Simulation of Piezoelectric Devices by Two- And Three-Dimensional Finite Elements

7/22/2019 Simulation of Piezoelectric Devices by Two- And Three-Dimensional Finite Elements

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Simulation of Piezoelectric Devices by Two- andThree-Dimensional Finite ElementsR E I N H A R DL E RCH. M E M B E R . IEW.

Abslract--A method for the analysis o f piezoelectric media hased o nfinite element calculations is presented in which the fundamental elec-troelastic equations gokerning piemelectric media are solved numeri-cally.Theresultsohta ined h this inite-elemen tcalculation chemeagree with theoretical and experimental data given in the iterature.The metho d is applied to the bihrational anal sis of piezoelectric sen-s o r s and actuators with arbitrary structure. Natural frequencies nithrelated eigenm odes of those devices as ell as their e.;ponses to Iarioustime-dependent nlechanical or electrical excitations are computed. Thetheoretically calculated mode shapes of piezoelectric transducers andtheir electrical impedances agree quantitativelj with our re5pective in-terferometric and electric measurements. The simulations are used tooptimize piezoelectric devices such as ultrasonic transducers for med-ical imaging. The method also provides deeper insight into the physicalmechanism5 of acoustic wabe propagation in piemelectric media.

1. INTRODUCTIONP E Z OE L E CT RICM A T E R I A L Sare widely used inelectromechanical sensors and actuators such as tele-phone handset transmitter and receiver insets, robotic sen -sors , ultrasonic transducers for medical imaging and non-destructive evaluation N D E , as well as transducers usedin the pperMHzange, e. g. . surface-acoustic-wave(SAW ) devices. In the past. the development of electro-acoustic transducers was primarily based on trial and er-ror, which s ime-consumingand hereforeexpensive.This kind o f development is not consisten t with modemindustrial engineerin g practice, which is to aid develop-ment by com puter simulatio ns for the theoretical predic-tion of thepropertiesexpected o esult from agiventransducer design.The main purposes of computersimulations in trans-ducer development are:

Optimization of transducer design without time-con-Evaluation of new materials i n device design,Deeper nsight nto hewavepropagation in piezo-suming experiments.

electric solids.The models commonly used to simulate the mechanical

and electrical behavior of piezoelectric ransducers gen-erally ntroduce implifyingassumptions thatareofteninvalid foractualdesigns.Thegeometriesof practicalManuscript receit)cd Deccrnbcr23. 1988; reviwd J u l j I O 1989 a n d Oc-The author is w i t h S l e r n c n \ AG. AFE - T PH 41. Postlach 3220 . D-8S lOIEE E I.og Nunlber 9034355.

tober 1 . 989. acctptcd October 2 5 . 19x9.Erlangen. Wes t Ge rma ny.

transducers reoften wo- (2- D) o r three-dimensional(3-D). The m ost popular models, such as Mason 's modelor theKLM-model 111-191, however,are only one-di-mensional ( 1 - D ) . For the 2-D or 3-D simulation of piezo-electric media the complete set of fundamental equationsgoverning piezoelectric media has to be solved. The finitedifference or finite-elementmethodsarehowever suffi-ciently general to handle these differential equa tions . Thefinite element method was preferred here because it is ca-pable of handlingcomplexgeometries. Hitherto mainlyresults of 2-D piezoelectric finite difference or finite ele-ment simulations have been reported in the literature [ 101-[ 141. The eometrical imensions of practicalrans-ducers,however,oftendemanda full 3-D description.Thu s we have implemen ted an analysis scheme for piezo-electric media with no restrictions other than linearity. Thcma.jor adva ntag es of our finite element calculation schem ecompared o other piezoelectric finite element software,e . g . , [ 151, are the availability of 2- D and 3-D piezoelec-tric finite elem ents a s well as the capability of computingtransient responses and of handling structures with non-uniform damping. Ou r finite element analysis scheme isthe first to allow the handlin g of different 2-D as well as3-D iezoelectricinitelementsortatic,igenfre-quency, armonic andransient nalysis. In transientanalysis the dampin g coefficients may differ from eleme ntto element,which is important f o r the computationofstructures with locallynonuniformdamping coefficientsas, for example, in array antennas wi th absorbing backingmaterials. With this analysis scheme piezoelectric med iawith anisotropic material tensors and almost any geome-try can be calculated. Telephone hand set transducers [ 161,array antennas for medical maging [ 17). acoustic delaylines [ 181 and SAW devic es [ 191 have already been suc-cessfully analyzed using this method. We will concentratebelow on the analysis of ultrasonic transduc ers as used inechographic systems.

11. T H E O K YF P I ~ : % O E I . I < C ' I R I CF I N I T EE L E M E N T SThe matrix eq uations ( 1 ) relating mechanical and elec-trical quantities in piezoelectricmediaare the basis orthederivation o f the finite elementmode l vectors andmatrices are printed in boldface):T = C'S e'E ( l a )D = eS + E ~ E ( l b )

0885-3010/90/0500-0233 01 OO 990 IEEE


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T vectorofmechanical tressesS vectorofmechanical trainsE vector of electric fieldD vector of dielectricdisplacementc E mechan ical tiffnessmatrix or onstant lectricE permittivitymatrix forconstantmechanical straine piezoelectricmatrix;uperscript ' meansrans-

field ES

posed.The electric field E is related to the electrical potential @by

E = -grad @ ( 2 )and hemechanicalstrain S to he mecha nical displace-ment U by

S = Buwhere in the Cartesian coordinates

B =

The elastic behavior of piezoelectric media is governedby N ewton's law:DIV T = pa 'u/a t ' ( 5 )

whereDIV is divergence of a dyadic [ 2 0 ] ndp is density of the iezoe lectricmedium;

whereas the electric behavior is described by M axw ell'sEquation considering that piezoelectric med ia are insulat-ing (no free volume cha rge):di t lD = 0. ( 6 )

