Research Article
Open Access
Piezoelectric Characterization with Mechanical
Excitation in PZT Bar with NonElectrode Boundary
Condition–k_{31} Mode Case
Maryam Majzoubi^{1}, Yuxuan Zhang^{1*}, Eberhard Hennig^{2}, Timo Scholehwar^{2} and Kenji Uchino^{1}
^{1}International Center for Actuators and Transducers, The Pennsylvania State University, State College, PA, 16802, USA
^{2}R&D Department, PI Ceramic GmbH, Lindenstrasse, 07589 Lederhose, Germany
*Corresponding author: Yuxuan Zhang, International Center for Actuators and Transducers, The Pennsylvania State University, 141 Energy and Environment Laboratory, State College, PA, 16801, USA, Tel:+8146990400; Email:
@
Received: March 07, 2018; Accepted: March 23, 2018; Published: April 10, 2018
Citation: Zhang Y, Majzoubi M, Kenji U, et al. (2018) Piezoelectric Characterization with Mechanical Excitation in PZT Bar with NonElectrode Boundary Condition–k
_{31} Mode Case. SOJ Mater Sci Eng 6(1): 110.
DOI:
http://dx.doi.org/10.15226/sojmse.2018.00151
We introduced an advanced piezoelectric characterization
method with mechanical excitation on partial electrode samples,
which can determine additional physical parameters the conventional
fullelectrode electrical excitation method cannot provide. A
nonelectrode sample with only 10% electrode at the center of a
rectangular k_{31} piezoelectric plate has the benefits for measuring the
extensive parameters directly and much precisely. In this paper, we
derived first the exact analytical solutions on the partial electrode
configuration, including the 10% mechanical excitation part in a
partial nonelectrode sample for calculating there sonance vibration
mode and mechanical quality factors of the composite bar sample
for obtaining the extensive elastic compliance and loss factor. The
result suggests that the almost accurate values can be obtained even
for a specimen with up to 40% center electrode area in the k_{31} mode
samples. Second objective is to compare three different simulations
(ATILA FEM, 6terminal Equivalent Circuit and Analytical Solution).
This mechanical excitation method has the benefit in measuring
physical parameters which cannot be obtained directly from the IEEE
Standard electrical excitation method.
Keywords: Direct loss measurement; intensive and extensive
parameters; resonance vibration mode; PZT; mechanical quality
factor;
Introduction
Miniaturized piezoelectric devices have been replacing
electromagnetic (EM) technologies such as motors and
transformers, since the same size of devices could generate 10
times higher power density in miniaturized transducer areas
[19]. The limitation of power in EM devices is mainly caused
by the heat generation due to the ohmic losses (i.e., Joule heat)
in the coils, dramatically increasing with reducing the lead wire
diameter [10]. On the other hand, further miniaturization of
piezoelectric devices is also restricted by heat generation due to
inherent hysteresis losses, primarily originated from microscopic
domain dynamics [1114]. Therefore, it is necessary to study
these losses as a function of input electric power under different
boundary conditions.
There are three types of losses in piezoelectric materials;
namely, elastic $tan\varnothing $
, dielectric ( $tan\delta $
), and piezoelectric losses
($tan\theta $
), which are respectively described as complex numbers in
the elastic compliance, permittivity, and piezoelectric constant in
the Equations (16) below [15,16]. Furthermore, each of these is
classified into two kinds of ‘intensive’ and ‘extensive’ parameters
[14,17]. Intensive parameters (externally controllable, electric
field E, stress T), unlike the extensive ones (internally determined
in a crystal, electric displacement D, strain S), do not depend on
the size or volume of the material theoretically, since they are
the ratio of two extensive properties [18,19]. The prime and
nonprime loss parameters in Equations (16) correspond to the
intensive and extensive losses, respectively.
$$\begin{array}{l}{s}^{E*}={s}^{E}(1j\mathrm{tan}\phi \text{'})\text{(1)}{c}^{D*}={c}^{D}(1+j\mathrm{tan}\phi )\text{(2)}\\ {\epsilon}^{T*}={\epsilon}^{T}(1j\mathrm{tan}\delta \text{'})\text{(3)}{\beta}^{S*}={\beta}^{S}(1+j\mathrm{tan}\delta )\text{(4)}\\ {d}^{*}=d(1j\mathrm{tan}\theta \text{'})\text{(5)}{h}^{*}=h(1+j\mathrm{tan}\theta )\text{(6)}\end{array}$$
s^{E} is the elastic compliance under constant electric field, and
ε^{T} is the dielectric constant under constant stress; while c^{D} is
the elastic stiffness under constant electric displacement (open
circuit), and β^{S} is the inverse dielectric constant under constant
strain (mechanically clamped condition). It should be noted
that intensive losses, generally, exhibit larger values than the
extensive ones [20,21]. Moreover, d is the piezoelectric constant,
and h is the inverse piezoelectric charge constant. In order to
have the numericallypositive loss tangents, consistent with the
experimental measurements, the sign of the imaginary part for
the intensive and extensive losses are considered to be negative
and positive accordingly [14,20,22].
