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35Electronic Journal «Technical acoustics» http://webcenter.ru/~eeaa/ejta/
2 (2002) 1.1-1.7 Dragan D. Mancic and Milan Dj. Radmanovic
Faculty of Electronic Engineering, University of Nis, Beogradska 14, 18000 Nis, Yugoslavia e-mail: dmancic@elfak.ni.ac.yu
Piezoceramic ring loaded on each face: a three-dimensional approach
Received 04.01.2002, published20.02.2002
An approximated 3D matrix model of piezoceramic ring loaded on each face is proposed. Ring is treated as a three-dimensional structure whose vibrations can be described by two coupled differential equations. Solution of this system are two orthogonal wave functions which depend only on one axis, corresponding to the propagation direction, and which satisfy the boundary conditions only in an integral form. With this model, which describes both the thickness and the radial modes, and the coupling between them, the piezoceramic element is schematized as a five-port system, with one electrical and four mechanical ports, one for each surface. The electrical impedance of the sample can also be easily computed. Also, the computed and experimental results are compared. Compared with numerical methods, the computation is time-saving.
1. INTRODUCTION
Piezoceramic disks with central hole (piezoceramic rings) are widely used as efficient vibratory sources in various fields of industrial application of high-power ultrasonics. The ring structure is determined by the necessity to prestress this in composite Langevin transducers for power applications. There are few new papers on this subject. Two and three-dimensional attempts have been made. 3D stress-free model of a ring to the piezoelectric material characterization is described in [1]. 2D matrix model of a thin ring capable of taking into account the interaction of the inner and outer lateral surfaces with the external medium is proposed in [2]. In [3] an approximated 3D matrix model of cylinder-shaped piezoceramic elements without central hole is described. This model is able to compute all the relations between the input applied voltage and the output forces or velocities on every external surface.
In this paper a more general form of approximated 3D model [3] is proposed for ring case, capable of taking into account the interaction of all the surfaces with external medium. The ring is modeled as a five-port system, with four mechanical ports, and one electrical port. With this approach, good agreement between the computed and measured electric impedance is found.
2. ANALYTICAL MODELLING
Consider a thickness poled piezoceramic ring with internal radius a1, external radius a2, and thickness 2b, fully electroded on its flat surfaces. The cylindrical coordinate system with the origin in the center of the ring, and dimensions are defined in Fig. 1a. Each surface of the ring is loaded by acoustic impedance Zi, where vi and Fi are velocities and forces on these surfaces oi (i=1, 2, 3, 4).
Figure 1. Loaded piezoceramic ring
a) Geometry and dimensions
b) 5-port network representation
The constitutive piezoelectric equations give the stresses (T) and electric field (E) inside the material in terms of strains (S) and electric displacement (D). As it is shown in [1, 3], due to the axial symmetry and metallizations, the set of constitutive equations to describe the vibration of piezoelectric ring is:
Trr = cu Srr + C12 S66 + Cl3 Szz - h31 Dz ;
Tee = C1D2Srr + CU See + Cl3Szz - h31 Dz ; Tzz = C1D Srr + Cd S„ + C33 Szz - h33Dz ;
(1)
Ez =-h31Srr
h33 Szz + Dz /g33,
where cD are the elastic stiffness constants; e3S3 is the clamped dielectric permittivity; hj are the piezoelectric tensor terms (i, j=1, 2, 3).
By assuming harmonic waves, the mechanical displacements in the r and z directions are represented by the two orthogonal wave functions [1, 2]:
ur = [AJ, (kr)+BY,(krr]; uz = [C sin (kzz )+D cos(kzz )\ejat,
where kr = olvr, kz = (o/vz, vr =^Jc^ /p and vz =-Jc33 / p are the wave numbers and the
phase velocities of the two uncoupled waves in the r and z directions, respectively; o is the angular frequency; p is the piezoceramic density; J1 and Y1 are the first and the second kind of Bessel’s functions of the first order.
The A, B, C and D constants are computed supposing that the outer surfaces are in contact with external media and assuming continuity of the velocities on these surfaces (ui = du/dt):
iir (ai) = vi eim; ur (a2) = v2 ejwt;
uz(b)=-v3eim; uz(-b)=v4eim.
We obtain:
A = Aivi+A2 v2; B=Bivi + B2 v2;
C =___v3 + v4 ; D = v4 - v3 (3)
2josin(kzb ) 2jocos(kzb)
where:
Yi (kra2) .
A
ja [Ji (ka )Yi (kra2)- Ji (kra2 )Yi (krai)] ’
A2 =_______________Y1 (krai )
B, =
ja [J1 (kra! )Y1 (kra2 ) - J1 (kra2 )Y1 (krai ) ’ __________________J1 (kra2 )__________.
1 jo [Ji (kra2 )Yi (krai ) - Ji (krai )Y1 (kra2 )] ]
B =______________J1 (krai )____________________________
2 j0 [J1 (kra2 )Y1 (krai ) - J1 (krai )Y1 (kra2 )] '
The external behavior of the element is computed imposing the continuity between the stresses and the forces on its surfaces in an integral form [4]:
\ Trr (ai )do = -F1; \ Trr (a2 )do = -F2;
1 02 (4)
f Tzz(b)do = -F,; r Tz(-b)do = -FA.