Equations (1)-(6) constitu te a com plete set of differentialequations which can be solved with appropriate mechan-ical (displacementsand orces) andelectrical potentialand charge) boundary conditions.An equivalent description of the abov e boun dary valueproblem is Hamilton'svariationalprinciple as extendedto piezoelectric media [ 2 ] , (221

6 L d r = Owhere the operator 6 denotes first-order variation and theLagrangian term L is determined by the energies availablein the piezoelectric medium:

( 7 )

L E,,, - E,, + E,, + W ( 8 )

with Elastic energy E,,E t = 1 S T d V

and Dielectric energy .EtlE l I \ D ~ d v

and Kinetic energy E,,,EL,,, = ; 1 pic d V

whereU is the vecto r of particlevelocityandV is the volume of the piezoelectric medium.

The energy I+ generated by external mechanical or electrical excitation is definedW = 11s u i f , d V + I uj f ,c lA

v ;I ,+q,sd A + c u'F,, - x @ Q p ( 12

1

wherefH vector of mechanical body forces [ N / m 3 ]fs vector of mechanicalsurface orces [ N / m ' ]FP vector of mechanical point forces [ NIA , area where forces are applied [m ' ]q surfacecharges [As/m' ]Q P pointcharges [ A s ]A , area where charges are applie d [ m' 1 .

In the finite element method he body to be computed subdivided nto small discrete elements, he so called fnite elements. The mec hanical displacements U and forceF as well as he electrical potential @ and charge Q ardetermined at the nodes of these elem ents. The values othese mecha nical and electrical quantities at an arbitraryposition on the elemen t are given by a linear combinatioof polynom ial interpolation functions N x . y , z ) and thenodal point values of these quantities as coefficients [23[24].For an element with I I nodes (n oda l coordinates: ( . xyf, z , ) ; = 1 . 2 . . . . , H ) thecontinuousdisplacementfunction u ( . u ,y , :) (vector of order three). for example,can be evaluated from its discrete nodal point vec tors afollows thequantities with t h e sign ' + ' * are the nodapoint values of one element) :

u ( . T , . Z ) = N f f ( . r , , Z ) U .rf,yf z f ) (13where

U is the vector of nodal point displacements (ord erN , , is the interpolation functions for the displacement.* n ) and


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2 2 5

All other mechanical and electrical quantities .Y are simi-larly interpolated with approp riate interpolation functionsN,. ith the interpolation functions for the displacement( N i l ) nd the electrical potential ( N + ) , (2 ) and ( 3 ) can bewritten:

E = grad + = grad ( N a & ) = - B a & ( l 4 a )S = Bu = BN,, l i = B,,U ( 1 4 b )

The subst i tut ion of the polynom ial interpolation functionsN , into (7) yields a set of linear differential equations thatdescribe one single piezoelectric finite element:u + d, , , , + k,,,, ' + k,,* = F B + F y + F P ( 15a )

k: ,+u + k + + 6 = Q , * + Q p (15b )where

ic, ii Vectors of nodalvelocities,accelerationsMechanical stiffness matrix:

k,,,,= 1B:,c"B, ,dl/. ( 16a)Mechanical damping m atrix:

d,,,, = CY' '1 pN: ,N, , ciV+ P B:,c"B, ,d l / . ( 1 6 b )

Piezoelectric coupling matrix:k,,@= 11 B:,e'B+ d V . ( 1 6 c )

Dielectric stiffness m atrix:kae = 1B ; c S B + d V . ( 1 6 d )

Mass matrix:m = p N ; , N , , V . ( 1 6 4

Mechanical body forces:

Mechanical surface forces:

Mechanical point forces:Electrical surface charges:

A

Electrical point charges:

wherea ' " ) , '' dampingcoefficients of element ( c )fk ' external body force at element c )f y externalurfaceorce at element c )F : ) external point force at element ( P )4: externalurfaceharge at element dQ:) external point charge at element ( e ) .

The damp ing behavior of the element is determined by thedamping matrix d ( , , , . hich can be introduced by standardfinite element techniques 1231. In the general case, thesematrices d,,,, an be assembled from the damping proper-ties of the structure, which a re usually frequency depen-dent . An arbi trary frequency dependence of the damping.however, equires more than twodampingcoefficients.Thiswould esult in a fu l l damping matrixand conse-quently in a significant amount of computational efiort, asreported in [ 2 3 ] . n practice it is conv enient therefore, toapproximate the damping behavior by (16 b). The rewith ,four types of ph ysical damp ing can be modeled, accord-ing to hevalues of thedampingcoefficients a and 6 .Theseare: l ) theundampedcase ( a = 0; p = 0 ) , 2 )st iffness-proport ional damping, i . e . , viscous damping, ( a= 0: p > 0) , 3 ) mass-proportionaldamping ( a > 0, p= 0 ) and 4)Rayleigh damping ( a > 0; p > 0 ) see also(251). The magnitud es of the Rayleigh coefficients cy andp depend on the energy dissipation characteristics of themodeled structure . Hysteret ic damping. for example, canalso be roughly approximated by suitable values of a andp. In order to handle structures with non-un iform damp -ing. the values of cy and p can be different from elementto elem ent. T ypical values of the Rayleigh coefficients forpiezoceramic materials operated at a frequency of 1 M H zare: CY = 7.5 and /3 = 2 X lo-'. For soundabsorbingmaterials we increase the values of CY and (3 so that criticaldamping is obtained.The subdivision of the area or body to be computed intofinite elements results in a mesh composed of numeroussing le elem ents. The com plete finite-elernent m esh o f apiezoelectric medium is mathematically described by a setof linear differential equations with symmetric band struc-ture .Here, thequantities U CD. FB, F*, Q s , and Qp arethe globally assembled field quantities and no longer ele-ment quantities as hosemarked by an .'* in (15) and(16):Mii + D,,,,u + K,,, ,u+ K, ,+@ = FR + F s + FP ( 1 7 a )

