Research Article Open Access
Piezoelectric Characterization with Mechanical Excitation in PZT Bar with Non-Electrode Boundary Condition–k31 Mode Case
Maryam Majzoubi1, Yuxuan Zhang1*, Eberhard Hennig2, Timo Scholehwar2 and Kenji Uchino1
1International Center for Actuators and Transducers, The Pennsylvania State University, State College, PA, 16802, USA
2R&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:+814-699-0400; E-mail: @
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 Non-Electrode Boundary Condition–k31 Mode Case. SOJ Mater Sci Eng 6(1): 1-10.
DOI: http://dx.doi.org/10.15226/sojmse.2018.00151
AbstractTop
We introduced an advanced piezoelectric characterization method with mechanical excitation on partial electrode samples, which can determine additional physical parameters the conventional full-electrode electrical excitation method cannot provide. A non-electrode sample with only 10% electrode at the center of a rectangular k31 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 non-electrode 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 k31 mode samples. Second objective is to compare three different simulations (ATILA FEM, 6-terminal 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 [1-9]. 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 [11-14]. 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 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bGaamyyaiaad6gacqGHfiIXaaa@3A61@ , dielectric ( tanδ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bGaamyyaiaad6gacqaH0oazaaa@3A8D@ ), and piezoelectric losses (  tanθ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamiDaiaadggacaWGUbGaeqiUdehaaa@3BC2@ ), which are respectively described as complex numbers in the elastic compliance, permittivity, and piezoelectric constant in the Equations (1-6) 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 non-prime loss parameters in Equations (1-6) correspond to the intensive and extensive losses, respectively.
s E* = s E (1jtanφ')   (1)             c D* = c D (1+jtanφ)     (2) ε T* = ε T (1jtanδ')    (3)             β S* = β S (1+jtanδ)       (4) d * =d(1jtanθ')        (5)             h * =h(1+jtanθ)          (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGZb WaaWbaaSqabeaacaWGfbGaaiOkaaaakiabg2da9iaadohadaahaaWc beqaaiaadweaaaGccaGGOaGaaGymaiabgkHiTiaadQgaciGG0bGaai yyaiaac6gacqaHgpGAcaGGNaGaaiykaiaabccacaqGGaGaaeiiaiaa bIcacaqGXaGaaeykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaadogadaah aaWcbeqaaiaadseacaGGQaaaaOGaeyypa0Jaam4yamaaCaaaleqaba GaamiraaaakiaacIcacaaIXaGaey4kaSIaamOAaiGacshacaGGHbGa aiOBaiabeA8aQjaacMcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabIcacaqGYaGaaeykaaqaaiabew7aLnaaCaaaleqabaGaamivaiaa cQcaaaGccqGH9aqpcqaH1oqzdaahaaWcbeqaaiaadsfaaaGccaGGOa GaaGymaiabgkHiTiaadQgaciGG0bGaaiyyaiaac6gacqaH0oazcaGG NaGaaiykaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGaaeykaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaeqOSdi2aaWbaaSqabeaacaWGtbGaaiOkaaaaki abg2da9iabek7aInaaCaaaleqabaGaam4uaaaakiaacIcacaaIXaGa ey4kaSIaamOAaiGacshacaGGHbGaaiOBaiabes7aKjaacMcacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG0aGaaeykaaqaaiaa dsgadaahaaWcbeqaaiaacQcaaaGccqGH9aqpcaWGKbGaaiikaiaaig dacqGHsislcaWGQbGaciiDaiaacggacaGGUbGaeqiUdeNaai4jaiaa cMcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae ynaiaabMcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaadIgadaahaaWcbeqaaiaacQ caaaGccqGH9aqpcaWGObGaaiikaiaaigdacqGHRaWkcaWGQbGaciiD aiaacggacaGGUbGaeqiUdeNaaiykaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOn aiaabMcaaaaa@C1CC@
sE is the elastic compliance under constant electric field, and εT is the dielectric constant under constant stress; while cD is the elastic stiffness under constant electric displacement (open circuit), and βS 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 numerically-positive 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 Qm is a sort of figure of merit for heat generation and device efficiency [23,24]; the larger Qm, the better in efficiency. It can be calculated both for the resonance (QA, A-type resonance), and anti-resonance (QB, B-type 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 (fa) and antiresonance (fb) frequencies roughly. And the 3dB bandwidth is the distance between the points having the frequencies 1/ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaai4lamaakaaapaqaa8qacaaIYaaaleqaaaaa@3879@ (or   2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcWaaOaaa8aabaWdbiaaikdaaSqabaaaaa@382F@ ) times the resonance (or anti-resonance) point in the admittance spectrum. The mechanical quality factor is then defined as follows, in which ( f a1 f a2 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamOza8aadaWgaaWcbaWdbiaadggacaaIXaaapaqabaGc peGaeyOeI0IaamOza8aadaWgaaWcbaWdbiaadggacaaIYaaapaqaba GcpeGaaiykaaaa@3E5D@ or ( f b1 f b2 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamOza8aadaWgaaWcbaWdbiaadkgacaaIXaaapaqabaGc peGaeyOeI0IaamOza8aadaWgaaWcbaWdbiaadkgacaaIYaaapaqaba GcpeGaaiykaaaa@3E5F@ attributes to the 3dB down/up full-bandwidths [22]:
