Research Journal of Applied Sciences, Engineering and Technology 4(14): 2176-2182, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 07, 2012
Accepted: April 04, 2012
Published: July 15, 2012
On Characterization and Equivalent Circuit of Triple-Layer Piezoelectric
Element in Thickness Vibration Mode
Li-Jiao Gong, Min Lu, Zhi-Chao Zhang and Xihuizi Liang
Mechanical and Electrical Engineering College, Shihezi University, Shihezi 832003, China
Abstract: This study presents the fabrication of triple-layer piezoelectric vibrator and analyzes its electrical
impedance. The characterization of the electrical impedance and equivalent circuit of the triple-layer
piezoelectric element in thickness vibration mode is discussed. The dielectric, elastic, piezoelectric and electro
mechanical coupling constants (kt) of the samples were evaluated from the experimental impedance data. A
simple but practical equivalent circuit expressed using only the electrical parameters of triple-layer piezoelectric
vibrator, was explored. Analysis based on this equivalent circuit can provide a good understanding of the
behavior of triple-layer piezoelectric vibrators as a function of frequency in the thickness mode. This will in
turn aid the design and analysis of piezoelectric transducers.
Key words: Ctrical impedance characterization, ele, equivalent circuit models, piezoelectric vibrator, PZT
INTRODUCTION
Piezoelectric materials are the most widely used
functional materials in smart structures. They have many
outstanding advantages that make them suitable for use in
transducers. These include small dimensions, simple
structures, low noise factors, excellent mechanicalelectrical coupling characteristics and frequency response
characteristics. These permit them broad applications in
the generation, damping and conversion of vibrations into
electrical energy and vice versa. Some novel devices
based on piezoelectric materials have been developed in
recent years for a wide range of electromechanical
applications (Steven et al., 2007; Shenck and Paradiso,
2001). The potential of triple-layer piezoelectric element
in energy harvesting applications has been investigated
through experiments (Li-jiao and Xing, 2010). Because
the electrical signals are involved in most piezoelectricbased systems, a good understanding of the electrical
characteristics of piezoelectric materials is essential to the
design and analysis of these systems. It is therefore
desirable to build an equivalent circuit for piezoelectric
materials.
On the one hand, the impedance properties of the
piezoelectric resonator can be represented by a lumpedparameter equivalent circuit. The Van Dyke circuit model
is widely used to represent the equivalent circuit of
lossless piezoelectric vibrators (Martin, 1954) and is
recommended by the IEEE Standard on Piezoelectricity
(1987). Sherrit et al. (1997) have used complex circuit
parameters to describe an improved equivalent circuit
model for an unloaded piezoelectric resonator in thickness
mode: by treating the elements of the circuit as complex
quantities, the losses of the piezoelectric resonator were
taken into account (Sherrit et al., 1997). Mingjie and Liao
(2004) proposed a kind of real lumped-parameter circuit
using resistance values to represent energy loss. On the
other hand, the impedance properties of the piezoelectric
resonator can be calculated from the materials point of
view. Some researchers have deduced the equation for
electrical impedance at thickness extensional piezoelectric
lossless resonance (Stewart and Binu, 2007). In the
derivation of the equation, the material constants were
assumed to be real. Several researchers have used
complex material constants in the function of impedance
to fit impedance spectrum and found these to have better
agreement with experimentally measured curves (IEEE
Standard on Piezoelectricity, 1987). In our previous study,
we investigated (Li-jiao and Xing, 2010) such a complex
model near an isolated resonance in piezoelectric
thickness mode; Triple-layer piezoelectric elements often
study as flexural piezoelectric vibrators. For this reason,
we also (Li-Jiao and Jiang-Quan, 2011) investigated the
equation for the equivalent admittance for the symmetrical
triple-layer flexural piezoelectric vibrators. The purpose
of this investigation is to determine under what
circumstances piezoelectric resonators in thickness mode
can be made to vibrate and under what circumstances
piezoelectric resonators undergo bending vibration.
In this study, the equivalent circuit of the triple-layer
piezoelectric vibrator in frequencies of several modes of
vibration at thickness extensional resonance is discussed.
