simulation of piezoelectric devices by two- and three...

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

REINHARDLERCH

Abslract--A method for the analysis of piezoelectric media hased o n finite element calculations is presented in which the fundamental elec-troelastic equations gokerning piemelectric media are solved numeri-callyTheresultsohtained h thisfinite-elementcalculationscheme agreewiththeoreticalandexperimentaldatagiven in the literature The method is applied to the bihrational analsis of piezoelectric sen-sors and actuators with arbitrary structure Natural frequencies nith related eigenmodes of those devices aswell as their reponses to Iarious time-dependent nlechanical or electrical excitations are computed The theoreticallycalculatedmodeshapes of piezoelectric transducersand their electrical impedances agree quantitativelj with our re5pective in-terferometric and electric measurements The simulations are used to optimize piezoelectric devices such as ultrasonic transducers for med-ical imaging The method also provides deeper insight into the physical mechanism5 of acoustic wabe propagation in piemelectric media

P 1 INTRODUCTION

IEZOELECTRICMATERIALSare widely used in electromechanical sensorsandactuators suchas tele-

phone handset transmitter and receiver insets robotic sen-sors ultrasonic transducers for medical imaging and non-destructiveevaluation NDE as well as transducers used in theupperMHzrange e g surface-acoustic-wave (SAW) devices In the past the development of electro-acoustic transducers was primarily based on trial and er-ror whichistime-consumingandthereforeexpensive This kind of development is not consistent with modem industrialengineeringpractice which is to aiddevelop-ment by computer simulations for the theoretical predic-tion of the propertiesexpectedtoresult from agiven transducer design

The main purposes of computer simulations in trans-ducer development are

Optimization of transducer design without time-con-suming experiments Evaluation of new materials in device design Deeperinsightintothewavepropagation in piezo-electric solids

The models commonly used to simulate the mechanical and electrical behaviorofpiezoelectric transducersgen-erallyintroducesimplifyingassumptions thatareoften invalid for actualdesignsThegeometriesof practical

Manuscript receit)cd Deccrnbcr23 1988 reviwd J u l j I O 1989 and Oc-tober 1 1989 acctptcdOctober 2 5 19x9

The author is w i t h Slerncn AG AFE-TPH 41 Postlach 3220 D-8SlO Erlangen West Germany

IEEE Iog Nunlber 9034355

M E M B E R IEW

transducersareoftentwo- (2-D) o r three-dimensional (3-D) The most popular models such as Masons model or theKLM-model 111-191 howeverare only one-dimensional ( 1 -D) For the 2-D or 3-D simulation of piezoelectric media the complete set of fundamental equations governing piezoelectric media has to be solved The finite difference or finite-elementmethodsarehowever suffi-ciently general to handle these differential equations The finite element method was preferred here because it is ca- pable of handling complexgeometriesHitherto mainly results of 2-D piezoelectric finite difference or finite ele-ment simulations have been reported in the literature [ 101-[ 141 Thegeometricaldimensions of practicaltrans-ducershoweveroftendemanda full 3-D description Thus we have implemented an analysis scheme for piezo-electric media with no restrictions other than linearity Thc major advantages of our finite element calculation scheme comparedtootherpiezoelectric finiteelementsoftware e g [ 151 are the availability of 2-D and 3-D piezoelec-tric finite elements as well as the capability of computing transientresponsesandofhandlingstructureswithnon-uniform damping Our finiteelementanalysisscheme is the first to allow the handling of different 2-D as well as 3-Dpiezoelectricfiniteelementsforstaticeigenfre-quencyharmonic andtransientanalysis In transient analysis the damping coefficients may differ from element to element which is important for the computation of structures with locally nonuniformdamping coefficients as for example in array antennas with absorbing backing materialsWith thisanalysis schemepiezoelectricmedia with anisotropic material tensors and almost any geome-try can be calculated Telephone handset transducers [ 161 arrayantennasformedicalimaging [ 17) acousticdelay lines [ 181 and SAW devices [ 191 have already been suc-cessfully analyzed using this method We will concentrate below on the analysis of ultrasonic transducers as used in echographic systems

11 THEOKY PI~OEIIltCIRICFINITEELEMENTSO F

The matrix equations ( 1 ) relating mechanical and elec-trical quantities in piezoelectricmediaare the basisfor thederivation of the finite element model(vectors and matrices are printed in boldface)

T = CS - eE ( l a )

D = eS + E ~ E ( l b )

0885-3010900500-0233$01OO 0 1990 IEEE

T vectorofmechanicalstresses S vectorofmechanicalstrains E vector of electric field D vector of dielectric displacement c E mechanicalstiffnessmatrixforconstantelectric

field E E permittivitymatrix forconstantmechanical strain

S e piezoelectricmatrixsuperscript meanstrans-

posed

The electric field E is related to the electrical potential by

E = -grad ( 2 )

andthemechanicalstrain S tothemechanicaldisplace-ment U by

S = Bu

where in the Cartesian coordinates

B =

Theelasticbehavior of piezoelectric media is governed by Newtons law

DIV T = pauat ( 5 )

where

DIV is divergence of a dyadic [20]and p is density of the piezoelectricmedium

whereas the electric behavior is described by Maxwells Equation considering that piezoelectric media are insulat-ing (no free volume charge)

ditlD = 0 ( 6 )

Equations (1)-(6) constitute a complete set of differential equations which can be solved with appropriate mechan-ical (displacements andforces) andelectrical(potential and charge) boundary conditions

An equivalent description of the above boundary value problem is Hamiltons variationalprinciple as extended to piezoelectric media [ 2 l ] (221

6 L d r = OS ( 7 ) where the operator 6 denotes first-order variation and the Lagrangian term L is determined by the energies available in the piezoelectric medium

L E - E + E + W ( 8 )

with Elastic energy E

E t = 12 S T d V

and Dielectric energy Etl

Ecl = I2 SS D ~ d v

and Kinetic energy E

EL = jS1pic dV

where

U is the vector of particle velocityand V is the volume of the piezoelectric medium

The energy I+ generated by external mechanical or elec-trical excitation is defined

W = 11s u i f d V + $I ujfclA v I

- S +qsdA + c uF - x Qp ( 12) 1

where

fH vector of mechanical body forces [ Nm3] fs vector of mechanicalsurfaceforces [ N m ] FP vectorofmechanicalpointforces [ NI A areawhereforcesareapplied [m ] qS surfacecharges [ Asm] QP pointcharges [As ] A areawherecharges areapplied [ m 1