Mechanical quality factor Q_{m} is a sort of figure of merit for
heat generation and device efficiency [23,24]; the larger Q_{m}, the
better in efficiency. It can be calculated both for the resonance (Q_{A},
Atype resonance), and antiresonance (Q_{B}, Btype resonance)
frequencies, from the 3dB bandwidth method in the admittance
or impedance curve. The maximum and minimum points in
the admittance curve attribute to the resonance (f_{a}) and antiresonance
(f_{b}) frequencies roughly. And the 3dB bandwidth is the
distance between the points having the frequencies
$1/\sqrt{2}$
(or $\sqrt{2}$
)
times the resonance (or antiresonance) point in the admittance
spectrum. The mechanical quality factor is then defined as
follows, in which $({f}_{a1}{f}_{a2})$
or $({f}_{b1}{f}_{b2})$
attributes to the 3dB
down/up fullbandwidths [22]:
$$\begin{array}{l}{Q}_{A}=\frac{{f}_{a}}{{f}_{a1}{f}_{a2}}\left(7\right)\\ {Q}_{B}=\frac{{f}_{b}}{{f}_{b1}{f}_{b2}}\left(8\right)\end{array}$$
In the conventional methods of measuring the properties of
piezoelectric materials in the k_{31} fundamental mode with a full
electrode configuration, only the intensive parameters can be
measured directly, and the extensive ones are derived indirectly
through the coupling factor, defined in Equation 10 as follows:
$${s}_{11}^{D}={s}_{11}^{E}\left(1{k}_{31}^{2}\right)\left(9\right)$$
$${k}_{31}^{2}=\frac{{d}_{31}^{2}}{{s}_{11}^{E}{\epsilon}_{33}^{T}}\text{}\text{(10)}$$
Furthermore, the same argument is true for the loss
calculation through the “K matrix”, which is involutory, exhibiting
a full symmetry relationship between the intensive and extensive
losses (i.e., K^{2} = I, or K = K^{1}) [21].
$$\left[\begin{array}{c}tan\delta \\ tan\varnothing \\ tan\theta \end{array}\right]=K\left[\begin{array}{c}tan\delta \text{'}\\ tan\varnothing \text{'}\\ tan\theta \text{'}\end{array}\right],K=\frac{1}{1{k}_{31}{}^{2}}\left[\begin{array}{ccc}1& {k}_{31}{}^{2}& 2{k}_{31}{}^{2}\\ {k}_{31}{}^{2}& 1& 2{k}_{31}{}^{2}\\ 1& 1& 1{k}_{31}{}^{2}\end{array}\right],\left(11\right)$$
The intensive dielectric loss can be calculated with a
capacitance meter at an offresonance frequency. According to
our recent research, the dielectric constant and loss determined
at an offresonance frequency are almost the same as determined
at the resonance frequency under small vibration level [29].
Using the conventional IEEE Standard k_{31} type sample with a full
electrode configuration, we could determine only the intensive
parameters and loss factors. The intensive mechanical loss is the
inverse of mechanical quality factor Q_{A} at the k_{31} resonance mode
(Equation 12), and then the intensive piezoelectric loss can be
derived from these two losses and the mechanical quality factor
Q_{B} at the antiresonance frequencies as in Equation 13 [21]:
$$tan{\varnothing}_{11}^{\text{'}}=\frac{1}{{Q}_{A,31}}\left(12\right)$$
$$tan{\theta}_{31}^{\text{'}}=\frac{1}{2}\left(tan{\delta}_{33}^{\text{'}}+tan{\varnothing}_{11}^{\text{'}}\right)+\frac{1}{4}\left(\frac{1}{{Q}_{A,31}}\frac{1}{{Q}_{B,31}}\right)\left(1+{\left(\frac{1}{{k}_{31}}{k}_{31}\right)}^{2}{\text{\Omega}}_{b,31}^{2}\right)\left(13\right)$$
Here ${\text{\Omega}}_{b,31}$
is the normalized angular frequency of the
antiresonance given by ${\text{\Omega}}_{b,31}=\frac{{\omega}_{b}l}{2{v}_{11}^{E}}$. Then the extensive (nonprime)
losses could be derived indirectly from the “K” matrix
(Equation 11).