We evaluate the best methods to determining the
parameters that characterize the electrical model of the
Corresponding Author: Li-Jiao Gong, Mechanical and Electrical Engineering College, Shihezi University, Shihezi 832003, China
2176
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
triple-layer piezoelectric vibrator in the thickness mode.
A triple-layer piezoelectric element made of PZT was
fabricated and its impedance properties were
characterized and studied. The dielectric, elastic,
piezoelectric, constants and the electromechanical
coupling constants (kt) of the samples will then be
extracted from the impedance data obtained from these
fabricated devices.
MATERIALS AND METHODS
Sample preparations: The structure of the triple-layer
piezoelectric element is illustrated as a piezoelectricmetal-piezoelectric sandwich structure in Fig. 1. It was
made of a metal layer covered symmetrically by two
transversely piezoelectric layers poled along the thickness
direction. The piezoelectric material was lead Zirconate
Titanate (PZT) (P-51, Haiying Enterprise Group Co. Ltd)
and the metal elastic layer was beryllium bronze for its
larger elastic modulus. The two piezoelectric layers were
bonded to the non-piezoelectric layer with polymeric
epoxy bonding materials and then cured at a temperature
of 80ºC for 2 h. The overall design of insulation was
carefully considered to prevent current leakage. The
geometric dimensions of the specimens measured are
given in Table 1 (length l, width w and thickness t). Key
material parameters are listed in Table 2.
Impedance measurements: An impedance analyzer
(Agilent 4294A) was used to measure and record the
impedance data and spectrum. The electrical impedance
measurements are performed with sweep voltage level in
the impedance analyzer set to 500 mV and sweeping
frequency signal in the 40Hz-10 kHz range. The upper
bound of the frequency range was set to 10 kHz so that
the bandwidth is wide enough to cover all possible
resonance frequencies of the specimens. The end of the
sample with leads was clamped in the vertical direction,
while the other end was left free. The leads were
connected to the Agilent 4294A by the 42941A probe kit.
The schematic view of the impedance measurement setup
is shown in Fig. 2. A personal computer was used to
calculate the circuit constants and the material parameters.
Retrospection on the theory of representing electrical
impedance curves of piezoelectric materials:
The equation for the electrical impedance described by
IEEE Std. 176-1987: IEEE Standard on Piezoelectricity
(1987) for the impedance equation for analyzing ideal
lossless piezoelectric materials in the thickness resonance
mode is given as follows:
Z (ω ) =
t
s
iAωε33
(1 −
kt2 tan[ω / (4 f p )]
ω / (4 f p )
)
(1)
Table 1: Physical dimensions of the sample
Dimension
lp
wp
tp
lm
Size (mm)
33
14
0.4
63
wm
14
tm
0.3
Table 2: Properties of the piezoelectric and beryllium bronze
Property
Symbol
Value
Displacement coefficient [×10G12C / N]
d33
400
gT33
2100
Relative dielectric constant
9.0
Elastic compliances constant [×10G12m2 / N] SD33
3
3
Dp
7.45
Density of piezo material [×10 kg / m ]
1.30
Young’s modulus of bronze [×1011N / m2] E
Dm
8.23
Density of bronze [×103kg / m3]
Fig. 1: Schematic drawing of the triple-layer piezoelectric
bender: 1: Piezoelectric material layer; 2: Metal
material; 3: Thin film electrodes; 4: Condutive adhesive
layer; p: The direction of polarization
Fig. 2: Schematic view of the impedance measurement
Here, t is the sample thickness, A is the electrode area, kt
is the thickness electromechanical coupling constant, gS33
is the clamped dielectric permittivity and fp is the parallel
resonance frequencies at which the real parts of the
impedance have a maximum. The parameters kt, gS33 and
fp were originally assumed to be real numbers in the
derivation of the resonator in Eq. (1). Sherrit et al. (1997)
and Kin et al. (1997) considered the constants kt, gS33 and
fp to be complex numbers in order to account for the
dielectric, elastic and piezoelectric losses (Sherrit et al.,
1997).