In thefiniteelementmethodthe body to be computed is subdividedintosmalldiscreteelements the so called fi-nite elements The mechanical displacements U and forces F as well as theelectricalpotential andcharge Q are determined at the nodes of these elements The values of thesemechanicalandelectricalquantities at an arbitrary position on the element are given by a linear combination of polynomial interpolation functions N ( x y z ) and the nodal point values of these quantities as coefficients [23] [24]For an element with I I nodes (nodal coordinates (xf yf z ) i = 1 2 H ) thecontinuousdisplacement function u ( u y ) (vector of order three) for example can be evaluated from its discrete nodal point vectors as follows(thequantities with the sign + are the nodal point values of one element)

u ( T J Z ) = N f f ( r y Z ) U(rfyf z f ) (13)

where

U is the vectorofnodalpointdisplacements (order 3 n ) and

N is the interpolation functions for the displacement

2 2 5

All other mechanical and electrical quantities Y are simi-larly interpolated with appropriate interpolation functions NWith the interpolation functions for the displacement ( N i l )and the electrical potential ( N + ) (2) and (3) can be written

E = -grad + = grad ( N a amp ) = -Baamp ( l4a)

S = Bu = BNli = BU (14b)

The substitution of the polynomial interpolation functions N into (7) yields a set of linear differential equations that describe one single piezoelectric finite element

mu + dU + kU + k6 = F B + F y + F P ( 15a)

k+u + k + + 6 = Q + Q p (15b)

where A A

ic ii Vectors of nodalvelocitiesaccelerations

Mechanical stiffness matrix

k= S1BcBdl ( 16a)

Mechanical damping matrix

d= CY15S pNN ciV

+ P BcBdl (16b)5 c Piezoelectric coupling matrix

k= BeB+ dV (16c)11S Dielectric stiffness matrix

kae = S51BcSB+ dV (16d)

Mass matrix

m = pNNdV ( 1 6 45 Mechanical body forces

Mechanical surface forces

Mechanical point forces

Electrical surface charges A

Electrical point charges

where

a ) p dampingcoefficients of element ( c )

fk external body force at element ( c ) f y externalsurfaceforce at element ( c )

F ) external point force at element ( P )

4 external charge at element ( dsurface Q) external point charge at element ( e )

The damping behavior of the element is determined by the damping matrix d( which can be introduced by standard finite element techniques 1231 In thegeneral case these matrices dcan be assembled from the damping proper-ties of the structure which are usually frequency depen-dent An arbitrary frequency dependence of the damping howeverrequires more than two dampingcoefficients Thiswouldresult in a fu l l damping matrix and conse-quently in a significant amount of computational efiort as reported in [ 2 3 ] In practice it is convenient therefore to approximate the damping behavior by (16b) Therewith four types of physical damping can be modeled accord-ing to thevalues of the dampingcoefficients a and 6 Theseare l ) theundampedcase ( a = 0 p = 0 ) 2) stiffness-proportional damping i e viscous damping ( a = 0 p gt 0) 3 ) mass-proportionaldamping ( a gt 0 p = 0) and 4)Rayleigh damping ( a gt 0 p gt 0 ) (see also (251) The magnitudes of the Rayleigh coefficients cy and p depend on the energy dissipation characteristics of the modeled structure Hysteretic damping for example can also be roughly approximated by suitable values of a and p In order to handle structures with non-uniform damp-ing the values of cy and p can be different from element to element Typical values of the Rayleigh coefficients for piezoceramic materials operated at a frequency of 1 MHz are CY = 75 and 3 = 2 X lo- For sound absorbing materials we increase the values of CY and (3 so that critical damping is obtained

The subdivision of the area or body to be computed into finite elements results in a mesh composed of numerous singleelementsThecomplete finite-elernent mesh of a piezoelectric medium is mathematically described by a set of linear differential equations with symmetric band struc-tureHere thequantities U CD FB F Q s and Qp are the globally assembled field quantities and no longer ele-ment quantities asthosemarked by an in (15) and (16)

Mii + Du + Ku+ K+ = FR + F s + FP (17a)

K L u + K++ = Qs + QP (17b)

According to the theory of conventional mechanical finite elements(seeforexample [23] 1241) the matricesand vectors describing the whole mesh (( 17)) result from as-sembling the vectors and matrices of the single elements ((15))of which the mesh is composed If the whole mesh contains n t n t nodes matrix equation (17a) will consist of 3 IZltand matrix equation (17b) of n linear equations This is because the mechanical description of a body re-

quiresthreevectorcomponents (egdisplacement) whereas for the description of a quasi-stationary electrical field a single scalar quantity (eg potential) is sufficient

Thesolution of (17) yieldsthemechanicaldisplace-ments U andelectricalpotentials in the piezoelectric medium The two sets of linear equations (17a) and (17b) are coupled by the matrix K+ and split into two separate matrixequations with diminishing piezoelectricity (pi-ezoelectricstresstensor e = 0 --$ K = 0 ) These two separate sets of equations describe respectively pure me-chanical finite element models already known from struc-tural mechanics [23] ((17a) with K+ = 0 ) and models of electrostatic field problems ( (17b) with Kl+ = 0 )

Besides these nodal or local results we further evaluate integral quantities such as the electrical input impedance or electromechanical coupling coefficient characterizing a piezoelectric transducer

Electromechanical Coupling Coeflcient The electromechanical coupling coefficient k is defined

in [2]

7k - G I

E Ed = __ ( 1 8 )

where E ismutual energy E is elastic energy and Ed is dielectric energy

In terms of piezoelectric finiteelementmatrices the three energies are written

E = f ( uK+ + K+u) (19a)

Ed = K+ (19c)

The magnitude of electromechanical coupling of a vibra-tionalmoderepresents thesignificanceof thatparticular mode compared to the other modes If the coupling of a certain mode is of the order of 50 or higher that mode will be strongly excited The larger the electromechanical coupling coefficient of the mode of interest the lower will be the insertion loss and the broader the bandwidth of the transducer

Electrical Impedunce Theelectricalimpedance is anothercharacterizing

quantitywhichcanmoreoverbeverifiedexperimentally without undue effort since impedance measurements can easily be carried out with a network analyzer by sweeping thefrequencyandrecordingthe real and imaginarypart of the impedance The input impedance of a piezoelectric transduceralsoreveals all theresonancesandantireso-nances of the device The resonances are the natural fre-quenciesforshort-circuitedelectrodeswhiletheanti-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 Thustheresonancefrequencies ( f ) arerepresented re-

spectively by the minima and the antiresonance frequen-cies ( f a ) by the maxima of the electrical input impedance Eventheelectromechanicalcouplingcoefficientcanbe determined from resonances and antiresonances using an approximated formula (21