K L u + K + + @ = Qs + Q P ( 1 7 b )According to the theory of conventional mechanical finiteelements (see, for exam ple, [23 ], 1241) the matrices andvectors describing the whole mesh (( 17)) result from as-sembling the vectors and matrices of the single elements((15))of which the mesh is compo sed. I f the whole meshcontains n t n t nod es, m atrix equation (17a) will consist of3 * IZ < nd matrix equation (17 b) of n,,,, l inear equations.This is because the mechanical description of a body re-


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quireshreeectoromponents (e .g . , i sp lacem ent) ,whereas for the description of a qua si-stationary electricalfield a single scalar quantity ( e .g ., potential) is sufficient.Thesolution of (17)yields hemechanicaldisplace-ments U andelectricalpotentials @ in thepiezoelectricmedium . The two sets of l inear equations (17a) and (17 b)are couple d by the matrix K,,+ and split into two separatematrix quations with diminishingpiezoelectr icity (pi-ezoelectricstress ensor e = 0 -- K, , , = 0 ) . These twoseparate sets of equations describe respectively pure m e-chanical finite element models already known from struc-tural mechanics [23] ((1 7a ) with K,,+ = 0 ) and models ofelectrostatic field problems ( (17b ) with K,l+ = 0 ) .Besides these nodal or local results we further evaluateintegral quantities, such as the electrical input impedanceor electromechanical coupling coefficient characterizing apiezoelectric transducer.Electromechanical Coupling Coeflcient

The electromechanical coupling coefficient k is definedin [2] :

7 G IE,, Edk - = __ ( 1 8 )

where E,,, is mutual energ y. E,, is elastic energy. and Edis dielectric energy.In terms of piezoelectric initeelementmatrices thethree energies are writtenE,, = f ( u'K,,+@ + @ ' K: ,+ u ) ( 1 9 a )

Ed = @'K*+@. ( 1 9 c )Th e magnitude of electromechanical coupling of a vibra-tional mode represen ts the significance of that particularmode compared to the other modes. If the coupling of acertain mode is of the order of 50% or higher, that modewill be strongly excited. The larger the electromechanicalcoupling coefficient of the mode of interest, the lower willbe the insertion loss and the broader the bandw idth of thetransducer.Electr ical Impedunce

Thelectricalmpedance is anotherharacterizingquantity which can m oreover be verified experimentallywithout undue effort, since impedance measuremen ts caneasily be carried out w ith a network analyzer by sweepingthe frequency and recording he real and maginary partof the impedan ce. The input impedance of a piezoelectr ictransduceralso eveals all the esonancesandantireso-nances of the device. The resonances are the natural fre-quencies or hort-circuitedelectrodes,while heanti-resonances are those for open-circuit conditions. The res-onances are excited by a pulse of the electrical potential,and the anti-resonances by a pulse of the electrical charge.Thus heresonancefrequencies ( f , ) arerepresented re-

spective ly by the minim a and the antiresonance frequen-cies ( f a ) by the maxima of the electrical input impedance.Even heelectromechanicalcouplingcoefficientcanbedetermined from resonances and antiresonances using a napproximated formula (21:

Equation (20) is strictlyvalidonly for pureone-dimen-sional vibration modes (21, e .g . , a pure th ickness mode,whereas the definition by ( 1 8) is exact without qualifyingassumptions. To compute he electr ical input mpedanceofpiezoelectric ransducer with finite elem ents, thetransducerhas obe excited by adelta unction of theelectrical harge at one lectrode,while the other isgrounded:Q ( [ ) = Q , J ( t > ( 2 1 1

where Q,, is the amplitude of charg e pulse and 6 t ) s theDelta unction.Theelectr ical m pedance Z ( o s thengiven by (22) (si nce I t ) = d Q ( r ) / d t ) :

wheref { + c , ( t ) is the Fourier transform of the electr icpotential at the excited electrode. I n this case, the appli-cation of a delta function charge is superior to a step function charge , since the comp utations for a delta pulse re-quire less data storage.Average Displacernmt

In practical transduce r developm ent it is useful to de-fine an integral quantity that characterizes the mech anicaloutput of the transduce r. On e integral result which can beused to quantify the mechanical response is, for examplethe average displaceme nt of a region of interest, e . g . , theaverage displacem ent of the sound emitting face of thetransducer.The veragedisplacem ent is computed bysumming the displacement amplitude spectra U , o f allthenodes i belonging to theregionof nterest. Th e re-sult ing average amplitude spectrum U, , , W ) then repre-sents the strength of the various vibrational modes i n re-spect to the mechanical output of the considered region:

Amp litude spectrum of mechanical displacementat node i Node i belongs to that face of ele-ment j which is a subare a of the region of in-terest .A A J Area of that face of el em en tj that is a subarea ofthe region of interestN,,, Nu mb er of nodes belonging to the considered ele-ment face


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N Number of elementsbelonging o he egion ofA,,,i Totalareaof regionof nterest.interest

111. I M P L F . M E N ' l 4 T I O N O F THl.. T H t : O R YTh e theory described above was implemented in FOR -TR AN routines upplemented oa initee lem en t con-puterprogramdeveloped at our computercenter.The

original version of this program was used to solve prob-lems of structural mecha nics. Up to now we hav e imple-mented he ollowing ypes of piezoe lectric initeele-ments: -D lane-strain lement ( 3 to 8 nodes).2-Dplane-stress element ( 3 to 8 nodes). axially symmetricalelement ( 3 to 8 nodes) and 3-D elements (4 to 27 nodes ) .Plane-strain conditions imply the absence of strain i n thethird (neglected)geometricaldimension (S , = Sh = 0:E , = 0 in (l a )) , while plane-stress implies the absence ofstress in that direction ( TI = T , = 0; D , = 0 ) . This phys-ically meansastructure with eithe ra very th in (plane-stress) or an infinitely long (plan e-stra in) third d imen sion.For the implementation of these elements the mechanical.electrical and piezoelectric anisotropies of the material areconside red in toto by using the ful l material tensors.For henumericalanalysis of piezoelectricfiniteele-ments, s tandard finite element equation solvers can be ap-plied because the matrix equation ((17)) o be solved ex-hibits symmetric band structure. As for onventionalmechanical finite elementsweapply, orexample, hesubsp ace iteration method [23] to calculate the natural fre-quencies and their related mode shapes. Further, the stan-dardNewm ark tep-by-step ntegrationmethod [23 ] isused o comp ute ransie nt responses o mecha nical (dis-placement or force) or electrical (charge or voltage) ex-citatio ns. An algorithm for solving (17) in complex for-mulation has also been implemen ted in order to computethe responses to harmonic excitation. In transient and har-monic analysis the damp ing coefficients may differ fromelement to elemen t. hisemandsmoreomputingpow er. for the matrix D,, (( 17)) has to be evaluated andstored separately. It is mor eove r a very important featurefor simulating structures in which dam ping is locally non-uniform. In none of our com putat ions did we experiencenumerical difficulties due to piezoelectric finite elements.Thus the implementation of the theory of piezoelectric fi-nite elements in other tandard finiteelement oftwarewould seem to be practicable without major problems.In finite-element analysis i t is furthermore necessary f o rappropriate pre- and postprocessing software to be avail-able for the convenient handling of he structures o beanalyze d. The preproce ssing software should support theinteractive generation of finite element meshesat a graphicworkstation . Once finite-elemen t analysis has been com -pleted by the kernel finite elemen t program . an appropri-ate postprocessing software is needed o convert he nu-meric values determined at the nodes of the finite elementmesh intographicaloutput.Since hepre- andpostpro-cessing software at present commercially available is u n -

able to handlepiezoelectricproblems. i t was necessaryfor us to develop appropriate software.Iv. c O N F I R M / \ T l O N OF T H k c A I . C I I L 4 ' I ' l O N S C . H l . M t 3

Boucher et t i l . have eported the simula tion o f piezo-electric cubes using a mixedfiniteelement-perturbationmethod [ 121. In Table I Boucher's theoretical and exper-imental esultsarecheckedagainstown 3-D finite ele-ment calc ulations as well as atestrelatedfiniteelementresults reported by Osterga ard ( 151.Esutnple 4

The resonances of an electromechanical Langevin-typetransducerwerealsocalculated w i t h our finite elementsoftware and compared with the results given in [ I S ] and(291 (Table 111). Like Kagawa 1291 we used 2-D axisym-metric elements, wherea s Ostergaard [ 151 modeled a smallsector of the axisymmetric rod with 3-D elem ent s.V . A P P L . I C A T I O NF T H ~ A L C C I I A T I O NC H F M E 0T R A N S D I J C E K SO R U L T K A S O N I CM A G I N GThe qualityofultrasonic mages is known odepend

greatly on the erformance of thelectromechanicaltransduce rs used. In order to improve transducer charac-teristics we analyzedpiezoelectricparallelepipedpiezo-ceramic bars as used in theultrasonicarrayantennas o fechographic systems. I n transducer development it is oftenassumed that the transducers vibrate like simple pistons.This s.howe ver. not correct in the following espects:First. he hicknessmode,which is closest o he idealpiston mode as to shap e. does not exhibit true piston be-havior at al l. Due to the strong lateral contraction o f con-


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3 5 -

3 0

? i

? 3-

0 5 I 1 I I I I1 3 4 5 6C m x l r r lh~ckncss

-+--'34 t---i--f06 t08 0

ventional piezoceramic m aterials, e.g. . Poi sso n's ratio= 0.4 for P Z T ( 3 = thicknessdimension; 2 = widthdimension ), the thickness modes of piezoceramic vibra-tors mostly exhibit significant displacements normal to thethickness dimension. The displacements along the width

dimension re onsequently onuniform.Second, stillother modes are often excited whose strength depends onthe transducer geometry. These modes areof parasitic na-ture and greatly differ from piston-like beha vior.A further. more general problem in the analysis of elec-tromech anical devices is that their vibrational modes canvery rarelybeassigned to puremodessuchas. orex-ample, the l-D thicknessmode.Theactuale igenmodesof complex vibrators are often a mixture of different puremodes. To obtain deeper insight into the physical mech-anisms of such vibrations we have computed he eigen-

modes of parallelepipedpiezoelectricbarswithgeome-triessimilar to thosegenerallyused in ultrasonicarrayantennas. These bars are ypically so long hat heirei-genmodes i n the length dimension appear far (at least bya factor of 10) below the frequency range of practical in-terest. Since these mode s and also their harmo nics are allweakly coupled. it is not nec essary to consider them be-low.However , it will beshown hat it is generally notpermis sible to neglect the length dimension on account ofits influence on the modes of interest.