However, since the dielectric loss measurement cannot be
very precise, in comparison with the mechanical loss, the error
propagation in the extensive loss calculation from the “K” matrix
is quite high, resulting in large uncertainty in extensive loss
values [20]. We would like to emphasize again that the precise
determination of both intensive & extensive losses is essential
to improve the simulation accuracy of high power density
piezoelectric devices, which is the motivation of our new method
by using a partial electrode configuration.
Our research target is to establish the mechanical excitation
method for characterizing piezoelectric properties for
supplementing the conventional electrical excitation methods.
(Figure 1) illustrates vibration modes of k_{31} type piezoelectric
plates: resonance, antiresonance modes of the electroded
plate, and resonance mode of the nonelectroded plate under
electric field (top) and mechanical excitation (bottom). In the
conventional fullelectrode case under electrical excitation, the
maximum and minimum admittance frequencies correspond
to the piezoelectric resonance and antiresonance frequencies
roughly. The same frequencies are obtained as the mechanical
resonance frequencies under mechanical excitation, with shortand
opencircuit conditions of top and bottom electrodes.
However, without electrodes, the resonance cannot be monitored
under electrical excitation method, while this resonance can be
measured with the mechanical excitation method, which can
provide the information on Dconstant parameters, such as ${s}_{11}^{D}$
and sound velocity v^{D}.
Figure 1: Equivalent electrode configuration and vibration modes of
k_{31} type piezoelectric plates: (a) resonance, (b) anti resonance modes
under electrical excitation, and (c) resonance mode under mechanical
excitation.
An advanced method of measuring the extensive (i.e., D
constant) elastic compliance and mechanical loss directly in the
k_{31} mode has been proposed in our recent work by using nonelectrode
samples which have only 10% electrodes at the middle
of the top and bottom surfaces, as shown in (Figure 2d) [25]. The
center electrode part acts similarly as an actuator to mechanically
excite the vibration in the whole sample (including the nonelectrode
parts), and simultaneously monitoring the responding
signal. The nonelectrode parts are the loads with the extensive
boundary condition (D constant, rather than E constant on the
electrode). The properties of the load material can be monitored
through the admittance/impedance spectrum of the small
center actuator part. The comparison between conventional
IEEE standard k_{31} mode piezoelectric measurement and our
novel partial electrode measurement method is illustrated in
(Table1). By using our advanced partial electrode mechanical
excitation method, both intensive and extensive parameters can
be measured without further derivation.
Table 1: Comparison between IEEE Standard electrical excitation method and partial electrode mechanical excitation method
Figure 2: Schematic view of (a) fullelectrode (FE), (b) partial electrode
– shortcircuit, (c) partial electrode – open circuit, and (d) nonelectrode
(NE) samples.
The partial electrode samples with electrode load (short and
opencircuit) were also prepared and measured as reference
data. The schematic view of (a) fullelectrode (FE), (b) partial
electrode – shortcircuit, (c) partial electrode – open circuit,
and (d) nonelectrode (NE) samples are shown in (Figure 2).
The preliminary results were examined by neglecting the 10%
actuator part difference and reported in [25]. This paper present
sa precise analysis on the elastic compliance (or resonance
frequency) and its corresponding loss (mechanical loss)in
this PZT composite bar with both intensive (electrode) and
extensive (nonelectrode) boundary conditions, and to verify the
feasibility of this methodology to more general sample geometry
applications.
The admittance spectra of the full electrode and three partial
electrode samples were measured as shown in (Figure 3). The fullelectrode
(FE) sample corresponds to a conventional rectangular
k_{31} mode sample. PEShort attributes to the pure Econstant
condition, namely the resonance mode. As demonstrated in
(Figure 3), the resonance frequency of this sample is almost the
same as the FE one, and its antiresonance frequency occurs at
a much lower frequency. This is because the electrode area for
actuator part in the PEShort sample covers only 10% of the total
length, and the load parts are still short circuit during the antiresonance
frequency. Therefore, its antiresonance frequency
happens between the resonance and antiresonance frequencies
of FE sample. On the contrary, the antiresonance frequency of
the PEOpen sample is the same as the one of the FE sample,
and its resonance frequency occurs in much higher frequency,
because only the center part is excited and the outside parts are
just mechanical loads with an open circuit, leading to apparently
a lower electromechanical coupling factor. This case corresponds
to the antiresonance mode. Since we consider the k_{31} mode and
also there is no electric field on the surface along the length
direction (i.e., Xaxis), the mechanical resonance of the PEOpen
happens between the pure E (i.e., PEShort) and D constant (i.e.,
NE) conditions.
Figure 3: Experimental Results  Admittance spectrum of the full and
partial electrode samples.
