Design,production and testing of PMN–PTelectrostrictive transducersJ.Coutte a,*,B.Dubus b,J.-C.Debus b,C.Granger b,D.Jones ca LAMH,Facult e des Sciences Appliqu e es,Universit e d’Artois,Technoparc Futura,62400B e thune,Franceb IEMN/ISEN Department,UMR CNRS8520,41Boulevard Vauban,59046Lille,Francec Defence Research Establishment Atlantic,9Grove Street,P.O.Box1012,Darmouth,Canada B2Y327AbstractLead magnesium niobate ceramics(PMN)are promising materials for application in thefield of high power transducers.The advantage of PMN materials are the large strains generated under moderate electricfield and the low hysteresis.The electrostrictive effect is non-linear,the corresponding physical constants depend on temperature and frequency and a DC electrical bias is required. These difficulties must be considered at the design stage.Afinite element model has been developed and validated in the ATILA code for non-linear static and time-domain analyses.These numerical modelings are used to design and test two Langevin-type electrostrictive transducers.Thefirst transducer is made of PMN–PT–La(90–10–1%)ceramics(TRS Ceramics),the second one of ESC1ceramics(Morgan Matroc).For given static mechanical prestresses,resonance frequencies and effective coupling coefficients are measured at different DC electricfields and temperatures.Ó2002Elsevier Science B.V.All rights reserved.Keywords:Transducers;Electrostriction;PMN ceramics;Finite element method1.IntroductionPiezoelectric ceramics are the dominant transduction material in sonars and ultrasonic applications.The analysis of the power limitations of transducers has been discussed by Woollett[1,2]and Berlincourt[3].Three types of limitations exist:(i)The mechanical limits are caused by different fac-tors such as fracture in the case of ceramic transduc-ers,or metal fatigue.(ii)The electrical limits are essentially concerned with the active material,such as insulation breakdown,de-polarization of materials operating at remanence,dis-tortion resulting from ferroelectric non-linearities.(iii)The thermal limit is the consequence of energy dissipation.The thermally limited power output de-pends not only on the dielectric losses but also on the mechanical losses and the heat transfer design of the transducer.These limitations are often due to the active material,i.e. piezoelectric ceramics.Thus,the development of new materials with high coupling constants,low losses and a higher limit of failure and fatigue could be an element of progress in the relevant technology.For these various reasons,we have studied and used the lead magnesium niobate-lead titanate ceramic(PMN–PT).The aim of this paper is to determine the optimal working condi-tions for a Langevin-type electrostrictive transducer in terms of effective coupling coefficient.Following this introduction,the second part of the paper summarises the properties of electrostrictive ceramics compared to conventional piezoelectric ceramics and the constitutive laws.Numerical modelings of electrostrictive material (briefly developed in the third part)have been made, validated and used in order to design two Langevin-type electrostrictive transducers which have been tested in the last part of this article.2.Lead magnesium niobate ceramics2.1.PropertiesIn electrostrictive materials such as PMN ceramics, strain is proportional to the square of the polarization.Ultrasonics40(2002)883–888*Corresponding author.Fax:+33-3-21-63-71-23.E-mail address:jocelyne.coutte@fsa.univ-artois.fr(J.Coutte).0041-624X/02/$-see front matterÓ2002Elsevier Science B.V.All rights reserved. PII:S0041-624X(02)00231-7They present a low hysteresis if the temperature is cor-rectly chosen (above the diffuse Curie transition:ferro-electric to paraelectric)and have strains roughly of an order of magnitude larger than those of the lead zirco-nate–lead titanate PZT ceramics (Fig.1).These large strains are due to their dielectric permittivities which are a factor of 10or more larger than those of the PZT ceramics.Handling the material non-linearities,the temperature dependence and the application of the elec-trical polarization remains a challenge,both at the de-signing and technological levels,which currently limits the use of PMN in acoustic transducers.2.2.Constitutive lawsThe constitutive laws of PMN ceramics are non-linear (the strain is proportional to the square of the polarization)and six different models [5–10]have been proposed in the literature.All these models correctly describe the behaviour of electrostrictive material.We have selected Hom’s model [10]because the parameters involved in this model are easy to determine from ex-periments.