Equation (20) is strictly validonly for pure one-dimen-sionalvibrationmodes (21 eg a pure thickness mode whereas the definition by ( 1 8) is exact without qualifying assumptionsTocompute theelectrical inputimpedance ofapiezoelectrictransducer with finite elements the transducerhastobeexcited by adeltafunction of the electricalcharge at oneelectrodewhile the other is grounded

Q ( [ )= Q J ( t gt (211 where Q is the amplitude of charge pulse and 6( t ) is the DeltafunctionTheelectricalimpedance Z ( o) is then given by (22) (since I ( t ) = d Q ( r ) d t )

wheref +c ( t ) is the Fourier transform of the electrical potential at the excited electrode In this case the appli-cation of a delta function charge is superior to a step func-tion charge since the computations for a delta pulse re- quire less data storage

Average Displacernmt In practical transducer development it is useful to de-

fine an integral quantity that characterizes the mechanical output of the transducer One integral result which can be used to quantify the mechanical response is for example the average displacement of a region of interest e g the averagedisplacement of the soundemittingface of the transducerTheaveragedisplacement is computed by summing the displacement amplitude spectra U( o)of all thenodes i belonging to the regionofinterest The re- sulting average amplitude spectrum U ( W ) thenrepre-sents the strength of the various vibrational modes in re-spect to the mechanical output of the considered region

Amplitude spectrum of mechanical displacement at node i Node i belongs to thatface of ele-ment j which is a subarea of the region of interest

AAJ Area of that face of elementj that is a subarea of the region of interest

N Number of nodes belonging to the considered ele-ment face

N Number of elements belongingtotheregion of interest

Ai Totalareaofregionofinterest

111 I M P L F M E N l 4 T I O N O F THl T H t O R Y

The theory described above was implemented in FOR-TRAN routines supplementedtoafiniteelement c o n -puterprogramdeveloped at our computercenterThe original version of this program was used to solve prob-lems of structural mechanics Up to now we have imple-mentedthefollowingtypes of piezoelectric finiteele-ments2-Dplane-strainelement ( 3 to 8 nodes)2-D plane-stresselement ( 3 to 8 nodes)axiallysymmetrical element ( 3 to 8 nodes) and 3-D elements (4 to 27 nodes) Plane-strain conditions imply the absence of strain in the third (neglected) geometricaldimension (S = Sh = 0 E = 0 in ( la ) ) while plane-stress implies the absence of stress in that direction ( TI = T = 0 D = 0 ) This physically meansastructure with either a very thin (plane-stress) or an infinitely long (plane-strain) third dimension For the implementation of these elements the mechanical electrical and piezoelectric anisotropies of the material are considered in toto by using the full material tensors

Forthenumericalanalysis of piezoelectricfiniteele-ments standard finite element equation solvers can be ap-plied because the matrix equation ((17))to be solved ex-hibits symmetric band structure As forconventional mechanical finite elementsweapplyforexamplethe subspace iteration method [23] to calculate the natural fre-quencies and their related mode shapes Further the stan-dardNewmarkstep-by-stepintegrationmethod [23] is usedto computetransientresponsestomechanical(dis-placement or force) or electrical (charge or voltage) ex-citations An algorithm forsolving (17) in complexfor-mulation has also been implemented in order to compute the responses to harmonic excitation In transient and har-monic analysis the damping coefficients may differ from element to elementThisdemandsmorecomputing power for the matrix D (( 17)) has to be evaluated and stored separately It is moreover a very important feature for simulating structures in which damping is locally non- uniform In none of our computations did we experience numerical difficulties due to piezoelectric finite elements Thus the implementation of the theory of piezoelectric finite elements in other standard finiteelementsoftware would seem to be practicable without major problems

In finite-element analysis i t is furthermore necessary for appropriate pre- and postprocessing software to be avail-able for theconvenienthandlingofthestructuresto be analyzed The preprocessing software should support the interactive generation of finite element meshes at a graphic workstation Once finite-element analysis hasbeen com-pleted by the kernel finite element program an appropri- atepostprocessingsoftware is needed toconvertthenu-meric values determined at the nodes of the finite element mesh into graphicaloutputSincethepre- andpostpro-cessing software at present commercially available is un -

able to handle piezoelectricproblems i t was necessary for us to develop appropriate software

Iv c O N F I R M T l O N OF THk cAICIIL4I lON S C H l M t lt

First four examples are reported that prove the validity of our calculation scheme by comparison with theoretical and experimental data given in the literature

E-rmtr1plc I

Jungermann e t N I havetheoretically and experimentally examined piezoelectricparallelepipedarraytrans-ducers (261 andcomputedtheresonanceandantireso-nancefrequencies o f theparallelepipedbars by using series-approximations In Table 1 the results are compared with our own piezoelectric finite element calculations

Eranlple 2

Thisexample was used to test ouraxisymmetricele-ment EerNisse 127) has presented numerical calculations for piezoceramic ( BaTi03) disks based on Taylor series approximations while Shaw 1281 has reported related ex-perimentalresultsThediagramofdispersionandthe shape of the sixth eigenmode of a BaTi03-disk I271 c o n -puted with our axisymmetric element are compared (Figs 1 and 2) with the results of EerNisse and Shaw

E-rarnpltgt 3

Boucher et t i l havereported the simulation o f piezo-electriccubes using a mixed finiteelement-perturbation method [ 121 In Table I1 Bouchers theoretical and exper-imentalresultsarecheckedagainstown 3-D finite element calculations as well as latestrelatedfiniteelement results reported by Ostergaard ( 151

Esutnple 4

The resonances of an electromechanical Langevin-type transducerwerealsocalculated with our finite element software and compared with the results given in [ I S ] and (291 (Table 111) Like Kagawa 1291 we used 2-D axisym-metric elements whereas Ostergaard [ 151 modeled a small sector of the axisymmetric rod with 3-D elements

V APPLICATION CALCCIIATION 10O F T H ~ SCHFME TRANSDIJCEKSFOR ULTKASONICI M A G I N G

The quality ofultrasonicimages is known todepend greatly on theperformance of theelectromechanical transducers used In order to improve transducer charac-teristics we analyzed piezoelectricparallelepipedpiezo-ceramicbars as used in theultrasonicarrayantennas o f echographic systems In transducer development it is often assumed that the transducersvibrate likesimplepistons Thisishowever not correct in the following respects Firstthethicknessmodewhich is closest tothe ideal piston mode as to shape does not exhibit true piston be-havior at all Due to the strong lateral contraction o f con-