It is notable that the NE piezoelectric behaves the Dconstant
performance (except for a small electrode portion for excitation,
which we neglected from the analysis in the previous paper
[25]). Its losses would also mostly attribute to the extensive
ones, which we could measure directly as the first trial. Since
the elastic compliance under extensive condition (Dconstant) is
lower than the intensive one (Econstant), which means having a
larger stiffness, higher resonance and antiresonance frequencies
are observed, in comparison to the FE sample and also the
other partial electrode samples. It is noteworthy again that the
resonance measurement on the nonelectrode sample is possible
only by using the mechanical excitation method.
In this paper, we describe first an accurate linear analytic
solution in a continuum media, which is simple to use for
calculating the resonance vibration mode and mechanical
quality factors in a nonelectrode sample. This is achieved by
solving constitutive and dynamic equations in a composite bar
by applying different boundary conditions, for the actuator and
load parts with various electrode areas. Second objective is to
compare three different simulations (ATILA FEM, 6terminal
Equivalent Circuit and Analytical Solution).
Theoretical Derivation
When the length L, width w and thickness b satisfy L >> w
>> b in a rectangular shape piezoelectric specimen, the two basic
constitutive equations in k_{31} mode piezoelectric materials are as
follows in a linear firstapproximation [28],
$${x}_{1}={s}_{11}^{E}{T}_{1}+{d}_{31}{E}_{3}\left(14\right)$$
$${D}_{3}={\epsilon}_{0}{\epsilon}_{33}^{T}{E}_{3}+{d}_{31}{T}_{1}\left(15\right)$$
The ${x}_{1}$
and ${T}_{1}$
are strain and stress in direction 1 (longitude L),
perpendicular to the poling direction. ${E}_{3}$
And ${D}_{3}$
is the electric field
and dielectric displacement in the polling direction 3 (thickness
b). The ${s}_{11}^{E}$
is the elastic compliance for both stress and strain in
direction 1 under the constant applied electric field. The ${\epsilon}_{0}{\epsilon}_{33}^{T}$
is
the permittivity for electric field and dielectric displacement in
direction 3 under constant external stress. By assuming the stress
to be zero in directions apart from the longitudinal vibration one
in a long thin plate (L >> w >> b, k_{31} mode), the dynamic equation
for extensional vibration is given as in Equation 16, in which $u$
and $x$
are the displacement and position in direction 1, and $\rho $
is
the material’s density[17],
$$\rho \frac{{\partial}^{2}u}{\partial {t}^{2}}=F=\frac{\partial {T}_{1}}{\partial x}\left(16\right)$$
By introducing the constitutive equations in the above
equation, and allowing ${x}_{1}=\frac{\partial u}{\partial x}$
, two different cases of boundary
conditions, electrode and nonelectrode ones, are considered.
Refer to (Figure 4). We introduce a parameter ‘a’ to represent the
electrode coverage rate on the plate specimen (a = 1 corresponds
to a full electrode configuration). Firstly, taking into account that
$\left(\frac{\partial {E}_{3}}{\partial x}\right)=0$
for the electrode part due to the equal potential of each
electrode, the following harmonic vibration equation is derived,
Figure 4: Nonelectrode sample with ‘a’ fraction electrode at the center.
$${\omega}^{2}\rho {s}_{11}^{E}u\left(x\right)=\frac{{\partial}^{2}u\left(x\right)}{\partial {x}^{2}}\left(17\right)$$
By substituting the general standingwave form of the
displacement equation as $u=u\left(x\right){e}^{j\omega t}$
, the following solution
for the displacement of the electrode part can be obtained, since
${u}_{e}\left(x\right)$
should be equal to zero at x = 0 (center of gravity):
$${u}_{e}\left(x\right)={A}_{e}sin\left(\frac{\omega}{{v}^{E}}x\right)\left(18\right)$$
Secondly, by allowing $\left(\frac{\partial {D}_{3}}{\partial x}\right)=0$
for the nonelectrode area,
the harmonic vibration equation and final solution can be derived
as (for $\leftx\right>aL/2$
), in which the displacement equation with
positive sign for B_{n} correspond to the positive position x.
$${\omega}^{2}\rho {s}_{11}^{E}\text{(1}{k}_{31}^{2})u\left(x\right)=\frac{{\partial}^{2}u\left(x\right)}{\partial {x}^{2}}\text{}\left(19\right)$$
$${u}_{n}\left(x\right)={A}_{n}sin\left(\frac{\omega}{{v}^{D}}x\right)\pm {B}_{n}cos\left(\frac{\omega}{{v}^{D}}x\right)\text{}\left(20\right)$$
The
${v}^{E}=\frac{1}{\sqrt{\rho {s}_{11}^{E}}}$
, and
${v}^{D}=\frac{1}{\sqrt{\rho {s}_{11}^{D}}}$
in these equations are the sound
velocities for the above two conditions. Here, and for the rest
of this paper, subscript ‘e’ or ‘n’ denotes the electrode or nonelectrode
parameters.