Q A = f a f a1 f a2  ( 7 ) Q B = f b f b1 f b2  ( 8 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaadgfapaWaaSbaaSqaa8qacaWGbbaapaqabaGcpeGaeyyp a0ZaaSaaa8aabaWdbiaadAgapaWaaSbaaSqaa8qacaWGHbaapaqaba aakeaapeGaamOza8aadaWgaaWcbaWdbiaadggacaaIXaaapaqabaGc peGaeyOeI0IaamOza8aadaWgaaWcbaWdbiaadggacaaIYaaapaqaba aaaOWdbiaacckadaqadaWdaeaapeGaaG4naaGaayjkaiaawMcaaaqa aiaadgfapaWaaSbaaSqaa8qacaWGcbaapaqabaGcpeGaeyypa0ZaaS aaa8aabaWdbiaadAgapaWaaSbaaSqaa8qacaWGIbaapaqabaaakeaa peGaamOza8aadaWgaaWcbaWdbiaadkgacaaIXaaapaqabaGcpeGaey OeI0IaamOza8aadaWgaaWcbaWdbiaadkgacaaIYaaapaqabaaaaOWd biaacckadaqadaWdaeaapeGaaGioaaGaayjkaiaawMcaaaaaaa@562C@
In the conventional methods of measuring the properties of piezoelectric materials in the k31 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 ( 1 k 31 2 )( 9 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadsea aaGccqGH9aqpcaWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aaba WdbiaadweaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTiaadUgapaWa a0baaSqaa8qacaaIZaGaaGymaaWdaeaapeGaaGOmaaaaaOGaayjkai aawMcaamaabmaapaqaa8qacaaI5aaacaGLOaGaayzkaaaaaa@47C9@ k 31 2 = d 31 2 s 11 E ε 33 T  (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGRbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaaikda aaGccqGH9aqpdaWcaaWdaeaapeGaamiza8aadaqhaaWcbaWdbiaaio dacaaIXaaapaqaa8qacaaIYaaaaaGcpaqaa8qacaWGZbWdamaaDaaa leaapeGaaGymaiaaigdaa8aabaWdbiaadweaaaGccqaH1oqzpaWaa0 baaSqaa8qacaaIZaGaaG4maaWdaeaapeGaamivaaaaaaGccaaMb8Ua aeiiaiaabIcacaqGXaGaaeimaiaabMcaaaa@4B9B@
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., K2 = I, or K = K-1) [21].
[ tanδ tan tanθ ]=K[ tanδ' tan' tanθ' ],K= 1 1 k 31 2 [ 1 k 31 2 2 k 31 2 k 31 2 1 2 k 31 2 1 1 1 k 31 2 ],( 11 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaWdaeaafaqabeWabaaabaWdbiaadshacaWGHbGaamOBaiab es7aKbWdaeaapeGaamiDaiaadggacaWGUbGaeyybIymapaqaa8qaca WG0bGaamyyaiaad6gacqaH4oqCaaaacaGLBbGaayzxaaGaeyypa0Ja am4samaadmaapaqaauaabeqadeaaaeaapeGaamiDaiaadggacaWGUb GaeqiTdqMaai4jaaWdaeaapeGaamiDaiaadggacaWGUbGaeyybIySa ai4jaaWdaeaapeGaamiDaiaadggacaWGUbGaeqiUdeNaai4jaaaaai aawUfacaGLDbaacaGGSaGaam4saiabg2da9maalaaapaqaa8qacaaI Xaaapaqaa8qacaaIXaGaeyOeI0Iaam4Aa8aadaWgaaWcbaWdbiaaio dacaaIXaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaaaakmaadmaa paqaauaabeqadmaaaeaapeGaaGymaaWdaeaapeGaam4Aa8aadaWgaa WcbaWdbiaaiodacaaIXaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaa aaGcpaqaa8qacqGHsislcaaIYaGaam4Aa8aadaWgaaWcbaWdbiaaio dacaaIXaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qa caWGRbWdamaaBaaaleaapeGaaG4maiaaigdaa8aabeaakmaaCaaale qabaWdbiaaikdaaaaak8aabaWdbiaaigdaa8aabaWdbiabgkHiTiaa ikdacaWGRbWdamaaBaaaleaapeGaaG4maiaaigdaa8aabeaakmaaCa aaleqabaWdbiaaikdaaaaak8aabaWdbiaaigdaa8aabaWdbiaaigda a8aabaWdbiabgkHiTiaaigdacqGHsislcaWGRbWdamaaBaaaleaape GaaG4maiaaigdaa8aabeaakmaaCaaaleqabaWdbiaaikdaaaaaaaGc caGLBbGaayzxaaGaaiilamaabmaapaqaa8qacaaIXaGaaGymaaGaay jkaiaawMcaaaaa@850F@
The intensive dielectric loss can be calculated with a capacitance meter at an off-resonance frequency. According to our recent research, the dielectric constant and loss determined at an off-resonance frequency are almost the same as determined at the resonance frequency under small vibration level [29]. Using the conventional IEEE Standard k31 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 QA at the k31 resonance mode (Equation 12), and then the intensive piezoelectric loss can be derived from these two losses and the mechanical quality factor QB at the anti-resonance frequencies as in Equation 13 [21]:
tan 11 ' = 1 Q A,31  ( 12 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bGaamyyaiaad6gacqGHfiIXpaWaa0baaSqaa8qacaaIXaGa aGymaaWdaeaapeGaai4jaaaakiabg2da9maalaaapaqaa8qacaaIXa aapaqaa8qacaWGrbWdamaaBaaaleaapeGaamyqaiaacYcacaaIZaGa aGymaaWdaeqaaaaak8qacaGGGcWaaeWaa8aabaWdbiaaigdacaaIYa aacaGLOaGaayzkaaaaaa@4781@ tan θ 31 ' = 1 2 ( tan δ 33 ' +tan 11 ' )+ 1 4 ( 1 Q A,31 1 Q B,31 )( 1+ ( 1 k 31 k 31 ) 2 Ω b,31 2 ) ( 13 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bGaamyyaiaad6gacqaH4oqCpaWaa0baaSqaa8qacaaIZaGa aGymaaWdaeaapeGaai4jaaaakiabg2da9maalaaapaqaa8qacaaIXa aapaqaa8qacaaIYaaaamaabmaapaqaa8qacaWG0bGaamyyaiaad6ga cqaH0oazpaWaa0baaSqaa8qacaaIZaGaaG4maaWdaeaapeGaai4jaa aakiabgUcaRiaadshacaWGHbGaamOBaiabgwGig=aadaqhaaWcbaWd biaaigdacaaIXaaapaqaa8qacaGGNaaaaaGccaGLOaGaayzkaaGaey 4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaisdaaaWaaeWaa8aa baWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaWGrbWdamaaBaaale aapeGaamyqaiaacYcacaaIZaGaaGymaaWdaeqaaaaak8qacqGHsisl daWcaaWdaeaapeGaaGymaaWdaeaapeGaamyua8aadaWgaaWcbaWdbi aadkeacaGGSaGaaG4maiaaigdaa8aabeaaaaaak8qacaGLOaGaayzk aaWaaeWaa8aabaWdbiaaigdacqGHRaWkdaqadaWdaeaapeWaaSaaa8 aabaWdbiaaigdaa8aabaWdbiaadUgapaWaaSbaaSqaa8qacaaIZaGa aGymaaWdaeqaaaaak8qacqGHsislcaWGRbWdamaaBaaaleaapeGaaG 4maiaaigdaa8aabeaaaOWdbiaawIcacaGLPaaapaWaaWbaaSqabeaa peGaaGOmaaaakiaabM6apaWaa0baaSqaa8qacaWGIbGaaiilaiaaio dacaaIXaaapaqaa8qacaaIYaaaaaGccaGLOaGaayzkaaGaaiiOamaa bmaapaqaa8qacaaIXaGaaG4maaGaayjkaiaawMcaaaaa@779E@
Here Ω b,31 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGPoWdamaaBaaaleaapeGaamOyaiaacYcacaaIZaGaaGymaaWd aeqaaaaa@3AAE@ is the normalized angular frequency of the antiresonance given by Ω b,31 = ω b l 2 v 11 E MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGPoWdamaaBaaaleaapeGaamOyaiaacYcacaaIZaGaaGymaaWd aeqaaOWdbiabg2da9maalaaapaqaa8qacqaHjpWDpaWaaSbaaSqaa8 qacaWGIbaapaqabaGcpeGaamiBaaWdaeaapeGaaGOmaiaadAhapaWa a0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaamyraaaaaaaaaa@4497@ . Then the extensive (non-prime)