3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6

C m x l r r lh~ckncss4

-+--t---i--f--t i 34 06 08 l 0

ventional piezoceramic materials eg Poissons ratio = 04 for PZT ( 3 = thicknessdimension 2 = width dimension ) the thickness modes of piezoceramic vibra-tors mostly exhibit significant displacements normal to the thickness dimension The displacements along the width

dimensionareconsequentlynonuniformSecond still other modes are often excited whose strength depends on the transducer geometry These modes are of parasitic na-ture and greatly differ from piston-like behavior

A further more general problem in the analysis of electromechanical devices is that their vibrational modes can very rarely beassigned to puremodessuchasforex-ample the l-D thicknessmodeTheactualeigenmodes of complex vibrators are often a mixture of different pure modes To obtain deeper insight into thephysicalmech-anisms of such vibrationswehavecomputedtheeigen-modes of parallelepiped piezoelectricbarswithgeome-triessimilar to those generallyused in ultrasonicarray antennasThese bars are typically so longthattheirei-genmodes in the length dimension appear far (at least by a factor of 10) below the frequency range of practical in-terest Since these modes and also their harmonics are all weakly coupled it is not necessary to consider thembe-lowHowever it will be shownthat it is generally not permissible to neglect the length dimension on account of its influence on the modes of interest

A Mechaniml Di~l~lar c~tnPt~t ~

First of all the lowest eigenmodesofparallelepiped piezoceramic bars have been analyzed by 2-D as well as 3-D simulations with respect to displacements and electric fields The permanent polarization of the bars as well as the applied external electric field are aligned in the thick-ness dimension The thickness modes of bars with width- to-thickness ( W T ) ratios of W T = 05 and W T = 20as well asthewidth-dilatationalmodefor W T = 20areshown in Figs 3-5 Theelasticdielectric and piezoelectric constants of the utilized piezoceramic ma-terial (Siemens-Vibrit-420) are givenin the appendix For narrow elements ( W T lt I ) the thickness mode always corresponds to the first and for wider bars ( W T gt I ) to the second natural frequency The mode switch occurs in the region W T = 1 sincetheW lt Tthecondition formechanicalresonance ( h2-resonance of a non-clamped bar) is first fulfilled in the larger thicknessdi-mension For W gt T the first resonance condition is met in the broader width dimension Consequently the thick-ness mode of bars with W gt T corresponds to the second eigenfrequency ForWT-ratiosgreater than 30 the thickness mode even corresponds to the third natural fre-quency As will be shown later the thickness mode is the mode of interest for ultrasonic imaging applications The resultsof the 3-D analysishavealsobeencomparedto related 2-D calculationsforwhich thelength L of the bars was assumed tobeinfiniteThenaturalfrequencies obtained with 2-D simulations typically differ by n o more than 1 fromthecorrespondingvalues of 3-D simulations as long as the length L of the bar is at least ten times greater than both its width and its thickness We discov-ered no differences between the 2-D mode shapes and the cross sections at I = L 2 of related 3-D modes ( in Figs 3(b)4(b) and 5(b)) Nevertheless theassumption that 2-Dsimulationscanadequatelydescribetheelasticde-formations of such vibratorsdoes not hold becausethe often considerable displacement gradients along the length dimension cannot be considered in 2-D calculations The analysis shows that thedisplacementsof suchbarseven the displacements of the thickness modes (conventional piezoceramicmaterialsassumed)are not constant along the lengthdimension (Fig 3(a) (c)) This is even true if the length of the bar is ten times greater than both its width and its thickness The observed displacement ripples along thelengthdimension (Fig 3 ) are of importance because they influence the emitted sound field

The computed eigenmode shapes of these bars have also been experimentally verified by laser interferometric mea-surementsThe normaldisplacementsweremeasured in thewidthdimensiononthetopelectrodeswhichwere polishedtoobtainhigherreflectivityforthelaser beam Computed and measured eigenmode shapes of piezocer-amic bars with various WT-ratios are compared in Fig 6 Foraconvenientcomparison of computed andmea-suredmodeshapesthefollowingprocedurewas chosen the computations were done by eigenfrequency analysis

MechanicalEqulpotentlal LlnesDlsplacements of the Electrcal FleH

- I

Mechal lcal Dlsplacements

( C l

whereas the measurements were performed in continuous-wave mode at the resonance frequencies to reproduce the mode shapes Theseresonancefrequenciescan be ob-

Mechanlcal Displacements

Mechancal Dlsolacements


Equlpotentlal Llnesof the Electrlcal Fleld

Geamely Walh 4mm Thickness-Zmm Lengll-50mm MateW S~emers ~br~l420

(h)

Mechanical Dlsplacemenrs

Equipotential Lines at the Electrlcal Fleld

Geometrf W1dth-4mm hckness-Pmm Lenglk-50mm Materla1 Semes-ibrl 420

( C )

Fig 5 Width-dilatational mode of a piezoceramic bar with W T = 20 ( a ) 3-D mode shape ( h ) Cross section at I = L = 25 mm (c ) Sagittal section tained from the input impedance results of Fig 1 1 Since

an eigenfrequency analysis delivers no absolute displace-ment values it was necessary to normalize measured and computeddisplacementsTheresults of Figs 6(a) (c) (d) were normalized with respect to the maximum values whereas those of Fig 6(b) were normalized with respect to the minimum value The normalization factors which wereevaluatedforeachmeasurementaregiven in the figure captions of Fig 6

B ElectromechunicalCoupling Thedependence of electromechanicalcouplingon

transducer geometry is often used to optimize design In Fig 7 the electromechanical coupling coefficients for the five lowest modes of a piezoceramic bar are displayed as a function of the W T ratio One discerns the maximum coupling of the thickness mode for W T = 06 At W T

