,FEE 'TR,\NSACTIUNS OH ULTI~ASONICS: F E R l t U E L E c r R I C S , AND FREQUENCY CONTROL: VOL. 49, N o . 5 . MAY 2002 573

Piezoelectrically Actuated Flextensional Micromachined Ultrasound

Transducers-I: Theory Gokhan Persin, Member, IEEE, a n d But rus T. Khuri-Yaknb, Fellow, IEEE

Abstract-This series of two papers considers piezoelec- trically actuated flextensional micromachined ultrasound transducers (PAFMUTs) and consists of theory, fabrica- tion, and experimental parts. The theory presented in this paper is developed for an ultrasound transducer applica- tion presented in the second part. In the absence of an- alytical expressions for the equivalent circuit parameters of a flextensional transducer, it is difficult to calculate its optimal parameters and dimensions and difficult to choose suitable materials. The influence of coupling between flex- ural and extensional deformation and that of coupling be- tween the structure and the acoustic volume an the dynamic response of piezoelectrically actuated flextensional trans- ducer are analyzed using two analytical methods: elassi- cal thin (Kirchhoff) plate theory and Mindlin plate theory. Classical thin plate theory and Mindlin plate theory are applied to derive two-dimensional plate equations for the transducer and to calculate the coupled electromechanical field variables such as mechanical displacement and elec- trical input impedance. In these methods, the variations across the thickness direction vanish by using the bending moments per unit length or stress resultants. Thus, two- dimensional plate equations for a step-wise laminated cir- cular plate are obtained as well as two different solutions t o the corresponding systems. An equivalent circuit of the transducer is also obtained from these solutions.

,

I. INTRODUCTION

HE TRANSDUCER design is based on a flextensional T transducer that excites the axisynimetric resomnt modes of a clamped circular plate. It is constructed by depositing a thin piezoelectric annular plate onto a thin, edge-clamped, circuler plate as shown in Fig. 1. An ac volt- age is applied across the piezoelectric material to set the compound plate into flexural vibration. We fabricated mi- croniachiried piezoelectrically actnated flextensional trans- ducers in a two-dimensional array by combining conven- tional integrated circuit (IC) manufacturing process tech- nology wit,h ZnO deposition [I]. Individual array elements, shown in Fig. 1, consist mainly of a circular plate attached

Manuscript received April 23, 2001; accepted November 12, 2001. This research was supported by the Defense Advanced Reseirrch

Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Grant No. F49620- 95-1-0525. This work made use of the National Nanofabrication Userr Net,work facilities funded by the Natiormi Science Foiindatiori under award number ECS-9731294.

G. P e r ~ i n is with ADEPTIENT, Los Altos, CA 94024 (e-mail: percin~adeptient.com)

B. T. Khuri-Yakub is with Edward L. Ginzton Laboratory, Stan- ford University, Stanford, CA 94305-4085.

CrlAu

Fig. 1. Configuration of the micromachined flextcnsional transducer.

to an annular disk of piezoelectric material, wliicli has op- timized dimensions. The arrays are made by using silicoii micromachining techniques and are capable of opcra,tion at high frequencies as well as low frequencies. Inherently, this approach offers tlic advantage of integrating trttns- ducers with transmitter and receiver electronics. Thus, we present arrays in which elements can be individiially addressed for ease of scanning and focusing by using on- board electronics. To date, however, flextensional tranduc- ers have not been used in ultrasonic applications because of their low frequency response (less than 100 kHz). Here, we present micromachined two-dimensional array flexten- sional unimorph ultrasonic tranducers that have typical operating resonance frequencies starting from 450 kHz to 4.5 M H z in air. I\~Iicromachined sma,ll dimensions such as diameter of the vibrating plate and depositing thin films result in higher frequency response along with increased reproducibility.

The transducer is designed to have maximum volume displacement of the plate at the resonant frequency. Anal- yses of similar devices such as those of Gerinano [Z], Denkmann et al. [3], Allaverdiev et al. [41, Vassergiser et al. [ 5 ] , Antonyak et al. [SI, Brailov et al. [7], Adelman et al. [SI, Okada et al. [Q], Aronov et al. [lo], [Ill; Rosato [ l Z ] , Grcenberg et al. [13], Stavsky et al. [14]: and R.ndgers [15] were helpful in identifying the important paramet,ers of the device. However, the coniplexity of the structure and the fact that the piezoelectric used is an annular disk rather than a full disk necessitates the use of a more complicated analysis to determine the resonant frequencies of t,he struc- ture, the input impedance of the transducer, and the nor- mal displacement of the surface.