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 k31 type piezoelectric plates: resonance, antiresonance modes of the electroded plate, and resonance mode of the non-electroded plate under electric field (top) and mechanical excitation (bottom). In the conventional full-electrode 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 open-circuit 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 D-constant parameters, such as s 11 D MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadsea aaaaaa@39B8@ and sound velocity vD.
Figure 1: Equivalent electrode configuration and vibration modes of k31 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 k31 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 non-electrode 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 k31 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) full-electrode (FE), (b) partial electrode – short-circuit, (c) partial electrode – open circuit, and (d) nonelectrode (NE) samples.
The partial electrode samples with electrode load (short- and open-circuit) were also prepared and measured as reference data. The schematic view of (a) full-electrode (FE), (b) partial electrode – short-circuit, (c) partial electrode – open circuit, and (d) non-electrode (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 (non-electrode) 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 k31 mode sample. PE-Short attributes to the pure E-constant 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 anti-resonance frequency occurs at a much lower frequency. This is because the electrode area for actuator part in the PE-Short sample covers only 10% of the total length, and the load parts are still short circuit during the antiresonance frequency. Therefore, its anti-resonance frequency happens between the resonance and anti-resonance frequencies of FE sample. On the contrary, the anti-resonance frequency of the PE-Open 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 anti-resonance mode. Since we consider the k31 mode and also there is no electric field on the surface along the length direction (i.e., X-axis), the mechanical resonance of the PE-Open happens between the pure E (i.e., PE-Short) 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 D-constant 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 (D-constant) is lower than the intensive one (E-constant), which means having a larger stiffness, higher resonance and anti-resonance 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 non-electrode 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 non-electrode 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, 6-terminal 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 k31 mode piezoelectric materials are as follows in a linear first-approximation [28],
x 1 = s 11 E T 1 + d 31 E 3  ( 14 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iaa dohapaWaa0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaamyraaaaki aadsfapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIaamiz a8aadaWgaaWcbaWdbiaaiodacaaIXaaapaqabaGcpeGaamyra8aada WgaaWcbaWdbiaaiodaa8aabeaak8qacaGGGcWaaeWaa8aabaWdbiaa igdacaaI0aaacaGLOaGaayzkaaaaaa@48F4@ D 3 = ε 0 ε 33 T E 3 + d 31 T 1  ( 15 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGebWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabg2da9iab ew7aL9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqaH1oqzpaWaa0 baaSqaa8qacaaIZaGaaG4maaWdaeaapeGaamivaaaakiaadweapaWa aSbaaSqaa8qacaaIZaaapaqabaGcpeGaey4kaSIaamiza8aadaWgaa WcbaWdbiaaiodacaaIXaaapaqabaGcpeGaamiva8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacaGGGcWaaeWaa8aabaWdbiaaigdacaaI1a aacaGLOaGaayzkaaaaaa@4C5A@
The x 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3828@ and T 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3828@ are strain and stress in direction 1 (longitude L), perpendicular to the poling direction. E 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGfbWdamaaBaaaleaapeGaaG4maaWdaeqaaaaa@37F7@ And D 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGfbWdamaaBaaaleaapeGaaG4maaWdaeqaaaaa@37F7@ is the electric field and dielectric displacement in the polling direction 3 (thickness b). The   s 11 E MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaam4Ca8aadaqhaaWcbaWdbiaaigdacaaIXaaapaqaa8qa caWGfbaaaaaa@3ADD@ is the elastic compliance for both stress and strain in direction 1 under the constant applied electric field. The ε 0 ε 33 T MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH1oqzpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeqyTdu2d amaaDaaaleaapeGaaG4maiaaiodaa8aabaWdbiaadsfaaaaaaa@3D50@ 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, k31 mode), the dynamic equation for extensional vibration is given as in Equation 16, in which u MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1baaaa@3710@ and x MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4baaaa@3713@ are the displacement and position in direction 1, and ρ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHbpGCaaa@37D6@ is the material’s density[17],
ρ 2 u t 2 =F= T 1 x  ( 16 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHbpGCdaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaa ikdaaaGccaWG1baapaqaa8qacqGHciITcaWG0bWdamaaCaaaleqaba WdbiaaikdaaaaaaOGaeyypa0JaamOraiabg2da9maalaaapaqaa8qa cqGHciITcaWGubWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbi abgkGi2kaadIhaaaGaaiiOamaabmaapaqaa8qacaaIXaGaaGOnaaGa ayjkaiaawMcaaaaa@4C25@
By introducing the constitutive equations in the above equation, and allowing x 1 = u x MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacqGHciITcaWG1baapaqaa8qacqGHciITcaWG4baaaa aa@3E59@ , two different cases of boundary conditions, electrode and non-electrode 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 ( E 3 x )=0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2kaadweapaWaaSba aSqaa8qacaaIZaaapaqabaaakeaapeGaeyOaIyRaamiEaaaaaiaawI cacaGLPaaacqGH9aqpcaaIWaaaaa@3F71@ for the electrode part due to the equal potential of each electrode, the following harmonic vibration equation is derived,
Figure 4: Non-electrode sample with ‘a’ fraction electrode at the center.
ω 2 ρ s 11 E u( x )= 2 u( x ) x 2  ( 17 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHsislcqaHjpWDpaWaaWbaaSqabeaapeGaaGOmaaaakiabeg8a YjaadohapaWaa0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaamyraa aakiaadwhadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da 9maalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaaki aadwhadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGa eyOaIyRaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiaacckada qadaWdaeaapeGaaGymaiaaiEdaaiaawIcacaGLPaaaaaa@5216@
By substituting the general standing-wave form of the displacement equation as u=u( x ) e jωt MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1bGaeyypa0JaamyDamaabmaapaqaa8qacaWG4baacaGLOaGa ayzkaaGaamyza8aadaahaaWcbeqaa8qacaWGQbGaeqyYdCNaamiDaa aaaaa@40A0@ , the following solution for the displacement of the electrode part can be obtained, since u e ( x ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1bWdamaaBaaaleaapeGaamyzaaWdaeqaaOWdbmaabmaapaqa a8qacaWG4baacaGLOaGaayzkaaaaaa@3B13@ should be equal to zero at x = 0 (center of gravity):
   u e ( x )= A e  sin( ω v E x ) ( 18 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaaiiOaiaadwhapaWaaSbaaSqaa8qacaWGLbaapaqabaGc peWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGbb WdamaaBaaaleaapeGaamyzaaWdaeqaaOWdbiaacckacaWGZbGaamyA aiaad6gadaqadaWdaeaapeWaaSaaa8aabaWdbiabeM8a3bWdaeaape GaamODa8aadaahaaWcbeqaa8qacaWGfbaaaaaakiaadIhaaiaawIca caGLPaaacaGGGcWaaeWaa8aabaWdbiaaigdacaaI4aaacaGLOaGaay zkaaaaaa@4FA6@