L E R C H SIMULATION OF PIEZOELECTRIC DtVICES BY - I ) A N D 3-1) FILII f kltlFh IS 24 I

I Theoretlcal result1 0 -

l P

F m resonance + Dtsplacemenl (narmalhed)

l 1 0

0 5

c

Second resonance f Dlrplacemenl (normallzed 1

l

Fig 7 Electromechanical coupling coefiicicnt o f long piezoelectric bars

= 06 the thickness mode is maximally excited while all othermodesarelargelysuppressedThis can be con-firmed by comparing the mode histogram for W T = 06 with any other for example W T = 20 (Fig 8) For W T = 20 thetotalenergy is split intoapproximately equalpartsamongseveralmodeswhereas for theopti-mum WT-ratio of 06 it predominantly concentrates on the thickness mode Usually the transduceris designed for vibration in one single mode as obtained for W T = 06 With this optimum W T-ratio array elements generate and detectultrasoundsignals with optimum efficiency since most of theelectricalenergy is converted intoanormal displacement of the sound emitting face (see also the shape of the thickness mode for W T = 05 in Fig 3(b)) Thus the thickness mode is the one of interest for imaging ap-plications

I 1110 1 1

000 1 2 0 3 6C 4 EO

C Dialtgrcrtnof Disprrsiotl For designing array transducers an exact knowledge of

the propagation velocities of the various vibrational modes is essential becausewithknownvelocitiesandspecified geometrical transducer dimensions the frequencies of the variousmodescanbeeasilydeterminedFiniteelement simulations are therefore used to calculate the diagram of dispersion for piezocerarnicmaterialsofpracticalinter-est Fig 9 shows a diagram of dispersion for the two low-estresonances of barsmadeof typicalPZT-SA-like pi-ezoceramicmaterialTheproductfrequency thick-ness expressed in ( M H z mm) equals half the velocity of sound expressed in ( lo3 ms )

In the following we declare that mode as main mode in whichexhibits the maximumaveragedisplacement U(W ) ((23)) of the sound-emitting front electrode The main mode is represented by the bold line in Figs 9 10 and 14 Forpiezoceramic barssuch as those considered here the thickness mode is always the main mode because thepermanentpolarizationofthepiezoceramicmaterial as well as theexcitingelectrical field are aligned in the thicknessdimensionFig 9 suggestssimplifiedmode shapes for the two lowest resonances of a long piezocer-amic bar One should however bear in mind that the real modeshapes of vibrators alwaysdifferfromthesepure modes For sometransducergeometries the differences are so greatthattheassignment of puremodeshapes would seem to be senseless

In a typical design for array transducers one first deter- mines in the manner already described the WT-ratio of the parallelepiped transducer elements required for max-imum electromechanical coupling Then the thickness of the vibrator is determined correspondingtothevelocity of the fundamental thickness mode (Fig 9) and the spec- ified operating frequency of the antenna The spectral po- sitions of the other parasitic modes are afterwards deter-mined with the aid of Fig 10 In Fig 10 thediagram in

Fig 9 is extended by higher resonancemodesTheve-locity of the pure l-D thicknessmode as obtainedfrom Masons model 1 1 1-[2] is also indicated in Fig 10 The differencebetweenthisvelocityandtheactualvelocities of the thickness mode shown in the diagram of dispersion demonstratesthat l -D models may prove ratherinade-quate for the simulation of actual transducer designs

DElectricalItnprdance

The electrical input impedances Z ( W ) of the bars were computed according to (22) and compared with measurements (Fig 1 1 ) Dampingvalues of 01 = 75 and 0 = 2 X lo- were used in calculationsacrossthefrequency range shown The close agreementbetweensimulation and measurements confirms once more the validity of our simulationprocess I t is evident that forsuch agood agreementbetweencalculationandexperimentthepre-cisedata of thematerial tensorsareclearlyessential In contrasttothesimpler ID models generally used for transducersimulationthefinite-elementanalysis is able to correctly compute all physically present modes

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

( a )

- 20

-30

- 40 -- 50 -- 60--iO --80- Theotellcal result

-90j

In the shown frequency band the narrow bar ( WT = 025 in Fig I l (a)) exhibits only a single strong thickness mode whereas its higher modes lie outside this frequency rangeFor W T = 05 (Fig 1 l)) the thicknessmode is again strongly coupled but the second and third vibra-tionalmodesalready appear in thefrequency band of i n -

terest As predicted by computations of the electrome-chanical coupling coefficient (Fig 7) these modes are only weakly coupled ForWT-ratioslarger 08 however thesemodesarestronglycoupled (Fig I I(c)) whereas the coupling of the first mode is slightly reduced This is oncemoreconsistent with the resultsobtainedforthe electromechanical coupling coefficient (Fig 7)

E Bllcking

The piezoelectric transducer elements of imaging array antennas are typically provided with a backing which has to be considered in any realistic simulation The functions of such a hacking are mechanical support and sound ab-sorption A backing damps resonances due to the transfer of acoustic energy to the sound-absorbing backing mate-rial The energy transfer is determined by the ratio of the acoustic impedances of the piezoceramicmaterial to the epoxy backing [1]-[9]Fig 12 shows the influence of the acoustic backing impedance on the mechanical output of an arraytransducerTheaveragedisplacement of the transducersfrontface is used as an integralresultto quantify the mechanical output of the transducer We ob-serve that thehacking influencesprimarily the thickness modeandto a farsmallerextentthewidth-dilational modeDueto thepreferredlateraldisplacementofthe width-dilatational mode only a small fraction of its energy is transferred to the backing The width-dilatational mode is a high-Q mode because most of its mechanical energy travels back and forth between the free sides of the ele-ment

Theelectrical inputimpedance of piezoceramic bars with backing likewise demonstrates the considerable dif-ference in dampingbehavior of the variousvibrational modesFig 13 shows the computedandthemeasured electrical impedance of such a bar The amplitudes of the various modes show the damping of the width-dilatational mode(lowesteigenmode)to beslight in relationtothe other modes for the reasons discussed above

Thediagram of dispersion for anarrayelement with backing also differs from that of the array element alone (Fig 14) The resonance frequency is approximately 5 lowerduetomassloadingandthemainmodeswitches hacktothe first natural frequency in thecaseoflarger WT-ratios This second switch occurs because the width-dilatationalmode(firsteigenmodefor W T gt I O ) is lessdamped than the thicknessmodefor the reasonsal-ready explained Due to the stronger damping theband-width of the thicknessmode is larger thanthatforthe width-dilatationalmodeThe WT-ratio at whichthe main mode switches back to the first eigenmode (Fig 14) depends on the acoustic impedance of the backing

In the presence of a backing we observe the mass-spring mode originally introduced by Larson (301This is an os- cillationofthealmostundeformedtransducerelement (mass)against themechanicallymuchmorecompliant

06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig

7 Electrical impedance ( R ) I I

I T = 2rnm -

mm

W L 4mm

100

v Theoretlcal result - 7

Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j

Fig 13 Electricalinput impedancc nf picroccramic bar with baching

backing (spring) The frequency of this mode can be determined by using an approximation formula given in [30] The exact finite element calculation confirms the essential