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A . M e c h a n i m l D i . ~ l ~ l a r . c ~ t n P t ~ t . ~First of all, the owest eigenm odesofparallelepipedpiezoceramic bars have been analyzed by 2-D as well as3-D simulations with respect to displacements a nd electricfields. The perma nent polarizatio n of the bars a s well asthe applied external electric field are align ed in the thick-ness dimension. The thickness modes of bars with width-to-thickness ( W / T ) ratios of W / T = 0.5 and W / T =

2.0 as well as hewidth-dilatationalmode or W / T =2.0areshown i n Figs. 3-5. Theelastic.dielectric andpiezoelectric constants, of the utilized piezoceramic ma-terial (Siemens-Vibrit-420 ) are given n the appendix. Fornar row e lem en ts ( W / T < I ) the thickness mode alwayscorresponds to the first and , for wider bars ( W / T > I ),to the second natural frequ ency . The mod e switch occursin the region W / T = 1. since he W < T he conditionformechanical esonance ( h/2-resonance of a non-clamped bar) is first fulfilled in the larger hicknessdi-mension. For W > T the first resona nce condit ion is metin the broader width dimension. Consequently the thick-ness mode of bars with W > T corresponds to the secondeigenfrequency . ForW/T-rat iosgreater than 3.0 thethickness mode even correspon ds to the third natural fre-quency. As will be show n later. the thickn ess mode is themode of interest for ultrasonic imagin g applications. Theresultsof the 3-D analysishavealsobeencompared orelated 2 -D calculations, for which the ength L of thebars was assumed o be nfinite. The natural frequenciesobtained with 2-D simulations typically differ by n o morethan 1 from hecorrespondingvalues of 3-D simula-tions as long as the length L of the bar is at least ten timesgreater than both its width and its thicknes s. We disco v-ered no differences between the 2 - D mode shapes and thecross sections at I = L / 2 of related 3-D modes ( i n Figs.3 (b ) 4 (b ) , and 5 (b ) ).Never theless , theassumption that2-Dsimulationscanadequatelydescribe heelasticde-formations of suchvibratorsdoes not holdbecause heoften considerable displacement gradients along the lengthdimension cannot be considered in 2-D calculations. Theanalysis shows that the displacements of such bars eventhe displacements of the thickness modes, (conventionalpiezoc eramic materia ls assume d) are not constantalongthe length dimension (Fig . 3(a ), (c)) . Th is is even true i fthe length of the bar is ten times greater than both its widthand its thickness. The observed displacement ipples alongthe ength dimension (F ig. 3 ) are of importance becausethey influence the emitted sound field.The computed eigenmode shapes of these bars have alsobeen experimentally verified by laser interferometric mea-surements. The normaldisplacementsweremeasured inthewidthdimensionon he opelectrodes.whichwerepolished o obtain higher reflectivity for he aser bea m.Computed and m easured eigenmode shapes of piezocer-amic bars with various W/T-ratios are compa red in Fig.6. Foraconvenientcomp arison of computedandmea-sured mode shapes he following procedure was chos en:the computations were done by eigenfrequ ency analysis,

Mechanicallsplacementsqulpotentlal Llnesof the Electrcal FleH

Me c h a l l c a l Dlsplacements

. .-

( C l

whereas the measurements were performed in continuous-wave mode at the resonance frequencies. to reproduce themode shapes.These esonance requenciescan be ob-


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Mechanlcal Displacements

Mechanlcal Displacements

Equlpotentlal Llnesof the Electrlcal Fleld

Mechancal Dlsolacements

tained from the input impedanc e results of Fig. 1 1 . Sincean eigenfrequency analysis delivers no absolute displace-ment values, it was necessary to normalize measured andcomputeddisplacements.The esults of Figs. 6 ( a ) . ( c ) ,(d ) were normalized with respect to the maximum values,whereas those of Fig. 6 ( b ) were normalized with respectto the minimum value. The normalization factors, whichwereevaluated oreachmeasurement.aregiven in thefigure captions of Fig. 6.

Geamely Walh 4mm Thickness-Zmm. Lengll-50mmMateW S~emers ' ~b r ~ l20( h )

Mechanical Dlsplacemenrs

Equipotential Lines at the Electrlcal Fleld

Geometrf W1dth-4mm hckness-Pm m Lenglk-50mmMaterla1 S eme,s- i 'br l 420( C )

Fig. 5 . Wid th -d i lata t iona l m ode of a piezoceramic bar w i t h W/T = 2.0( a ) 3-D mode shape. ( h ) Cross section at = L / ? = 2.5 m m . ( c ) Sagittsection.

