Abstract— In this paper, an electromechanical model for ring-type piezoelectric transformer (PT) has been proposed based on Hamilton’s principle. To establish the model, vibration characteristics of the piezoelectric ring with free boundary conditions are analyzed in advance. Based on the vibration analysis results of the piezoelectric ring, operating frequency and vibration mode of the PT are chosen. Then, an electromechanical model for the PT was obtained, which can be used to simulate the coupled electromechanical system for the transformer, such as voltage step-up ratio, input impedance, output impedance, input power, output power, and efficiency. Thus, the optimal load resistance and the maximum efficiency for the PT will be introduced in this paper.
I. INTRODUCTION
HE idea of a piezoelectric transformer was first implemented by Rosen in 1956 [1]. It used the coupling effect between electrical and mechanical energy of piezoelectric materials. A sinusoidal signal is used to excite mechanical vibrations by the inverse piezoelectric effect via the driver section. Due to the direct piezoelectric effect, an output voltage can be induced in the generator part. The PT offers many advantages over the conventional electromagnetic transformer such as high power-to-volume ratio, electromagnetic field immunity, and nonflammable.
Due to the demand on miniaturization of power supplying systems of electrical equipment, the study of PT has become a very active research area in engineering. In the literature [2], [3], many piezoelectric transformers have been proposed and a few of them found practical applications. Apart from switching power supply system, a Rosen-type PT has been adopted in cold cathode fluorescent lamp inverters for liquid-crystal display. The PT with multilayer structure to provide high-output power may be used in various kinds of power supply units. Recently, PT of ring or disk shapes have been proposed and investigated [4,5]. Their main advantages are simple structure and small size. In comparing with the structure of a ring and a disk, the PZT ring offers higher electromechanical coupling implies that a ring structure is more efficient in converting mechanical energy to electrical energy, and vice versa, which is essential for a high performance PT.
Different from all the conventional PT, the ring-type PT
The author is with the Mechanical Engineering Department, Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan (corresponding author to provide e-mail: [email protected]).
requires only a single poling process and a proper electrode pattern, and it was fabricated by a PZT ring by dividing one of the electrodes into two concentric circular regions. Because of the mode coupling effect and the complexity of vibration modes at high frequency, the conventional lumped-equivalent circuit method may not accurately predict the dynamic behaviors of the PT.
In this study, an electromechanical model for ring-type PT is obtained based on Hamilton’s principle. In order to establish the model, vibration characteristics of the piezoelectric ring with free boundary conditions are analyzed in advance, and the natural frequencies and mode shapes are obtained. In addition, an equivalent circuit model of the PT is obtained based on the equations of the motion for the coupling electromechanical system. Furthermore, the voltage step-up ratio, input impedance, output impedance, input power, output power, and efficiency for the PT will be conducted. Then, the optimal load resistance and the maximum efficiency for the PT will be calculated in this paper.
Fig. 1. Structure of ring-type piezoelectric transformer. II. THEORETICAL ANALYSIS
A. Vibration Analysis of the Piezoelectric Ring
Fig.1 shows the geometric configuration of a ring-type PT with external radius Ro, internal radius Ri and thickness h. The ring is assumed to be thin, h << Ri. The cylindrical coordinate system is adopted where the r-θ plane is coincident with the mid-plane of the undeformed ring, and the origin is in the center of the ring. The piezoelectric ring is polarized in the thickness direction, and two opposite surfaces are covered by electrodes.
Modeling and Analysis on Ring-type Piezoelectric Transformer
Shine-Tzong Ho
T
The constitutive equations for a piezoelectric material with crystal symmetry class C6v can be expressed as
+ = z r r zr z z r E E E E E E E E E E E E r zr z z r E E E d d d d d s s s s s s s s s s s s θ θ θ θ θ θ θ τ τ τ σ σ σ γ γ γ ε ε ε 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 15 33 31 31 66 44 44 33 13 13 13 11 12 13 12 11 , (1a) + = z r T T T r zr z z r z r E E E d d d d d D D D θ θ θ θ θ ε ε ε γ γ γ σ σ σ 33 11 11 33 31 31 15 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , (1b) where σr, σθ, σz, τθz, τzr, τθr are the components of the stress, εr,
εθ, εz, γθz, γzr, γθr are the components of the strain, and all the components are functions of r, θ, z, and t. sE
11, E s12, E s13, E s33, E