(1)With a moderate electric field,the polarization is linear to the electric field.The dielectric impermeability matrix [b T ]is also a constant and diagonal matrix.The coefficients of the tensor [g ]denotes the ratio of the strain to the electrical excitation at zero stress:½g ðQ ;D Þ ¼Q 11D 1Q 12D 1Q 12D 10ðQ 11ÀQ 12ÞD 3ðQ 11ÀQ 12ÞD 2Q 12D 2Q 11D 2Q 12D 2ðQ 11ÀQ 12ÞD 30ðQ 11ÀQ 12ÞD 1Q 12D 3Q 12D 3Q 11D 3ðQ 11ÀQ 12ÞD 2ðQ 11ÀQ 12ÞD 10264375ð1ÞAs the dielectric constant of PMN–PT is very large,the polarization P and the electric displacement D are considered as equal.Q ij are the electrostrictive constants of the condensed tensor.(2)With a high electric field,the induced polarization begins to saturate and [b T ]is henceforth a function of the polarization.Following the model proposed by Hom and Shankar [10]for PMN,we write:b T P s ;k ;D ðÞ¼1k j D j atanh j D j P sð2Þj D j is the magnitude of the electric displacement and P sis the spontaneous polarization.Finally,choosing the electric displacement and the stress as the independent state variables,the constitutive equations are written:S ¼½s D T þ½g ðQ ;D Þ tD E ¼À2½g ðQ ;D Þ T þb T P s ;k ;D ðÞÂÃD&ð3Þ[s D ]is the elastic compliance tensor at constant excita-tion,S and T are the strain and stress condensed tensors,E and D are the electric field and electric displacement vectors.k and P s are constants depending on the tem-perature.3.Finite element formulation 3.1.FormulationTwo models are developed in the ATILA code to model electrostrictive materials.Non-linear static [11]and time-domain analyses [12]are considered.The prin-ciple of virtual work is used to get the finite element formulation.The set of equations for an electrostrictive structure can be written for transient analysis as follows:½M ½0 ½0 ½0 !€U€/&'þ½K uu ðD Þ ½K u /ðD Þ 2½K t u /ðD Þ ½K //ðD Þ !U /&'¼F ÀQ &'ð4ÞFor static analysis,the second order time derivative must be eliminated from Eq.(4).This equation is non-linear,because D is a function of U and /,vectors of the displacement and the electric potential respectively.[K uu ],[K u /],[K //]and [M ]are the classical stiffness,pi-ezoelectric,dielectric and consistent mass matrices of a piezoelectric finite element model.To solve Eq.(4),a direct iterative procedure is used for the static analysis and a transient analysis by time steps is used for the dynamic analysis.Time discretiza-tion is performed using a central difference scheme.The solution is obtained by iteratively solving a linear set of equations until they converge.Static analysis of an electrostrictive bar has been used to validate these numerical models [11].A goodagree-ment between numerical and measured results on a PMN–PT–La bar[13]has been obtained on the whole range of applied electricfields and prestresses.3.2.Application to electrostrictive transducer:theoretical evaluation of the effective coupling coefficientThe aforementioned formulation is used to evaluate the effective coupling coefficient of electrostrictive trans-ducers.The classical method based on eigenvalue com-putation with short-circuit and open-circuit conditions cannot be used because Eq.(4)is non-linear.Two types of excitations are considered:•A step in voltage which generates a vibration of the structure at constant electricfield E.The correspond-ing natural frequency is named f E.•A step in charge which generates a vibration of the structure at constant electric displacement D.The corresponding natural frequency is named f D.Knowing f E and f D,the effective coupling coefficient for a structure is calculated from the expression:k2 eff ¼f2DÀf2Ef2Dð5ÞFor a simple geometry of vibrator[12](e.g.length ex-pander bar with parallelfield),Eq.