B. ElectromechunicalCouplingThe epend ence of electromechanicalouplingntransducer geome try is often used to optim ize design . IFig. 7the electromechan ical coupling coefficients for the

five lowest modes of a piezoceramic bar are displayed asa funct ion of the W / T rat io . One discerns the maximumcoupling of the th ickness mode for W / T = 0.6. At W / T


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L E R C H . S I M U L A T I O N O F P I E Z O E L E C T R I C DtVICES B Y ? - I A N D 3-1) FILI'I f; k.l.t\lFh IS 24I

I 1 0 - Theoretlcal result

l

+ Dtsplacemenl (na rma lhed) F m : resonancel 1 0

0 5

c

Second resonancef Dlrplacemenl (normal lzed 1

l

Fig. 7 . Electromechanical coupling coefiicicnt o f long piezoelectric bars

= 0.6 the thickness mode is maxim ally excited. while allothermodes re argely uppressed.This can be con-firmed by comparing the mode histogram for W / T = 0.6with any other, for e x am p le . W / T = 2.0 (Fig . 8). ForW / T = 2 .0 the otalenergy is split ntoapproximatelyequalpartsamongseveralmodes,whereas fo r theopti-mum W/T-ra t io o f 0.6 it predominantly concentrates onthe thickness mode. Usually the transducer s designed f o rvibration in one sing le mode as obtained for W / T = 0.6.With this optimum W /T-ratioarray elements generate anddetectultrasoundsignals with optimum efficiency. sincemost of the electrical energy is converted nto a normaldisplacement of the sound emitting face (see also the shapeof the thickness mode for W / T = 0.5 in Fig. 3(b ) ) . Th u sthe thickness mode is the one of interest fo r imaging ap-plications.


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I 1110 1 1

. .000 1 2 0 3 6C 4 EO

C . Dia


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00 0 4 0 8 1 2 l 2 0Freque-cy (MHz)__

( a )

- 20-30- 40 -- 50 -- 60--iO -80- Theotel lca l esult

-90j

In the shown frequency band the narrow bar ( W , / T =0.25 in Fig . I l( a )) exhibits only a single strong thicknessmode , where as its higher modes lie outside this frequencyr an g e. F o r W / T = 0.5 ( F ig . 1 l@ )) the hickness modeis again strongly coupled. but the second and third vibra-

terest. A s predicted by computations of theelectrome-chanical coupling coefficient (Fig. 7 ) these modes are onlyweakly coupled.ForW/T -ratios arger 0.8. however.thesemodesarestronglycoupled (Fig. I I(c)) , whereasthe couplin g of the first mod e is slightly reduced. Th is isoncemoreconsistent with the esultsobtained or heelectromechanical coupling coefficient (F ig. 7) .E. Bllcking

The piezoelectric transducer elements of imaging arrayantennas are typically provided with a backing, which hasto be considered in any realistic simulation. The functionsof such a hacking are mechanical support and sound ab-sorption. A backing damps resonances due to t he transferof acoustic energy to the sound-absorbing backing mate-rial. The energy transfer is determined by the ratio of theacoustic impedances of the piezoceramic material to theepoxy backing [ 1]-[9]. ig . 12 show s the influence of theacoustic backing impedance on the mechanical output ofan array ransducer.The veragedisplacem ent of thetransduc er's ront ace is used a s an integral esult oquantify the mechanical output of the transducer. We ob-serve that the hacking influences primarily the thicknessmode nd o a far mallerextent hewidth-dilationalmode.Due o thepreferred ateraldisplacementof hewidth-dilatational mode only a small fraction of its energyis transferred to the backing. The w idth-dilatational modeis a high-Q mode because most of its mechanical energytravels back and forth between the free sides of the ele-ment.Theelectrical input mpedance of piezoceramicbarswith backing likewise demonstrates the considerable dif-ference in dampingbehavior of thevariousvibrationalmodes.Fig. 13 shows thecomputedand hemeasuredelectrical impedance of such a bar. The amplitudes of t h evarious modes show the damping of the width-dilatationalmode ( lowest eigenmode) o beslight in relation o heother modes for the reasons discussed above.Thediagram of dispersion for anarrayelement withbacking also differs from that of the array elem ent alone(Fig . 14). The resonance frequency is approximately 5 %lower due o mass oading and he main mode switcheshack o he first natural frequency in the case of argerW /T-rati os. This seco nd switch occurs because the width-dilatationalmode firste igen mod e o r W /T > I . O ) isless damped than the thickness mode for the reasons al-ready explained. Due to the stronger damp ing the band-width of the hicknessmode is larger han hat or hewidth-dilatationalmode.The W /T-ra tio at whichhemain mode switches back to the first eig enm ode (F ig. 14)depends on the acoustic impedance of the backi ng.

In the presence of a backing we observe the mass-springmode originally introduced by Larson (301.This is an os-cillationof healmostundeformed ransducerelementtionalmodesalready appear in the requency band of i n - (mass)against themechanicallymuchmorecompliant


12/15

Fig.

06Cl-

0 4 0 -

0 ~ 1 I I I0 5C 1 O C5 C ? C O ? S C 300 350

7 I IElectrical impedance ( RI T = 2rnm -mm

W 4mm100

Measuremerllv Theoretlcal result-.