Therefore, the strain and stress for the electrode and nonelectrode
parts can be inferred as:
$${x}_{1e}\left(x\right)=\frac{\partial {u}_{e}}{\partial x}={A}_{e}\frac{\omega}{{v}^{E}}cos\left(\frac{\omega}{{v}^{E}}x\right)\text{}\left(21\right)$$
$${x}_{1n}\left(x\right)=\frac{\partial {u}_{n}}{\partial x}={A}_{n}\frac{\omega}{{v}^{D}}cos\left(\frac{\omega}{{v}^{D}}x\right){B}_{n}\frac{\omega}{{v}^{D}}sin\left(\frac{\omega}{{v}^{D}}x\right)\left(22\right)$$
$${T}_{1e}\left(x\right)=\frac{{x}_{1e}}{{s}_{11}^{E}}\left(\frac{{d}_{31}}{{s}_{11}^{E}}\right){E}_{3e}\left(23\right)$$
$${T}_{1n}\left(x\right)=\frac{{x}_{1n}}{{s}_{11}^{E}}\left(\frac{{d}_{31}}{{s}_{11}^{E}}\right){E}_{3n}\left(x\right)\left(24\right)$$
${\text{E}}_{3\text{e}}$
, the electric field of the electrode area, is a constant value
with respect to position which is known and can be calculated
from the applied voltage to the sample. ${\text{D}}_{3\text{n}}$
is also a constant
value, since the nonelectrode part cannot provide free charge,
therefore $\nabla \text{D}=\text{\rho}=0$
. Furthermore, from $\text{D}={\text{\epsilon}}_{0}\text{E}+\text{P}$
and the
fact that for no electrode condition depolarization field E is
proportion to the negative polarization (P), therefore, ${\text{D}}_{3\text{n}}=0$,
and by giving ${\text{E}}_{3\text{n}}\left(\text{x}\right)=\frac{{\text{d}}_{31}}{{\text{\epsilon}}_{0}{\text{\epsilon}}_{33}^{\text{X}}}{\text{T}}_{1\text{n}}\left(\text{x}\right)+\frac{{\text{D}}_{3\text{n}}}{{\text{\epsilon}}_{0}{\text{\epsilon}}_{33}^{\text{X}}}==\frac{{\text{d}}_{31}}{{\text{\epsilon}}_{0}{\text{\epsilon}}_{33}^{\text{X}}}{\text{T}}_{1\text{n}}\left(\text{x}\right)$
we have,
$${T}_{1n}\left(x\right)=\frac{{x}_{1n}}{{s}_{11}^{E}\left(1{k}_{31}^{2}\right)}=\frac{{x}_{1n}}{{s}_{11}^{D}}\left(25\right)$$
There are three unknown parameters in the above equations,
namely: ${\text{A}}_{\text{e}}$
, ${\text{A}}_{\text{n}}$
, and ${B}_{\text{n}}$
, which can be solved by introducing three
boundary conditions as follows:
1. Continuation of displacement at
$x=\pm \frac{aL}{2}$
(the interface
of electrode and nonelectrode parts). The strain can exhibit
discontinuity at $x=\pm \frac{aL}{2}$
2. Continuation of stress at $x=\pm \frac{aL}{2}$
3. Stress being zero at the plate end $x=\pm \frac{L}{2}$
The parameter “a” was introduced to describe the electrode
coverage rate (0 < a < 1). These can be summarized in the
following matrix relationship, and the solution for unknown
parameters are given in Equation (2729),
$$\left[\begin{array}{ccc}\mathrm{sin}\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)& \mathrm{sin}\left(\frac{\omega}{{v}^{D}}\frac{aL}{2}\right)& \mathrm{cos}\left(\frac{\omega}{{v}^{D}}x\frac{aL}{2}\right)\\ \frac{\omega}{{v}^{E}}\mathrm{cos}\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)& \frac{\omega}{{v}^{D}}\frac{1}{1{k}_{31}^{2}}\mathrm{cos}\left(\frac{\omega}{{v}^{D}}\frac{aL}{2}\right)& \frac{\omega}{{v}^{D}}\frac{1}{1{k}_{31}^{2}}\mathrm{sin}\left(\frac{\omega}{{v}^{D}}\frac{aL}{2}\right)\\ 0& \frac{\omega}{{v}^{D}}\frac{1}{1{k}_{31}^{2}}\mathrm{cos}\left(\frac{\omega}{{v}^{D}}\frac{L}{2}\right)& \frac{\omega}{{v}^{D}}\frac{1}{1{k}_{31}^{2}}\mathrm{sin}\left(\frac{\omega}{{v}^{D}}\frac{L}{2}\right)\end{array}\right]\left[\begin{array}{c}{A}_{e}\\ {A}_{n}\\ {B}_{n}\end{array}\right]=\left[\begin{array}{c}0\\ {d}_{31}{E}_{3e}\\ 0\end{array}\right]\text{}(26)$$