P e m Elernems

l

E D W Racklna

Fig IS Mas-pring mode o f piemccramic transduccrarray with backlng

validity of this formula but the exactly computed frequen-ciesaregenerallyabout 30 lower The mode shape of a typical mass-springmode is shown in Fig 15 This mode is slightly damped and always hasalower fre-quency than the main transducer mode It may disturb the sonographicimage by adding clutter 1301 whichcan however be avoided by using high-absorptionbacking material

F Electric r r t d Mechmicwl Cross Colrplirlg

Cross-coupling between the transducer elements of an ultrasonicarrayantenna degrades the sonographic image scanned by the antennaKinoandDeSilets haveshown that cross-coupling between neighboring elements should be below -30 dB in antennas for which a wide angle of acceptance is required [311 but conclude that reasonably satisfactory results can even be obtained with cross-coupling in the range of -25 to -20 dB Finite element sim-ulationsare used to analyze thephysicalmechanismof cross-coupling in array antennas

The purely electriccross-couplingbetweentwoele-ments of anultrasonicphased-arrayantennawassimu-latedwithsuppressedmechanicaldegrees of freedom Fig 16 shows the cross section of the transducer config-uration for different saw-cuts In each of the three systems a pair of PZT-transducers is provided withepoxyback-ing The upper electrodes of both vibrators are electrically groundedwhilethelowerelectrode of each left-hand transducer is energized by an electric potentialof 1 V The voltage appearing across the lower electrode of each right-handtransducer is induced by electric cross-cou-plingFig 16 shows the crosstalkforthree different

LERCH SIMULATIOh OF PIEZOF1ECTRIC DEVICFS B Y -DA K D 3-D FINITE ELEMENTS 245

I

Fig 16 Electrlc croswxupling between transducers of arrayantenna

depths of saw-cut which reaches a maximum of -43 dB betweenthetwoneighbouringvibratorsas in the case where the backing is without a saw-cut If the saw-cut is as deep as the vibrator is thick the cross-coupling due to electrical effects almost vanishes (crosstalk -61 dB)

In a further computer operation the cross-coupling due toelectrical andmechanicaleffectswascomputedThe responseofapulse-excitedtransducer(element n in Fig 17) is for this purpose calculated and compared with the responses of the neighbouring elements (elements n + 1 to n + 4) Resultsareobtainedforantennas with cut and non-cutmatchinglayersWefurthercom-pare configurations in which the saw-cut ends at the top of the backing withthose in which the saw-cut in the backing is as deepasthevibrator is thick During pro-duction these cuts may be filled with the epoxy adhesive used to fix the matching layer to the top of the piezocer-amic transducers This case was also analyzed (Fig 17) Fromtheresultsobtainedcross-coupling may be con-cluded to be present at most in the case of non-cut match-ing layers Thisindicates thatcross-couplingderives mainlyfrommechanical wave propagation in the match-ing or protective layer In-depth investigations show sev-eraldifferentvibrational modes mainly Lamb modes to be involved in mechanical cross-couplingThe results (Fig 17) show a saw-cut in the backingmaterial to lead toasubstantialreduction in cross-coupling only in the case of cut matching layers If the matching layer is cut theelectriccross-couplingthroughthebackingmaterial will also be a major source of coupling Mechanical wave propagationthroughthebackingishoweveronlysig-nificant when a low-absorption backing material is used

In view of the results reported by Kino and DeSilets the simulation furthermore predicts that the filling of saw-cuts by epoxy materialintroducesthe risk of visibleimage degradation due to strong mechanical cross-coupling es-pecially if theapplicationrequiresawideangle of ac-ceptance

Finally the influence of saw-cut geometryoncross-coupling is analyzed Fig 18 showsthecross-coupling betweentwoneighbouringarray elements with cut and non-cut matching layers The results are shown as a func-

Number of element 4

Fig 17 Overall(electricalandmechanical)cross-couplingbetween transducers of array antennas

-70- --- sawcut wldth vibrator ~ d l l - - 0 2 saw-cul wdlh vlbralof width 03

-80-t 0 1125 0 5 075 10

Sa+cul depth vlbralor 1hlkness-

Fig 18 Influences o f raw-cutgeometry on cross-coupllng

tion of thesaw-cutdepth with the saw-cutwidthaspa-rameterCross-coupling is seen to beinfluencedgreatly by the saw-cut depth only in the case of a cut matching layerwhereasalreadyexplained it is primarily deter-mined by electrical crosstalkWefurtherdiscerned that the cross-coupling is only partially influenced by the saw-cut width

VI CONCLUSION

A finite-elementcalculationschemeforthe2-Dand 3-D simulation of anisotropic piezoelectric media is presented Using this method the natural frequencies and re-lated eigenmodes as well as the dynamic responses to me-chanicalandelectricalexcitationscanbecomputedfor piezoelectric transducers of almost any geometry The validity of the simulation scheme has been confirmed by data reported in theliteratureas well as by in-house experi-ments

This finite-element analysis method allows the solution of numerous problems encountered in piezoelectric trans-ducer design One of the main problems that arise in pi-ezoelectric sensor design is the simultaneous appearance of various Vibrational modes with quite different physical characteristics In many casesthesemodescan only be

sufficiently described by 3-D analysisasshownhere I n theoreticalmodeling i t shouldalwaysbeborne in mind that the standard 1 -D models are only applicable i f two or the three geometrical dimensions can be neglected They will not yield correctresults if differenttypes of vibra-tionalmodes appearsimultaneouslyThe finiteelement method however yields correct results even if more than one type of vibrational mode is essential for the operation of the transducer The simulations allow a deeper under-standing of thephysicalmechanisms ofacoustic wave propagation in piezoelectric sensors and actuators We use suchsimulations to optimizetransducerdesignwithre-spect to efficiency bandwith angle of acceptance cross-coupling etc

Theimportance of these computersimulations isstill growing in transducerdevelopmentas well as in many otherbranches of technology Thisdevelopment is sup-ported by the continuouslyincreasingpower of modern computerequipmentFutureperspectives in this sector can be seen in the use of such simulations in combination with appropriate computer-aided design (CAD)

