Thickness-shear vibration of a quartz plate connected to piezoelectric plates and electric field sensing

Ultrasonics 51 (2011) 131–135

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Thickness-shear vibration of a quartz plate connected to piezoelectric plates and electric field sensing Yunying Zhou a, Weiqiu Chen a, Jiashi Yang b, Jianke Du c,* a

Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, PR China Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526, USA c Department of Mechanics and Engineering Science, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, PR China b

a r t i c l e

i n f o

Article history: Received 10 June 2009 Received in revised form 29 June 2010 Accepted 2 July 2010 Available online 7 July 2010 Keywords: Vibration Quartz Piezoelectric Sensor

a b s t r a c t We study thickness-shear vibration of a quartz plate connected to two piezoelectric ceramic plates with initial deformations caused by a biasing electric field. The theory for small deformations superposed on finite biasing deformations in an electroelastic body is used. It is shown that the resonant frequencies of the incremental thickness-shear vibration of the quartz plate vary with the biasing electric field. The biasing electric field induced frequency shift depends linearly on the field. Therefore this effect may be used for electric field sensing. The dependence of the electric field induced frequency shift on various material and geometric parameters is examined. When the electric field is of the order of 100 V/mm, the relative frequency shift is of the order of 105. The case when the piezoelectric plates are replaced by piezomagnetic plates is also investigated for magnetic field sensing, and similar results are obtained. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Piezoelectric resonators are vibrating crystals as key elements of electric circuits called oscillators which have been used for a long time for time-keeping as well as frequency generation and operation. Oscillators are important components of telecommunication devices, satellites, radars, and many other electronic equipment. These equipment are often used in harsh environment and on objects in motion. For these applications the frequency stability of the crystal resonators against environmental effects like a temperature change or accelerations is desired [1]. Recently the application of frequency shifts in crystal resonators as pressure, force, acceleration and temperature sensors is growing [2,3]. For sensor applications the resonant frequencies of resonators need to be made sensitive to environmental changes for high sensitivity. In principle resonators sensors can be made to measure many kinds of mechanical, thermal, electric, and magnetic effects. The most commonly used material for crystal resonators and sensors is quartz. While possessing many advantages over other materials, quartz has very weak piezoelectric coupling. When a quartz resonator is placed in an electric field, the electric field induced frequency shift in the resonator is relatively small. This effect is measurable and useful in material characterization [4–6], but it is relatively weak and is not ideal for electric field sensing.

* Corresponding author. E-mail address: [email protected] (J. Du). 0041-624X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2010.07.002

In this paper we propose a new structure with components made from both quartz and polarized ceramics which have strong piezoelectric coupling. We use the strong piezoelectric coupling of polarized ceramics to enhance the response of a quartz resonator to an applied electric field. This suggests the possibility of a quartz electric field sensor with improved sensitivity. A theoretical analysis on thickness-shear vibrations of a quartz plate connected to two ceramic plates under an electric field is performed. The theory of small fields superposed on finite biasing fields in an electroelastic body is used. The analysis shows that the proposed structure can operate as an electric field sensor. The effects of various physical and geometric parameters are examined. When the piezoelectric plates are replaced by piezomagnetic plates, the same analysis still applies when the piezoelectric constants are replaced by piezomagnetic constants. Results on frequency shifts due to a magnetic field are also presented. 2. A constrained crystal plate Consider a quartz crystal plate connected to two ceramic plates with in-plane poling as shown in Fig. 1. The plates are joined by rigid end walls. The structure is long in the X3 direction and Fig. 1 shows a cross section only. There are two air gaps among the plates so that the crystal and ceramic plates do not interact. The crystal plate has two electrodes represented by the thick lines at the top and bottom of the plate for vibration excitation. The operating mode of the crystal plate resonator is the thickness-shear mode

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Y. Zhou et al. / Ultrasonics 51 (2011) 131–135

E0 X2

t Ceramic Gap Quartz

P X1 2b

Gap Rigid wall

Ceramic

P 2a

yi ðX; tÞ ¼ dia ½xa ðXÞ þ ua ðX; tÞ;

Fig. 1. A quartz plate connected to two ceramic plates.