$${A}_{n}=\frac{{d}_{31}{E}_{3e}sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)sin\left(\frac{\omega}{{v}^{D}}\frac{L}{2}\right)}{\frac{\omega}{{v}^{E}}cos\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)cos\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)\frac{\omega}{{v}^{D}\left(1{k}_{31}^{2}\right)}sin\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)}\left(27\right)$$
$${B}_{n}=\frac{{d}_{31}{E}_{3e}sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)cos\left(\frac{\omega}{{v}^{D}}\frac{L}{2}\right)}{\frac{\omega}{{v}^{E}}cos\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)cos\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)\frac{\omega}{{v}^{D}\left(1{k}_{31}^{2}\right)}sin\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)}\left(28\right)$$
$${A}_{e}=\frac{{d}_{31}{E}_{3e}cos\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)}{\frac{\omega}{{v}^{E}}cos\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)cos\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)\frac{\omega}{{v}^{D}\left(1{k}_{31}^{2}\right)}sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)sin\left(\frac{\omega}{{v}^{D}}\frac{\left(1a\right)L}{2}\right)}\left(29\right)$$
In order to derive the admittance equation, the current of the
electrode (only on the center electrode part) is required which
can be calculated from,
$$\begin{array}{l}I=j\omega {{\displaystyle \iint}}^{\text{}}{D}_{3e}dxdy=2j\omega W\{\underset{0}{\overset{\frac{aL}{2}}{{\displaystyle \int}}}{D}_{3e}dx\}=\\ 2j\omega W\{\underset{0}{\overset{\frac{aL}{2}}{{\displaystyle \int}}}({d}_{31}(\frac{1}{{s}_{11}^{E}}{A}_{e}\frac{\omega}{{v}^{E}}cos\left(\frac{\omega}{{v}^{E}}x\right)\left(\frac{{d}_{31}}{{s}_{11}^{E}}\right){E}_{3e})+{\epsilon}_{0}{\epsilon}_{33}^{T}{E}_{3e})dx\}=\\ 2j\omega W\left(\frac{{d}_{31}{A}_{e}}{{s}_{11}^{E}}sin\left(\frac{\omega}{{v}^{E}}\frac{aL}{2}\right)+{\epsilon}_{0}{\epsilon}_{33}^{T}\left(1{k}_{31}^{2}\right){E}_{3e}\frac{aL}{2}\right)\left(30\right)\end{array}$$
Then, by defining the 3 e
${E}_{3e}=\frac{V}{b}$
, and $Y=\frac{I}{V}$
considering the
losses and complex values as in Equation (16), the admittance
equation is deduced:
$$\begin{array}{l}Y=2jW\frac{{d}_{31}^{*}}{b{s}_{11}^{E}{}^{*}}\frac{{d}_{31}^{*}cos\left(\frac{\omega}{{v}^{D}{}^{*}}\frac{\left(1a\right)L}{2}\right)}{\frac{1}{{v}^{E}{}^{*}}cos\left(\frac{\omega}{{v}^{E}{}^{*}}\frac{aL}{2}\right)cos\left(\frac{\omega}{{v}^{D}{}^{*}}\frac{\left(1a\right)L}{2}\right)\frac{1}{{v}^{D}{}^{*}\left(1{k}_{31}^{*}{}^{2}\right)}sin\left(\frac{\omega}{{v}^{E}{}^{*}}\frac{aL}{2}\right)sin\left(\frac{\omega}{{v}^{D}{}^{*}}\frac{\left(1a\right)L}{2}\right)}\\ sin\left(\frac{\omega}{{v}^{E}{}^{*}}\frac{aL}{2}\right)+j\omega {\epsilon}_{0}{\epsilon}_{33}^{T}{}^{*}\left(1{k}_{31}^{*}{}^{2}\right)\frac{aLW}{b}\text{(31)}\end{array}$$
Note that we introduced ‘*’ for the dielectric, elastic and
piezoelectric parameters in order to introduce three losses to
calculate the admittance curves accurately. ${v}^{E}{}^{*}$
, ${v}^{D}{}^{*}$
, and k_{31}*
were calculated by putting these complex parameters in their
corresponding equations, i.e. ${v}^{E}=\frac{1}{\sqrt{\rho {s}_{11}^{E}{}^{*}}},{v}^{D}=\frac{1}{\sqrt{\rho {s}_{11}^{D}{}^{*}}},$
and
${k}_{31}^{*}{}^{2}=\frac{{d}_{31}^{*}{}^{2}}{{s}_{11}^{E}{}^{*}{\epsilon}_{33}^{T}{}^{*}}$.