APPENDIX Material data of the used piezoelectric material VIBRIT

420

Density p = 7600 kgm-3

Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1

Piezoelectric constants

0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0

Dielectric constants

O 0 7 2

ACKNOWLEDGMENT Theauthor is grateful to Prof Dr H Ermert(Ruhr

UniversityBochum)Dipl-Ing W FriedrichDr H

Kaarmann Dr P Kraemmer B Sachs and Dr H von Seggern for fabricatingtransducersperformingexperi-ments and for valuable and stimulating discussions

REFERENCES I l] W P Mason Eccrro-Mechtrrllctrl 7rtrrrsrlrtcerv ond WNW F i h m

third ed PrincetonNJD van Nostrand 1948 121 D A Berllncourt D R Curran H Jatle Piezoelectricand pi-

ezomagneticmaterials In Prricu 4 ~ 0 ~ 4 s r i c v o l 1 Part A New YorkAcademicPress1964pp233-256

131 R Krimholtz D A Leedom G LMatthaei New equivalentcir-cuits forelementarypiezoelectrictransducers E I P ~ O J I L c f t vol 6p3981976

141 G Kossofi The effects of hacking and matching o n the pertormancc of piezoelectric ceramic transducers lEEE Trltl11sSo~nc Ltrrlo i r o i SU-13pp20-30Mar1966 E K Sittig Effects of bondingandelectrodelayersonthe trans-mission parameters o f piezoelectric transducers used in ultrasonic dip-ita1 delay lines lEEE Truru Sotrics U r u o t r vo l SU-16pp 2-IO Jan1969 J Souquet P Defranould J Dehois Design o f low-loss wide-hand ultrasonictransducersfornoninvasive medical application f E E E 7rcrrrr S o r u c r U h r ~ ~ m r SU-26 pp75-81Mar1974 v o I T M Reeder D K Wllson Characteristic o l microwave acourtlc transducer for volungtc waveexcitatmn IEEE Trcrm Microwrrw Theory Tcchrr v o l MTT-17pp927-941 N o v 1969 C S Desilets J DFraaer G S Kino The design o f etficient broad-bandpiezoelectrictransducers IEEE Trtr~r~Sfmi( Lrrr- r o l l v o l su-2s pp 115-125 1978 R Lerch Simula~ion rtm Ultraschall-Wnndlern A c ~ r r r i c t r c o l 57 pp 205-217 1985 Y Kagawaand T YarnahuchiFinite elernent simulation of ~ W O -

dimensionalelectromechanicalresonators IEEE 7r~rmS o l i t s U- trusorr v o l SU-21 pp275-283 Oct 1974 H Allik K M Wehman J T Hunt Vlbrational rcsponw ofsonar transducers using piezoelectric finite elements JAcrmcr h An v o l 56pp1782-17911974 D Boucher M Lagier and C MaerfeldComputation of the v i - brational modes for piezoelectric array tranducei-s using a mixed ti-niteelement-perturbationmethod lEEE Trm S o r l i c v Ulrrtrsot l VOISU-28 pp 318-330 1981 M Naillon R Couraant and F Besnler 4nalyis 01 piczocleclrlc structures by a finite element method Arrtr Ectmur v o l 25 pp 341-3621983 E Langrr Anregung und 4uabreitung clektroakustischer Wellen in piezoelektrischen Kristallen thesis University 0 1 Vienna Vienna 1986 D F05tergaardand T P Pawlak Three-dimensional tinite cle-menta for analyzingpiezoelectric htructure5 i n ProltIEEE Uifrtr- s o n S ~ n r p Williamsburg V A 1986pp639-642 R LerchModerntechniques i n electroacoutictransducer d e w -opment in Proc 9th CWI~A c o r r v r i r s BudapestHungaryMay 1988 - -Berechnung des Schvingungsverhaltensplezoclcktrischer Korper mit einemVektorprozessorSiemensForchungs- und Entwicklungsberichte 15 BerlinSpringer-Verlag1986 pp 234-2 3 8 W Friedrich R Lerch K Pretcle and R SolJnerSirnulations ofpiczoelcctrlc lamb wave delay lines using a finite element Inethod f E E t Trot UlrOon F]-lt n o 3 pp 748-254corir3Ol 3 May 1990 - Finiteelementa~rnulatrons 01- S4W-tmnsduccrs i n Proc lEEE Conf Srf Crrcuirs HelsinkiFinland June 1988 pp 1399-1403 A Korn and T M Korn Mtrfhcrntr t ic t r H t r r r t h o k j i r k ier l r i r t c trr t i Etrqitwcrs New York McGraw-Hill 1Y68 H F TierstenHamiltonprinciple For linear piezoelectric nlc-dm in Prot IEEE 1967 p 1523 H Allik and T J R Hughes Flnite element method lor piczoelec-tric vibration h r J Nurrt Metlr E t ~ g v o l 2 pp 151-157 1970 K 1 Batheand E Wllson V~rnrrric~rMrrhods i n Firic E l c r ~ r r r Antryvi Englewood Clitis NJ Prentice-Hall1976

247 LERCHSIMULATION OF PIEZOELECTRICDEVICES B Y 2-D AND 3-D FIVITEELEMENTS

[24] 0 C Zienkiewicz The Finite EIetnerzr Merhod i n Etyqtwering Sci-m c r New York McGraw-Hill1971

1251 H Kardestuncer Finirr Elrmcwt Handbook New YorkEd McGraw-Hill1987 p 4-77

1261 R LJungermann P Benett A R Selfridge B T Khuri-Yakub and G S KinoldquoMeasurement of normalsurfacedisplacementfor the characterization of rectangularacousticarrayelementsrdquo J Acousr Soc A m vol 76 1984 pp 5 16

1271 P EerNisseldquoVarlationalmethodforelectroelastic vibration analy- sisrdquo f E E E T r a m Sot1ics Ulrruson vol SU-14 pp 153-160 1967

1281 E A Shaw ldquoOn the resonant vibrations of thickbariumtitanate disksrdquo J Acoust Soc A m vol 28pp38-501956

(291 Y Kagawaand T YamabuchildquoFiniteelementapproachfora PI-ezoelectric circular rodldquo lEEE Truns Sonics Utrcr ~on vol SU-23 pp379-3851976

[30] J DLarson ldquoA new vibration mode in tallnarrowpiezoelectrlc elementsrdquo in Proc JEEESytnp 1979 pp 108-1 13

1311 G S Kino and C S DeSilets ldquoDesign of slotted transducer arrays withmatchedbackingsrdquo (ilrrason Imaging pp 189-209 1979

ReinhardLerch (rdquo85) wasborn i n WestGer-manyin 1953Hereceived his masters in 1977 and his PhD degree in 1980 all in electrical en-gineering from the University of Darmstadt West Germany