The coupling factor can be determined from
frequencies fs and fp in (1) as follows:
kt2 =
π fs
2 fp
tan(
π f p − fs
2
fp
)
(2)
The equivalent circuit with lumped circuit
parameters: The Van Dyke circuit model, as shown in
Fig. 3a, is widely used to represent the impedance
properties of piezoelectric excited vibrators near isolated
resonance by a lumped-parameter equivalent circuit
(Sherrit et al., 1997; IEEE Standard on Piezoelectricity,
2177
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
L1 =
1
4π 2 f S2C1
(6)
The complex parallel and series resonance
frequencies of the circuit shown in Fig. 3b can be
calculated using Eq. (7) and (8):
fs =
Fig. 3: The impedance properties of a piezoelectrically excited
vibrator can be represented near an isolated resonance
by a lumped-parameter equivalent circuit, (a) The Van
Dyke circuit model: The values of the circuit elements
are real, (b) A complex circuit with three complex
parameters
1987). The Van Dyke model uses four real circuit
parameters, C0, C1, L1 and R1, to represent the impedance
of a free-standing piezoelectric resonator around
resonance. This is recommended by the IEEE standard on
piezoelectricity. The close proximity in frequency of
several modes of vibration may be represented by adding
additional R-L-C branches in parallel to the R1-L1-C1
branch shown. The motional resistance R1 in the
equivalent circuit represents the mechanical dissipation of
the piezoelectric resonator. The motional inductance L1 in
the equivalent circuit represents the inertia of the
piezoelectric resonator. The motional resistance C1 is
relevant to the spring constant of the piezoelectric
vibrator.
Complex circuit model as proposed by Sherrit et al.
(1997): Sherrit et al. (1997) proposed the complex circuit
model shown in Fig. 3b Kin et al. (1997). The model
contains three circuit elements, C0, C1 and L1, which are
defined as complex. It takes into account the dielectric,
elastic and piezoelectric losses of piezoelectric material.
The model applies to unloaded piezoelectric resonators in
the thickness, thickness shear and length extensional
modes. According to Sherrit’s theory, the circuit
parameters can be expressed as functions of the material
constants as shown in formulas (3)-(6). Thus the material
constants can be extracted by solving these equations, if
the circuit parameters are known:
C0 + C1 =
C1 =
C0 =
s
ε33
A/t
1 − kt2
2
2
s
ε33
A / t f p − fs
1 − kt2
s
ε33
A/t
1 − kt2
f p2
− C1
1
fp =
2π L1
C1C0
C1 + C0
(7)
(8)
Once the complex frequencies fs and fp are calculated,
the electromechanical coupling constant kt can be
determined from Eq. (2). The dielectric permittivity will
take the following complex-valued form:
S
ε33
= (t / A) (1 − kt2 ) (C1 + C0 )
(9)
Resonance equations in thickness mode: The electric
field can be coupled to the mechanical stress in a material
at a given frequency T by driving the material electrically
at T (Stewart and Binu, 2007). The resonance equations
can be derived from the constitutive equation, wave
equation and boundary conditions. The resonance
equations, then, can reflect the relationship between the
physical dimension and material constants for the
piezoelectric resonator and the frequency T.
Consider a poled piezoelectric material plate of
electrode area A and thickness t. If it is poled in parallel to
the thickness direction and perpendicular to the electrode
planes and its lateral dimensions are much larger than its
thickness, then the thickness mode impedance equation is
given as a function of the frequency T as follows (Stewart
and Binu, 2007):
⎛
⎛
ρ ⎞⎟ ⎞⎟
⎜
tan⎜⎜ tπ f
D ⎟⎟
⎜
c33
⎝
⎠
t
⎜ 1 − kt2
⎟
Z (ω ) =
s ⎜
⎟
iAωε33
ρ
tπ f
⎜
⎟
D
⎜
⎟
c33
⎝
⎠
(10)
where, kt2 is defined as:
(3)
(4)
1
2π L1C1
s D
2
kt2 = e33
/ (ε33
c33 )
(11)
where, e33 is the piezoelectric stress constant. The elastic
stiffness cD33 may be determined by measurement of fpi
IEEE Standard no. 177 (1966):
(5)
D
c33
= 4 ρ ( f pi t / i ) 2
2178
(12)
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
model shown in Fig. 3a. Based on the model, we can fit
the experimental impedance spectrum by choosing the
values of the electronic components. The close proximity
in frequency of several modes of vibration may be
represented by adding additional R-L-C branches in
parallel to the R1-L1-C1 branch as shown.