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lEEE TRANSACTIONS ON ULTRASONICS. E 574

11. THREE-DIMENSIONAL COUPLED ELECTROMECHANICAL SYSTEM

Because we are interested here in obtaining plate dif- ferential equations for axisymmetric vibrations only, we consider the problem in cylindrical coordinates. The lin- ear piezoelectric strain constitutive relations that provide the coupling between the mechanical and electrical fields are given by

SI = d I j E , + sBTj Di = t;Ej + dijTj (1)

where T J , Sr, Di, and E3 are the components of stress, strain, electric displacement, and electric field, respec- tively. Material properties are the elastic compliance con- stants (zero electric field) s f J , the piezoelectric strain con- stants dri z d j I , and the dielectric permittivity (zero stress) E;. In addition to these relations, there are two ba- sic field equations, the equation of motion and the strain- mechanical displacement relation,

where p is the m a s density and ii is the mechanical displacement. Because most piezoelectric materials have hexagonal symmetry, this paper contains the solutions for the hexagonal and isotropic systems. The reader should note that these methods can be applied to other geome- tries in different coordinate systems. Note that [ S I J ] , [ d i J ] , and [ ~ j ] matrices in cylindrical coordinates are the same as those in rectangular coordinntes in the case of hexagonal and isotropic systems, Le., dr5 = ds5 and E:? = Fi- nally, the linear piezoelectric strain constitutive relations

'ERROELECTRICS, AND FREQUENCY CONTROL, VOL. 48, NO. 5 , hilAY 2002

4 piezoelec~ic layer

Fig. 2. The geometry of the device.

111. STEP-WISE LAMINATED PLATE ANALYSIS

In classical linear thin (Kirchhoff) plate theory, there are a number of assumptions that are necessary to reduce the three-dimensional equations to a two-dimensional set that can be solved. The basic assumptions made in the derivation of the equations for the bending of thin plates are S,, is zero, which in turn means that the transverse deflection w is a function of T and 4 only. Therefore, S,, will be neglected in three-dimensional coupled electrome- chanical equations. Moreover, S,, and S4z are zero, and T,, is also zero in the stress-strain relations.

In addition, for the flexurally vibrated circular plate, we assume the thickness to be so small that the change of stress is negligible along the z direction. Because the stresses are zero on both surfaces, we can set

T,, = T,, = T+ = 0. (4)

Furthermore, because the motion is entirely in t and radial directions,

u, = 0 and T,.4 = 0. ( 5 )

For the region II shown in Fig. 2, because a field is applied only along the z direction,

D, = Do = 0 and E, = E4 = 0. (6)

The remaining coupled electromechanical equations in (3) then become

S,, = SFIT,, + ~ f z T 4 g + dziEz

Sb4 = sfJ',, + sflT40 + dziEz (7)

(3) Dz = dziTy, + dziT44 + &E*

where

(8) a%, 1 aT T

S,, = - and S$g = -u,.

Because T,, and E, are zero, transverse shear deformation S,, is also zero, i.e.,

In the presence of mechanical losses, c w s are complex numbers and will be replaced by e 6 + jwqFJ. The corre- sponding sFJs can be found by inverting the cfJ matrix. In the presence of electrical losses, bers. As a result of these complex representations, there

are also complex Therefore, the classical kinematic relationships for the flex- ural motion Of the Plate are Obtained =

(9) will not be any change in the analysis developed in the aw

oi = W(T, t ) and or = -z-. following sections. ar

PEHFlN AND KHURI-YAKUB: PAFUUT: THEORY 575

By nsing the previous equations, we obtain the following for the piezoelectric material

and

where slip = sfl, slzP = sf2, IC, is the planar coupling factor, E, is Young's modulus, up is Poisson's ratio, and

Moreover, the electric field variables have the following relationships:

V . D = 0 i.e., DZ+ = 0,

= I

Similar equations for the non-piezoelectric carrier plate can be derived as follows:

For the regions I and 111 shown in Fig. 2 , the bending moments per unit length are given by

M,'= 7 TEzdz M i = 7 T$dz. -hm/2 --hmL12 (14)

For the region I I , the bending rnonients per unit length are given by

where

is the flexural rigidity of regions I and I I I . In a similar way, the bending moments per unit length for region f I are obtained in terms of the curvatures, the deflection, and the excitation voltage as follows:

where D I ~ and UII are the flexural rigidity and the equiv- alent Poisson's ratio for region I I , respectively.

The equation of motion for the axisynimetric vibrations takes the following form:

1 Tw;, + ;(T'T - T4+) + Z z , z = -P+J'U~ 0

1 7 7 . Tzz,, + Tr,,, + - = -PW2uz. (19) T

The term pw2U,, inplane radial inertia, can be neglected. This quasi-static approximation is not valid at higher fre- quencies. By using the equation of motion, we obtain the following for regions I and 111:

where &, = pmh,, and [TZz]<.> = P, where PejWt is the pressure difference between top and bottom surfaces of the plate. Note that the plate can tell only the difference between normal tractions on the upper and lower surfaces as a result of this analysis. Similar expressions can be found for region 11 a s follows:

where &,, = pph, + pmh,. DI and DII arc the corre- J J

1 2 z1 (15) sponding flexural rigidities for the single-laycred regions ( I and 111) and the multi-layered region (11), respectively. As a result, (20) and (21) take the following forms:

1 1 1"

Mi' = /T&zdz + S T G z d z . il 10

By substituting in the previous equations while assuming , an eJwt time dependence, the bending moments per unit

length for regions I and 111 are obtained in terms of the

P DI P

DIr

(V2 + k,2) ( 0 2 - k:) WI - - = 0

( 0 2 + k ; I ) ( 0 2 - !&) WI, - - = 0 (22)

curvatures and the deflection as follows:

2 T dr (16) where k: = u f i . k:, = u&> and Vz = P+

- I d - l d - F z ( r g ) . Solutions to (22 ) are in the following

"

with the boundary conditions of

Wiir = 0 ~ V I I l r = 0 at r = rm

Wii = l'i'rii Wiib7 = ~ V I I I , ~ a t r = rP, M;i = M;'i Q:' = Q;" at T = rp2 Wrr = Wr at r = T PI

b y = PI; Qf' = Q: a t r = rP, W I I , ~ = l ' i ' ~ ,~

where Q: = s T,,dz for i = I, 11, III. Note that, be-

cause Y and K tend to infinity as r tends to zero, and Wi is finite a t the center of the plate, Wi does not coii- tain terms with Y and K . In addition, these boundary conditions represent a clamped-edge boundary condition. WIII = 0 and hf;" = 0 are set for a simply supported edge boundary condition. Qi" = 0 and PI,!ir = 0 are set for a free edge bouiidary condition. In the presence of mechanical losses; k1 and krr are complex numbers. Be- cause the functions Jn(kr ) , Y,(kr), I , L ( k ~ ) , and K,(kr) can have complex variables: the previously mentioned for- mulation does riot have any change. Solutions to (22) can be generalized to step-wise mult,iple layer laminated an- nular plates. For cxaniple, the geometry of the transducer may also contain the top arid bottom electrode layers; or, equivalently. region 11 may also contain tlie top and bot- tom electrode layers; arid region I may also contain a hole.

A . Equivalent Circuit

<=>,

The electreacoustic two-port network equations corrc- sponding to the network in Fig. 3 can be written in the following form:

(24) I = YllV + Y12P B = g21V + g22P

where I is the electric displacement current on tlie piezo- electric material, 5 is tlie volnrne velocity of the vibrating plate, V is the electric voltage across the piezoelectric ma- terial, P is the pressure applied to the transducer, and Z L is the acoustical load impedance caused by the radiation from a vibrating plate.

To study the electro-acoustical efficiency, the equivalent circuit of the transducer shown in Fig. 4 is used, where Z A is the acoustical impedance of the transducer, CO is the clamped electrical capacitance of the compound plate, and N A is the electroacouatical conversion efficiency.

Fig. 3. Equivalent twO-port network

Fig. 4. Equivalent circuit

The electric displacement current in the piezoelectric material, I, is given by

I = j2nw D,rdr. ( 2 5 ) TPl 7

The volume velocity of the vibrating plate, 5: is given b)

The equivalent circuit and expressions for calculating its paranieters can be wed to estimate the input impedance, the sensitivity, the efficiency, and a nnniber of other characteristics of the transducer. For example, the input impedance of the transducer is given by

(27) V ?/22ZL + 1 Zl, Y - = I Y l l + ZL(?/111122 - ?/12?/a1)

- where Z L = 3, Z, is the acoustical impedance of the niedinm, and S, = 'TT$ is the surface area of the trans- ducer. In these equations, the radiation impedance of a circular pist.on should be used for 7~ to obtain more ac- curate results.

T

IV. APPLICATION OF MINDLIN PLATE THEORY

It is well known that the classical thin plate theory sat- isfactorily predicts actual behavior only for the first few flexural modes of motion of a plate whose thickness is small

PEirqlN AND KHURI-YAKUR: PAFMUT: 'TIIEORY 577

ness. To compensate for this error, blindlin [le] proposed a shear correction factor to bc applied to the shcar force. Besides, Mindlin modified the sixth assumption so that the effect of rotatory inertia (or inplane radial inertia) is in- cluded. Thus, the influence of coupling hetwecn Hexure and shear is taken into account hy iriclusion of rotatory inertia (or inplane radial inertia) arid shear deformation terms i n the equations. Mindlin plate theory is further modified hy including Tzz im the cquations to consider the relatively large pressure difference het,weeri upper and lower snrfaces of the plate. In the Mindlin plate theory. the displaccment components are assnmed to be given by

u? = z u(r; t ) uz = W(T, t ) . (28)

u ( T , ~ ) can be thought of as the local rotation (change of slope) in the T direction of lines originally normal to the neutral plane hefore deformation. That is, the rota- tion u(r, t ) is due to bending. The deflection of tlie neutral plane w(r, t ) is then composed of two parts: one caused by bending a id the other caused by shear deformation. The remaining coupled electromechanical equations in (3) thcn become

S,, = sflT,, + ~fZT44 + s~ZT,, + d,iEz

S4g = &Tm + ~flT++ + sfZTzZ + dziEz

2S,, = sf4Trz (29) D, = tT,Ez + d,iT,, i dziTm* + d X Z z

where

7 - r I + v - Fig. 5. Equivalent bhree-port nctwork

TABLE I PHYSTCAL DiMENSlONS OF 'THE SIMULATED LARGE-SCALE DEVICE

Dimension Value

Radius of brass carrier plate Inner radius of pieauelectric Murata SW matcrivl Outer radius of piezoelectric R4urdta SW material Thickness of brass carrier platc Thickness uf piezoeiectric Murata SW Material

4 " 1 mni 3 nun

25 prir 20 p m

compared with its lateral dimensions. For the higher Hexu- ral modes, the inHuence of coupling to the thickness-shear mode of rriotion becomes increasingly impostant. Hence: the classical theory, in which this effect is neglected, ceases to yield reliable information. Thc assumption of the clas- sical thcory regarding normals to tlie neutral remaining normal to the daformcd plane amounts to neglcctimg the effect of transverse shear deformation. This effect: t.ogether with the rotatory inertia (or inplane radial inert,ia) effect, hecoines important when the plate is relatively thick or when accnrate solutions for higher modes of vibration are desired. To allow for the effect of transvcrse shear deforma- tion, the tlicory relaxcs the normality assumption so that normals to thc undeformed neutral plane remain straight and unstretched in length but not necessarily normal to the deformed iieutra,l plane. This assumption implies a non- zero transverse slicar strain, but it also leads to the statical riolat,ion of zero shear stress at thc free surfaces because the shear stress becomes constant through the plate thick-

Dimension Value Radius of the silicon nitride carrier plate Radius of the orificr? Inncr radius of piezoelectric zinc oxide Outer radius of piezoelectric zinc oxide Thickness of carrier plate silicon nitride Thickness of piezoelectric zinc oxide Thickness of gold electrodes

50 pm

15 pm 4U fim

0.3 pm 0.3 pm 0.1 & I n

3 pm

By using the previous cquations, we obtain the following for the piezoelectric material:

T,P, = G, ( u + g), and

Similar eqnations can he obtained for the non-piezoelectric material as follows:

578 IEEE TRANSACTIONS ON u x r n A s o N i c s , FEKROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO. 6 , MAY zooz

~iectrical inDut imcedance measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..:: : : - .n air

. < . . . . . . . . . . . . . . . . . . . . .

I ........ ....... ................... ; .................. i ................... i ................... 1 os

- 9 2108

3 .................... ' " ' t , ; ; I I . . ! :. : : . : i i . ~ ... . : . . . . . . . . . .:.:: . . . . !. ,:,:':;I . . . . . . . . . . . . . . . . . . . . .) . . . . . . . . . . . . . . . . . . .......................................................... ~ ....................................... 10 20 30 40 50

1 ooo

Frequew (W

Elaclfkal input Impedance measurement 0, 1

J 0 10 20 30 40 50

-20' ' Frequew

EIBC1ricaI input impedance simulion

I 10 20 30 40 50

Frequency (kHr)

Elearical input impedance simulatbn

0 10 20 30 40 Freguew ( k W

Fig. 6. Largescale device electrical input impedance measurement and simulation.

where G, = s,,, 1 and G - - are the shear moduli, shearing forces per unit length for regions I and I I I : VI2 - s44,

and T,, is approximated by

(34) In a similar way, we obtain the following for region 11 2,4 - SLP, + (Pu - 4 ) z Tzz = tu - a

- FII I I M , =DII P,L and Pl are the pressures a t the top and bottom surfaces . .

- Frr I I of the plate, respectively, and zu and t i are the coordinates of the top and bottom surfaces of the plate, respectively. Mr$ =DII

(36) By using the previous equations, we obtain the follow-

ing bending moments per unit length and the transverse

579 PERClX AND KHUHI-YAKUB: PAFHUT: THEORY

Average dlSplaCBment sfmulatlon

J 10 20 30 40 M)

Frequency (kHz)

Fig. 7. Large-scale device average displacement and mode shape simulations.

Mode shapes in air

........ ; ..........

0.5 1 1.5 2 2.5 3 3.5 4 Radial distanm (mm)

Klmhhon Dlale theory. no eiection hob. no radlaUon Impedance

. . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . .

Frequeniy (MM)

1 I 1 2 3 4 5 6

104' 0

Freauency (MHz)

Fig. 8. Simulation example of a rnicronischiried device without ejection hole by using classical thin (Kirchhoff) plate theory and not using radiation impedance

where K' is the shear correction factor to compensate for the error in assuming a constant shear stress throughout the plate thickness. It is introduced for the dispersion curves to match those of the three-dimensional theory at the thickness-shear frequencies.

The equation of motion can he written as

A)(zB - .I91

and

112 By using the equation of motion, the differential equation corresponding to the forced flexural vibrations in regions Br = K2Gmhm, Brr = K'(Gmhm + G P ~ P ) , xz = 12.

580 I E E E TRANSACTIONS ON ULTRASONICS; FERROELECTRICS, AND FREQUENCY CONTROL, VOI.. 48, NO. 5 . MAY 2002

Kirchhoff piate thBOIY, no eie*imn hole, with radialion impedance

. . . . . . . . . . 10"

. . . . . . . . . .

Frequency (MHz)

Fig. 9. Simulation example of a micromachined device without ejecti a circular piston.

Kirchhoff dale meow. wim eiectlon hole. with radiation lmoedanca

. . . . . . . . .

- 0 1 2 3 4 5 6 Frequency (MHz)

on

Kirchlmff plate theow. no ejeotion hole, wlh radiation imcedance 1 0"

8 2 10-2

I 5 10.'

H $104

3

E - E

I 0 1 2 3 4 5 6

Frequency (MHz)

hole by using classical thin plate theory and radiation impcdance of

Kirchhoff platetheow, with eieciion hole, wim radiation impedance

1 1 2 3 4 5 6

Frequency (MHz)

Fig. 10. Simulation cxarnple of a micromachined device with 6 - / ~ m diameter ejection hole using classical thin plattc theory and ridiation impedance of a circular piston.

I and 111 is obtaincd as follows: where

or, equivalently, it can be written as Eq. (38) can also be written in the following compact forrn:

PERGIN AND KBURI-YAKUB: PAFMUT: THEORY

whcre

k: = 2 2 [Rl + S I + 4-1 I - - 2 R i + S r - JG iK i i 7 ]

l 2 - s: The general solution to (39) can be expressed as

A siniilar differential equation corresponding to the forced flexural vibrations in region If is obtained as fol- lows:

V4Wii + ~ : , ( R I I + S I I ) V ~ W I I +J:r(RriSrr6:, - 1)U'rr

(41) Pu - 9

D I I + (RrrSrr6fr - 1)- = 0

where

Eq. (41) can be also written in thc following compact form:

where

! Rri + Srr + J(Rri ~ Srr)' + 46c4

RII + SII - J(Rrr - Srr)' + 46f: The general solution to (42) can be expressed as

( 0 2 + l;r)WI12 = 0.

Solutions to (42) and (39) in regions I, 11, and IfI are in the following forms:

681

Mode shapes in air

I 10 20 30 40 50

Radial distanca (run)

Fig. 11. Simulated mode shapes of a micrornachiried device with 6- (mi diameter ejection hole using classical thin plate theory.

with the boundary conditions of

W1rr = 0 Ul l I = 0 at r = r, Wrr = Wrrr Urr = Urir at r = rpz

M,!I = MLrr Q;I = Q I r r at r = vp2

Wrr = WI Urr = Ur at T = vp1

M," = M,! Qir = QI at r = rpl.

Note that, because Y tends to infinity as r tends to zero and becausc WI is finite at the center of the plat?, Wr does not contain terms with Y . In addition, these boundary conditions represent a clamped-edge boundary condition; Wrrr = 0 and A4;Ir = 0 are set for a siniply supported edge boundary condition. Qf" = 0 and M;" = 0 are set for a free-edge boundary condition. In the presence of mechanical losses, kr , 11, krr, and l r r are coniplex numbers. Because the functions .Jn(kr) and Y,(kr) can have complex variables, this formulation does not change.

A . Equivalent Circuit

The clectro-acoustic three-port network equations cor- responding to the network in Fig. 5 can be written in the following form:

1 = YllV + Y l Z P , + Y l S P l

(45) - 21 = Y?lv + ?/22pu + Y 2 3 9

-V = Y31v f Y32Pu + V 3 3 9

where I is the electric displacement currcnt on the piezo- electric material, V is the volume velocity of the vibrating plate, V is the electric voltage across the piezoelectric.ma- terial, P, is the pressure applied to the top surface of the transducer, and 4 is t,he pressure applied to the bottom

582 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, A N 0 FREQUENCY CONTROL, VOL. 49, NO. 5 , MAY 2002

I

. . . . . . . . IO"

210' ' 3 2

8 E 10'9

ilO'.

10"

...... I ........ .............. i ............... : ................ i ............... .: ............... .- 0 1 2 3 4 5 6

Frequency (MM)

Fig. 12. Simulated equivalent circuit parameters of a micromachined thwry.

MlndUn plate wry, win ejection hale, wkh IadlaUOn Impedance

Fig. 13. Simulation example of a micromachined device with 6 p m diar impedance of a circular piston.

surface of the transducer. Because the motion is entirely flexural, the velocity of the bottom surface of the plate is the negative of the velocity of the top surface. I t is clear from the previous equations that y31 = -YZI , y32 = -7~22,

and y33 = -y23. In addition, if the assumption of the clas- sical plate theory, which states that T,, vanishes through the thickness of the plate, is made, then y13 = -ylz and Y23 = -y22.

The equivalent threeport network and expressions for calculating its parameters can be used to estimate the in- put impedance, the sensitivity, the efficiency, and a number of other characteristics of the transducer. For example, the

............... ............... 8 .... ......... ............... 8 ... .......... i ...............

................................ il..... ......... < ................ i ............... .i ............... 0 1 2 3 4 5 6

Frequency (MHz)

device with 6 p m diameter ejection hale using classical thin plate

Mtrdlin plate theory. wkh e m n hole. wkh tadlation Impedance 10" y . . . . . . 8 . . ~ . . . . . . . . . . . . . . ::.. . . :. :. ::' :: . ::. ,:.; . . . . . .

..., ::.. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./ . . ..; . . . . ...... ....... ............... : ............... .i ............... : ............... .i ............... I

-- 0 1 2 3 4 5 6 Frequency (MM)

neter ejection hole using modified Mindlin plate theory and radiation

input impedance of the transducer is given by

V I

z. - _ = 271 -

-

(46) 1 + YZZZu y23Zl

Y l l +Zu(yllyZZ ylZZ/Zl) - z1(ylly23 y13yZ1)

where zu and zl are the acoustical load impedance of the media that are in contact with the top and bottom surfaces of the transducer, respectively.

V. SIMULATION EXAMPLES

By using the model developed in this paper, the elec- trical input impedance, average displacement, and mode

PERCiN AND KHURI-YAKUB' PAFMUT THEORY

shape simulations for large-scale device dimensions and materials given in Table I are presented in Fig. 6 and 7. As shown in Fig. 6, the measured real and imaginary parts of the electrical input impedance resemble those obtained by the model. The model is also capable of predicting elec- trical input impedance and average displacement of the fluid-loaded device, and the corresponding simulations are also presented in Fig. 6 and 7. As presented in these plots, the resonant frequencies are decreased by fluid loading on one side of the plate.

In the following figures, simulation examples, using the theory developed in this paper, are presented for the mi- cromachined device dimensions, and materials are given in Table 11. In Fig. 8, simulation results of a micromachined device without an ejection hole (orifice) by using classical thin (Kirchhoff) plate theory and by not using the radia- tion impedance definition are given. In Fig. 9, the simu- lation results of the same configuration, except using the radiation impedance definition, are given. Note that using the radiation impedance definition changes the low fre- quency response, Le., it introduces relatively wide band electrical resonance at lower frequencies for the water- loaded case. This is because of the fact that the propaga- tion wavenumber for acoustic waves in water is relatively small. The simulation results in Fig. 10 correspond to the case where the ejection hole is also included in the calcu- lations. Addition of the ejection hole does not change the simulation results significantly, because the radius of the orifice is much smaller compared with the overall radius of the device. In Fig. 11 and 12, the mode shapes, the acous- tical impedance, and the electro-acoustical conversion ef- ficiency are shown for the case in Fig. 10. Fig. 13 shows the simulation results by using the modified Mindlin plate theory developed in this paper. These results are given for the cases in which the device operates in air, has one side loaded with water, and has two sides loaded with water. As Seen in Fig. 13, the average displacement decreases more in the case of two sides, loaded with water.

583

REFERENCES

(11 G. Peqin and B. T. Khuri-Yakub, "Piezoelectrically actuated Hextensional micromachined ultrasound transducers-11 Fabri- cation and Experiments," lEEE Trans. Ultmson., Fewe lec t . , Frq. Contr. , VOI. 49, no. 4; pp. 585-595, Apr. 2002.

121 C. P. Germano, "Flexure mode piezoelectric transducers," IEEE Tmns. Audio Electmaeoust., vol. AU-19, pp. 6-12, Mar. 1973.

[3] W. J. Denkmann, R. E. Nickell, and D. C. Stickler, "Analy- sis of structural-acoustic interactions in metal-cera& transduc- ers: IEEE %ns. Audio Electmacoust., vol. AU-21, no. 4, pp. 317-324, Aug. 1973.

(41 A. M. Allaverdiev, N. B. Akhmedav, and T. D. Shermergor, "Coupled Rexurel-shear oscillations of stepwiselayered piezoce ramic disk transducers," Sou. Appl. Mech., "01. 23, pp. 465471, May 1987. M. E. Vassergiser, A. N. Vinnichenko, and A. G. Dorosh, "Calcu- lation and investigation of Hexural-mode piezoelectric disk t r a n s ducers on a passive substrate in reception and radiation,'' Sou. Phys. A c o w t . , vol. 38, no. 6, pp. 558-561, No".-Dec. 1992. Y. T. Antonyak and M. E. Vassergiser, "Calculation of the char- acteristics of a membrane-type flexural-mode piezoelectric trans- ducer," Sou. Phys. Acoust., "01. 28, no. 3, pp. 17&180, May- Jun. 1982. E. S. Brailov and M. E. Vassergiser, "Estimation of the char- acteristics of a disk-shaped Hexural-mode bimorph cell," Sou. Phys. Acoust., vol. 26, no. 4, pp. 325-328, Jd-Aug. 1980.

1.81 N. T. Adelman and Y. Stavskv. "Flexural-extensional behav-

(51

[6]

[7]

VI. CONCLUSION

In summary, classical thin plate theory and Mindlin plate theory are applied to derive two-dimensional plate equations for the transducer and to calculate the cou- pled electromechanical field variables such as mechanical displacement. These methods use classical kinematic re- lations for the coupled electromechanical field to reduce three-dimensional field equations to two-dimensional plate equations. As a result, two different exact solutions to the corresponding systems are obtained. An equivalent circuit of the transducer is also obtained from these solutions.

ACKNOWLEDGMENT

This research was conducted during G. Perqin's Ph.D. study at Stanford University.

" , ,~ , ~

ior of composite piezoelectric circular plates," J. Aeoust. Soc. Amer. , vol. 67, no. 3, pp. 819-822, Mar. 1980.

191 H. Okada, h4. Kurosawa, S. Ueha, and M. Masuda, "New air- borne ultrasonic transducer with high output sound pressure level," Jpn. 3. Appl. Phys., pt. 1, vol. 33, no. 5B, pp. 3040-3044, May 1994.

[lo] B. S. Aronov and L. B. Nikitin, "Calculation of the flexural modes of piemceramic plates," Sou. Phys. Aconst., vol. 27, no. 5, pp. 382-387, Sep:Oct. 1981.

[ I l l B. S. Aronov and G. K. Skrebnev, "Effect of the attachment con- ditions of the piezoceramic plates on the parameters of circular laminated electromechanical transducers: Sou. Phys. Acoust., vol. 29, no. 2, pp, 8 e 9 2 , Mar.-Apr. 1983.

