0.148mm 0.158mm

7 tcr 60 (A 100 (A

8 Window — —

9 Hair line — —

10 tall 1000 (A 2000 (A

11 tt 1000 (A 5000 (A

S. Lee et al. / Vacuum 75 (2004) 57–69 63

photolithography with side-wall electrodes andinterconnections defined by a stencil mask asoutlined in a previous research paper [17] and

subsequently assembled and evaluated in the sameway as that described in a previous research paper[17]. The SEMCO tuning fork resonators with acomplete surface, side-wall electrodes and inter-connections are shown in Fig. 8 and a tuning forksample complete with packaging is shown in Fig. 9.Tine length and tine width of 2.43 and 0.26mmwere finally selected for actual fabrication of thetuning fork as illustrated in Fig. 4, simultaneouslytaking the major design parameters affecting thecrystal impedance into account as discussed inSection 4.4.

4. Results and discussion

4.1. Analytical modeling

The results of the analytic solution of theequation of motion for the deflection of acantilever beam are shown in Fig. 6. Theresonance frequency modeled by this method isgenerally lower than that expected by Sezawa’sapproximation. This is probably attributed to theboundary condition of the cantilever beam modelwhere vibration of the tuning fork base is nottaken into account. More specifically, the analy-tical modeling corresponds to L1 ¼ 0 in Fig. 2 and,according to Sezawa’s theory, the resonance

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Table 3

Design of experiment table following Taguchi’s method [19]: see Tables 1 and 2 and Fig. 4 for symbols of design parameters and

schematics, respectively.

Case Parameter

y x Rarc asw dl We te win hl ts tt1 2 3 4 5 6 7 8 9 10 11

1 1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 2 2 1 1 2 2

3 1 1 2 2 2 1 1 1 1 2 2

4 1 2 1 2 2 1 2 1 1 1 2

5 1 2 2 1 2 2 1 1 1 2 1

6 1 2 2 2 1 2 2 1 1 1 1

7 2 1 2 2 1 1 2 1 1 2 1

8 2 1 2 1 2 2 2 1 1 1 2

9 2 1 1 2 2 2 1 1 1 1 1

10 2 2 2 1 1 1 1 1 1 1 2

11 2 2 1 2 1 2 1 1 1 2 2

12 2 2 1 1 2 1 2 1 1 2 1

Fig. 8. Sample tuning fork resonator chips demonstrating the

geometries and electrodes under consideration: etching aniso-

tropy is shown in circled region.

Fig. 9. Mounting structure of tuning fork resonator.

S. Lee et al. / Vacuum 75 (2004) 57–6964

frequency increases with L1 [11]. In view of this,the resonance frequency modeled by the analyticalmethod is expected to be lower than thatcalculated by Sezawa’s approximation except atL1 ¼ 0 where the resonance frequencies calculatedby this theory and Sezawa’s approximation areidentical [11]. The analytical method can only beused on a limited number of geometries [10] andthe resonance frequency is calculated as a functionof tine width and tine length in Fig. 6. Therefore,the analytical method is simpler than the FEM tomodel tuning fork crystals. However, approxima-tions are often needed for the analytical modelingapproach to be manageable and the analyticexpression should be refined by using FEManalysis to properly simulate part of the geometry,electro-mechanical and other relevant physicaleffects of the piezoelectric quartz crystals [10].Besides, some difficulties arise in the calculation ofthe temperature vs. frequency behavior using theanalytical method: (1) The tuning fork dimensionand quartz density depend on temperature, whichis written mathematically as follows:

Tf ¼Df

f0

� �¼ F ðTcð¼ F ðy; cÞÞ;Tr;TlÞ;

T ; f ; f0 are the temperature, frequency, resonancefrequency, respectively. Subscripts of T meancauses of temperature deviation for quartz:

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Table 4

Solving for g value satisfying Eq. (8) by a trial and error

method.