We used a lumped Van Dyke’s model with three
resonance circuits to fit the measured specimen. The
impedance of the resonant circuits is as follows:
6
Impedance value (Ohm)
10
5
10
4
10
3
10
2
0
0
10
0 00
9 00
8 00
70
00
60
00
50
00
40
00
30
00
0
1 00
0
20 0
0
10
Z=
Impedance phrase (degree)
-20
-30
-40
(13)
1
iωCo +
1
R1 + i (ωL1 −
1
ωC1
+
)
1
R2 + i (ωL2 −
1
ωC2
+
)
1
R3 + i (ωL3 −
1
ωC3
)
-50
-60
-70
-80
0
900
10
000
0
800
70
00
60
00
50
00
40
00
0
100
0
200
0
30
00
-90
Fig. 4: Impedance and phase angle of the specimen vs.
frequency. The magnitude of the impedance is plotted
on a logarithmic scale against the frequency (top) and
the impedance phase versus the frequency (bottom). The
solid line is the measured curve of the impedance
magnitude and phase of the specimen. The dash-dot line
was generated using Eq. (13) fitted to the experimental
data
RESULTS AND DISCUSSION
In order to evaluate the electrical impedance
characteristics and the equivalent circuit of triple-layer
piezoelectric transducers, we applied various previously
described theories to generate impedance spectra. The
dielectric, elastic, piezoelectric and the electromechanical
coupling constants of the sample were evaluated from the
impedance data. We compared the spectrum generated
against measured spectra. The measured impedance
results are shown as a solid line in Fig. 4. From this
figure, we can divide the impedance curve into two parts,
the resonant frequency portion and the non-resonant
portion. We see that the experimental curves of the triplelayer piezoelectric transducer have three distinct
resonance modes.
Representation of the measured spectrum by the Van
Dyke circuit model: For transducer applications, it is
more practical to fit the impedance plots to lumped circuit
models to fit the impedance curve in the vicinity of the
resonant frequency and predict the electrical behavior of
the resonator. We first consider the Van Dyke circuit
The results for the impedance of the circuit model are
shown as the dash-dot line in Fig. 4. When determining
the values of the electronic components, the dominant
component C0 should be first considered according to the
impedance magnitude. After the value of C0 is chosen, the
values of C1, C2 and C3 can be determined according to
the appearance of the impedance spectrum around the
appropriate resonance. The component Ci (i = 1, 2, 3) has
significant effect on the total impedance in the vicinity of
appropriate resonance frequency bandwidth. The value of
the components Li (i = 1, 2, 3) is relevant to Ci (i = 1, 2, 3)
and is considered according to the appropriate resonant
frequency. The value of component Ri (i = 1, 2, 3) is
adjusted according to impedance phase in the vicinity of
the appropriate resonance frequency bandwidth. The
values of parameters of the components are listed in
Table 3. We can see that a lumped Van Dyke’s model
with several resonances fit the experimental impedance
magnitude well but does not agree with the impedance
phase curve below the first resonant frequency within the
non-resonant range. From the impedance phase response,
we can see that the impedance phase curve within the
non-resonant range is close to -90º but not equal to -90º.
This means that the feature below the first resonant
frequency cannot be considered pure capacitance but
rather must be considered resistance capacitance. It is
clear that there is little energy dissipation.