(5)can be seen as an extension of Ikeda’s concept developed in thefield of piezoelectric delay lines[14].4.Design of Langevin-type electrostrictive transducers 4.1.ObjectivesThese numerical models are used to design and test two Langevin-type electrostrictive transducers.The main objective is to compare the effective coupling coefficients of these two transducers with a reference piezoelectric transducer.Evaluation is made at different DC electric field,temperatures and mechanical prestresses.4.2.Geometry of the transducersThefinal geometry results from an optimization of the transducer in terms of effective coupling coefficient and radiating acoustic power for in-air or in-water ap-plication[15].Several points should be emphasized:(i)the material constituting the end-masses must bestiffenough to keep a high effective coupling coeffi-cient;(ii)the radiated power is mechanically limited and therefore the prestress must be as large as possible (this compressive stress remains limited by mechani-cal failure of the active material during prestress application);(iii)for a given prestress,the optimum bias voltage varies between0.8and1.2MV/m,the corresponding effective coupling coefficient being around0.45.Consequently,two Langevin-type transducers have been built.Thefirst transducer is made of ESC1ce-ramics manufactured by the Morgan Matroc company (Fig.2a),the second one of PMN–PT–La(90–10–1%) ceramics manufactured by TRS Ceramics(Fig.2b).They consist of four rings and a central supporting elec-trode prestressed by a bolt between two steel-made end-masses.Part of the rod has been kept visible in order to modify the prestress value.The two electrostrictive transducers and the reference piezoelectric transducer have the same volume of active material and the same size.4.3.Experimental set-upTo measure the f p and f s frequencies(used to calcu-late the effective effective coupling coefficient)at various static electricfield,temperatures and prestresses,the experimental procedure is asfollows:Fig.2.(a)Langevin-type transducer with ESC1ceramics(Morgan Matroc).(b)Langevin-type transducer with PMN–PT–La(90–10–1%) ceramics(TRS Ceramics).J.Coutte et al./Ultrasonics40(2002)883–888885(i)Application of the mechanical prestress.(ii)Immersion of the transducer in an oil bath to avoid flashover.Note that in this set-up,the transducer is not perfectly isolated (acoustic energy is radiated into oil).The experimental conditions for determination of the effective coupling coefficient are therefore not exactly verified.(iii)Installation in an oven to regulate the tempera-ture of the bath.(iv)Frequencies of maximum conductance and of maximum resistance are measured for several values of electric field at different temperatures.Mea-surements on the transducer are made with an im-pedancemeter (protection circuitry is required for impedancemeter).(v)Repeat the procedure with increased prestress.Fig.3shows the scheme of the experimental set-up.4.4.ResultsFigs.4and 5show respectively the conductance (de-noted G )and the resistance (denoted R )of ESC1trans-ducer for different static electric fields.f s and f p are defined respectively as the frequencies of maximum conductance and of maximum resistance.The relative difference in the frequencies f s and f p depends on both the material coupling factor and the resonator geometry.These two frequencies allow the calculation of the ef-fective coupling coefficient from the equation:k 02eff¼f 2p Àf 2s f 2p ð6ÞAn increase in the DC electric field produces a decrease in the f s frequency and a slight increase of the f p fre-quency.This result is consistent with the initial hy-pothesis (Eq.