1121 F. J. Rosato, "Lagrange equations applied to flexural mode transducers," J. Acoust. Soc. Amer . , "01. 57, no. 6, pt. 1, pp. 1397-1401, Jun. 1975.

[I31 J. B. Greenberg and Y. Stavsky, "Flexural vibrations of certain full and annular composite orthotropic plates," 3. Acoust. SOC. Amer. , vol. 66, no. 2, pp. 501-508, Aug. 1979.

[14] Y. Stavsky and R. Loewy, "Axisymmetric vibrations of isotropic composite circular plates," J. Acoust. Soc. Amer. , vol. 49, no. 5, pt. 2, pp, 1542-1550, May 1971.

[IS] A. J. Rudgers, "Acoustical behavior of thick, compasit,e, fluid- loaded plates, calculated using Timoshenko-Mindlin plate the- ory," Naval Research Laboratory, Washington, D.C., Rep. no. 8317, No". 16, 1979.

(161 R. D. Mindlin, "Influence of rotatory inertia and shear on flexu- ral motions of isotropic, elastic plates," J. Appl. Meehan. ?Inns. ASME, vol. 73, pp. 31-38, 1951.

[17] H. F. Tiersten and R. D. Mindlin, "Forced vibrations of piezo- electric crystal plates," Q. Appl. Math., vol. 20, no. 2, pp. 107- 119, Jul. 1962.

Giikhan Perpin (S'92-M'OO) was born in Ezine, Canakkale, Turkey. He received the B.S. degree in electrical and electronics en- gineering from Bilkent University, Ankara, Turkey in 1994. He received the M.S. degree in electrical engineering in 1996 and the Ph.D. degree in electrical engineering with minor subject in engineering-economic systems and operations research in 2002, both from Stan- ford University, California. He worked in Nu- merical Technologies, Inc., San Jose, Califor- nia, as a senior software engineer to develop

584 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND rREQlJENCY COSTROL, VOL. 49. N O 5, MAY 2002

optical microlithography software. He is cofounder, president, and chief technical officer of ADEPTIENT, Los Altos, California. He has years of experience in the areas of fluid ejection, rnicrofluidics, micromachining, micromachined electromechanical systems, ultra- sonic actuators, ultrasound transducers, biomedical imaging, soft- ware development, sensors, and actuators. Dr. Perch is a riremher of IEEE, IEEE Uitrasonics, Ferroelectrics, and Frequency Control Society, IEEE Electron Devices Society, the Electrochemical Soci- ety, and American Vacuum Society. His personal interests include Sufism, the Bektaslii Order of Dervishes, photography, swimming, hiking, goorrnct cooking, and collecting coins and stamps.

Butrus T. Khuri-Yakuh (S'7CK3'73-h.1'76- SM'87-F'96) was born in Beirut, Lebanon. He received the B.S. degree in 1970 from the American University of Beirut. the M.S. de- gree i n 1972 from Dartmouth College, and the Ph.D. deaiee in 1975 from Stanford Univer- - sity, all in electrical engineering. He joined the research staff a t the E. L. Ginaton Labora tory of Stanford University in 197G as a re-

ik,; 1.j , ~: ',\ search associate. He was promoted to Senior Research Associate in 1978 and to Professor of Electrical Engineering (Research) in 1982.

He has served on many university committees in the School of Engi- neering and the Department of Electrical Engineering

-4

Presently, he is the Depnty Director of the E. L. Ginaton L a b e ratory. Professor Khuri-Yitkuh has been teaching both at the grad- uate and undergraduate levels for over 15 years, and his current research interests include in situ acoustic sensors (temperature, film thickncss, resist cure, etc.) for monitoring and control of integrated circuits manufacturing processes, micromachining silicon to make acoustic materials and devices such as airborne and water imnier- sion ultrasonic transducers and arrays, and fluid ejectors: and the field of ultrasonic nondestructive evaluation and acoustic inraging and microscopy.

Professor Khuri-Yakuh is a fellow of the IEEE, a senior member of the Acoustical Society of America, and a mcniher of Tau Beta Pi. He is Associate Editor of Research in Nundestmctive Euahation, a journal of thc American Society for Nondcstroctive Testing. Profes- sor Khuri-Yakub has authored over 300 publications and has hcen principal inventor or cc-inventor of 52 issued patents. He received the Stanford University School of Engineering Distinguished Advi- sor Award (June 1987) and the Medal of the City of Bordeaux for contributions to NDE (1983).