s44, s66E are the compliance constants, d15, d31, d33 are the
piezoelectric constants, T
11
ε , T
33
ε are the dielectric constants,
Dr, Dθ, Dz are the components of the electrical displacement, and Er, Eθ, Ez are the components of the electrical field. The piezoelectric material is isotropic in the plane normal to the z-axis. The charge equation of electrostatics is represented as
0
1
1
=
∂
∂
+
+
∂
∂
+
∂
∂
z
D
D
r
D
r
r
D
z r rθ
θ . (2)The electric field-electric potential relations are given by:
r
E
r∂
∂
−
=
ϕ
,θ
ϕ
θ=
−
r
∂
∂
E
1
,z
E
z∂
∂
−
=
ϕ
, (3)where φ is the electrical potential. The differential equations of equilibrium for three-dimensional problems in cylindrical coordinates are: 2 2
1
t
u
r
z
r
r
r r zr r r∂
∂
=
−
+
∂
∂
+
∂
∂
+
∂
∂
τ
σ
σ
ρ
θ
τ
σ
θ θ , (4a) 2 22
1
t
u
r
z
r
r
r z r∂
∂
=
+
∂
∂
+
∂
∂
+
∂
∂
θ θτ
θτ
θρ
θθ
σ
τ
, (4b) 2 21
t
u
r
z
r
r
z zr z z zr∂
∂
=
+
∂
∂
+
∂
∂
+
∂
∂
σ
τ
ρ
θ
τ
τ
θ , (4c)where ur(r,θ,z,t), uθ(r,θ,z,t), uz(r,θ,z,t) are the displacements of the ring in the radial, tangential, and transverse direction, respectively; and ρ is the material density. The strain-displacement relations for three-dimensional problems in cylindrical coordinates are given by
r
u
r r∂
∂
=
ε
,θ
ε
θ θ=
+
∂
∂
u
r
r
u
r1
,z
u
z z∂
∂
=
ε
, (5a)r
u
r
u
u
r
r rθθ
θ θγ
−
∂
∂
+
∂
∂
=
1
, (5b)θ
γ
θ θz=
∂
∂
+
∂
∂
zu
r
z
u
1
, (5c)z
u
r
u
z r zr∂
∂
+
∂
∂
=
γ
. (5d)Because the piezoelectric disk is thin and the deformation is small, the kirchoff assumption is made. The kirchoff assumptions are as follows:
r
t
r
w
z
t
r
u
t
z
r
u
r∂
∂
+
=
(
,
,
)
(
,
,
)
)
,
,
,
(
0 0θ
θ
θ
(6)r
t
r
w
r
z
t
r
v
t
z
r
u
∂
∂
+
=
(
,
,
)
(
,
,
)
)
,
,
,
(
0 0θ
θ
θ
θ (7))
,
,
(
)
,
,
,
(
r
z
t
w
0r
t
u
zθ
=
θ
(8)where u0, v0, w0 represent the radial, the tangential, and the
transverse displacements of the middle surface of the plane, respectively. After inserting (6)-(8) into (5a),(5b), the strain-displacement relations can be obtained as
2 0 2 0 r w z r u r ur r ∂ ∂ + ∂ ∂ = ∂ ∂ = ε , (9a) ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ + = ∂ ∂ + = θ θ θ θ ε θ θ 0 0 0 0 1 1 1 w r r w z v r r u u r r ur , (9b) θ θ θ θ θ γ θ θ θ ∂ ∂ − ∂ ∂ ∂ ∂ + ∂ ∂ ∂ + − ∂ ∂ + ∂ ∂ = − ∂ ∂ + ∂ ∂ = 0 2 0 0 2 0 0 0 1 1 1 w r z w r r z r w r z r v r v u r r u r u u r r r . (9c)
Since the ring is thin, stress σz can be neglected relative to the other stresses, and strain γθz, γzr can also be neglected. Thus, the constitutive equations of (1a),(1b) can be simplified as z E Er r s E d s (1 ) 11(1 ) 31 2 11
ν
ν
νε
ε
σ
θ − − − + = , (10) z E E r E s d s (1 ) 11(1 ) 31 2 11ν
ν
ε
νε
σ
θ θ = +− − − , (11) ) 1 ( 2 11ν
γ
τ
θ θ = E r+ r s , (12) z T r z d E D = 31(σ
+σ
θ)+ε
33 , (13)where
ν
=
−
s
12E/
s
11E is the Poisson’s ratio. In the piezoelectric transformer, the radial extensional vibration can be generated by driving the input electrode with AC voltage. The radial extensional vibration is supposed to be axisymmetric, and the radial extensional displacement of the middle plane can be assumed to be
u
r(
r
,
t
)
=
U
(
r
)
e
iωt. (14)The stress-displacement relations for the extensional vibration are given by
(c) 3rd vibration mode
Fig.4 Vibration modes of piezoelectric transformer. V. CONCLUSION
In this paper, an electromechanical model for ring-type PT is presented. Based on the electromechanical model, an equivalent circuit of the PT is shown. Also, the voltage step-up ratio, input impedance, output impedance, and output power of the PT are shown, and the optimal load resistance and the maximum efficiency for the PT have been obtained. In the last, some simulated results of the electromechanical model are compared with the experimental results for verification. The model presented here lays foundation for a general framework capable of serving a useful design tool for optimizing the configuration of the PT.
REFERENCES
[1] C. A. Rosen, “Ceramic Transformers and Filters,” Proceedings of Electronic Comp. Symp., pp.205-211, 1956.
[2] R. P. Bishop, “Multi-Layer Piezoelectric Transformer,” US Patent No. 5834882, 1998.
[3] Y. Sasaki, K. Uehara and T. Inoue, “Piezoelectric Ceramic Transformer Being Driven with Thickness Extensional Vibration,” US Patent No.5241236, 1993.
[4] P. Laoratanakul, A. V. Carazo, P. Bouchilloux and K. Uchino, “Unipoled Disk-type Piezoelectric Transformers,” Jpn. J. Appl. Phys., vol. 41, pp.1446-1450, 2002.
[5] J. H. Hu, H. L. Li, H. L. W. Chan, C. L. Choy, “A Ring-shaped Piezoelectric Transformer operating in the third sysmmetric extensional vibration mode,” Sensor and Actuators, A, 88, pp.79-86, 2001.
[6] N. W. Hagood, W. H. Chung and A. V. Flotow, “Modeling of Piezoelectric Actuator Dynamics for Active Structural Control,” J. of Intell. Mater. Syst. And Struct., vol. 1, pp. 327-354, 1990.
Fig.5 Input and output impedances as a function of frequency.
Fig.6 Calculated input impedance at different load resistances.
Fig.7 Measured input impedance at different load resistances.
Fig.8 Experimental setup
Fig.9 Voltage step-up ratio as at different load resistances.