The fit by complex circuit model: Sherrit et al. (1997)
pointed out that the circuit model of a piezoelectric
vibrator near an isolated resonance can be characterized
by its complex material constants as shown in Fig. 3b. To
model this piezoelectric structure, we used the complex
circuit model shown in Fig. 5, in which the circuit
elements Li and Ci are complex. The impedance of an
isolated resonant circuit is defined as follows:
2179
Zn= sLn+1/(sCn), n=1, 2, 3, s = iT
(14)
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
Fig. 5: The complex circuit model for the piezoelectric
structure. The values of all the circuit element are
complex numbers
The total impedance can then be calculated using the
following equation:
Z(s) =1/(sC0+1/(Z1)+ 1/(Z2)+ 1/(Z3))
(15)
To fit the experimental impedance spectrum, the
method of parameter selection is similar to that used
previously. However, the real value must be chosen
before the imaginary value. The circuit constants for this
simulation are listed in Table 3. The impedance of the
6
Impedance value (Ohm)
10
5
10
4
10
3
10
2
0
0
10
000
9 00
8 00
70
00
60
00
50
00
40
00
30
00
0
200
0
1 00
0
10
Frequency (Hz)
0
-10
-20
-30
-40
-50
-60
-70
0
900
10
000
0
800
70
00
60
00
40
00
50
00
0
-80
-90
100
0
200
0
30
00
Impedance phrase (degree)
Table 3: Values of the components in the equivalent circuit and the
material parameters of the sample. The various curves
generated with these constants are shown in Fig. 4, 5, 6
and 7
Constant
Value (real and imaginary)
Van Dyke model
C0 (F)
15.63×10G9
C1 (F)
0.368×10G9
L1 (H)
69.142
R1 (S)
39.219×103
C2 (F)
0.886×10-9
L2 (H)
3.571
R2 (S)
2.791×103
C3 (F)
0.563×10G9
L3 (H)
3.026
R3 (S)
5.913×103
Sherrit’s complex circuit model
C0 (F)
16.065×10G9 (1-0.0168i)
C1 (F)
0.368×10G9 (1-0.091i)
L1 (H)
69.142 (1+0.035i)
C2 (F)
0.886×10G9 (1-0.0466i)
L2 (H)
3.571 (1+0.0191i)
C3 (F)
0.563×10G9 (1-0.0419i)
L3 (H)
3.026 (1+0.00131i)
Resonance frequency
fs1 (Hz)
995.06 + 2.7752i
fp1 (Hz)
1006.4 + 2.7232i
fs2 (Hz)
2827.2 + 3.8833i
fp2 (Hz)
2904.2 + 3.7629i
fs3 (Hz)
3852.6 + 7.8152i
fp3 (Hz)
3919.6 + 7.7846i
Material parameters
gS33(FmG1)
3.805×10G8 (1-0.01643i)
kt
0.1660 (1-0.0361i)
D (kg/m3)
76710
CD33(N/m2)
3.7579×104 (1+0.0542i)
Frequency (Hz)
Fig. 6: Impedance and phase angle of the specimen vs.
frequency. The magnitude of impedance is plotted on a
logarithmic scale against the frequency (top) and the
impedance phase versus the frequency (bottom). The
solid line is the measured curves of the impedance
magnitude and phase of the specimen. The dashed line
was generated using Eq. (15) and the model shown as
Fig. 5
complex circuit model is shown as the dashed line in
Fig. 6. The response curves of the impedance value and
phase of the specimen are shown as the solid line in
Fig. 6. As shown in Fig. 6, the proposed complex circuit
model, on the basis of Sherrit’s complex circuit model, is
appropriate for the transducer within frequencies of
several modes of vibration. However, we need further
study on how to select suitable parameters.
