1. ARTICLE IN PRESS
Vacuum 75 (2004) 57–69
Analytical and finite element method design of quartz tuning
fork 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 Korea
b
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 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 Â 10À2 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
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 the
legal 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 the
0042-207X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.vacuum.2003.12.156

2. ARTICLE IN PRESS
58 S. Lee et al. / Vacuum 75 (2004) 57–69
quartz-driven wristwatch but also in the portable following sections with primary focus on proper
and personal communication equipments. Tuning design to obtain a desirable resonance frequency
fork-type quartz crystals (32.768 kHz) are of of 31.964 kHz. 31.964 kHz was chosen as the first
special interest here because they are widely used target frequency considering a frequency increase
as sleep-mode timing pulse generator of Qual- of about 25,000 ppm during the subsequent laser
comms mobile station modem-3000t series cen- trimming of the tine tip electrodes to the exactly
tral 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 using
such as code division multiple access (CDMA), photolithography with side-wall electrodes and
personal communication system (PCS), and global interconnections defined by a stencil mask and the
system for mobile communication (GSM). The assembled tuning forks evaluated to compare
tuning fork-type quartz crystals are favored modeled resonance frequencies with experimental
because 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-laser
battery power consumption and (2) minimal trimming resonance frequency of 31.964 kHz using
frequency change with temperature and time after theoretical modeling and actual fabrication of test
thermal or mechanical shock. samples using photolithography. It is sincerely
Resonance frequency, crystal impedance, static hoped that ordinary readers understand this
and motional capacitances are important crystal unique piezoelectric device that recently emerges
parameters of tuning fork-type crystals. These as a key electronic part for use in mobile and
crystal parameters depend on various design personal telecommunication units.
parameters, for example, shape and thickness of
tuning fork blanks and electrodes [5], manufactur-
ing considerations such as etching anisotropy at
2. Modeling of tuning fork crystals
the biforkation point, and other factors [6].
Although significant design parameters contribut-
2.1. Analytical solution of a cantilever beam
ing to the resonance frequency and the crystal
impedance were already statistically analyzed
Tuning fork crystals have been mathematically
using finite element method (FEM) analysis [7],
analyzed as a cantilever beam vibrating in a
further literature search [5–12] revealed that
flexural mode [9,10,12–14] and an analytical
similar FEM analysis of device characteristics
solution of the equation of motion for tuning
has not been comprehensively made of static
forks has been obtained with pertinent boundary
capacitance C0 of quartz tuning fork resonators
conditions. The flexural mode vibration of a
and the individual contribution of design para-
tuning fork crystal is modeled by a cantilever
meters to C0 is to be detailed. It was also revealed
beam with one end clamped and the other end free
from the extensive literature search [5–12] that a
as shown in Fig. 1. A vibrating beam of uniform
comparison has to be made in a more compre-
cross-section and stiffness with this boundary
hensive manner among tuning fork resonance
condition is rather easily dealt with analytically
frequencies calculated by analytic cantilever beam
[9–10,12–14] and resonance frequency is obtained
model, FEM analysis, and Sezawa’s approxima-
from analytic solution as follows:
tion where the effect of clamped position of tuning
fork base is taken into account. sffiffiffiffiffiffiffiffiffi
To this effect, research began with Samsung m2 2x0 1
f ¼ pffiffiffi 2
ð1Þ
Electro-Mechanics Co. Ltd. (SEMCO) and those 2p2 3 ð2y0 Þ rs22
aforementioned modeling methods of resonance
frequency and FEM analysis on sensitivity of Resonance frequencies and other important
static capacitance C0 for various tuning fork functional relationships can thus been calculated
design parameters were to be presented in the for various tine-width to tine-length ratios. For