From1977 to 1981he wasengaged in the de-velopment of a new type of audio transducer based onpiezoelectricpolymerfoilsat the Institute of Electroacoustics at Darmstadt UniversitySince 1981 he is employed attheSIEMENS-Research Center in Erlangen West Germany where he has

implementednewcomputertoolssupporting thedevelopmentofpiezo-electric transducers His latest work is concerned with finite-element sim-ulations of piezoelectrictransducerswhichare used in medical imaging and communication engineering

Dr Lerch is member of the Acoustical Society of America In 1982 he got the Award of the German Nachrichtentechnische Gesellschaft for his work about piezoelectric audio transducers

T vectorofmechanicalstresses S vectorofmechanicalstrains E vector of electric field D vector of dielectric displacement c E mechanicalstiffnessmatrixforconstantelectric

field E E permittivitymatrix forconstantmechanical strain

S e piezoelectricmatrixsuperscript meanstrans-

posed

The electric field E is related to the electrical potential by

E = -grad ( 2 )

andthemechanicalstrain S tothemechanicaldisplace-ment U by

S = Bu

where in the Cartesian coordinates

B =

Theelasticbehavior of piezoelectric media is governed by Newtons law

DIV T = pauat ( 5 )

where

DIV is divergence of a dyadic [20]and p is density of the piezoelectricmedium

whereas the electric behavior is described by Maxwells Equation considering that piezoelectric media are insulat-ing (no free volume charge)

ditlD = 0 ( 6 )

Equations (1)-(6) constitute a complete set of differential equations which can be solved with appropriate mechan-ical (displacements andforces) andelectrical(potential and charge) boundary conditions

An equivalent description of the above boundary value problem is Hamiltons variationalprinciple as extended to piezoelectric media [ 2 l ] (221

6 L d r = OS ( 7 ) where the operator 6 denotes first-order variation and the Lagrangian term L is determined by the energies available in the piezoelectric medium

L E - E + E + W ( 8 )

with Elastic energy E

E t = 12 S T d V

and Dielectric energy Etl

Ecl = I2 SS D ~ d v

and Kinetic energy E

EL = jS1pic dV

where

U is the vector of particle velocityand V is the volume of the piezoelectric medium

The energy I+ generated by external mechanical or elec-trical excitation is defined

W = 11s u i f d V + $I ujfclA v I

- S +qsdA + c uF - x Qp ( 12) 1

where

fH vector of mechanical body forces [ Nm3] fs vector of mechanicalsurfaceforces [ N m ] FP vectorofmechanicalpointforces [ NI A areawhereforcesareapplied [m ] qS surfacecharges [ Asm] QP pointcharges [As ] A areawherecharges areapplied [ m 1

In thefiniteelementmethodthe body to be computed is subdividedintosmalldiscreteelements the so called fi-nite elements The mechanical displacements U and forces F as well as theelectricalpotential andcharge Q are determined at the nodes of these elements The values of thesemechanicalandelectricalquantities at an arbitrary position on the element are given by a linear combination of polynomial interpolation functions N ( x y z ) and the nodal point values of these quantities as coefficients [23] [24]For an element with I I nodes (nodal coordinates (xf yf z ) i = 1 2 H ) thecontinuousdisplacement function u ( u y ) (vector of order three) for example can be evaluated from its discrete nodal point vectors as follows(thequantities with the sign + are the nodal point values of one element)

u ( T J Z ) = N f f ( r y Z ) U(rfyf z f ) (13)

where

U is the vectorofnodalpointdisplacements (order 3 n ) and

N is the interpolation functions for the displacement

2 2 5





mu + dU + kU + k6 = F B + F y + F P ( 15a)

k+u + k + + 6 = Q + Q p (15b)

where A A



k= S1BcBdl ( 16a)


d= CY15S pNN ciV




Mass matrix






where








K L u + K++ = Qs + QP (17b)






in [2]

7k - G I

E Ed = __ ( 1 8 )



E = f ( uK+ + K+u) (19a)

Ed = K+ (19c)






















E-rmtr1plc I


Eranlple 2


E-rarnpltgt 3


Esutnple 4




3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6


-+--t---i--f--t i 34 06 08 l 0








- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2























2 2 5





mu + dU + kU + k6 = F B + F y + F P ( 15a)

k+u + k + + 6 = Q + Q p (15b)

where A A



k= S1BcBdl ( 16a)


d= CY15S pNN ciV




Mass matrix






where








K L u + K++ = Qs + QP (17b)






in [2]

7k - G I

E Ed = __ ( 1 8 )



E = f ( uK+ + K+u) (19a)

Ed = K+ (19c)






















E-rmtr1plc I


Eranlple 2


E-rarnpltgt 3


Esutnple 4




3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6


-+--t---i--f--t i 34 06 08 l 0








- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2



























in [2]

7k - G I

E Ed = __ ( 1 8 )



E = f ( uK+ + K+u) (19a)

Ed = K+ (19c)






















E-rmtr1plc I


Eranlple 2


E-rarnpltgt 3


Esutnple 4




3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6


-+--t---i--f--t i 34 06 08 l 0








- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2
































E-rmtr1plc I


Eranlple 2


E-rarnpltgt 3


Esutnple 4




3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6


-+--t---i--f--t i 34 06 08 l 0








- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2























3 5 -

3 0-

i-

3-

0 5 I 1

1 3

I

4 I I I 5 6


-+--t---i--f--t i 34 06 08 l 0








- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2



























- I


( C l







(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2




























(h)




( C )







l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2

























l P


l 1 0

0 5

c


l



I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2























I 1110 1 1

000 1 2 0 3 6C 4 EO









( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2
























( a )

- 20

-30


-90j



E Bllcking





06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2























06Cl-

0 4 0 -

0 C~C- 1 I I I

05C 1 O C C O S C 300 3501 5 C

Fig


I T = 2rnm -

mm

W L 4mm

100


Measuremerll

10 I 01 0 5 0 9 1 3 17 21

Frequency (MHz j



P e m Elernems

l

E D W Racklna







I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2
























I






Number of element 4



-80-t 0 1125 0 5 075 10




VI CONCLUSION






420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2


























420


Mechanical moduli

p 4 9 101 98 00 00 001 149 98 00 00 00 I

I 2 2 00 I L 2- 4 1


0 0 0 0 117 0-( 0 0 0 117 0 0

-5 4 -54 135 0 0 0


O 0 7 2























simulation of piezoelectric devices by two- and three...

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