The fit obtained using the equation for electrical
impedance described by IEEE Std. 176-1987: The
frequencies of the complex numbers and complex material
constants were calculated using (2), (8) and (9). The
calculated material constants are listed in Table 3. We
propose the Eq. (16) for the thickness mode in frequency
of several modes of vibration. The predicted impedance
spectra generated by the impedance function (16) are
shown as the dotted line in Fig. 7. The fit is good away
from resonance and poor near resonance. This indicates
that the Eq. (16) can be used to describe the impedance
characteristics. The material constants determine the
2180
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
6
6
10
Impedance value (Ohm)
Impedance value (Ohm)
10
5
10
4
10
3
10
5
10
4
10
3
10
2
0
10
000
0
0
0
00
100
9 00
8 00
0
7 00
0
0
0
40 0
6 00
0
30 0
Frequency (Hz)
-45
-45
0
0
0
0
7 00
8 00
9 00
00
0
6 00
100
0
5 00
0
40 0
-85
-90
0
Frequency (Hz)
-75
-80
0
0
9 00
10
0 00
0
8 00
70
00
60
00
40
00
50
00
20 0
0
30
00
1 00
0
-85
-90
-70
30 0
-75
-80
-55
-60
-65
0
-70
-50
20 0
-55
-60
-65
1 00
Impedance phrase (degree)
-50
0
Impedance phrase (degree)
5 00
0
20 0
0
Frequency (Hz)
1 00
0
2
10
9 00
8 00
70
00
60
00
50
00
40
00
30
00
0
200
0
1 00
0
10
Frequency (Hz)
Fig. 7: Impedance and phase angle of the specimen vs.
frequency. The magnitude of the impedance is plotted
on a logarithmic scale versus the frequency (top) and the
phase of the impedance is plotted versus the frequency
(bottom). The solid line is the measured curves of the
impedance magnitude and phase of the specimen. The
dotted line is generated using Eq. (16)
frequency characteristics and the resonance frequency can
be used to calculate the material constants and provide the
impedance curve of the device. The material constants,
then, can be calculated using the measured value of the
resonance frequency of the piezoelectric device:
[
Z (ω ) =
t
s
iAωε22
]
⎤
⎡
tan ω / (4 f p1 )
⎥
⎢1 − k 2 (
+
t
⎥
⎢
ω / (4 f p1 )
⎥
⎢
⎢ tan[ω / (4 f p2 )] tan[ω / (4 f p 3 )] ⎥
+
)
⎥
⎢
ω / (4 f p3 ) ⎥⎦
⎢⎣ ω / (4 f p2 )
Fig. 8: Impedance and phase angle of the specimen vs.
frequency. The magnitude of the impedance is plotted
on a logarithmic scale against the frequency (top) and
the impedance phase is plotted versus the frequency
(bottom). The solid line is the measured curves of the
magnitude of the impedance and the phase of the
specimen. The dashed line was generated using Eq. (17)
shown in Fig. 8. Figure 8 shows the fit to the experimental
impedance data. The fit is good both away from resonance
and near resonance. The impedance function
demonstrated the relationship between material
parameters such as the dielectric, elastic and
electromechanical coupling constants and geometric
dimension and the frequency T for these fabricated
piezoelectric devices:
(16)
Representing the measured spectrum by resonance
equations in the thickness mode: The complex material
constants were calculated using (2), (8), (9) and (12). The
calculated material constants are listed in Table 3. The
elastic stiffness CD33 was obtained from the fundamental
mode fp of the samples in (12). We propose Eq. (17) as a
resonance equation of several modes of vibration in the
thickness mode. The measured impedance spectra and the
simulations obtained by the impedance function (17) are
⎛
⎛
⎞⎞
⎛
⎞
⎜
⎜ tan⎜ tπ f ρ ⎟
⎟⎟
D ⎟
⎜
⎜
⎜
⎟⎟
c
⎝
33 ⎠
⎜
⎜
⎟⎟
+
⎜
⎜
⎟⎟
ρ
⎜
tπ f D
⎜
⎟⎟
t
c
⎜
⎟⎟
33
Z (ω ) =
1 − kt2 ⎜
S ⎜
⎜
⎟⎟
iAωε33
⎛ πf ⎞ ⎟ ⎟
⎛ π f ⎞
⎜
⎜
⎟
⎜
⎟
⎜
tan
tan
⎜
⎜ 2f ⎟ ⎟⎟
⎜ 2f ⎟
⎜
⎝ p3 ⎠ ⎟ ⎟
⎝ p2 ⎠
⎜
⎜
+
⎜
πf
πf
⎜
⎟⎟
⎜
⎟
⎜
2 f p2
2 f p3 ⎟⎠ ⎠
⎝
⎝
(17)
Through evaluation of the four different equivalent
circuit models against experimental impedance data, we
can draw the following conclusions. For the above two
2181
Res. J. Appl. Sci. Eng. Technol., 4(14): 2176-2182, 2012
kinds of circuit models, the model whose circuit elements
are complex is more effective than the model whose
circuit elements are real. Analysis based on this
equivalent circuit can provide a good understanding of the
behavior of triple-layer piezoelectric vibrator as a function
of frequency in the thickness mode. This will in turn aid
the design and analysis of piezoelectric transducers.