3. ARTICLE IN PRESS
S. Lee et al. / Vacuum 75 (2004) 57–69 59
Fig. 1. Coordinate system of cantilever beam in flexural mode
vibration.
mathematical details leading to Eq. (1), also see
Refs. [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 crystal
tuning fork cantilever beam vibrating in a flexural
mode, the tuning fork base has been assumed to be
Also see Ref. [11] for relevant mathematical
non-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, A2
both the vibration mode of the base of tuning fork
and A3 in flexural vibration as shown in Fig. 2 are
and the influence of clamped position of the base
expressed by
on resonance frequency from different analytical
viewpoints, the right half section of a quartz q2 y1 q4 y1
crystal tuning fork has been approximated to an rA1 þ E1 I1 4 ¼ 0;
qt2 qx1
L-shaped bar, in which the right half section of
tuning 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 qx2
to the base (designated by the beams A1 and A2) q2 y3 q4 y3
and the bar corresponding to the arm (designated rA3 þ E3 I3 4 ¼ 0:
qt2 qx3
by the beam A3). The beam A1 is joined to the
beam A2 and the beams A1, A2, and A3 are If we write y1=u1 cos pt, y2=u2 cos pt, and
considered to be in bending vibration as illustrated y3=u3 cos pt, then
in Fig. 2. The configuration of Fig. 2 was chosen to
simulate actual mounting structure of the quartz d 4 u1
¼ l4 u1 ;
1 ð3Þ
tuning fork resonators as explained in detail in dx4 1
Section 3. The resonance frequency of the vibrat-
ing tuning fork system depicted in Fig. 2 was
d 4 u2
obtained from Sezawa’s theory of Ref. [11] as ¼ l4 u2 ;
2 ð4Þ
follows: dx4 2
sffiffiffiffiffiffiffiffiffi
g2 E3 I3 d 4 u3
f ¼ 2
ð2Þ ¼ l4 u3 ;
3 ð5Þ
2pL3 rA3 dx4 3

4. ARTICLE IN PRESS
60 S. Lee et al. / Vacuum 75 (2004) 57–69
where expressed in terms of g:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
rA1 p2 4 A1 E3 I3 L1 A 1 E 3 I3
l4 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 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l4 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 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l4 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 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
simplified by the use of the following boundary A3 E3 I3
Z¼ :
condition notations following the procedure of A2 E2 I2
Ref. [11]:
Substituting these notations a; b; x; Z; l1 ; l2 ; and
l1 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 thus
rA3 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 base
obtained: 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
 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 bÞŠ frequencies.
2
If we let the length of the base w1 (=w2 ) be
 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 bÞŠ 1 þ cosh g cos g ¼ 0: ð8Þ
 l5 fðcos g þ cosh gÞ2
3
þ ðsin g À sinh gÞðsinh g þ sin gÞg ¼ 0; ð7Þ 2.3. FEM analysis
where In actual design of tuning fork crystals, how-
ever, other important design parameters must also
cos a be considered such as geometry of tuning fork
X 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 information
eigenvalue g; the signs a; b; x; Z; l1 ; l2 ; and l3 are on geometry of tuning fork blanks and electrodes

5. 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 carried
out by harmonic analysis [6,7,12,18]. Considering
solid material with losses, stress, electric potential
distribution and equivalent circuit parameters of
Fig. 3 could also be obtained by FEM analysis as
described elsewhere [6,7,18]. Therefore, in the
present research, various tuning fork design
parameters and their levels have been laid out by
well-known Taguchi’s design of the experiment
method [19]. Design parameters for FEM analysis
are schematically illustrated in Fig. 4 and are listed
along with levels according to L27 (313) matrix of
Taguchi’s method [19] in Table 1. In the present
paper, FEM modeling was used for both reso-
nance frequency and static capacitance C0 : sensi-
tivity analysis was subsequently carried out only
for C0 using statistical F-test method [20,21]
to distinguish relevant design parameters and
their individual contribution to the static capaci-
tance C0 :
Tuning fork crystals were theoretically modeled
in 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 of
the static capacitance C0 for various design
parameters were calculated. For the analysis by
the FEM, the tuning fork half blank was divided
into 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) electrode
the bare quartz portion and 176 elements in the dimensions and (c) cross-section of tuning fork tines across
electrode portion. Of 262 elements in the bare A–A0 .
quartz portion, 168 elements are in the arm
portion and 94 elements are in the base portion
as 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 for
Fig. 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