03 04
Equations (1), (2), (3) and (4), relations Srr =dur / d r , See =ur / r, Szz = duz / d z, and the classical relation between the current I and the electrical displacement Dz = I/(jwn{al - af )), lead to 5^5 linear system relating the electrical (voltage V and current I) to the mechanical variables (forces F and velocities Vi) in the frequency domain (Fig. 1b):
" F "
F2
F3 =
F4
V
-21
-13
-23
-23
-25
-35
(5)
where:
= ( fe -c{ I1 + (-0^ o(kra, )+B,Yo(kra, ))]} i =
jœ
zj =-
4nkraibcl1
jœ
[AjJo (krai )+ j (krai )]
1,2
(i>j ) = (1,2) and (2,1)
z
z
z
z
z
v
12
13
13
15
v
z22 z23
2
z
z
z33 z34
z
v
13
23
35
3
v
z34 z33
z
z
z35 z35
z
15
25
55
z
2naiC13 . z = 4naibh31 . i = 12
jœ jœS
cD
z
33
jœtan(2kzb) 34 jœsin (2kzb)
z35 =
h33 _ 1
jœ jœCo
where S = n(a2 -a12) is surface area of the ring and C0 = £3S3S /2b is the so-called clamped capacity. Voltage Vis computed by integrating Ez not only along z, but also along r axis [4], in order to make V independent of r, because the surfaces o3 and o4 are metallized and therefore equipotential. In the model the lossless material is considered.
Finally, from equation (5) all the transfer functions of the ring can be determined for different inner-to-outer radius ratios (a1/a2), and different outer radius-to-thickness ratios (a2/2b).
3. EXPERIMENTAL RESULTS AND NUMERICAL ANALYSIS
Computed and experimental results were carried out using a PZT4 piezoceramic material [5]. The effective electromechanical coupling coefficient keff is probably the most important parameter, as it gives an indication of the performance potential of the transducer. kf lies between 0 and 1, and for a good transducer is anywhere from 0.5 upwards. Effective coupling factor can be computed by applying the classical expression: f - f2
k 2 J p J s
keff = 72
J p
where fp and f can be assumed to be the frequencies of maximum and minimum impedance magnitude respectively, for the lowest order (fundamental) mode.
Fig. 2 shows keff for PZT4 ring as a function of a1/a2 and a2/2b. It can be seen from Fig. 2, that when a1/a2 decreases from 1 to 0, kf increases. Decreasing a2/2b, kf increases to a maximum for a2/2b~1, and then decreases.
In order to evaluate the capability of the model, we also compute input electrical impedance versus frequency. Fig. 3 shows experimental and simulated input electrical impedance (zin = V/I) versus frequency for two different rings, loaded by air. In each case the
piezoceramic element is driven into vibration by the application of an ac voltage across electrodes coated on major surfaces perpendicular to the z poling axis. Experimental impedance curves of piezoceramic samples were measured with the frequency-sweeping apparatus (HP 4194A Network Impedance Analyzer). The form of the impedance curves and the computed resonance frequencies are very close to the experimental ones. The first radial R1 and thickness T1 modes are those mainly used in practical applications. Our model predicts with sufficient accuracy these modes. Results shown in Fig. 3 should not be used for very fine comparisons of measured and theoretical results, because of limited measurement accuracies,
and since only typical values for the material constants are used [5]. However, general trends can be observed.
(a)
Figure 3. Magnitude of the input electrical impedance versus frequency for PZT4 rings: = 20l°g(„ [q] 50 +1).
Ri (/=1,2,3) are the radial modes and Ti is the thickness mode
a) a2=10 mm, a1=4 mm, 2b=2 mm
b) a2=38 mm, a1=13 mm, 2b=6.35 mm
4. CONCLUSION
With our model, the external behavior of the element in the frequency domain can be described by a 5*5 matrix from which all the transfer functions of the element can be easily computed. Frequency spectrum, displacements and effective electromechanical coupling factor, valid for any a1/a2 and a2/2b ratios, can also be computed by this analysis. When the inner radius vanishes (a1^0), the electric impedance of the ring coincides with those of a disk model [3], because the ring degenerates into a disk with the same radius. Therefore, proposed model extends this 3D model. Our model can be extended to model of more complex structure, like classical Langevin transducer.
REFERENCES
[1] Brissaud M. Characterization of piezoceramics. IEEE Trans. Ultrason., Ferroelect., Freq. Control. Nov. 1991, vol. 38, no. 6, pp. 603-617.
[2] Iula A., Lamberti N., Pappalardo M. A model for the theoretical characterization of thin piezoceramic rings. IEEE Trans. Ultrason., Ferroelect., Freq. Control. May 1996, vol. 43, no. 3, pp. 370-375.
[3] Iula A., Lamberti N., Pappalardo M. An approximated 3-D model of cylinder-shaped piezoceramic elements for transducer design. IEEE Trans. Ultrason., Ferroelect., Freq. Control. July 1998, vol. 45, no. 4, pp. 1056-1064.
[4] Hayward G., Gillies D. Block diagram modeling of tall, thin parallelepiped piezoelectric structures. J. Acoust. Soc. Amer. Nov. 1989, vol. 86, no. 5, pp. 16431653.
[5] Five piezoelectric ceramics. Bulletin 66011/F, Vernitron Ltd., 1976.