Further study is needed to better understand these
proposed circuit models. We will look into ways of
determining the values of the complex parameter circuit
elements Li and Ci and into how to create simulation
models using electronic simulating programs.
CONCLUSION
The equivalent circuit models of the triple-layer
piezoelectric vibrator were studied and we found that the
impedance information can be evaluated and the relative
material properties of the triple layer piezoelectric
vibrator can be calculated. The impedance curve
spectrums are fitted with various models. The results of
this fitting show that the impedance curves produced by
complex circuit models effectively match the impedance
measured for piezoelectric samples.
ACKNOWLEDGMENT
This study described in this study was supported by
a grant from the National Natural Science Foundation of
China (Project No. 50907042). Li-Jiao et al. (2011) was
elected chosen by the Shihezi University “263 talent
program” based on his reliable and impressive study as an
investigator. We express our gratitude to the Aeronautical
Science Key Laboratory for Smart Materials and
Structures and to Nanjing University of Aeronautics and
Astronautics in Nanjing for providing us with laboratory
equipment.
REFERENCES
IEEE Standard on Piezoelectricity, 1987. IEEE Standard
176-1987. Institute of Electrical and Electronic
Engineers.
Kin, W.K., H.L.W. Chan and L.C. Chung, 1997.
Evaluation of the material parameters of piezoelectric
materials by various methods. IEEE T. Ultrason.
Ferr., 44(4).
Li-Jiao, G. and S. Xing, 2010. Study of piezoelectric
cantilever for power generation. Adv. Tech. Elec.
Eng. Energ. China, 29(1): 54-57.
Li-Jiao, G., and L. Jiang-Quan, 2011. Research on
impedance characterization of triple-layer
piezoelectric bender. Appl. Mech. Mater. 50-51: 3236.
Li-Jiao, G., L. Jiang-Quan and W. Yu-Shan, 2011.
Analysis of the admittance of symmetrical triplelayer flexural piezoelectric vibrator. 2011
International Symposium on Applications of
Ferroelectrics and 2011 International Symposium on
Piezoresponse Force Microscopy and Nanoscale
Phenomena in Polar Materials, Canada.
Martin, G., 1954. Determination of equivalent circuit
constants of piezoelectric resonators of moderately
low Q by absolute-admittance measurements. J.
Acoust. Soc. Am., 26(3): 413-420.
Mingjie, G. and W.H. Liao, 2004. Studies on the Circuit
Models of Piezoelectric Ceramics. IEEE Proceedings
of International Conference on Information
Acquisition, pp: 26-31.
Shenck, N. and J. Paradiso, 2001. Energy scavenging
with shoe-mounted piezoelectrics. IEEE Micro,
21(3): 30-42.
Sherrit, S., H.D. Wiederick, B.K. Mukherjee and
M. Sayer, l997. An accurate equivalent circuit for the
unloaded piezoelectric vibrator in the thickness
mode. J. Phys. D: Appl. Phys., 30: 2354-2363.
Stewart, S. and K.M. Binu, 2007. Characterization of
Piezoelectric Materials for Transducers. Arxiv
preprint arXiv: 0711.2657-arxiv.org.
Steven, R., H. Anton and A. Sodano, 2007. A review of
power harvesting using piezoelectric materials
(2003-2006). Smart Mater. Struct., 16: 1-21.
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