3. Equations for small deformations superposed on finite biasing deformations Consider an electroelastic body. We distinguish the following three states of the body (see Fig. 2). In the reference state the body is undeformed and is free of electric fields. A generic point at this state is denoted by X with Cartesian coordinates XK. The Cartesian tensor notation is used. The mass density is q0. In the initial state the body is deformed finitely and statically, and carries finite static electric fields. The deformation and fields at this state are called the initial or biasing fields. The position of the material point asso-

Then it can be shown that the equations governing the incremental fields u and /1 are [9]

€a ; K 1K a;K ¼ q0 u

ð2Þ

D1K;K ¼ 0;

where q0 is the mass density, and the incremental stress tensor and the incremental electric displacement are given by the following incremental constitutive relations:

K 1Lc ¼ GLcMa ua;M þ RMLc /1;M ;

ð3Þ

D1K ¼ RKLc uc;L  LKL /1;L :

Eq. (3) shows that the incremental stress tensor and electric displacement vector depend linearly on the incremental displacement and potential gradients. GKaLc, RKLc, and LKL are called the effective or apparent elastic, piezoelectric, and dielectric constants. They depend on the initial deformation xa(X) and the initial electric potential /0(X). In many applications, the biasing deformations and fields are also infinitesimal. In this case, only their first-order effects on the incremental fields need to be considered. For small biasing fields it is convenient to introduce the following small displacement vector w of the initial deformation

xa ¼ daK X K þ wa :

ð4Þ

Then, neglecting terms quadratic in the gradients of w and /0, the effective material constants take the following form [1]:

GK aLc ¼ cK aLc þ ^cK aLc ;

RKLc ¼ eKLc þ ^eKLc ;

LKL ¼ eKL þ ^eKL ;

ð5Þ

where

^cK aLc ¼ T 0KL dac þ cK aLN wc;N þ cKNLc wa;N þ cK aLcAB S0AB þ kAK aLc E0A ;   ^eKLc ¼ eKLM wc;M  kKLcAB S0AB þ bAKLc E0A þ e0 E0K dLc  E0L dK c  E0M dMc dKL ;   ^eKL ¼ bKLAB S0AB þ vKLA E0A þ e0 S0MM dKL  2S0KL ;

Reference w (Biasing)

ð1Þ

/ðX; tÞ ¼ /0 ðXÞ þ /1 ðX; tÞ:

with only one displacement component in the X1 direction [7]. The electrodes on the crystal plate are shorted for the free-vibration frequency analysis in this paper. Therefore there can be no electric field in the quartz plate. When the structure is under an applied biasing electric field E0 in the X1 direction, the electric field in the ceramic plates is the same as the biasing electric field because of the continuity of the tangential electric field at the surfaces of the ceramic plates. The electric field in the ceramic plates causes constrained expansion (contraction) in the ceramic and quartz plates due to the end constraints. This induces stresses and strains in the quartz plate and produces frequency shifts in the quartz resonator. This frequency shift may be considered for measuring the biasing electric field. To understand and describe this effect and to predict the sensitivity we need to study thickness-shear vibrations of a crystal plate with initial stresses and strains due to an applied electric field and the end constraints, for which the theory of small fields superposed on finite biasing fields in a piezoelectric body is necessary and is summarized in the next section.

3

ciated with X is given by x = x(X) or xc = xc(X). Greek indices are used for the initial state. The electric potential in this state is denoted by /0(X). x(X) and /0(X) satisfy the static equations of nonlinear electroelasticity [8]. In the present state, time-dependent, small, incremental deformations and electric fields are applied to the deformed body at the initial state. The final position of X is given by y = y(X, t), and the final electric potential is /(X, t). y(X, t) and /(X, t) satisfy the dynamic equations of nonlinear electroelasticity [8]. For small incremental deformation and fields, we denote the incremental displacement by u(X, t) and the incremental potential by /1(X, t). u and /1 are assumed to be infinitesimal. We can write y and / as

Initial

S0AB ffi ðwA;B þ wB;A Þ=2; E0K ¼ /0;K :

X

x

u (Incremental)

y

Present 2

1 Fig. 2. Reference, initial and present states of an electroelastic body.

ð6Þ T 0KL ;

S0KL ,

E0K

In (6), and are the initial stress, strain, and electric field. cKaLc, eKLc, and eKL are the usual elastic, piezoelectric, and dielectric constants (second-order constants). cKaLcAB, kAKaLc, bAKLc, and vAKL are the third-order elastic, first odd electroelastic, electrostrictive, and third-order electric susceptibility tensors [1]. It is important to note that the third-order material constants are necessary for a complete description of the lowest order effects of the biasing fields.