6. ARTICLE IN PRESS
62 S. Lee et al. / Vacuum 75 (2004) 57–69
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 2.358 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 1.5
Face 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 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.
following a general scheme of commercially and important design parameters. Tine face and
available 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 commonly
nated so that tuning fork samples could be more designated as electrode thickness (tall ). Design
effectively fabricated using photolithography: the parameters of tuning forks were thus finally laid
thickness of tuning fork blank, tuning fork side out in an L12 ð211 Þ matrix following Taguchi’s
notch radius, misalignment of face top and bottom method [19] as shown in Table 2 to fabricate
electrodes and cutting angle were excluded as tuning fork samples. Using Table 2 as a design
significant design parameters affecting resonance basis, L12 design of experiment table was also
frequency. Instead, tine width asymmetry (asw), prepared as shown in Table 3 according to
tine electrode length (dl), and chromium adhesion Taguchi’s method [19] and 12 different tuning
layer thickness (tcr ) were added as other relevant fork samples were fabricated at SEMCO using

7. 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 for
Resonance 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. 925
Fig. 6. Resonance frequency values for various tuning fork
dimensions 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 is
Fig. 7. Design parameters affecting static capacitance: each generally lower than that expected by Sezawa’s
factor is defined in Fig. 4 and Table 1. (a) Factors affecting approximation. This is probably attributed to the
static capacitance and (b) effects of various design parameters. boundary condition of the cantilever beam model
where vibration of the tuning fork base is not
photolithography 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

8. ARTICLE IN PRESS
64 S. Lee et al. / Vacuum 75 (2004) 57–69
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 tt
1 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
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 physical
geometries 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:

9. ARTICLE IN PRESS
S. Lee et al. / Vacuum 75 (2004) 57–69 65
c; y; r; and l are stiffness coefficient, cutting angle, Table 4
density, 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 r
The effects of overtone mode and details of tuning
0.1 1.005004 0.995004 1.999983
fork geometry cannot be properly modeled using 0.2 1.020067 0.980067 1.999733
the analytical method, either. 0.3 1.045339 0.955336 1.99865
0.4 1.081072 0.921061 1.995734
4.2. 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 1.9318
of motion for the vibrating tuning fork system 0.9 1.433086 0.62161 1.890821
depicted in Fig. 2 are also incorporated in Fig. 6. 1 1.543081 0.540302 1.83373
In Fig. 6, the resonance frequency is calculated as 1.1 1.668519 0.453596 1.756834
1.2 1.810656 0.362358 1.656105
a function of tine width and tine length based on
1.3 1.970914 0.267499 1.527217
Sezawa’s approximation of Ref. [11]. The observed 1.4 2.150898 0.169967 1.365582
discrepancy between Sezawa’s approximation and 1.5 2.35241 0.070737 1.166403
the experimentally measured resonance frequen- 1.6 2.577464 À0.0292 0.924739
cies indicates that the base of a quartz crystal 1.7 2.828315 À0.12884 0.635587
1.8 3.107473 À0.2272 0.293976
tuning fork behaves more rigidly than the flexural 1.81 3.137051 À0.23693 0.256742
bar model of Fig. 2 [22]. As discussed in the next 1.82 3.166942 À0.24663 0.21893
section, the configuration of Fig. 2 was modeled to 1.83 3.19715 À0.25631 0.180536
simulate actual mounting of the quartz tuning fork 1.84 3.227678 À0.26596 0.141554
resonators. The tuning fork resonator base part is 1.85 3.258528 À0.27559 0.101981
1.86 3.289705 À0.28519 0.061812
placed onto the two adhesive-dabbed shelves of
1.87 3.32121 À0.29476 0.021042
the ceramic package base as shown in Fig. 9. In 1.871 3.324379 À0.29571 0.016932
this case, it is strongly inferred that the nodal 1.872 3.327551 À0.29667 0.012816
points (x1 ¼ L1 in Fig. 2(b)) of the tuning fork 1.873 3.330726 À0.29762 0.008693
resonator are fixed to the ceramic package base. 1.874 3.333905 À0.29858 0.004565
1.875 3.337087 À0.29953 0.000431
There are neither displacements nor vibrations at
1.8751 3.337405 À0.29963 1.68E-05
the 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 À0.00081
the ceramic package base at their nodal points. 1.8754 3.338361 À0.29992 À0.00122
However, the flexural vibration of the base as per 1.8755 3.338679 À0.30001 À0.00164
1.8756 3.338998 À0.30011 À0.00205
Fig. 2(b) is not significantly contributing to the
1.8757 3.339316 À0.3002 À0.00247
resonance frequency as shown by the disparity 1.8758 3.339635 À0.3003 À0.00288
between the resonance frequencies calculated by 1.8759 3.339954 À0.30039 À0.0033
Sezawa’s theory and the experiment (y bar of
Fig. 6) following previous arguments. Because of
the finite length of L1 taken into account in The solution of g ¼ 1:875 thus obtained was
Sezawa’s approximation, the resonance frequency inserted into Eq. (2) along with other tuning fork
modeled by Sezawa’s theory is always higher than design parameters, Young’s modulus, moment of
that calculated by the cantilever beam model inertia of the A3 beam, and other relevant material
following 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, frequencies
Sezawa’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