100.1 0 5 0 9 1 3 1.7 2.1Frequency (MHz

Fig. 13. Electrical nput impedancc nf picroccram ic bar with bach ing .

backing (spring). The frequen cy of this mode can be de-termined by using an approximation form ula given in [30],The exact finite element calculation confirms the essential

P e m

l

E D W Racklna

Elernems

Fig. IS. M a s - \ p r i n g mode o f p iemccramic a r rayransduccr wback lng .

validity of this formula but the exactly computed frequencies are generally about 30% lower . The mode shape a typicalmass-springmode is shown in Fig. 15. Thmode is slightlydamped and alwayshasa owe r frquency than the main transducer mode. It may distu rb thsonographicmage by adding lutter 1301, which anhowever be avoided by usinghigh-absorptionbackingmaterial.F. Electric r r t d Mechmicwl Cross Colrplirlg

Cross-coupling between the transducer elemen ts of anultrasonic array antenna degra des the sonog raphic imagscanned by the antenna. Kino and DeSilets haveshownthat cross-coupling between neighboring elemen ts shouldbe below -30 d B in antennas for which a wide angle acceptance is required [3 11. but c onc lude that reasonablsatisfactory results can even be obtained with cross-copling in the range of -25 to -20 dB. Finite element simulationsare used to analyze hephysicalmechanismofcross-coupling in array antennas.Th e purely lectric ross-couplingbetween wo le-ments of anultrasonicphased-arrayantennawassimu-latedwith uppressedmechanicaldegrees of f reedomFig. 16 shows the cross section of the transducer co nfig-uration for different saw -cuts . In each of the three systema pair of PZT-transduce rs is provided with epoxy back-ing. The upper electrodes f both vibrators are electricalgrounded,while he owerelectrod e of each eft-handtransducer is energized by an electricpotentialof 1 VThe voltage appearing across the lower electrode of eacright-hand ransducer is induced by electri ccross-cou-pling.Fig . 16 shows t h e crosstalk orhree differen


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LERCH S I M U L A T I O h OF PIEZOF.1.ECTRIC D EV ICF , S BY ?-D K D 3-D F I N I T E E L E M E N T S 245

' I .

F i g . 16. Electr lc croswxupling between ransducers of array antenna

depths of saw-cut, which reaches a maximum of 43 dBbetween he woneighbouringvibratorsas in thecasewhere the backing is without a saw-cut. If the saw-cut isas deep as the vibrator is thick. the cross-coupling due toelectrical effects almost vanishes (crosstalk: -61 d B ) .In a further computer operation the cross-coupling dueto electrical andmechanicaleffectswascomputed.Theresponse of a pulse-excited ransducer (element n in

Fig. 17) is for this purpose calculated and comp ared withthe responses of the neighbouring elem ents (elements n+ 1 to ' ' n + 4 . Resultsareobtainedforantennaswith cut and non-cut matching ayers. We further com-pare configurations in which the saw-cu t ends at the topof thebackin g with hose in which thesaw-cut in thebacking is as deep as he vibrator is thick. During pro-duction these cuts may be filled with the epoxy adhesiveused to fix the match ing layer to the top of the piezocer-amic transducers. This case was also analyzed (F ig. 17).From he esultsobtainedcross-coupling may becon-cluded to be present at most in the case of non-cut match-ing layers.Thisndicates that ross-coupling erivesmainly from mechanical wave propaga tion in the match-ing o r protective layer. In-depth investigations show sev-eral different vibrational mod es, mainly Lam b mode s. tobe involved in mecha nical ross-coup ling.The results(Fig . 17) show a saw-cut in the backing material to leadtoasubstantial eduction in cross-couplingonly in thecase of cut match ing layers. If the match ing layer is cutthe electric cross-coupling hrough he backing materialwill also be a major source of coupling. Mechanical wavepropagation hrough hebacking s.however,onlysig-nificant when a low-absorption backing material is used .In view of the results reported by Kino and DeSile ts thesimulation furthermore predicts that the filling of saw-cutsby epoxymaterial ntroduces he risk of visible magedegradation due to strong mec hanical cross-coupling, es-pecially if theapplication equiresawideangle of ac-ceptance.Fina lly, the nfluence of saw-cutgeometryoncross-coupling is analyzed.Fig. 18 shows hecross-couplingbetween woneighbouringarray elem ents with cutandnon-cut matching layers. The results are shown as a func-

Number of elementFig. 17. Overal lelectricalndmechanical)ross-couplingetweentransducers of array antennas.

-70-

-80-t--- awcut wldth vibrator ~ d l l - - 0 2saw-cul w d l h vlbralof width. 03

0 15 0 5 075 10Sa+cul depth vlbralor 1h:kness-

Fig. 18. Influences o f raw-cut geometry on cross-coupllng

tion of the saw-cut depth with the saw-cut width as pa-ramete r. Cross-cou pling is seen o be nfluenced greatlyby the saw-cut depth only in the case of a cut matchinglayerwhereasalreadyexplaine d it is primarilydeter-mined by electrical crosstalk. We further discerned thatthe cross-co upling is only partially influenced by the saw-cut width.