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4. Vibration analysis The reference configuration of the quartz plate in Fig. 1 is shown in Fig. 3a. The biasing deformation is due to a constrained expansion which is shown in Fig. 3b. The incremental motion is the thickness-shear vibration shown in Fig. 3c. We study plane strain motions independent of X3, with o/oX3 = 0 and u3 = 0. We study these three states separately below.

The constrained expansion of the three connected plates in Fig. 1 under an applied electric field is a standard problem of statically indeterminate structures in mechanics of materials [12]. The solution can be obtained in a straightforward manner. The stress and strains in the crystal plate are given by

T 01 ¼ c11

w11 E0 t ; c11 b þ c11 t

w11 E0 t ; c11 b þ c11 t c21 0 ¼ S02 ¼  S : c22 1

ð12Þ

w1;1 ¼ S01 ¼

4.1. The biasing extensional deformation Consider a crystal plate of rotated Y-cut quartz which is of symmetry class 32 [10]. The biasing fields are assumed to be small and are governed by the linear theory of piezoelectricity. Since the electrodes are shorted and there is no biasing electric field in the quartz plate, the relevant constitutive relations for plane-strain deformations are

T 01 ¼ c11 S01 þ c12 S02 ;

ð7Þ

T 02 ¼ c12 S01 þ c22 S02 ;

where under the compact matrix notation [10], T 01 and T 02 are the extensional stresses in the X1 and X2 directions and S01 and S02 the corresponding strains. c11, c12 and c22 are elastic constants. The plate is assumed to be thin with T 02 ¼ 0 (stress relaxation for thin plates [10]) from which we can solve for S02 from (7)2 and substitute the resulting expression into (7)1 to obtain the following relaxed constitutive relation for the crystal plate

T 01 ¼ c11 S01 ;

ð8Þ

w2;2

4.2. The incremental thickness-shear vibration Since the biasing deformations are uniform and the plate is assumed to be long and thin, for the incremental motion we neglect edge effects and consider motions independent of X1. We study thickness-shear vibration in the X1 direction with

u1 ¼ u1 ðX 2 ; tÞ;

u2 ¼ u3 ¼ 0:

The relevant stress components for the incremental fields have the following expressions:

h i K 121 ¼ c66 þ ð2c66 þ c661 ÞS01 þ c662 S02 u1;2 ¼ c66 ð1 þ 2aE0 Þu1;2 ;   K 122 ¼ c261 S01 þ c262 S02 u1;2 ;   K 123 ¼ c461 S01 þ c462 S02 u1;2 ; ð14Þ

where

c11 ¼ c11  c212 =c22 :

ð9Þ

The relevant constitutive relation for the extension of the ceramic layers is [11]

T 011 ¼ c11 S01  w11 E0 ;

ð10Þ

where



2

c11 ¼ cE33  cE13 =cE11 ; w11 ¼ e33  e31 cE13 =cE11 :

ð11Þ

The modified material constants in (11) are the result of the same stress relaxation of T 02 ¼ 0 as in the process of obtaining (8).

2a ¼

  1 c21 w11 t 2c66 þ c661  c662 : c66 c22 c11 b þ c11 t

ð15Þ

Since the third-order elastic constants are usually larger in value than the second-order elastic constants [13], K 122 and K 123 are not necessarily small compared to K 121 . For orthorhombic; tetrago  nal 422, 4 mm, 42m and 4/mmm; trigonal 32, 3 m and 3m; hexag onal 622, 6 mm, 6m2 and 6/mm; and cubic as well as isotropic materials [14]

c261 ¼ c262 ¼ c461 ¼ c462 ¼ 0;

ð16Þ

K 123

and hence ¼ ¼ 0. This includes, e.g., the quartz plates studied in this paper. The equations of motion are

X1

2b

where under the compact matrix notation, c261, c262, c461, and c462 are the third-order elastic constants, and

K 122

X2

€1 ; K 121;2 ¼ q0 u

2a

K 122;2 ¼ 0;

K 123;2 ¼ 0:

ð17Þ

The analysis below is parallel to that in [15] for thickness-shear vibrations of a quartz plate under a thermal bias. Eqs. (17)2,3 are trivially satisfied when K22 and K23 vanish. Eq. (17)1 takes the following form:

(a) T 012b

ð13Þ

X2

T 012b

X1

€1 : c66 ð1 þ 2aE0 Þu1;22 ¼ q0 u

ð18Þ

We are interested in odd thickness-shear modes which are excitable by a thickness electric field. Therefore we let