Gamma cosh r cos r 1þ cosh r cos r

0.1 1.005004 0.995004 1.999983

0.2 1.020067 0.980067 1.999733

0.3 1.045339 0.955336 1.99865

0.4 1.081072 0.921061 1.995734

0.5 1.127626 0.877583 1.989585

0.6 1.185465 0.825336 1.978407

0.7 1.255169 0.764842 1.960006

0.8 1.337435 0.696707 1.9318

0.9 1.433086 0.62161 1.890821

1 1.543081 0.540302 1.83373

1.1 1.668519 0.453596 1.756834

1.2 1.810656 0.362358 1.656105

1.3 1.970914 0.267499 1.527217

1.4 2.150898 0.169967 1.365582

1.5 2.35241 0.070737 1.166403

1.6 2.577464 �0.0292 0.924739

1.7 2.828315 �0.12884 0.635587

1.8 3.107473 �0.2272 0.293976

1.81 3.137051 �0.23693 0.256742

1.82 3.166942 �0.24663 0.21893

1.83 3.19715 �0.25631 0.180536

1.84 3.227678 �0.26596 0.141554

1.85 3.258528 �0.27559 0.101981

1.86 3.289705 �0.28519 0.061812

1.87 3.32121 �0.29476 0.021042

1.871 3.324379 �0.29571 0.016932

1.872 3.327551 �0.29667 0.012816

1.873 3.330726 �0.29762 0.008693

1.874 3.333905 �0.29858 0.004565

1.875 3.337087 �0.29953 0.000431

1.8751 3.337405 �0.29963 1.68E-05

1.8752 3.337724 �0.29972 �0.0004

1.8753 3.338042 �0.29982 �0.00081

1.8754 3.338361 �0.29992 �0.00122

1.8755 3.338679 �0.30001 �0.00164

1.8756 3.338998 �0.30011 �0.00205

1.8757 3.339316 �0.3002 �0.00247

1.8758 3.339635 �0.3003 �0.00288

1.8759 3.339954 �0.30039 �0.0033

S. Lee et al. / Vacuum 75 (2004) 57–69 65

c; y; r; and l are stiffness coefficient, cutting angle,density, and length, respectively. These tempera-ture effects cannot be properly modeled analyti-cally to refine frequency–temperature curves. (2)The effects of overtone mode and details of tuningfork geometry cannot be properly modeled usingthe analytical method, either.

4.2. Sezawa’s theory

The results of Sezawa’s theory of the equationof motion for the vibrating tuning fork systemdepicted in Fig. 2 are also incorporated in Fig. 6.In Fig. 6, the resonance frequency is calculated asa function of tine width and tine length based onSezawa’s approximation of Ref. [11]. The observeddiscrepancy between Sezawa’s approximation andthe experimentally measured resonance frequen-cies indicates that the base of a quartz crystaltuning fork behaves more rigidly than the flexuralbar model of Fig. 2 [22]. As discussed in the nextsection, the configuration of Fig. 2 was modeled tosimulate actual mounting of the quartz tuning forkresonators. The tuning fork resonator base part isplaced onto the two adhesive-dabbed shelves ofthe ceramic package base as shown in Fig. 9. Inthis case, it is strongly inferred that the nodalpoints (x1 ¼ L1 in Fig. 2(b)) of the tuning forkresonator are fixed to the ceramic package base.There are neither displacements nor vibrations atthe nodal points of the quartz crystal resonators[23] and the quartz tuning forks are mounted tothe ceramic package base at their nodal points.However, the flexural vibration of the base as perFig. 2(b) is not significantly contributing to theresonance frequency as shown by the disparitybetween the resonance frequencies calculated bySezawa’s theory and the experiment (y bar ofFig. 6) following previous arguments. Because ofthe finite length of L1 taken into account inSezawa’s approximation, the resonance frequencymodeled by Sezawa’s theory is always higher thanthat calculated by the cantilever beam modelfollowing the arguments of Ref. [11].

To calculate the resonance frequency of Eq. (2)as a function of tine width and tine length usingSezawa’s theory [11], g value satisfying Eq. (8) wasobtained by a trial and error method (Table 4).

The solution of g ¼ 1:875 thus obtained wasinserted into Eq. (2) along with other tuning forkdesign parameters, Young’s modulus, moment ofinertia of the A3 beam, and other relevant materialconstants of the a-quartz and the resonancefrequency was subsequently calculated (Table 5).For the purpose of comparison, frequenciescalculated by analytical cantilever beam modeling,FEM analysis, and Sezawa’s approximation are

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Table 5

Calculation of the resonance frequency (2).

f Gamma 2x0 (m) 2y0 (m) t (m) E3 I3 Density (kg/m3) A3

34797.17 1.8751 0.00026 0.00256 0.00013 7.81E+10 1.9E�16 2650 3.38E�08

32395.31 1.8751 0.0003 0.00285 0.00013 7.81E+10 2.93E�16 2650 3.9E�08

38619.92 1.8751 0.00026 0.00243 0.00013 7.81E+10 1.9E�16 2650 3.38E�08

38924.68 1.8751 0.0003 0.0026 0.00013 7.81E+10 2.93E�16 2650 3.9E�08

Fig. 10. Electric potential distribution across a beam cross-

section.