Secondly, by allowing ( D 3 x )=0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2kaadseapaWaaSba aSqaa8qacaaIZaaapaqabaaakeaapeGaeyOaIyRaamiEaaaaaiaawI cacaGLPaaacqGH9aqpcaaIWaaaaa@3F70@ for the non-electrode area, the harmonic vibration equation and final solution can be derived as (for | x |>aL/2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaabdaWdaeaapeGaamiEaaGaay5bSlaawIa7aiabg6da+iaadgga caWGmbGaai4laiaaikdaaaa@3E82@ ), in which the displacement equation with positive sign for Bn correspond to the positive position x.
ω 2 ρ s 11  E (1 k 31 2 )u( x )= 2 u( x ) x 2   ( 19 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHsislcqaHjpWDpaWaaWbaaSqabeaapeGaaGOmaaaakiabeg8a YjaadohapaWaa0baaSqaa8qacaaIXaGaaGymaiaacckaa8aabaWdbi aadweaaaGccaqGOaGaaeymaiabgkHiTiaadUgapaWaa0baaSqaa8qa caaIZaGaaGymaaWdaeaapeGaaGOmaaaakiaacMcacaWG1bWaaeWaa8 aabaWdbiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaapeGa eyOaIy7damaaCaaaleqabaWdbiaaikdaaaGccaWG1bWaaeWaa8aaba WdbiaadIhaaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWa aWbaaSqabeaapeGaaGOmaaaaaaGccaGGGcGaaeiiamaabmaapaqaa8 qacaaIXaGaaGyoaaGaayjkaiaawMcaaaaa@5A71@ u n ( x )= A n  sin( ω v D x )± B n  cos( ω v D x )  ( 20 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1bWdamaaBaaaleaapeGaamOBaaWdaeqaaOWdbmaabmaapaqa a8qacaWG4baacaGLOaGaayzkaaGaeyypa0Jaamyqa8aadaWgaaWcba Wdbiaad6gaa8aabeaak8qacaGGGcGaam4CaiaadMgacaWGUbWaaeWa a8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadAhapaWaaW baaSqabeaapeGaamiraaaaaaGccaWG4baacaGLOaGaayzkaaGaeyyS aeRaamOqa8aadaWgaaWcbaWdbiaad6gaa8aabeaak8qacaGGGcGaam 4yaiaad+gacaWGZbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHjpWD a8aabaWdbiaadAhapaWaaWbaaSqabeaapeGaamiraaaaaaGccaWG4b aacaGLOaGaayzkaaGaaiiOaiaabccadaqadaWdaeaapeGaaGOmaiaa icdaaiaawIcacaGLPaaaaaa@5CF9@
The   v E = 1 ρ s 11 E MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamODa8aadaahaaWcbeqaa8qacaWGfbaaaOGaeyypa0Za aSaaa8aabaWdbiaaigdaa8aabaWdbmaakaaapaqaa8qacqaHbpGCca WGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaaaa beaaaaaaaa@40F6@ , and v D = 1 ρ s 11 D MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWdamaaCaaaleqabaWdbiaadseaaaGccqGH9aqpdaWcaaWd aeaapeGaaGymaaWdaeaapeWaaOaaa8aabaWdbiabeg8aYjaadohapa Waa0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaamiraaaaaeqaaaaa aaa@3FD0@ 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 ( x )= u e x = A e ω v E cos( ω v E x ) ( 21 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaiaadwgaa8aabeaak8qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8 qacqGHciITcaWG1bWdamaaBaaaleaapeGaamyzaaWdaeqaaaGcbaWd biabgkGi2kaadIhaaaGaeyypa0Jaamyqa8aadaWgaaWcbaWdbiaadw gaa8aabeaak8qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWd amaaCaaaleqabaWdbiaadweaaaaaaOGaam4yaiaad+gacaWGZbWaae Waa8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadAhapaWa aWbaaSqabeaapeGaamyraaaaaaGccaWG4baacaGLOaGaayzkaaGaae iOamaabmaapaqaa8qacaaIYaGaaGymaaGaayjkaiaawMcaaaaa@5878@ x 1n ( x )= u n x = A n ω v D cos( ω v D x ) B n ω v D sin( ω v D x ) ( 22 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaiaad6gaa8aabeaak8qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8 qacqGHciITcaWG1bWdamaaBaaaleaapeGaamOBaaWdaeqaaaGcbaWd biabgkGi2kaadIhaaaGaeyypa0Jaamyqa8aadaWgaaWcbaWdbiaad6 gaa8aabeaak8qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWd amaaCaaaleqabaWdbiaadseaaaaaaOGaam4yaiaad+gacaWGZbWaae Waa8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadAhapaWa aWbaaSqabeaapeGaamiraaaaaaGccaWG4baacaGLOaGaayzkaaGaey OeI0IaamOqa8aadaWgaaWcbaWdbiaad6gaa8aabeaak8qadaWcaaWd aeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbiaads eaaaaaaOGaam4CaiaadMgacaWGUbWaaeWaa8aabaWdbmaalaaapaqa a8qacqaHjpWDa8aabaWdbiaadAhapaWaaWbaaSqabeaapeGaamiraa aaaaGccaWG4baacaGLOaGaayzkaaGaaiiOamaabmaapaqaa8qacaaI YaGaaGOmaaGaayjkaiaawMcaaaaa@6996@ T 1e ( x )= x 1e s 11 E ( d 31 s 11 E ) E 3e  ( 23 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaaGymaiaadwgaa8aabeaak8qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaiaadwgaa8aabeaaaOqaa8qa caWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaa aaaOGaeyOeI0YaaeWaa8aabaWdbmaalaaapaqaa8qacaWGKbWdamaa BaaaleaapeGaaG4maiaaigdaa8aabeaaaOqaa8qacaWGZbWdamaaDa aaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaaaaaaGccaGLOaGa ayzkaaGaamyra8aadaWgaaWcbaWdbiaaiodacaWGLbaapaqabaGcpe GaaiiOamaabmaapaqaa8qacaaIYaGaaG4maaGaayjkaiaawMcaaaaa @5415@ T 1n ( x )= x 1n s 11 E ( d 31 s 11 E ) E 3n ( x ) ( 24 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaaGymaiaad6gaa8aabeaak8qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaiaad6gaa8aabeaaaOqaa8qa caWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaa aaaOGaeyOeI0YaaeWaa8aabaWdbmaalaaapaqaa8qacaWGKbWdamaa BaaaleaapeGaaG4maiaaigdaa8aabeaaaOqaa8qacaWGZbWdamaaDa aaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaaaaaaGccaGLOaGa ayzkaaGaamyra8aadaWgaaWcbaWdbiaaiodacaWGUbaapaqabaGcpe WaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaGGGcWaaeWaa8aa baWdbiaaikdacaaI0aaacaGLOaGaayzkaaaaaa@56D6@
E 3e MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGfbWdamaaBaaaleaapeGaaG4maiaabwgaa8aabeaaaaa@38DD@ , 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. D 3n MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGebWdamaaBaaaleaapeGaaG4maiaab6gaa8aabeaaaaa@38E5@ is also a constant value, since the non-electrode part cannot provide free charge, therefore D=ρ=0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHhis0caqGebGaeyypa0JaaeyWdiabg2da9iaaicdaaaa@3C70@ . Furthermore, from D= ε 0 E+P MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGebGaeyypa0JaaeyTd8aadaWgaaWcbaWdbiaaicdaa8aabeaa k8qacaqGfbGaey4kaSIaaeiuaaaa@3CC9@ and the fact that for no electrode condition depolarization field E is proportion to the negative polarization (-P), therefore, D 3n =0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGebWdamaaBaaaleaapeGaaG4maiaab6gaa8aabeaak8qacqGH 9aqpcaaIWaaaaa@3ABF@ , and by giving E 3n ( x )= d 31 ε 0 ε 33 X T 1n ( x )+ D 3n ε 0 ε 33 X == d 31 ε 0 ε 33 X T 1n ( x ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGfbWdamaaBaaaleaapeGaaG4maiaab6gaa8aabeaak8qadaqa daWdaeaapeGaaeiEaaGaayjkaiaawMcaaiabg2da9iabgkHiTmaala aapaqaa8qacaqGKbWdamaaBaaaleaapeGaaG4maiaaigdaa8aabeaa aOqaa8qacaqG1oWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaabw 7apaWaa0baaSqaa8qacaaIZaGaaG4maaWdaeaapeGaaeiwaaaaaaGc caqGubWdamaaBaaaleaapeGaaGymaiaab6gaa8aabeaak8qadaqada WdaeaapeGaaeiEaaGaayjkaiaawMcaaiabgUcaRmaalaaapaqaa8qa caqGebWdamaaBaaaleaapeGaaG4maiaab6gaa8aabeaaaOqaa8qaca qG1oWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaabw7apaWaa0ba aSqaa8qacaaIZaGaaG4maaWdaeaapeGaaeiwaaaaaaGccqGH9aqpcq GH9aqpcqGHsisldaWcaaWdaeaapeGaaeiza8aadaWgaaWcbaWdbiaa iodacaaIXaaapaqabaaakeaapeGaaeyTd8aadaWgaaWcbaWdbiaaic daa8aabeaak8qacaqG1oWdamaaDaaaleaapeGaaG4maiaaiodaa8aa baWdbiaabIfaaaaaaOGaaeiva8aadaWgaaWcbaWdbiaaigdacaqGUb aapaqabaGcpeWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaaaaa@690B@ we have,
T 1n ( x )= x 1n s 11 E ( 1 k 31 2 ) = x 1n s 11 D  ( 25 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaaGymaiaad6gaa8aabeaak8qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8 qacaWG4bWdamaaBaaaleaapeGaaGymaiaad6gaa8aabeaaaOqaa8qa caWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaa GcdaqadaWdaeaapeGaaGymaiabgkHiTiaadUgapaWaa0baaSqaa8qa caaIZaGaaGymaaWdaeaapeGaaGOmaaaaaOGaayjkaiaawMcaaaaacq GH9aqpdaWcaaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaaigdacaWG UbaapaqabaaakeaapeGaam4Ca8aadaqhaaWcbaWdbiaaigdacaaIXa aapaqaa8qacaWGebaaaaaakiaacckadaqadaWdaeaapeGaaGOmaiaa iwdaaiaawIcacaGLPaaaaaa@56E7@
There are three unknown parameters in the above equations, namely: A e MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbWdamaaBaaaleaapeGaaeyzaaWdaeqaaaaa@381C@ , A n MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbWdamaaBaaaleaapeGaaeOBaaWdaeqaaaaa@3825@ , and B n MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaaeOBaaWdaeqaaaaa@3828@ , which can be solved by introducing three boundary conditions as follows:

1. Continuation of displacement at x=± aL 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0JaeyySae7aaSaaa8aabaWdbiaadggacaWGmbaa paqaa8qacaaIYaaaaaaa@3CC8@ (the interface of electrode and non-electrode parts). The strain can exhibit discontinuity at x=± aL 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0JaeyySae7aaSaaa8aabaWdbiaadggacaWGmbaa paqaa8qacaaIYaaaaaaa@3CC8@

2. Continuation of stress at x=± aL 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0JaeyySae7aaSaaa8aabaWdbiaadggacaWGmbaa paqaa8qacaaIYaaaaaaa@3CC8@

3. Stress being zero at the plate end x=± L 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0JaeyySae7aaSaaa8aabaWdbiaadggacaWGmbaa paqaa8qacaaIYaaaaaaa@3CC8@

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 (27-29),
[ sin( ω v E aL 2 ) sin( ω v D aL 2 ) cos( ω v D x aL 2 ) ω v E cos( ω v E aL 2 ) ω v D 1 1 k 31 2 cos( ω v D aL 2 ) ω v D 1 1 k 31 2 sin( ω v D aL 2 ) 0 ω v D 1 1 k 31 2 cos( ω v D L 2 ) ω v D 1 1 k 31 2 sin( ω v D L 2 ) ][ A e A n B n ]= [ 0 d 31 E 3e 0 ] (26) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaWdaeaafaqabeWadaaabaWdbiGacohacaGGPbGaaiOBamaa bmaapaqaa8qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdam aaCaaaleqabaWdbiaadweaaaaaaOWaaSaaa8aabaWdbiaadggacaWG mbaapaqaa8qacaaIYaaaaaGaayjkaiaawMcaaaWdaeaapeGaeyOeI0 Iaci4CaiaacMgacaGGUbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaH jpWDa8aabaWdbiaadAhapaWaaWbaaSqabeaapeGaamiraaaaaaGcda WcaaWdaeaapeGaamyyaiaadYeaa8aabaWdbiaaikdaaaaacaGLOaGa ayzkaaaapaqaa8qacqGHsislciGGJbGaai4BaiaacohadaqadaWdae 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aalaaapaqaa8qacqaHjpWDa8aabaWdbiaadAhapaWaaWbaaSqabeaa peGaamiraaaaaaGcdaWcaaWdaeaapeGaamyyaiaadYeaa8aabaWdbi aaikdaaaaacaGLOaGaayzkaaaapaqaa8qacaaIWaaapaqaa8qadaWc aaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbi aadseaaaaaaOWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaigdacqGH sislcaWGRbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaaik daaaaaaOGaci4yaiaac+gacaGGZbWaaeWaa8aabaWdbmaalaaapaqa a8qacqaHjpWDa8aabaWdbiaadAhapaWaaWbaaSqabeaapeGaamiraa aaaaGcdaWcaaWdaeaapeGaamitaaWdaeaapeGaaGOmaaaaaiaawIca caGLPaaaa8aabaWdbiabgkHiTmaalaaapaqaa8qacqaHjpWDa8aaba WdbiaadAhapaWaaWbaaSqabeaapeGaamiraaaaaaGcdaWcaaWdaeaa peGaaGymaaWdaeaapeGaaGymaiabgkHiTiaadUgapaWaa0baaSqaa8 qacaaIZaGaaGymaaWdaeaapeGaaGOmaaaaaaGcciGGZbGaaiyAaiaa c6gadaqadaWdaeaapeWaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaam ODa8aadaahaaWcbeqaa8qacaWGebaaaaaakmaalaaapaqaa8qacaWG mbaapaqaa8qacaaIYaaaaaGaayjkaiaawMcaaaaaaiaawUfacaGLDb aadaWadaWdaeaafaqabeWabaaabaWdbiaadgeapaWaaSbaaSqaa8qa caWGLbaapaqabaaakeaapeGaamyqa8aadaWgaaWcbaWdbiaad6gaa8 aabeaaaOqaa8qacaWGcbWdamaaBaaaleaapeGaamOBaaWdaeqaaaaa aOWdbiaawUfacaGLDbaacqGH9aqpcaGGGcWaamWaa8aabaqbaeqabm qaaaqaa8qacaaIWaaapaqaa8qacaWGKbWdamaaBaaaleaapeGaaG4m aiaaigdaa8aabeaak8qacaWGfbWdamaaBaaaleaapeGaaG4maiaadw gaa8aabeaaaOqaa8qacaaIWaaaaaGaay5waiaaw2faaiaabccacaGG OaGaaGOmaiaaiAdacaGGPaaaaa@DE53@ A n = d 31 E 3e  sin( ω v E aL 2 ) sin( ω v D L 2 ) ω v E  cos( ω v D ( 1a )L 2 ) cos( ω v E aL 2 ) ω v D ( 1 k 31 2 )  sin( ω v D ( 1a )L 2 ) sin( ω v E aL 2 ) ( 27 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbWdamaaBaaaleaapeGaamOBaaWdaeqaaOWdbiabg2da9maa laaabaGaamiza8aadaWgaaWcbaWdbiaaiodacaaIXaaapaqabaGcpe Gaamyra8aadaWgaaWcbaWdbiaaiodacaWGLbaapaqabaGcpeGaaiiO aiaadohacaWGPbGaamOBamaabmaapaqaa8qadaWcaaWdaeaapeGaeq yYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbiaadweaaaaaaOWa aSaaa8aabaWdbiaadggacaWGmbaapaqaa8qacaaIYaaaaaGaayjkai aawMcaaiaacckacaWGZbGaamyAaiaad6gadaqadaWdaeaapeWaaSaa a8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWcbeqaa8qaca WGebaaaaaakmaalaaapaqaa8qacaWGmbaapaqaa8qacaaIYaaaaaGa ayjkaiaawMcaaaqaamaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadA hapaWaaWbaaSqabeaapeGaamyraaaaaaGccaGGGcGaam4yaiaad+ga caWGZbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aabaWdbi aadAhapaWaaWbaaSqabeaapeGaamiraaaaaaGcdaWcaaWdaeaapeWa aeWaa8aabaWdbiaaigdacqGHsislcaWGHbaacaGLOaGaayzkaaGaam itaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacaGGGcGaam4yaiaa d+gacaWGZbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aaba WdbiaadAhapaWaaWbaaSqabeaapeGaamyraaaaaaGcdaWcaaWdaeaa peGaamyyaiaadYeaa8aabaWdbiaaikdaaaaacaGLOaGaayzkaaGaey OeI0YaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWc beqaa8qacaWGebaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcaWGRb WdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaaikdaaaaakiaa wIcacaGLPaaaaaGaaiiOaiaadohacaWGPbGaamOBamaabmaapaqaa8 qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqa baWdbiaadseaaaaaaOWaaSaaa8aabaWdbmaabmaapaqaa8qacaaIXa GaeyOeI0IaamyyaaGaayjkaiaawMcaaiaadYeaa8aabaWdbiaaikda aaaacaGLOaGaayzkaaGaaiiOaiaadohacaWGPbGaamOBamaabmaapa qaa8qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaa leqabaWdbiaadweaaaaaaOWaaSaaa8aabaWdbiaadggacaWGmbaapa qaa8qacaaIYaaaaaGaayjkaiaawMcaaaaadaqadaWdaeaapeGaaGOm aiaaiEdaaiaawIcacaGLPaaaaaa@A3CE@ B n = d 31 E 3e  sin( ω v E aL 2 ) cos( ω v D L 2 ) ω v E cos( ω v D ( 1a )L 2 ) cos( ω v E aL 2 ) ω v D ( 1 k 31 2 ) sin( ω v D ( 1a )L 2 ) sin( ω v E aL 2 ) ( 28 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaamOBaaWdaeqaaOWdbiabg2da9maa laaabaGaamiza8aadaWgaaWcbaWdbiaaiodacaaIXaaapaqabaGcpe Gaamyra8aadaWgaaWcbaWdbiaaiodacaWGLbaapaqabaGcpeGaaiiO aiaadohacaWGPbGaamOBamaabmaapaqaa8qadaWcaaWdaeaapeGaeq yYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbiaadweaaaaaaOWa aSaaa8aabaWdbiaadggacaWGmbaapaqaa8qacaaIYaaaaaGaayjkai aawMcaaiaacckacaWGJbGaam4BaiaadohadaqadaWdaeaapeWaaSaa a8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWcbeqaa8qaca WGebaaaaaakmaalaaapaqaa8qacaWGmbaapaqaa8qacaaIYaaaaaGa ayjkaiaawMcaaaqaamaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadA hapaWaaWbaaSqabeaapeGaamyraaaaaaGccaWGJbGaam4Baiaadoha daqadaWdaeaapeWaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaamODa8 aadaahaaWcbeqaa8qacaWGebaaaaaakmaalaaapaqaa8qadaqadaWd aeaapeGaaGymaiabgkHiTiaadggaaiaawIcacaGLPaaacaWGmbaapa qaa8qacaaIYaaaaaGaayjkaiaawMcaaiaacckacaWGJbGaam4Baiaa dohadaqadaWdaeaapeWaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaam ODa8aadaahaaWcbeqaa8qacaWGfbaaaaaakmaalaaapaqaa8qacaWG HbGaamitaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacqGHsislda WcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWd biaadseaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTiaadUgapaWaa0 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aadaqhaaWcbaWdbiaaiodacaaIXaaapaqaa8qacaaIYaaaaaGccaGL OaGaayzkaaaaaiaadohacaWGPbGaamOBamaabmaapaqaa8qadaWcaa WdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbiaa dweaaaaaaOWaaSaaa8aabaWdbiaadggacaWGmbaapaqaa8qacaaIYa aaaaGaayjkaiaawMcaaiaacckacaWGZbGaamyAaiaad6gadaqadaWd aeaapeWaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaa Wcbeqaa8qacaWGebaaaaaakmaalaaapaqaa8qadaqadaWdaeaapeGa aGymaiabgkHiTiaadggaaiaawIcacaGLPaaacaWGmbaapaqaa8qaca aIYaaaaaGaayjkaiaawMcaaaaadaqadaWdaeaapeGaaGOmaiaaiMda aiaawIcacaGLPaaaaaa@9C80@
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,
I=jω D 3e dxdy=2jωW{ 0 aL 2 D 3e  dx}= 2jωW{ 0 aL 2 ( d 31 ( 1 s 11 E A e ω v E cos( ω v E x )( d 31 s 11 E ) E 3e )+ ε 0 ε 33 T E 3e ) dx}= 2jωW( d 31 A e s 11 E sin( ω v E aL 2 )+ ε 0 ε 33 T ( 1 k 31 2 ) E 3e aL 2 )( 30 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaadMeacqGH9aqpcaWGQbGaeqyYdC3damaavacabeWcbeqa aiaaygW7a0qaamXvP5wqSX2qVrwzqf2zLnharyGqHrxyUDgaiuaape Gaa8hlIaaakiaadseapaWaaSbaaSqaa8qacaaIZaGaamyzaaWdaeqa aOWdbiaadsgacaWG4bGaamizaiaadMhacqGH9aqpcaaIYaGaamOAai abeM8a3jaadEfacaGG7bWaaybCaeqal8aabaWdbiaaicdaa8aabaWd bmaalaaapaqaa8qacaWGHbGaamitaaWdaeaapeGaaGOmaaaaa0Wdae aapeGaey4kIipaaOGaamira8aadaWgaaWcbaWdbiaaiodacaWGLbaa paqabaGcpeGaaiiOaiaadsgacaWG4bGaaiyFaiabg2da9aqaaiaaik dacaWGQbGaeqyYdCNaam4vaiaacUhadaGfWbqabSWdaeaapeGaaGim aaWdaeaapeWaaSaaa8aabaWdbiaadggacaWGmbaapaqaa8qacaaIYa aaaaqdpaqaa8qacqGHRiI8aaGccaGGOaGaamiza8aadaWgaaWcbaWd biaaiodacaaIXaaapaqabaGcpeGaaiikamaalaaapaqaa8qacaaIXa aapaqaa8qacaWGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWd biaadweaaaaaaOGaamyqa8aadaWgaaWcbaWdbiaadwgaa8aabeaak8 qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqa baWdbiaadweaaaaaaOGaam4yaiaad+gacaWGZbWaaeWaa8aabaWdbm aalaaapaqaa8qacqaHjpWDa8aabaWdbiaadAhapaWaaWbaaSqabeaa peGaamyraaaaaaGccaWG4baacaGLOaGaayzkaaGaeyOeI0YaaeWaa8 aabaWdbmaalaaapaqaa8qacaWGKbWdamaaBaaaleaapeGaaG4maiaa igdaa8aabeaaaOqaa8qacaWGZbWdamaaDaaaleaapeGaaGymaiaaig daa8aabaWdbiaadweaaaaaaaGccaGLOaGaayzkaaGaamyra8aadaWg aaWcbaWdbiaaiodacaWGLbaapaqabaGcpeGaaiykaiabgUcaRiabew 7aL9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqaH1oqzpaWaa0ba aSqaa8qacaaIZaGaaG4maaWdaeaapeGaamivaaaakiaadweapaWaaS baaSqaa8qacaaIZaGaamyzaaWdaeqaaOWdbiaacMcacaGGGcGaamiz aiaadIhacaGG9bGaeyypa0dabaGaaGOmaiaadQgacqaHjpWDcaWGxb WaaeWaa8aabaWdbmaalaaapaqaa8qacaWGKbWdamaaBaaaleaapeGa aG4maiaaigdaa8aabeaak8qacaWGbbWdamaaBaaaleaapeGaamyzaa WdaeqaaaGcbaWdbiaadohapaWaa0baaSqaa8qacaaIXaGaaGymaaWd aeaapeGaamyraaaaaaGccaWGZbGaamyAaiaad6gadaqadaWdaeaape WaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWcbeqa a8qacaWGfbaaaaaakmaalaaapaqaa8qacaWGHbGaamitaaWdaeaape GaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkcqaH1oqzpaWaaSbaaSqa a8qacaaIWaaapaqabaGcpeGaeqyTdu2damaaDaaaleaapeGaaG4mai aaiodaa8aabaWdbiaadsfaaaGcdaqadaWdaeaapeGaaGymaiabgkHi TiaadUgapaWaa0baaSqaa8qacaaIZaGaaGymaaWdaeaapeGaaGOmaa aaaOGaayjkaiaawMcaaiaadweapaWaaSbaaSqaa8qacaaIZaGaamyz aaWdaeqaaOWdbmaalaaapaqaa8qacaWGHbGaamitaaWdaeaapeGaaG OmaaaaaiaawIcacaGLPaaadaqadaWdaeaapeGaaG4maiaaicdaaiaa wIcacaGLPaaaaaaa@D55E@
Then, by defining the 3 e   E 3e = V b MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamyra8aadaWgaaWcbaWdbiaaiodacaWGLbaapaqabaGc peGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadAfaa8aabaWdbiaadk gaaaaaaa@3E22@ , and Y= I V MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadMeaa8aabaWd biaadAfaaaaaaa@3ADE@ considering the losses and complex values as in Equation (1-6), the admittance equation is deduced:
Y=2jW d 31 * b s 11 E * d 31 * cos( ω v D * ( 1a )L 2 ) 1 v E * cos( ω v E * aL 2 ) cos( ω v D * ( 1a )L 2 )  1 v D * ( 1 k 31 * 2 ) sin( ω v E * aL 2 )sin( ω v D * ( 1a )L 2 ) sin( ω v E * aL 2 )+jω ε 0 ε 33 T * ( 1 k 31 * 2 ) aLW b  (31) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaadMfacqGH9aqpcaaIYaGaamOAaiaadEfadaWcaaWdaeaa peGaamiza8aadaqhaaWcbaWdbiaaiodacaaIXaaapaqaa8qacaGGQa aaaaGcpaqaa8qacaWGIbGaam4Ca8aadaqhaaWcbaWdbiaaigdacaaI Xaaapaqaa8qacaWGfbaaaOWdamaaCaaaleqabaWdbiaacQcaaaaaaO WaaSaaaeaacaWGKbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWd biaacQcaaaGccaWGJbGaam4BaiaadohadaqadaWdaeaapeWaaSaaa8 aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWcbeqaa8qacaWG ebaaaOWdamaaCaaaleqabaWdbiaacQcaaaaaaOWaaSaaa8aabaWdbm aabmaapaqaa8qacaaIXaGaeyOeI0IaamyyaaGaayjkaiaawMcaaiaa dYeaa8aabaWdbiaaikdaaaaacaGLOaGaayzkaaaabaWaaSaaa8aaba Wdbiaaigdaa8aabaWdbiaadAhapaWaaWbaaSqabeaapeGaamyraaaa k8aadaahaaWcbeqaa8qacaGGQaaaaaaakiaadogacaWGVbGaam4Cam aabmaapaqaa8qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWd amaaCaaaleqabaWdbiaadweaaaGcpaWaaWbaaSqabeaapeGaaiOkaa aaaaGcdaWcaaWdaeaapeGaamyyaiaadYeaa8aabaWdbiaaikdaaaaa caGLOaGaayzkaaGaaiiOaiaadogacaWGVbGaam4Camaabmaapaqaa8 qadaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWG2bWdamaaCaaaleqa baWdbiaadseaaaGcpaWaaWbaaSqabeaapeGaaiOkaaaaaaGcdaWcaa WdaeaapeWaaeWaa8aabaWdbiaaigdacqGHsislcaWGHbaacaGLOaGa ayzkaaGaamitaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacqGHsi slcaGGGcWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadAhapaWaaWba aSqabeaapeGaamiraaaak8aadaahaaWcbeqaa8qacaGGQaaaaOWaae Waa8aabaWdbiaaigdacqGHsislcaWGRbWdamaaDaaaleaapeGaaG4m aiaaigdaa8aabaWdbiaacQcaaaGcpaWaaWbaaSqabeaapeGaaGOmaa aaaOGaayjkaiaawMcaaaaacaWGZbGaamyAaiaad6gadaqadaWdaeaa peWaaSaaa8aabaWdbiabeM8a3bWdaeaapeGaamODa8aadaahaaWcbe qaa8qacaWGfbaaaOWdamaaCaaaleqabaWdbiaacQcaaaaaaOWaaSaa a8aabaWdbiaadggacaWGmbaapaqaa8qacaaIYaaaaaGaayjkaiaawM caaiaadohacaWGPbGaamOBamaabmaapaqaa8qadaWcaaWdaeaapeGa eqyYdChapaqaa8qacaWG2bWdamaaCaaaleqabaWdbiaadseaaaGcpa WaaWbaaSqabeaapeGaaiOkaaaaaaGcdaWcaaWdaeaapeWaaeWaa8aa baWdbiaaigdacqGHsislcaWGHbaacaGLOaGaayzkaaGaamitaaWdae aapeGaaGOmaaaaaiaawIcacaGLPaaaaaaabaGaam4CaiaadMgacaWG UbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHjpWDa8aabaWdbiaadA hapaWaaWbaaSqabeaapeGaamyraaaak8aadaahaaWcbeqaa8qacaGG Qaaaaaaakmaalaaapaqaa8qacaWGHbGaamitaaWdaeaapeGaaGOmaa aaaiaawIcacaGLPaaacqGHRaWkcaWGQbGaeqyYdCNaeqyTdu2damaa BaaaleaapeGaaGimaaWdaeqaaOWdbiabew7aL9aadaqhaaWcbaWdbi aaiodacaaIZaaapaqaa8qacaWGubaaaOWdamaaCaaaleqabaWdbiaa cQcaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTiaadUgapaWaa0baaS qaa8qacaaIZaGaaGymaaWdaeaapeGaaiOkaaaak8aadaahaaWcbeqa a8qacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaa8aabaWdbiaadggaca WGmbGaam4vaaWdaeaapeGaamOyaaaacaqGGcGaaeikaiaabodacaqG XaGaaeykaaaaaa@CB3B@
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 * MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWdamaaCaaaleqabaWdbiaadweaaaGcpaWaaWbaaSqabeaa peGaaiOkaaaaaaa@392B@ , v D * MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWdamaaCaaaleqabaWdbiaadseaaaGcpaWaaWbaaSqabeaa peGaaiOkaaaaaaa@392A@ , and k31* were calculated by putting these complex parameters in their corresponding equations, i.e.   v E = 1 ρ s 11 E * ,  v D = 1 ρ s 11 D * , MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamODa8aadaahaaWcbeqaa8qacaWGfbaaaOGaeyypa0Za aSaaa8aabaWdbiaaigdaa8aabaWdbmaakaaapaqaa8qacqaHbpGCca WGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaadweaaaGc paWaaWbaaSqabeaapeGaaiOkaaaaaeqaaaaakiaacYcacaGGGcGaam ODa8aadaahaaWcbeqaa8qacaWGebaaaOGaeyypa0ZaaSaaa8aabaWd biaaigdaa8aabaWdbmaakaaapaqaa8qacqaHbpGCcaWGZbWdamaaDa aaleaapeGaaGymaiaaigdaa8aabaWdbiaadseaaaGcpaWaaWbaaSqa beaapeGaaiOkaaaaaeqaaaaakiaacYcaaaa@4F50@ and k 31 * 2 = d 31 * 2 s 11 E * ε 33 T * MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGRbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaacQca aaGcpaWaaWbaaSqabeaapeGaaGOmaaaakiabg2da9maalaaapaqaa8 qacaWGKbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaacQca aaGcpaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaapeGaam4Ca8aada qhaaWcbaWdbiaaigdacaaIXaaapaqaa8qacaWGfbaaaOWdamaaCaaa leqabaWdbiaacQcaaaGccqaH1oqzpaWaa0baaSqaa8qacaaIZaGaaG 4maaWdaeaapeGaamivaaaak8aadaahaaWcbeqaa8qacaGGQaaaaaaa aaa@4AB7@ .
Experimental and Analytical Results and Discussions
The non-electrode samples were made from the commercial ‘hard’ bulk PZT rectangular plates, PIC144 [PI Ceramic GmbH, Lederhose, Germany] with 40 x 5 x 1 mm3, 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 full-electrode and non-electrode 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 (non-prime)