(b)

u1 ðX 2 ; tÞ ¼ uðtÞ sin

X2

T 012b

X1 T 012b

(c) Fig. 3. Reference (a), initial (b), and present (c) configurations of the quartz plate.

np X2; 2b

n ¼ 1; 3; 5 . . . ;

ð19Þ

which satisfies the boundary conditions

K 121 ðX 2 ¼ bÞ ¼ 0:

ð20Þ

Substituting (19) into (18), we obtain the following ordinary differential equation for u(t):

€ þ x20 ð1 þ 2aE0 Þu ¼ 0; u

ð21Þ

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Y. Zhou et al. / Ultrasonics 51 (2011) 131–135

where we have denoted

x20 ¼

2

n

p c66 2

2

4b q0

ð22Þ

:

We note that x0 is the resonant frequency of the nth odd thickness-shear mode of the crystal plate when the biasing fields are not present. Letting

uðtÞ ¼ U expðixtÞ

ð23Þ

in (21), where U is a constant, we obtain the resonant frequency as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x ¼ x0 1 þ 2aE0 ffi x0 ð1 þ aE0 Þ;

ð24Þ

which implies the following frequency shift due to the biasing electric field

x  x0 ¼ aE0 : x0

ð25Þ

4.3. A numerical example Consider plates of Y-cut quartz and langasite which are common materials for crystal resonators. For quartz plate at hR = 25 °C we have [10,13]

determine the biasing fields, and the second- or higher-order effects of the biasing fields need to be considered [16]. Then a nonlinear relationship between the frequency shift and the biasing fields will be predicted. Such a calculation requires knowledge of the fourth-order material constants which at present are not available. Therefore the range of the linear output cannot be determined from the present analysis. For a moderate electric field of 105 V/m or 100 V/mm, the relative frequency shift is of the order of 105 which is a significant frequency shift in a crystal resonator and is measurable. Fig. 4 shows that thicker piezoelectric plates imply higher sensitivity as expected. The results for a Y-cut langasite resonator are shown in Fig. 5, which are similar to those of a quartz resonator except that the frequency shift is negative. The material constants for langasite are from [17].

5. Connected quartz and piezomagnetic plates The above analysis is also valid when the piezoelectric ceramic plates are replaced by piezomagnetic plates. We only need to change the piezoelectric constants to piezomagnetic constants in the above equations. Such a structure can function as a possible magnetic field sensor. The results are shown in Figs. 6 and 7. The

q0 ¼ 2651 kg=m3 ; c66 ¼ 39:88; c21 ¼ 6:99; c11 ¼ c22 ¼ 86:74;

c111 ¼ 210;

c112 ¼ 345;

c222 ¼ 332 GPa;

ð26Þ

and the following relations among the third-order elastic constants exist [14]

c661 ¼ ð2c111  c112 þ 3c222 Þ=4; c662 ¼ ð2c111  c112  c222 Þ=4:

ð27Þ

For the plate thickness we chose h = 0.9696 mm so that for the thickness-shear vibration without biasing deformations the fundamental thickness-shear frequency (n = 1 in (22)) is x0/2p = 106 Hz. We plot (25) in Fig. 4. A linear relationship between the frequency shift and the electric field is predicted, which is ideal for electric field sensing. This linearity is the consequence of the theory employed, i.e., the biasing field is assume to be small and is obtained by the linear theory, and only the first-order effect of the biasing field on the incremental vibration is considered. For large biasing fields, the nonlinear theory of electroelasticity [8] is needed to

Fig. 5. Frequency shift versus electric field (Y-cut langasite).

Fig. 4. Frequency shift versus electric field (Y-cut quartz).

Fig. 6. Frequency shift versus magnetic field (Y-cut quartz).

Y. Zhou et al. / Ultrasonics 51 (2011) 131–135

135

Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060335107), and the National Basic Research Program of China (No. 2009CB623204). References

Fig. 7. Frequency shift versus magnetic field (Y-cut langasite).

piezomagnetic material used is CoFe2O4 whose material constants are given in [18]. 6. Conclusion The resonant frequencies of a quartz resonator connected to two piezoelectric plates change when an electric field is applied. When the electric field is of the order of 100 V/mm, the relative frequency shift is of the order of 105. This effect can be used to make electric field sensors. When the piezoelectric plates are replaced by piezomagnetic plates, similar effects are shown theoretically which can be used to make magnetic field sensors. Acknowledgement The work was supported by the National Natural Science Foundation of China (Nos. 10725210, 10832009, and 10772087), the

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