10. ARTICLE IN PRESS
66 S. Lee et al. / Vacuum 75 (2004) 57–69
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 1.9EÀ16 2650 3.38EÀ08
32395.31 1.8751 0.0003 0.00285 0.00013 7.81E+10 2.93EÀ16 2650 3.9EÀ08
38619.92 1.8751 0.00026 0.00243 0.00013 7.81E+10 1.9EÀ16 2650 3.38EÀ08
38924.68 1.8751 0.0003 0.0026 0.00013 7.81E+10 2.93EÀ16 2650 3.9EÀ08
also comprehensively tabulated in Fig. 6 along
with the experimentally measured ones.
4.3. FEM analysis
FEM modeling results for resonance frequency
are also incorporated into Fig. 6 and sensitivity
analysis results of static capacitance C0 for various
tuning fork design parameters are illustrated in
Fig. 7 which shows that the most significant
factors affecting static capacitance C0 are tine
surface electrode width and tine width. The
statistical F-test procedure leading to the sensitiv-
ity analysis of static capacitance C0 for various
tuning 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 tine
cross-section and vibration mode of a tuning fork
blank are obtained following the methods outlined
in 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 a
mechanical deformation can create large voltages
when the applied harmonic voltage reaches its the tine width and length via the modal analysis
maximum. FEM can thus be used to study and this is very close, but not equal, to that defined
physical, piezoelectric and other electro-mechan- as the frequency at which the imaginary part of the
ical 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 configuration
with other methods [10]. FEM analysis of reso- are also considered in the FEM analysis and the
nance frequency is subsequently made, the results resonance frequency calculated by FEM more
shown in Fig. 6 and compared with analytical accurately approximates experimental results
modeling 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 tine
the FEM analysis results but the lower frequency width and the tine tip electrode thickness are
of the FEM (Atila) results should be accounted major factors affecting the resonance frequency of
for. In the analytical method, the resonance tuning fork crystals [6,7,12,17]: the resonance
frequency of the tuning forks was calculated from frequency is proportional to the tine width and