Fig. 11. Vibration mode of a tuning fork blank.

S. Lee et al. / Vacuum 75 (2004) 57–6966

also 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 similarto that described in a previous research paper [7].

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 inFigs. 10 and 11. Fig. 10 clearly illustrates that amechanical deformation can create large voltageswhen the applied harmonic voltage reaches itsmaximum. FEM can thus be used to studyphysical, piezoelectric and other electro-mechan-ical effects of the quartz tuning fork crystals thatare difficult and laborious to analyze and visualizewith other methods [10]. FEM analysis of reso-nance frequency is subsequently made, the resultsshown in Fig. 6 and compared with analyticalmodeling and Sezawa’s theory results. Reasonableand consistent agreement showed the validity ofthe FEM analysis results but the lower frequencyof the FEM (Atila) results should be accountedfor. In the analytical method, the resonancefrequency of the tuning forks was calculated from

the tine width and length via the modal analysisand this is very close, but not equal, to that definedas the frequency at which the imaginary part of thedynamic deflection has its maximum [10]. Besides,the tuning fork shape and electrode configurationare also considered in the FEM analysis and theresonance frequency calculated by FEM moreaccurately approximates experimental results(marked by y bar) at Rxy = 0.107 as in Fig. 6.

From FEM analysis, it was shown that the tinewidth and the tine tip electrode thickness aremajor factors affecting the resonance frequency oftuning fork crystals [6,7,12,17]: the resonancefrequency is proportional to the tine width and

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inversely proportional to tine tip electrode thick-ness. Increase of the tine width by 10 mm increasedthe resonance frequency by about 1.265 kHz and1000 (A increase of the tine tip electrode thicknessreduced the resonance frequency by about 118Hz.Also, the resonance frequency is inversely propor-tional to the square of the tine length. Therefore,the precise control of the tine width is crucial toobtaining the desired resonance frequency oftuning fork crystals.

Although the FEM analysis of Section 2.3 iscapable of specifying the direction and magnitudeof the tuning fork base displacement and, accord-ingly, of giving calculated resonance frequencies inreasonably better agreement with experimentallymeasured ones than those calculated by thecantilever beam model as shown in Fig. 6, itcannot specify the vibration mode for the dis-placement of the tuning fork base. Therefore, theresonance frequency of a quartz tuning forkcrystal was further analyzed in Section 2.2 usingSezawa’s theory [11], considering vibration of bothtuning fork tine and base. However, resonancefrequency calculated by Sezawa’s theory is alsohigher than the experimentally measured reso-nance frequency and it is also strongly inferredthat the base of a quartz crystal tuning forkbehaves more rigidly than the flexural bar modelof Fig. 2 [22].

4.3.2. Sensitivity analysis of static capacitance C0

for various tuning fork design parameters

In the FEM analysis of Ref. [7], resonancefrequency is modeled from detailed information onthe geometry of tuning fork blanks and electrodes.The dependence of sensitivities for other crystalparameter C0 on various design parameters canthus be comprehensively analyzed in the same wayas that described in the previous research paper [7].Therefore, FEM enables a more versatile analysisas to the effects of tuning fork design parameterson crystal performance. In the present research,various tuning fork design parameters and theirlevels have thus been laid out by the well-knownTaguchi’s design of experiment method [7,19].Design parameters for FEM analysis of the staticcapacitance C0 are also schematically illustrated inFig. 4 and are listed in Table 1 along with three

levels according to L27ð313Þ matrix of Taguchi’smethod as outlined in the previous research [7].Vibration mode analysis was carried out for eachcase and the sensitivity analysis was subsequentlyperformed using the same statistical F-test method[7,19–21] to distinguish relevant design parametersand their individual contribution to the staticcapacitance C0: FEM modeling results of Fig. 7show that the most significant factors affectingstatic capacitance is tine face electrode width (we)and tine width (x). Although quartz crystal blankthickness (t) and misalignment of face top andbottom electrodes (Gb) give minor contributions of5 and 3%, respectively, to the static capacitanceC0; these were excluded as significant designparameters affecting static capacitance C0:

4.4. Fabrication and test of manufactured tuning

fork samples

Tine length and tine width of 2.43 and 0.26mmwere finally selected for actual fabrication of thetuning fork as illustrated in Fig. 4. From previousdiscussions, it is clear that a precise process controland a reproducible tine width formation arerequired for an additional fine-tuning of theresonance frequency by subsequently controllingthe tine tip electrode thickness. Variations infrequency and crystal impedance are summarizedin Table 6 along with vacuum levels of thepackages. These experimentally measured reso-nance frequency values are collectively depicted inFig. 6 as y bar which represents maximum 32.357and minimum 30.759 kHz values listed in Table 6.

The resonance frequency values of SEMCOsamples in Table 6 are less than the targetfrequency value of 31.964 kHz by about 0.6 kHzat 3� 10�2 Torr. It is evident that the presenttuning fork sample design has to be modified andthe tine width must be increased by 5–6 mm. Thedifference among tuning fork resonance frequen-cies calculated by analytic cantilever beam model,FEM analysis, Sezawa’s approximation and mea-sured by experiments is already accounted for inthe previous sections. However, the crystal im-pedance is another important crystal parameterand the major design parameters affecting thecrystal impedance have to be adjusted as well.

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Table 6

Variation of frequency and crystal impedance with increasing vacuum level. CI and fR represent crystal impedance and resonance

frequency, respectively.

Vacuum level (Torr) SEMCO sample #1 SEMCO sample #2 SEMCO sample #3

CI (kO) 7.6� 102 800 — —

1.0 104 154 —

3� 10�2 82 127 —

3� 10�5 74.4 125 80

fR (kHz) 7.6� 102 — — —

1.0 31.384 32.357 —

3� 10�2 31.228 31.462 —

3� 10�5 30.759 31.500 30.700

S. Lee et al. / Vacuum 75 (2004) 57–6968

Since the resonance frequency and the crystalimpedance are controlled rather independently ofeach other by different design parameters, themost suitable combination of design parametersmust be selected, following the arguments of Refs.[6,7,10,12,17]. The tine length and tine width of2.43 and 0.26mm were thus finally selected foractual fabrication of the SEMCO tuning forksamples.

5. Summary

The resonance frequency of tuning fork crystalswas obtained by the analytical solution of theequation of motion with pertinent boundaryconditions, Sezawa’s theory and FEM analysis.Comparison was made among tuning fork reso-nance frequencies experimentally measured andcalculated by analytic cantilever beam model,FEM analysis, and Sezawa’s approximation wherethe effect of clamped position of tuning fork baseis taken into account. From the FEM analysisresults, actual tuning fork crystals were fabricatedusing photolithography and oblique evaporationby a stencil mask. A resonance frequency close to31.964 kHz was aimed following the FEM results,but otherwise a general scheme of commerciallyavailable 32.768 kHz tuning fork resonators wasfollowed. The difference among resonance fre-quencies modeled by various methods and experi-mentally measured was discussed. The analyticalcantilever beam modeling is simpler than bothSezawa’s theory and FEM analysis. However, the

tuning fork shape and the electrode configurationare also considered in the FEM analysis and theresonance frequency is calculated more accuratelyby FEM. The difference between modeling andexperimentally measured resonance frequency isattributed to the error in the exactly manufactur-ing tuning fork tine width by photolithography.The dependence of sensitivities for other crystalparameter C0 on various design parameters wasalso comprehensively analyzed using FEM andTaguchi’s design of experiment method. However,the tuning fork design using FEM modeling mustbe modified comprehensively to optimize variousdesign parameters affecting both the resonancefrequency and other crystal parameters, mostimportantly crystal impedance.

Acknowledgements

Korean Ministry of Education and HumanResources Development is gratefully acknowl-edged for support by Brain Korea (BK) 21 projectthrough Korea Research Foundation. This workwas supported by the Multilayer and Thin FilmProducts Division of Samsung Electro-MechanicsCo. Ltd., Korea. The authors gratefully acknowl-edge the assistance of H.W. Kim and D.Y. Yangfor modeling and analysis and of D.J. Na, C.H.Jung, and J.P. Lee for fabrication of tuning forksamples. J.-H. Moon and S.-H. Yoo of AjouUniversity are also gratefully acknowledged forartworks, preparation of the mathematical for-mulae, and helpful discussions.

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