(3)):s D is constant rather than s E .Figs.6and 7display the small-signal effective cou-pling coefficient versus the DC electric field at different temperatures.The effective coupling coefficient of the reference PZT transducer is equal to 0.156.The highest effective coupling coefficient is obtained at ambient temperature for the ESC1transducer with a maximum of 0.27and at 0°C for the TRS transducer with amaximum of 0.32.At 10MPa prestress,a saturation of the effective coupling coefficient appears below 24°C for the ESC1transducer,10°C for the TRS transducer.The different behaviour in temperature between bothtran-886J.Coutte et al./Ultrasonics 40(2002)883–888ducers can be attributed to the composition of the active material,as the temperature of maximum permittivity is controlled by the PT content in the PMN–PT material. Fig.8shows the effective coupling coefficient for theTRS transducer with a prestress of20MPa.The de-crease of the effective coupling coefficient with increased prestress predicted by models[15,16]is verified,partic-ularly for high electricfield.No saturation is observed at 20MPa prestress.Fig.9displays a comparison between the experi-mental and numerical results on the electrostrictive transducers at ambient temperature.For the numerical results,no prestress is taken into account.A good match is obtained for TRS transducer considering that mea-surement made with10MPa prestress should provide higher effective coupling coefficient[16].The data set used in this case were extracted from a complete char-acterization of the material made at various electrical field and mechanical prestresses.For ESC1material, only partial data from the manufacturer were available. The numerical computation of ESC1transducer effec-tive coupling coefficient does not provide satisfactory results even if the general trend is predicted:computed values should be higher than measured ones because prestress is not taken into account,saturation appears with a lower electricfield than expected.This last comparison emphasizes the need for a complete char-acterization of PMN–PT ceramics(under DC electric field and mechanical prestress)to enable quantitative numerical computation of transducercharacteristics.5.ConclusionAs the coupling coefficient of PMN–PT ceramics in 33mode saturates at around0.45,the effective coupling factor of electrostrictive ultrasonic transducers is ex-pected to be a key problem.If the choices of constitutive materials,mechanical prestress,DC electrical bias and temperature are optimized,effective coupling factors around0.3can be obtained.However,the design of such transducers will remain difficult as long as the complete characterization of PMN–PT materials from manufac-turers,under combined high stress and high electric field,are unavailable.References[1]R.S.Woollett,Theoretical power limits of sonar transducers,I.R.E.Int.Con.Rec.6(1962)90–94.[2]R.S.Woollett,Power limitations of sonic transducers,IEEE SU15(1968)218.[3]D.A.Berlincourt,D.R.Curran,H.Jaffe,in:W.P.Mason(Ed.),Physical Acoustics Principles and Methods1(A),Ed.Academic Press,New York,1964.[4]D.Damajanovic,R.E.Newhnam,Electrostrictive and piezoelec-tric materials for actuator applications,J.Intel.Mat.Struct.3 (1992)190–208.[5]ngevin,J.Phys.(France)4(1905)678.[6]W.P.Mason,Piezoelectric Crystals and Their Application toUltrasonics,D.Van Nostrand Company,Princeton,1964.[7]X.D.Zhang,A.Rodgers,A macroscopic phenomological formu-lation for coupled electromechanical effects in piezoelectricity, J.Intell.Mater.Syst.Struct.4(1993)307–316.[8]J.C.Piquette,S.E.Forsythe,Generalized material model for leadmagnesium niobate(PMN),J.Acoust.Soc.Am.104(1998) 2763–2772.[9]S.Sherrit,G.Catoiu,R.B.Stimpson,B.K.Mukherjee,Modelingand characterization of electrostrictive ceramics,S.P.I.E.3324 (1998)161–172.[10]C.L.Hom,N.Shankar,A fully coupled constitutive model forelectrostrictive ceramic materials,J.Intel.Mat.Struct.5(1994) 795–801.[11]J.-C.Debus,B.Dubus,J.Coutte,Finite-element modeling of leadmagnesium niobate electrostrictive materials:static analysis,J.Acoust.Soc.Am.103(6)(1998)3336–3343.[12]J.Coutte,J.-C.Debus, B.Dubus,R.Bossut,Finite-elementmodeling of lead magnesium niobate electrostrictive materi-als:dynamic analysis,J.Acoust.Soc.Am.109(4)(2001)1403–1411.[13]E.A.McLaughlin,J.M.Powers,M.B.Moffett,R.S.Janus,Characterization of PMN–PT–La for use in high-power electro-strictive projectors,J.Acoust.Soc.Am.100(1996)2729. 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