Efficiency of excitation of piezoceramic transducer at antiresonance frequency

Transcription

1 fficiency of excittion of piezocemic tnsduce t ntiesonnce fequency Mezheitsky A.V. fficiency of excittion of piezocemic tnsduce t ntiesonnce fequency dpted fom I Tns. Ultson. Feoelect. Feq. Cont. vol. 49 N 4 pp Abstct. The efficiency of piezocemic tnsduces excited t both the esonnce nd ntiesonnce fequencies ws investigted. