INTRODUCTION TO ACOUSTICS - ULTRASOUND

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1 INTRODUCTION TO ACOUSTICS - ULTRASOUND Rudof Báek (bek@fe.cvut.cz). Introduction This text covers the propgtion of utrsonic wves in soids nd iquids. It is shown wht kinds of wves cn propgte in vrious types of mteris, how they re refected from nd trnsmitted through the boundries between two mteris, how they re excited, nd how their properties re reted to the mss, density, nd esticity of the mteri invoved. We sh be concerned with the properties of piezoeectric trnsducers for converting eectric into mechnic energy. Therefore, the formus for their conversion efficiency nd eectric impednce s function of frequency, nd for their puse response chrcteristic wi be derived. To do this, we need to know how wves propgte in nonpiezoeectric nd piezoeectric mteris. In gener, the tretment of utrsonic wve propgtion in soids is compicted by the fct tht soids re not isotropic. Thus the prmeters of the coustic wve must be expressed in terms of tensor quntities nd retions between them. For simpicity, we sh ssume tht the wves of interest re either pure ongitudin or pure sher wves, nd tht physic quntities cn be expressed in one-dimension form. We therefore restrict our nysis in this chpter to pne wve propgtion, which reduces the probem from three dimensions to one. The resuts we obtin re identic in form to gener, more compete tretment for isotropic mteris, nd re vid even for propgtion of wves ong xes of crystine mteri... Utrsound Wves in Nonpiezoeectric Mteris: One-Dimension Theory Let us first define the two bsic types of wves tht re importnt in utrsonic wve propgtion. The first is ongitudin wve, in which the motion of prtice in the coustic medium is ony in the direction of propgtion. Thus when force is ppied to the coustic medium, the medium expnds or contrcts in the z direction, s shown in Fig...(). The second type of wve is sher wve, in which the motion of prtice in the medium is trnsverse to the direction of propgtion, s iustrted in Fig...(b). There is no chnge in voume or density of the mteri in sher wve mode, s shown in Fig...(b). In gener, the utrsonic wves tht cn propgte through soid medium my combine sher nd ongitudin motion. Fig...() Longitudin wve propgtion; (b) sher wve propgtion (fu cube not shown).

2 We sh define the bsic wve eqution for utrsonic propgtion in the ongitudin cse. The resuts obtined re identic in form to those for sher wve propgtion. Stress. The force per unit re ppied to soid is ced the stress. In the one-dimension cse, we sh denote it by the symbo T. A force ppied to soid, stretches or compresses it. We first consider sb of mteri of infinitesim ength, s shown in Fig... Fig...() iustrtes the ppiction of ongitudin stress, nd Fig...(b) sher stress. The stress T(z) is defined s the force per unit re on prtices to the eft or to the right of the pne z. We note tht the ongitudin stress is defined s positive if the extern stress ppied to the right-hnd side of the sb is in the +z direction, whie the extern stress ppied to the eft-hnd side of the sb is T in the z direction. If the stress is tken to be positive in the +x or +y directions, these definitions so ppy to sher stress. The net difference between the extern stresses ppied to ech side of the sb is ( T/). Thus the net force ppied to move unite voume of the mteri retive to its center of mss is T/. Fig... ) Longitudin stress in sb of mteri of thickness ; b) sher stress Dispcement nd strin. Suppose tht in the one-dimension cse, the pin z in the mteri is dispced in the z direction by ongitudin stress to pne z = z + u, s shown in Fig...(). The prmeter u is ced the dispcement of the mteri nd in gener is function of z. At some other point in the mteri z +, the dispcement u chnges to u + δu. If the dispcement u is constnt through the entire mteri, the mteri hs simpy undergone buk trnstion. Such gross movements re of no interest to us here. We re interested in the vrition of prtice dispcement s function of z. A Tyor expnsion cn be used to show tht, to the first order, the chnge in u in ength is δu, where u δ u = = S (..) The frction extension of the mteri is defined s

3 u S = = δ u (..) The prmeter S is ced the strin. We my so consider the one-dimension cse of sher motion where the mteri is dispced in the y direction by wve propgting in the z direction. Then, s shown in Fig...(b), prtice in the position z z is dispyed to point z z + y u, where y nd z re unit vectors in the y nd z directions, respectivey. The sme tretment used for ongitudin motion hods for the sher wve cse. We define sher strin s S = u/ = u/. Here the ony difference is tht the dispcement u is in the y direction. The digrm shows tht there is no chnge in the re of the rectnge s sher motion distorts it. Longitudin motion, however, chnges the cube voume by δu A, where A is the re of the x, y fce. Thus the retive chnge in voume is u/ = S. Hook s w nd esticity. Hook s w sttes tht for sm stresses ppied to onedimension system, the stress is proportion to the strin, or T = cs (..3) where c is the estic constnt of the mteri. The prmeters T nd c woud be tensors in the gener system, but cn be represented by one component for one-dimension ongitudin or sher wve propgtion. Eqution of motion. Consider now the eqution of motion of point in the mteri when sm time-vribe stress is ppied to it. From Newton s second w, the net trnstion force per unit re ppied to the mteri is ( T/), so the eqution of motion must be T, (..4) = ρ u = ρ v Here v is defined s the prtice veocity of the mteri nd ρ is its mss density in the sttionry stte. is Conservtion of mss. In sm ength (in one-dimension cse), the chnge in veocity δ v From (..) v z =. (..5) S δv = δu = t t. (..6) Combine (..5) nd (..6) S v =, (..7) t which mens tht incresing of deformed mtter in time is equ to incoming mount of deformed mtter. (..7) is essentiy the eqution of conservtion of mss of the mteri for ongitudin wves. The wve eqution nd definition of propgtion constnt. From (..3), (..4) nd (..7), nd put T S ρ T = ρ = t c t The soution for the stress hs the form ( soution re of the form u v = : t. (..) j( t z e ω ± β ) Ft z V j t ± ). For wve with ω nd time dependence s e ω, the, where negtive sign corresponds to forwrd wve in z direction 3

4 where ρ ω β = ω( ) =, (..) c V V c ( ) ρ = is the wve veocity nd β is ced the propgtion constnt of the wve. For the forwrd direction propgtion it foows V c v V S T = = (..3) or, the mgnitude of the strin S is the rtio of the prtice veocity to the sound wve veocity of the medium S v V =. (.3.5) Exmpes: wter retivey esy to compress ρ=0 3 kg/m 3 (sm) c= N/m V=.5 km/s spphire very rigid ρ= kg/m 3 c= N/m V =. km/s mets V = 5 km/s Be extremey ight nd rigid ρ= kg/m 3 V =.9 km/s Led ρ= kg/m 3 V =.96 km/s Gs V = 330 m/s Energy. The tot stored energy per unit voume is the sum of the two components: ) the estic energy due to force ppied to dispce the mteri Wc = TS. ) the kinetic energy due to motion of the medium Wv = ρv. j t For pne wve whose component vry s e ω (s the nogy with (EM) eectromgnetic theory) is vid ) The verge estic energy per unit voume Wc = Re( TS ) = Re( css ), (..4) 4 4 3) The verge kinetic energy Wv = Re( ρ vv ), (..5) 4 4

5 The tot verge energy per unit voume in n coustic wve is: W = Re( ρ vv + TS ) = Re( TS ). (..6) 4 Simiry, the power fow per unit re in the coustic wve P ( = v T). (..7) A The verge vue of vt during the RF cyce is sometimes ced intensity of wve. For ossess medium (v, T re in phse, P is re) ( P = v T ) A= V W A. (..8) Poynting s theorem. The power nd energy in coustic wve obey conservtion ws simir to those of Poynting s theorem in EM theory. To prove the conservtion ws, we write (..4) in the form T = jωρv (..9) nd (..7) in the form v jω = jω S = T. c (..0) By mutipying (..9) by v nd the compex conjugte of (..0) by T nd dding, it foows: Re( P ) = A ( T v ) = jω( ρvv jωts ) A = cs Re( P ) = A ( T v ) = jω( ρvv css ) A where T is used. We suppose tht c is re, so The right hnd side of (..) is imginry it foows tht Re( P ) = 0, (..). (..). (..3) This retion is nogous to Poynting s theorem in EM theory nd shows tht Re(P ) is constnt (i.e., the power in the wve is conserved). Since P is re (for propgting wve), the verge stored kinetic energy per unit voume is equ to the verge estic energy ρvv = css. Acoustic osses. When the system cn dissipte energy so tht it is not purey estic nd modue c is compex, there is oss term ike tht due to conduction current in EM theory. The viscous forces between neighboring prtices with different veocities re mjor cuse of coustic wve ttenution in soids nd iquids. There re ddition viscous stresses T η on prtices v Tη η S = = η, (..5) t where η is the coefficient of viscosity. From (..3) nd (..5) foows for the tot stress: S T = cs + η. (..6) t The eqution of motion hs the sme form s for ossess medium (see (..4)): T v = ρ t. (..7) Just s in EM theory, the ttenution due to oss cn be ccuted by Poynting s theorem. Foowing the derivtion of (..) nd (..3) nd ssuming e jω t time dependence, Poynting s theorem hs n ddition term ssocited with viscosity: Re( P ) = ηω SS A, (..8) 5

6 where A is the re of the bem. The viscosity gives rise to oss term tht vries s the squre of the frequency. If we regrd this oss term s sm, it foows from (..6) nd (..8) tht P css W VA = =. (..9) After substitution this into (..8) with respect to (..) P ηω ηω = =. (..30) 3 P Vc V ρ z The soution of this eqution is P = Pe α 0, where α is the ttenution constnt of the wve nd P 0 is constnt. For α we obtin ηω α =. (..3) 3 V ρ The ttenution of the wve due to viscous osses vries s the squre of the frequency nd inversey s the cube of the veocity. As sher wves hve veocity bout ½ V L, we might expect the sher wve ttenution per unit ength considerby rger. Exmpes wter α MHz = 0. db/m... ow frequency wves propgte over ong distnces α GHz = db/m = 0. db/μm... coustic microscopes - ony distnces bout 0μm There re mny other sources of oss in re mteris: ) Therm conduction. When mteri is compressed dibticy, the temperture is incresing. When mteri is expnded dibticy, the temperture is decresing. Therm conduction contributes to oss of energy higher ttenution in mets thn in insutors. Therm ttenution is proportion to ω. ) Attenution due to scttering by finite-size grins or by disoctions in soid. Exmpes Spphire hrd, high quity singe-crystine mteri cn be used for dey ines up to 0GHz, ttenution is bout 40 db/cm, α/λ is db (λ 0GHz is bout μm). Rubber viscous mteri high osses t f > few khz, mke good sound bsorbers. Acoustic impednce. Anogy to EM theory the specific coustic impednce is Z T v =. (..3) For pne wve trveing in the forwrd direction z (subscript F) we define chrcteristic impednce TF Z0 ( ρc) V ρ v = = =. (..33) F For bckwrd direction (subscript B) Z T B = = V ρ = Z 0. (..34) vb We cn compre (..9) with V = jωli nd (..0) with I = jωcv, (..35), (..36) 6

7 so ced trnsmission-ine equtions. It describe Fig...3, where L is series inductnce per unit ength, C is the shunt cpcitnce per unit ength, I is current nd V votge. A wve propgting ong this ine hs propgtion constnt β = ω( LC) nd impednce Z0 ( ) Repce V T I v L =. C L ρ C c It foows tht the sme trnsmission ine circuit is vid for coustic wve s we used in EM theory for trnsmission ine. Fig...3 Trnsmission ine. Let s define refection coefficient of n coustic wve in terms of the stress. This gives retion excty equivent to the simir retion in EM theory. Consider wve refected normy from the interfce between two medi of different Z 0 Z 0, Fig...4. Fig...4 ) Wve refected normy from the interfce between two medi; b) Stress puse refected normy from the interfce between two medi. Symbo mens dependence on z. Thn T nd v wi be continuous t the interfce (Z=0). We cn write for eft-hnd side: ˆ j β z j β T z = TFe + TB e (..37) jβz jβz vˆ = vfe + vbe. (..38) Refection coefficient t the pne Z=0 et s define 7

8 T B Γ=. (..39) T F From (..33) nd (..34) we obtin ˆ ( jβz jβ z T = T e +Γe ) F T ˆ F jβ z jβ z v ( e e ) Z0 (..40) = +Γ. (..4) In the region to the right of the interfce there is ony one wve ˆ j z T = T e β (..4) F T ˆ F jβ z v = e. (..43) Z0 The boundry condition t the pne Z=0 is continuity of T v. It eds to Z Γ= Z Z Z 0 0. (..44) The stress trnsmission coefficient is defined T τ T = = +Γ = T Z F 0 F Z0 + Z. (..45) 0 nd the power trnsmission coefficient is defined TF PF Z0 τ P = = = Γ. (..46) PF TF Z0 This prmeter determines how efficienty power is trnsmitted from one medium to nother. Note tht the mismtches in coustic re genery much more severe thn they re in the EM. Exmpes Spphire Z 0 = kg/m s for ongitudin wve Wter Z 0 = kg/m s Air Z 0 is sm The good mtching between medi is very importnt. We use λ/4 (qurter wve) mtching sections just s in optic or microwve ines. Consider the gener cse when yer (Z 0, β, thickness ) is pced in contct with semi-infinite medium (Z L ). From (..40) nd (..4) try to find the impednce t (z=-). This impednce is Z IN jβ jβ T( ) e +Γe ZIN = = Z0, (..47) jβ jβ v( ) e Γe where β is the propgtion constnt in the yer. It foows from (..44) nd (..47), Z cos β+ jz sin β. (..48) L 0 ZIN = Z0 Z 0 cos β + jz L sin β By using n intermedite yer, the input impednce cn be chnged to different vue. If the yer is λ/4 thick ( β = π /) then Z0 Z IN =. (..49) Z L For λ /: β = π ZIN = ZL. Widey different impednces cn be mtched propery if it is used λ/4 yer of suitbe impednce Z 0, but ony t one frequency. The rger the rtio Z IN /Z L or Z L /Z in, the nrrower the bndwidths of the impednce mtch. Techniques so exist for mtching with mutipe λ /4yers to improve the bndwidth. Refection of n coustic puse by n interfce. 8

9 We consider stress puse ˆ z TF (, t z) = F( t ), V incident on the interfce Z=0, Fig...4b. This wi give rise to refected puse of the sme form. Thus the tot stress T nd veocity v in the medium re ˆ z T (, t z ) = F ( t ) F ( t ) V +Γ + z V (..50) z z vˆ (, t z) = F( t ) Γ F( t+ ). (..5) Z0 V V By foowing excty the sme procedure s before, we rrive of the sme vue of the refection coefficient (..44). Suppose tht incident stress is in squre puse s iustrted in Fig...5. The refection coefficient is negtive, so the refected puse hs opposite sign. In prticur Z 0 = 0 (ir) we obtin from (..44) Γ =. The tot stress t the interfce wi be zero. The veocity of the returned echo is of the sme sign s the incident wve s veocity or the refected stress, so v B is doubed t n ir interfce. Simiry in Fig...5.b.! Second cse: perfecty rigid - Z0 =, Γ= TB =0! Fig...5 Utrsonic puse incidence to the interfce between two medi; ) Z0 Z0 ; Z Z. b) 0 0 9

10 . PIEZOELECTRIC MATERIALS.. Piezoeectric phenomen A piezoeectric phenomenon ws discovered in 880 by brothers Curie on tourmine. They found eectric chrge on the surfce of some crysts s resut of their deformtion - direct piezoeectric phenomen Fig... The vice vers effect is vid too, see Fig...b. When the eectric fied is ppied to such mteri, it chnges its mechnic dimensions. Conversey, n eectric fied is generted in piezoeectric mteri tht is strined. Fig... ) Direct (deformed eement cuses eectric votge) nd b) Reverse Piezoeectric Effect (eectric fied cuses chnge in ength of the eement) Piezoeectric effect is inherent to some cryst mteris. Some of those re pyroeectric they chnge the eectric dipoes with chnge of temperture. Sm prt of pyroeectrics is ced ferroeectrics. They content eectric dipoes orgnized to domins, which cn be porized to one direction. Retion between these three mteris is expressed in Fig... Fig... Retion between piezo, pyro fero-eectric mteris. Piezocermic is the most usbe piezoeectric mteri in prctice. It is rugged, chemicy resistive nd chep. The shpe nd dimension of piezocermic trnsducers for excittion nd detection of utrsound cn be esiy mtched to the specific go. The cermics mde from PbZrO 3 (PZT) hve good piezo-effect nd withstnd high temperture. They re dieectrics with n symmetric tomic ttice. They cn be used s trnsducers, s sensors of force, strin nd pressure nd in sign processing s fiters nd resontors... Piezoeectric cermic Piezocermic contents from oxides PbZrO 3 nd PbTiO 3. The most frequent rtio is 48% PZ nd 5% PT with the best quity for utrsonic utiiztion, Fig

11 Fig...3 Digrm of phse stte. Anti-ferroeectric orthorhombic fied (mmm) Ferroeectric ow-temperture trigon fied (3m) 3 Ferroeectric high-temperture trigon fied (3m) 4 Ferroeectric tetrgon fied (4mm) 5 Pr-eectric cubic fied (m3m) From the Fig...3 critic temperture (Curie) θ c cn be seen, where cermic is trnsferred from ferroeectric to preectric phse nd vice vers. The extent of this temperture is from 50 C to 360 C. For higher temperture thn θ c the het energy cuses the sme probbiity of pcing of Ti 4+ ions. These ions re situted in the center of cube, Fig...4, with Pb + in corners nd O - in centers of ws. For ower temperture thn θ c, Ti 4+ is trnsferred from the center of the cube, Pb + s we. Ony O - hods the sme position, Fig...4b. The eementry ces re osing the center of symmetry nd goes to tetrgon re 4 in Fig...3. A eementry ces in which ions move in one direction crete so ced domins. They exhibit dipoe moment. The dipoe directions in voume hve rndom orienttion, Fig...5. Therefore the resuting dipoe moment of the medium is zero. Fig...4 Cryst Structure of Piezoeectric Cermic. A + - e.g. Pb, B; B 4+ - e.g. Ti, Zr. ) Temperture higher thn Curie cubic structure b) Temperture ower thn Curie tetrgon structure The dipoes my be igned by ppying strong E t temperture ner Curie point. This is ced s poing, Fig...5b. Idey, fter poing the domins re igned with ech other. The eectric fied for the cermic poriztion moves from to 4 kv/mm. After removing E the poriztion is most conserved, Fig...5c. Together with poriztion, deporiztion exists too. It cn be reced by temperture, eectric fied, or mechnic stress. The piezoeectric properties re chnged. If temperture goes up, the inner energy is incresing nd body dipoe moment goes down. After crossing Curie point, the domins orienttion is ost nd cermic is chnged to nonpiezoeectric one.

12 Fig...5 Poriztion of Piezoeectric Cermic. ) rndom orienttion b) poriztion due to eectric fied E c) conserved poriztion fter removing E From Fig...6b is cer tht poriztion is connected with body deformtion. The hysteresis curve is the resut. It is the mutu dependence between strin S nd eectric intensity E. In Fig...6 the process of poriztion P on E is expressed. The re of hysteresis oop is proportion to n energy which is necessry for over poriztion of unit voume t piezoeectric. Fig...6 ) Infuence of E to poriztion P nd b) E to S. The piezocermic cn be divided to soft one, very esy porized - their permittivity is high nd hysteresis re is big. They re suitbe for sensors nd ctutors. For the hrd one the chnge of poriztion is not esy, they hve nrrow oop nd sm permittivity nd they re used for resontors nd utrsonic trnsducers...3 Piezoeectric crysts

13 Some crysts such s qurtz cn be piezoeectric without being ferroeectric. They hve most threefod symmetry-xis, Fig...7. It represents 3 dipoes ign t 0 to ech other. The sum of the dipoe moments of ech vertex is zero. When n E is ppied in z direction, the three dipoes tend to expnd or contrct by different distnces in the z direction. So the cryst cn deveop net stress. Simiry, when mteri is strined in the z direction, it cn deveop net dipoe moment, so it is piezoeectric mteri. Such mteris hve much smer piezo-stress constnt thn do ferroeectrics. Fig...7 Piezoeectric cryst tht is not ferroeectric () The unstressed cryst hs threefod symmetry xis. The three rrows represent pnr group of ions with tripe chrge, positive t ech vertex, nd singe negtive t the rrowheds. The sum of the three dipoe moments is zero. (b) When the smpe is stressed, sum of dipoe moments is finite, no onger threefod symmetry..4 Constitutive retion Sh we imgine projection of toms of tetrgon ttice of porized ferroeectric cryst, Fig...4b, to one surfce. We get the periodic tomic system Fig...8. in which the equiibrium spcing between neighboring rows of toms in the z direction re nd. The spcing in x nd y is. The dipoe moment per unit voume: q( ) dipoe strength of unit ce P = = (..) ( + ) voume of unit ce where +q; -q re chrges of toms. Consider the effect of strin on poriztion. When + nd +, the P is chnged by P. We cn write = S nd = S nd from (..) expnsion to the first order in strin q q P = S = S P = e S, (..) + + where e is ced piezoeectric stress constnt nd S is the mcroscopic strin in the mteri. The tot chnge in eectric dispcement in the presence of n E D = ε E+ P (..3) or S D = ε E+ es (..4) S where ε is the permittivity with zero or constnt strin. D depends on S nd E, Fig

14 Fig...8 Ferroeectric crysts () unstrined (b) strined. We sh determine the T due to E. The Lorentz force per unit re is +qe / nd qe /. The stresses in the regions of ength nd re qe T = (..5) nd qe T =. (..6) Therefore the verge stress in the medium due to E T + T q( ) TE = = E = e E. (..7) + + The tot stress ppied to medium is sum of externy ppied stress T nd the intern stress T E due to E. By ppiction of Hook s w, E T + TE = c S, (..8) or E T = c S ee, (..9) E where c is estic constnt in the presence of constnt or zero E. The eqution (..4) nd (..9) re the piezoeectric constitutive retions. Exmpe: Estimte the vue of e by tking: = Å, = 4 Å, = 3 Å nd q= eectron chrge Resut: e=0.6 C/m Atoms hve mutipe chrges, so e is considerby higher. For cermics (PZT/5H zirconium titnte) e z3 =3.3 C/m...5 Effect of piezoeectric couping on wve propgtion in medium of infinite extent Consider wve propgtion in piezoeectric medium of infinite extent. The wve motion is thn one-dimension. There is no free chrge within the medium nd D in the z direction is D z = 0. (..0) This impies tht D z is constnt does not vry with z, though it my vry with time. The tot dispcement current density is D id =, (..) t 4

15 which must be either uniform with z, or zero. In trnsducer with met eectrodes i D psses between eectrodes, s it does in ny cpcitor. In piezoeectric medium of infinite extent we woud expect i 0 D=0 in medium. D Piezoeectricy stiffened estic constnt. Let s sove effective vue of c with D=0 (or c D ) nd determine propgtion constnt β of wve in infinite piezoeectric medium. Writing D=0 in eqution (..4) nd substituting resut in (..9), we find es E = (..) S ε E e T = c ( + ) S = c D S. (..3) E S c ε Thus it is s if the piezoeectric medium hs n effective esticity D E c = c ( + K ), (..4) where c D is the stiffened estic constnt nd e K = (..5) E S c ε is the piezoeectric couping constnt. The definition of K my be done from different views, s the energetic definition or for fied ppied in rbitrry direction in nisotropic piezoeectric mteri. The propgtion constnt in piezoeectric medium. We use β, V for stiffened propgtion constnt nd veocity. From (..) nd D c ω β = ω( ) = (..6) ρ V ( ) + K D E c c V = = = V + K = + ρ ρ ( ) V( K ). (..7) The stiffened veocity is wys rger thn the equivent veocity in nonpiezoeectric mteri or in 4 piezoeectric medium with E=0. Exmpes - mteris such s InSb, GAs with K 0 opposite to piezocermic, LiNbO3 with K 0,5. Stress free dieectric constnt. Let s consider the properties of finite-ength medium of infinite cross-section. D my be finite nd non-zero. Thn we cn define the strin free ε S S (put S=0 into Eq...4), ε = D/ E, nd in T the sme wy ε (ε under stress free condition). Putting T=0 to (..9), e S = E. (..8) E c On substituting (..8) to (..4) we find T S D = ε E = ε ( + K ) E. (..9) Therefore T S ε = ε ( + K ). (..0) Thus the ε T is rger thn ε S. In re mteris we must know components T or D in finite piezoeectric medium. 5

16 .4 PIEZOELECTRIC TRANSDUCERS.4. Introduction Convert eectric energy into mechnic nd vice vers. Low frequency ppictions: microphones nd oudspekers. Higher frequency ppictions: piezoeectric trnsducers. They cn be used t medium frequency (-50 MHz) in qurtz cryst resontors nd fiters, nd so t much higher frequencies s buk nd surfce wve trnsducers, from few khz to microwve rnge. In Fig..4. is simpe configurtion for dey ine (bout 00 MHz), mde of spphire (Fig..4.). A met fim is deposited on either end to crete two met eectrodes. Then fim of piezoeectric mteri ZnO is deposited on ech eectrode by sputtering or evportion. Thickness of ZnO yer is chosen to be between qurter nd hf wveength. Down on the surfce of this yer is id met fim top eectrode, thickness sm frction of λ. At one end potenti is ppied between the two eectrodes to excite ongitudin coustic wve in dey ine. On the other end the wve is detected by the simir trnsducer. The eectric impednce of trnsducer wi depend on: thickness nd coustic impednce of the eectrodes, piezoeectric mteri nd the nture of the substrte mteri. To mke brodbnd dey ine, the thick of mtching yer is λ/4. Fig..4.b describes the ir-bcked trnsducer to excite wve in wter. Piezocermic: Z 0 = kg m - s -, wter: Z 0 =,5 0 6 kg m - s -. There is : mismtch in impednce. This eds to resonnt chrcteristic with coustic quity coefficient Q~30. After using λ/4-mtching intermedite yer with Z 0 =(Z 0piezo. Z 0wter ) /, we get broder-bnd chrcteristic with better power trnsmission. The resonnce cn so be brodened by using cousticy oosey mteri with impednce comprbe to tht of the PKM (PZT), bounded to the other side of the trnsducer - Fig..4.c. met eectrode dey ine rod piezoeectric mteri met eectrode met eectrodes mtching yer bcking mteri piezoeectric mteri wter wter piezoeectric yer met eectrodes Fig..4. ) Dey ine used t UHF frequencies. A mteris re typicy deposited by vcuum deposition on the dey-ine rod. b) Air-bcked piezoeectric trnsducer with mtching yer on its front surfce, for exciting wve in wter. c) Piezoeectric trnsducer oded on its bck surfce, for exciting wve in wter. 6

17 .4. The Trnsducer s Three-Port Network Let s consider uniform trnsducer with cross-section dimensions of mny λ, nd eectrodes norm to z direction, Fig..4.. Eectrodes short out the fied: E x =0, E y =0. According to the symmetry, ony ongitudin wves re generted. A prmeters ike S, E, D, v, u, T hve component ony in the z direction (see eq...4,..9). We now regrd the trnsducer s threeport bck box. We define F t the surfce of the trnsducer s we do votge in eectric circuits nd the prtice veocity v s we do current in eectric circuits. Using the nottion shown in Fig..4.3 for three-port network, nd tht shown in Fig..4.3b for the physic trnsducer, we cn find n equivent circuit. The extern force ppied to piezoeectric mteri t the surfce F = AT, (.4.) where A is the re of the trnsducer nd T is the intern stress. Fig..4. Piezoeectric resontor of ength with eectrodes on opposite surfces. eectric port coustic port coustic port Fig..4.3 ) Trnsducer, regrded s three-port bck box. b) Retion of three-port nottion to the physic prmeters of the trnsducer. The theory of trnsducers re bsed on the ide tht prtice veocity v ~ current I, mechnic stress T ~ votge V nd +v mens the direction inwrd to the piezoeectric mteri. Boundry condition t coustic ports: 7

18 F= A T( ) v= v( ) F = A T( ) v = v( ). (.4.) From (..4) nd (..), the retion between T nd v within the trnsducer dt = jωρv (.4.3) dz nd dv = jω S, (.4.4) dz nd from (..), the tot current through the trnsducer I3 = jω AD. (.4.5) The votge cross the trnsducer V 3 = Edz. (.4.6) Since current is conserved, D must be uniform with z. Eiminting E from (..4) to (..9) nd with using (..3) for finite D : D T = c S h D, (.4.7) where h is known s trnsmitting constnt: e h = (.4.8) ε S nd e c c c K c ε D E E ( ) ( ) E S = + = +. (.4.9) After eiminting T nd S from (.4.3), (.4.4), (.4.5) nd (.4.7) we hve the wve eqution for v: dv ω ρ + v = 0. (.4.0) D dz c This hs the soution jβz jβ z v= vf e + vb e (.4.) nd jβz jβz T = TF e + TB e hd, (.4.) where subscript F nd B denote forwrd nd bckwrd propgtion wves. β mens stiffened constnt of propgtion, ρ / β = ω( ) (.4.3) D c D / Z0 = ( ρ c ) (.4.4) nd with (..33) TF = Z0 vf (.4.5) nd TB = Z0 vb. (.4.6) Using the boundry conditions, eq. (.4.) nd (.4.), vsin β ( z+ ) + vsin β ( z+ ) v = sin β Substituting (.4.7) to (.4.)-(.4.7) it foows. (.4.7) 8

19 F Z cot β Z cos ecβ h/ ω v F = j Z cos ecβ Z cot β h/ ω v V h/ ω h/ ω / ωc I C C C C 3 0 3, (.4.8) where the cmped (zero strin) cpcitnce of the trnsducer is ε S A C0 = (.4.9) For consistence with definition of eectric impednce, we define coustic impednce of n A re of piezoeectric mteri ZC = Z0 A, (.4.0) where Z c hs dimension force/veocity or kg/s, nd Z 0 hs dimension pressure/veocity or kg/m s. Impednce Z c is sometimes ced rdition impednce. Exmpe: Eectric input impednce of trnsducer Let s determine the eectric input impednce of trnsducer terminted by coustic od impednce Z nd Z using (.4.8). We define the rdition impednces of the ods (ooking outwrd from the trnsducer) s (Fig..4.3b): nd Z Z F AT ( /) v v( /) = = F AT ( /) (.4.) = =. (.4.) v v(/) Using Z nd Z in (.4.8) yieds the eectric input impednce of the trnsducer V 3 j( Z+ Z) ZC sinβ ZC( cos β) Z3 = = + k T, (.4.3) I 3 jωc 0 ( ZC + ZZ ) sin β jz ( + Z) ZC cos β β where K kt = (.4.4) + K where K is previousy derived (..5) the piezoeectric couping constnt nd c c D = + K =, (.4.5) E kt K is kt K. k T is nother expression of K. For.4.3 Mson Equivent Circuit Wi be shown, tht the mtrix formu (.4.8) resuts in the Mson equivent circuit of Fig h v + v ) is The trnsformed potenti doesn t pper cross termins or. A potenti - ( jω deveoped cross the negtive cpcity C 0, which cnces out the potenti cross C 0, Fig

20 Fig..4.4 Mson series equivent circuit. N = hc0 = ec0 / ε S = ea/ ; Z = Z0A C From (.4.8) consider vue of F hi3 F = jzcvcot β jzcvcos ecβ+ jω (.4.6) When I 3 = 0, the first two terms (.4.6) write in the form of n impednce mtrix of type in Fig..4.5b Z = jzc cot β Z = jzc cos ecβ (.4.7-8) nd β Z Z = jzc(cos ecβ cot β) = jzc tn β Z Z = jzc(cos ecβ cot β) = jzc tn. (.4.9) There is n extr potenti hi 3 /jω in series with the potentis generted by v nd v. Let s consider h I3 V3 = ( v+ v) + jω jωc 0. (.4.30) vue of V 3 The st term in (.4.30) is votge cross the cpcitor C 0. The first term is votge proportion to equivent current (v + v ), fowing into the trnsformer of vue N:, where / S N = hc0 = ec0 ε = ea/ woud introduce trnsformed current (v+ v )/N into the right side of the h circuit nd the current woud then deveop potenti ( v + v )cross C 0. jω h The trnsformed potenti V 3 does not pper cross termins or. potenti - ( v + v ) is jω deveoped cross the negtive cpcitor - C 0, in Fig..4.4, which cnces out the potenti cross C 0. This equivent circuit so gives the st term in F. The fin Mson equivent circuit is in Fig

21 Fig..4.5 Redwood equivent circuit. ) Coxi trnsmission ine of impednce Z C. b) T-network equivent of coxi ine. c) Redwood equivent circuit, derived from the Mson mode of Fig Exmpe: Cmped trnsducer If the trnsducer is rigidy hed so tht v =v =0, the mechnic termintion impednces re infinite. It foows from (.4.8) or the Mson equivent circuit Fig h F = j I3 ω h F F ea F = j I3 = = hc0 = N = (.4.3) ω V V V = j I 3 3 ωc Redwood Equivent Circuit It cn be derived from the Mson mode Fig The T-network Z, Z, Z of the Mson circuit cn be represented by trnsmission ine of impednce Z C, s in Fig..4.5,b. This mens we cn write the Mson mode in the form in Fig..4.5c, Redwood mode. Trnsmission ine is e.g. coxi ine, whose outer shied is connected to the trnsformer. Redwood mode is usefu for deing with short puse excittion of trnsducers - when puse ength is smer thn the dey time of n coustic wve pssing through the trnsducer. The input impednce ~ Z C nd is esy to determine the puse response. Exmpe: Open-circuited puse-excited cousticy mtched receiving trnsducer

22 Trnsducer open-circuited (I 3 =0). Votge cross the trnsformer is zero becuse the two C 0 nd -C 0 in series, form short circuit. Suppose tht the trnsducer is excited t its eft hnd side with veocity puse v (t). The puse propgtes ong the trnsmission ine nd gives rise to puse v (t) v() t = v( t T) (.4.3) nd T = / V is its trnsit time ong the ine. The current fowing into C 0 I = N( v+ v) = N[ v( t) v( t T) ]. (.4.33) This current cretes chrge dq=c 0 dv=i dt. From which the output votge or t N V = v () t v ( t T) ]dt [ 3 C0 0 t (.4.34) [ () ( )]. (.4.35) V = h v t v t T dt 3 0 If v (t) hs form of δ -function veocity puse, V 3 (t) wi be squre-topped puse of ength T s is iustrted in Fig veocity puse output votge Fig..4.6 Output votge of open-circuited receiving trnsducer, terminted in mtching coustic impednce when excited by δ -function veocity puse..4.5 Impednce of n Unoded Trnsducer Consider n unoded trnsducer in ir (with thin eectrodes). In this cse, F =F =0 or Z =Z =0. From Eq. (.4.3) tnβ V3 Z = 3 ( kt ) I = 3 jωc. (.4.36) 0 β This eqution shows tht the equivent circuit of the trnsducer shown in Fig..4.7 cn be represented by the cmped cpcity of the trnsducer in series with the motion impednce Z (i.e. Z is the coustic contribution to the eectric impednce) defined by the retion tnβ k T Z = jωc. (.4.37) 0 β Fig..4.7 Equivent circuit of piezoeectric trnsducer.

23 The trnsducer exhibits pre resonnce with n infinite eectric impednce (ike pre resonnce circuit) t frequencies where the trnsducer is n odd number of hf-wve-engths ong (i.e., where β = ( n + ) π ). The corresponding resonnt frequencies re given by the retion π (n+ ) V ω 0n = (.4.38) For simpicity, we sh c the owest-order pre resonnce (n=0) ω 0. The trnsducer exhibits zero eectric impednce t frequency ω ner the (n=0) pre resonnce. Ner this frequency ω, the trnsducer behves ike n inductnce nd cpcitnce in series, nd so exhibits series resonnce. At this frequency ω, the trnsducer impednce is Z 3 =0 nd from (.4.36) tnβ =. β kt (.4.39) It foows from eq. (.4.38) nd (.4.39) tht πω tn ω0 = πω k T. (.4.40) ω 0 We cn, in principe, determine k T from (.4.40) by mesuring ω ndω 0. Equivent circuit of n unoded trnsducer. It is convenient to express the impednce Z in the form of equivent umped circuits tht correspond to the fundment resonnce nd higher-order resonnces of the resontor. The function tn x hs poes t x=(n+) π. This ows us to obtin prti frction, or Mittg- Leffer expnsion, in the form x tn x =. (.4.4) n= 0 π n+ ) x We cn express the motion impednce Z in simir form Z = jωc k eff, n, (.4.4) 0 n ω ω0n where k eff,n is n effective couping coefficient for the n-th mode, defined by the retion 8 k eff, n = k T. (.4.43) ( n + ) π ) The couping coefficients to the higher-order modes of the resontor k eff,n f off with n. This impies tht it is possibe to excite resontor t n odd hrmonics of its fundment resonnt frequency, though the effective couping coefficient for this higher-order mode is smer then for the fundment mode. Couping to higher-order modes is often very convenient. This mkes it possibe, for instnce, to work t frequency of 00 MHz with qurtz or LiNbO 3 resontor whose fundment frequency is in the 0 MHz rnge. In this cse, the trnsducer hs thickness on the order of 0, to 0, mm, mking it esier to hnde thn fundment mode resontor, which woud be ony 5 μm thick. This technique so tend to give higher Q or nrrower bndwidth thn thin fundment mode resontor, whose pped surfces might contribute some power oss. Let us derive umped equivent circuit for the trnsducer. Using (.4.4) we write the impednce of resontor s 3

24 Z eff, n 3 = jωc 0 n ω ω0n k. (.4.44) From this we cn get equivent circuit shown in Fig Fig..4.8 Compete series equivent circuit incuding resonnces. Ech pre circuit hs resonnt frequency ω 0n. 4