11. ARTICLE IN PRESS
S. Lee et al. / Vacuum 75 (2004) 57–69 67
inversely proportional to tine tip electrode thick- levels according to L27 ð313 Þ matrix of Taguchi’s
ness. 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 subsequently
reduced the resonance frequency by about 118 Hz. performed using the same statistical F-test method
Also, the resonance frequency is inversely propor- [7,19–21] to distinguish relevant design parameters
tional to the square of the tine length. Therefore, and their individual contribution to the static
the precise control of the tine width is crucial to capacitance C0 : FEM modeling results of Fig. 7
obtaining the desired resonance frequency of show that the most significant factors affecting
tuning 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 blank
capable of specifying the direction and magnitude thickness (t) and misalignment of face top and
of the tuning fork base displacement and, accord- bottom electrodes (Gb ) give minor contributions of
ingly, of giving calculated resonance frequencies in 5 and 3%, respectively, to the static capacitance
reasonably better agreement with experimentally C0 ; these were excluded as significant design
measured ones than those calculated by the parameters affecting static capacitance C0 :
cantilever beam model as shown in Fig. 6, it
cannot specify the vibration mode for the dis- 4.4. Fabrication and test of manufactured tuning
placement of the tuning fork base. Therefore, the fork samples
resonance frequency of a quartz tuning fork
crystal was further analyzed in Section 2.2 using Tine length and tine width of 2.43 and 0.26 mm
Sezawa’s theory [11], considering vibration of both were finally selected for actual fabrication of the
tuning fork tine and base. However, resonance tuning fork as illustrated in Fig. 4. From previous
frequency calculated by Sezawa’s theory is also discussions, it is clear that a precise process control
higher than the experimentally measured reso- and a reproducible tine width formation are
nance frequency and it is also strongly inferred required for an additional fine-tuning of the
that the base of a quartz crystal tuning fork resonance frequency by subsequently controlling
behaves more rigidly than the flexural bar model the tine tip electrode thickness. Variations in
of Fig. 2 [22]. frequency and crystal impedance are summarized
in Table 6 along with vacuum levels of the
4.3.2. 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 32.357
frequency 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 SEMCO
The dependence of sensitivities for other crystal samples in Table 6 are less than the target
parameter C0 on various design parameters can frequency value of 31.964 kHz by about 0.6 kHz
thus be comprehensively analyzed in the same way at 3 Â 10À2 Torr. It is evident that the present
as that described in the previous research paper [7]. tuning fork sample design has to be modified and
Therefore, FEM enables a more versatile analysis the tine width must be increased by 5–6 mm. The
as 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 in
Taguchi’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 parameter
capacitance C0 are also schematically illustrated in and the major design parameters affecting the
Fig. 4 and are listed in Table 1 along with three crystal impedance have to be adjusted as well.

12. ARTICLE IN PRESS
68 S. Lee et al. / Vacuum 75 (2004) 57–69
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
2
CI (kO) 7.6 Â 10 800 — —
1.0 104 154 —
3 Â 10À2 82 127 —
3 Â 10À5 74.4 125 80
fR (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 30.700
Since the resonance frequency and the crystal tuning fork shape and the electrode configuration
impedance are controlled rather independently of are also considered in the FEM analysis and the
each other by different design parameters, the resonance frequency is calculated more accurately
most suitable combination of design parameters by FEM. The difference between modeling and
must 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 crystal
samples. 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 resonance
was obtained by the analytical solution of the frequency and other crystal parameters, most
equation 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 Acknowledgements
calculated by analytic cantilever beam model,
FEM analysis, and Sezawa’s approximation where Korean Ministry of Education and Human
the 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 project
results, actual tuning fork crystals were fabricated through Korea Research Foundation. This work
using photolithography and oblique evaporation was supported by the Multilayer and Thin Film
by a stencil mask. A resonance frequency close to Products Division of Samsung Electro-Mechanics
31.964 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. Yang
available 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 fork
quencies modeled by various methods and experi- samples. J.-H. Moon and S.-H. Yoo of Ajou
mentally measured was discussed. The analytical University are also gratefully acknowledged for
cantilever beam modeling is simpler than both artworks, preparation of the mathematical for-
Sezawa’s theory and FEM analysis. However, the mulae, and helpful discussions.

13. ARTICLE IN PRESS
S. Lee et al. / Vacuum 75 (2004) 57–69 69
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