Experimental and Analytical Results and Discussions
The nonelectrode samples were made from the commercial
‘hard’ bulk PZT rectangular plates, PIC144 [PI Ceramic GmbH,
Lederhose, Germany] with 40 x 5 x 1 mm^{3}, and the essential
material properties as listed in the (Table 2), which were further
used for analytical simulation. The permittivity and its dielectric
loss were measured with an LCR meter at 100Hz frequency
[SR715, Stanford Research Systems, Inc., Sunnyvale, CA]. The
other parameters were derived directly or indirectly from the
admittance spectrum, measured with the Precision Impedance
Analyzer [4294A, Agilent Technologies, Santa Clara, CA] [25].
The picture of the fullelectrode and nonelectrode samples are
shown in (Figure 5)
The admittance spectra for the different fractions of the
center electrode (‘a’ values) can be calculated from the Equation
31, and shown in (Figure 6). For instance, a = 1 is the typical FE
Table 2: piezoelectric ceramic material properties
Properties 
Real Parameter 
Intensive Loss (prime) 
Extensive Loss (nonprime) 
Dielectric 
${\text{\epsilon}}_{33}^{\text{T}}/{\text{\epsilon}}_{0}=1073$

${\text{tan\delta}}_{33}^{\text{'}}=2.3\times {10}^{3}$

${\text{tan\delta}}_{33}^{}=2.1\times {10}^{3}$

Elastic 
${\text{s}}_{11}^{\text{E}}=11.7\times {10}^{12}\left[{\text{m}}^{2}/\text{N}\right]$

$\text{tan}{\varnothing}_{11}^{\text{'}}=7.19\times {10}^{4}$

$\text{tan}{\varnothing}_{11}^{}=5.8\text{x}{10}^{4}$

Piezoelectric 
${\text{d}}_{31}=103\times {10}^{12}\left[\text{C}/\text{N}\right]$

${\text{tan\theta}}_{31}^{\text{'}}=2.2\times {10}^{3}$

${\text{tan\theta}}_{31}^{}=6.6\times {10}^{4}$

Density 
$\text{\rho}=8080\left[\text{kg}/{\text{m}}^{3}\right]$

 
 
Figure 5: (a) fullelectrode (FE), (b) partial electrode – shortcircuit,
(c) partial electrode – open circuit, and (d) nonelectrode (NE) samples
(a = 0.1).
Figure 6: Admittance spectra for different fractions of the middle.
Figure 7: Experimental results and analytical solution admittance spectra comparison for FE and NE samples.
sample, and a = 0.1 is the partial NE one with 10% electrode in
the middle.
(Figure 7) demonstrates the comparison between the
analytical solution (b) and experimental results (a) admittance
spectra, which verifies a good agreement in terms of the
resonance frequencies and admittance values.
(Figure 8) illustrates the effect of the actuator portion ‘a’
on the resonance frequencies and mechanical quality factors
obtained from the 3dB bandwidth method of the admittance
spectrum.
As demonstrated in (Figure 7,8), the resonance frequency
f_{a} (maximum peak in admittance curve), and Q_{A} value shift to
lower monotonically with increasing the electrode portion.
However, it is worth to note that the antiresonance frequency f_{b}
(minimum peak in admittance curve) and Q_{B} value do not change
Figure 8: Effect of the center electrode area on the resonance and antiresonance frequencies and their corresponding mechanical quality factors.
dramatically up to a = 0.4. This result is important, since fb and
QB of nonelectrode sample are used for calculating the extensive
elastic compliance s_{11}^{D}, and mechanical loss tan . A nonelectrode
sample with up to 40% electrode area can give sufficiently
accurate results for these values.
The Finite Element Method (FEM) analysis using FEM ATILA
software [Version: ATILA ++ 3.0.27, and GID 12.0.9, Micro
mechatronics, Inc., State College, PA] was conducted in [25], which
were in good agreements with the experimental measurements.
The FEM analysis is a practical tool for calculating the modes
of vibrations, or strain and stress distributions for complex
geometries. However, it is not an inexpensive software, nor easy
one to use for electrical behavior characterization, in general
[26,27]. We also reported a new sixterminal Mason’s equivalent
circuit (EC) model for nonelectrode samples, by considering
all three losses [26]. We demonstrated to obtain the voltage
distribution in the nonelectrode (NE) configuration with the
equivalent circuit simulation. Each of the nonelectrode load part
was segmented into 20 elements along length. NE configuration
elements behave under Dconstant though the voltage could not
be measured. The Econstant element was integrated in between
the Dconstant elements to measure the voltage distribution.