V I . C O N C L U S I O NA finite-elementcalculation cheme o r he2-Dand3-D simulation of anisotropic piezoelectric media is pre-sente d. Using this method the natural frequencies and re-lated eigen mo des as well as the d yna mic responses to me-chanicalandelectricalexcitationscanbecomputed orpiezoelectric transducers of almost any geometry. The va-lidity of the sim ulation schem e has been confirmed by data

reported in the iteratureas well as by in-hou seexperi-ments.This finite-element analysis method allows the solutionof numerous problems encountered in piezoelectric trans-ducer des ign. One of the main problems that arise in pi-ezoelec tric sensor design is the simultaneous appearanceof various Vibrational modes with quite different physicalcharacteristics. In many cases hesemodescan only be


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sufficiently described by 3-D analysis as show n here. I ntheoreticalmodeling i t shouldalwaysbeborne in mindthat the standard 1 -D models are only applicable i f two orthe three geometrical dimensions can be neglected. Theywill not yield correct results if different ypes of vibra-tionalmodes appearsimultaneously.The finiteelementmethod, however, yields correct results, even if more thanone type of vibrational mode is essential for the operationof the transducer. The simulations allow a deeper under-standing of thephysicalmechanisms ofacoustic wavepropagation in piezoelectric sensors and actuators. We usesuchsimulations to optimize ransducerdesignwithre-spect to efficiency, bandwith, angle of acceptance. cross-coupling. e tc .The mportance of these compu ter simulations is stillgrowing in transducerdevelopmentas well as in manyother branches of technology. This developmen t is sup-ported by the continuou sly ncreasing pow er of moderncomputerequipment.Futurepersp ectives in this ectorcan be seen in the use of such simulations in combinationwith appropriate computer-aided design (C A D ).A P P E N D I XMaterial data of the used piezoelectric material VIBRIT

420.Density: p = 7600 kg/m- .

Mechanical modulip 4 . 9 10.1 9.8 0.0 0.0 0.0114.9 9.8 0.0 0.0 0.0I 2 . 2 0.0 IL 2 - 4 1

Piezoelectric constants:0 0 0 0 11.7 0.-( 0 0 0 11.7 0 0

- 5 . 4 -5.4 13.5 0 0 0Dielectric constants:

\ O 0 7 . 2 /A C K N O W L E D G M E N T

Theauthor is grateful o Prof . Dr . H . Ermert RuhrUniversi ty,Bochum).Dipl .-Ing. W . Friedrich,Dr. H .

Kaarmann, Dr . P. K raemmer, B. Sachs, and Dr. H . vonSeggern for abricating ransducers,performingexperi-ments and for valuable and stimulating discussions.REFERENCES

l] W . P .Mason. E/cc.rro-Mec.htrrllc.trl 7'rtrrrsrlrtcer.v ond W NW F i h mthird ed. Prince ton.NJ:D. van Nostrand. 1948.121 D . A . Berllncourt, D . R . Curran. H . Ja t l e . Piezoelectric nd piezomagneticmaterials. In P/r\.ricu/ 4 ~ 0 ~ 4 , s r i c . \ . o l . 1. Part A . NewYork:Academic Press.1964.pp.233-256.131 R . Krimholtz. D. A . Leedom, G . L . Mat thaei . N e w equivalent cir-cuits forelementarypiezoelectric ransducers. E I P ~ O J I .c f t . vol6. p . 398. 1976.141 G . Kossofi The effects of hacking and matching o n the per tormancof piezoelectric ceramic transducers. l E E E Tr


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L E R C H .S I M U L A T I O N O F P I E Z O E L E C T R I CD E V I C E S B Y 2 - D A N D 3-D FIVITEE L E M E N T S 247[24] 0 . C . Zienkiewicz, The Finite EIetnerzr Merhod i n Etyqtwering Sci -

m c r . New York:McGraw-Hill , 971.12.51 H . Kardestuncer.d.. Finirr Elrmcwt Handbook . New York:McGraw-H ill , 1987. p . 4-77.1261 R . L.Jungermann. P. Benett , A. R . Selfridge. B. T. Khuri-Yakub,and G . S . Kino, Measurement of normal surface displacement forthe characterization of rectangular coustic rray lements. J .Acousr . Soc. A m . , vol. 76. 1984, pp. 5 16.1271 P . EerNisse. Varlational method orelectroelastic vibration analy-s is . f E E E T r a m Sot1ic.s Ulrruson. . vol. SU-14. pp. 153-160, 1967.1281 E. A . Shaw, On the resonantvibrations of thickbarium itanated isks , J . A c o u s t . Soc. A m . v o l . 28, pp. 38-50, 1956.

(291 Y . Kagawaand T. Yamabuchi.Finiteelementapproach ora PI-ezoelectric circular rod. l E E E Truns. S o n i c s U / t r c r . ~ o n . ,ol. SU-23.pp.379-385.1976.[30] J . D .Larson. A new vibrationmode in tall,narrowpiezoelectrlce lements , in Proc J E E E S y t n p . . 1979. pp. 108-1 13.1311 G . S . Kino and C. S . DeSilets. Design of slotted ransducer arrayswith matched backings. ( i l rrason. Imag ing. pp. 189-209. 1979 .

Reinhard Lerch (85) was born i n West Ger-many n 1953.He eceived his masters i n 1977and his Ph.D. degree i n 1980, all i n electrical en-gineering, from the University of D armstadt, W estGermany.From 1977 to 1981he was engaged in the de-velopm ent of a new type of audio transducerbasedon piezoelectric polymer foils at the Institute ofElectroacoustics at DarmstadtUniversity.Since1981 he is employedat heSIEMENS-ResearchCenter in Erlangen, West German y. where he hasimplementednewcomputer oolssupporting thedevelopmentofpiezo-electric transducers. His latest work is concerned with finite-element sim-ulations of piezoelectric ransducers, which are used i n medical magingand communication engineering.Dr. Lerch is mem ber of the Acoustical Society of America. In 1982 hegot the Award of the German Nachrichtentechn ische Gesellschaft for hiswork ab out piezoelectric audio transducers.

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Simulation of Piezoelectric Devices by Two- And Three-Dimensional Finite Elements