Dielectric

ε 33 T / ε 0 =1073 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG1oWdamaaDaaaleaapeGaaG4maiaaiodaa8aabaWdbiaabsfa aaGccaGGVaGaaeyTd8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacq GH9aqpcaaIXaGaaGimaiaaiEdacaaIZaaaaa@412C@

tanδ 33 ' =2.3× 10 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacaqG0oWdamaaDaaaleaapeGaaG4maiaa iodaa8aabaWdbiaabEcaaaGccqGH9aqpcaaIYaGaaiOlaiaaiodacq GHxdaTcaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaIZaaa aaaa@4568@

tanδ 33 =2.1× 10 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacaqG0oWdamaaDaaaleaapeGaaG4maiaa iodaa8aabaaaaOWdbiabg2da9iaaikdacaGGUaGaaGymaiabgEna0k aaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHiTiaaiodaaaaaaa@44BC@

Elastic

s 11 E =11.7× 10 12 [ m 2 /N ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGZbWdamaaDaaaleaapeGaaGymaiaaigdaa8aabaWdbiaabwea aaGccqGH9aqpcaaIXaGaaGymaiaac6cacaaI3aGaey41aqRaaGymai aaicdapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaiaaikdaaaGcdaWa daWdaeaapeGaaeyBa8aadaahaaWcbeqaa8qacaaIYaaaaOGaai4lai aab6eaaiaawUfacaGLDbaaaaa@498B@

tan 11 ' =7.19× 10 4 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacqaHfiIXpaWaa0baaSqaa8qacaaIXaGa aGymaaWdaeaapeGaae4jaaaakiabg2da9iaaiEdacaGGUaGaaGymai aaiMdacqGHxdaTcaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsisl caaI0aaaaaaa@4668@

tan 11 =5.8x 10 4 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacqaHfiIXpaWaa0baaSqaa8qacaaIXaGa aGymaaWdaeaaaaGcpeGaeyypa0JaaGynaiaac6cacaaI4aGaaeiEai aaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHiTiaaisdaaaaaaa@43E4@

Piezoelectric

d 31 =103× 10 12 [ C/N ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGKbWdamaaBaaaleaapeGaaG4maiaaigdaa8aabeaak8qacqGH 9aqpcqGHsislcaaIXaGaaGimaiaaiodacqGHxdaTcaaIXaGaaGima8 aadaahaaWcbeqaa8qacqGHsislcaaIXaGaaGOmaaaakmaadmaapaqa a8qacaqGdbGaai4laiaab6eaaiaawUfacaGLDbaaaaa@47AF@

tanθ 31 ' =2.2× 10 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacaqG4oWdamaaDaaaleaapeGaaG4maiaa igdaa8aabaWdbiaabEcaaaGccqGH9aqpcaaIYaGaaiOlaiaaikdacq GHxdaTcaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaIZaaa aaaa@4569@

tanθ 31 =6.6× 10 4 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0bGaaeyyaiaab6gacaqG4oWdamaaDaaaleaapeGaaG4maiaa igdaa8aabaaaaOWdbiabg2da9iaaiAdacaGGUaGaaGOnaiabgEna0k aaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHiTiaaisdaaaaaaa@44C8@

Density

ρ=8080[ kg/ m 3 ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbpGaeyypa0JaaGioaiaaicdacaaI4aGaaGimamaadmaapaqa a8qacaqGRbGaae4zaiaac+cacaqGTbWdamaaCaaaleqabaWdbiaaio daaaaakiaawUfacaGLDbaaaaa@41FA@

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-

Figure 5: (a) full-electrode (FE), (b) partial electrode – short-circuit, (c) partial electrode – open circuit, and (d) non-electrode (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 fa (maximum peak in admittance curve), and QA value shift to lower monotonically with increasing the electrode portion. However, it is worth to note that the anti-resonance frequency fb (minimum peak in admittance curve) and QB value do not change
Figure 8: Effect of the center electrode area on the resonance and anti-resonance frequencies and their corresponding mechanical quality factors.
dramatically up to a = 0.4. This result is important, since fb and QB of non-electrode sample are used for calculating the extensive elastic compliance s11D, and mechanical loss tan . A non-electrode 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 six-terminal Mason’s equivalent circuit (EC) model for non-electrode samples, by considering all three losses [26]. We demonstrated to obtain the voltage distribution in the non-electrode (NE) configuration with the equivalent circuit simulation. Each of the non-electrode load part was segmented into 20 elements along length. NE configuration elements behave under D-constant though the voltage could not be measured. The E-constant element was integrated in between the D-constant 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

fa-FE

40.69 ± 0.04

40.62

40.64

40.66

fb-FE

42.31 ± 0.04

42.26

42.31

42.32

fa-NE

42.43 ± 0.04

42.16

42.26

42.30

fb-NE

42.85 ± 0.04

42.68

42.71

42.75

QA-FE

1390 ± 28

1390

1355

1390

QB-FE

1650 ± 33

1650

1567

1656

QA-NE

1690 ± 30

1629

1626

1648

QB-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 non-electrode PZT rectangular plate in the k31 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 non-electrode sample. From the analytical solution, we found that a non-electrode 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 non-electrode 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 N00014-17- 1-2088.
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