The finite elements are connected in parallel as illustrated
in (Figure 9). The maximum voltage occurs at the resonance
frequency in this simulation. The results showed an improvement
comparing with a conventional equivalent circuit with only one
elastic loss parameter, and were generally in consistent with the
experimental measurements. However, some mechanical quality
parameters were calculated slightly lower than the experimental
method [26]. It is essential to note that, for simplification, the
EC considers discrete linear LCR components, while the FEM
considers limited number of segmented parts for piezoelectric
devices, resulting in approximated solutions for their behavior,
which may be less accurate. Furthermore, the equivalent circuit
method is still complicated from the viewpoint of the circuit
design implementation in the required software.
Figure 9: EFinite element configuration of right (left) load to simulate the voltage distribution in NE configuration
The resonance frequencies and mechanical quality factors
comparison for the analytical approach and experimental
results, as well as the FEM ATILA simulation and EC method
are summarized in (Table 3) for both FE sample and NE one
with a = 0.1 (10%). The EC result was cited from [26], and was
simulated with the proposed equivalent circuit by MATLAB
[Version R2016a, The Math Works, Inc., Natick, Massachusetts].
On the other hand, the ATILA simulation results in this table were
calculated from FEM ATILA simulation [Version: ATILA ++ 3.0.27,
and GID 12.0.9, Micro mechatronics, Inc., State College, PA] and
improved from [25] by integrating the material loss parameters
with at least two digits of accuracy. As is shown, the analytical
method gives the most accurate results, close to the experimental
measurement values.
Finally, the voltage and strain distributions along the length
versus the position x at a frequency where the maximum strain
happens (mechanical resonance frequency) for the NE sample
(a = 0.1) is demonstrated in (Figure 10) as an extension of the
analytical solution, which has the same trend as FEM ATILA
simulation results as in [25]. In FEM simulations, the maximum
strain value is almost 1.3 times higher, and the maximum voltage
value is 0.6 times lower than the analytical calculation.
Table 3: Comparison of the analytical solution with experimental results and other methods for FE and NE (with a = 0.1) samples
Samples 
Experimental 
FEM ATILA simulation 
Equivalent Circuit 
Analytical Solution 
f_{a}FE 
40.69 ± 0.04 
40.62 
40.64 
40.66 
f_{b}FE 
42.31 ± 0.04 
42.26 
42.31 
42.32 
f_{a}NE 
42.43 ± 0.04 
42.16 
42.26 
42.30 
f_{b}NE 
42.85 ± 0.04 
42.68 
42.71 
42.75 
Q_{A}FE 
1390 ± 28 
1390 
1355 
1390 
Q_{B}FE 
1650 ± 33 
1650 
1567 
1656 
Q_{A}NE 
1690 ± 30 
1629 
1626 
1648 
Q_{B}NE 
1770 ± 35 
1731 
1708 
1738 
Figure 10: Voltage and strain distributions along length for the NE sample (a = 0.1) – Analytical calculation.
Conclusion
In conclusion, this paper introduced an advanced piezoelectric
characterization method with mechanical excitation on partial
electrode samples, which can determine additional physical
parameters the conventional IEEE Standard electrical excitation
method cannot provide. In order to measure the side mechanical
load, the admittance/impedance of the small center actuator part
is monitored. First, an accurate analytical solution for deriving the
resonance frequencies, mechanical quality factors and vibration
mode in a partial nonelectrode PZT rectangular plate in the k_{31}
mode This was achieved by means of the dynamic and constitutive
equations, and solving the displacement and stress distributions
with different boundary conditions for the electrode and nonelectrode
parts. This is important in the sense of calculating the
extensive elastic compliance and mechanical loss, which can
theoretically be measured from the mechanical excitation of a
complete nonelectrode sample. From the analytical solution,
we found that a nonelectrode sample with up to 40% electrode
area can exhibit precise piezoelectric parameters including loss
factors. Therefore, our experimental measurement calculations
with neglecting the actuator part difference for the nonelectrode
sample with a = 10% in our previous paper [25] had almost
accurate results. Those were in consistent with the “K” matrix
indirect solution, and therefore, verify the validation of the
indirect method, apart from its higher standard deviation.
Second, the new analytical method, compared with the FEM
ATILA software and EC method, is more precise and also simple
to use. This method can be further generalized for calculating
the resonance frequencies and mechanical losses in complex
structures with different boundary conditions and modes.
Acknowledgements
The authors acknowledge the support by funding from The
Office of Naval Research (ONR) Grant Number: ONR N0001417
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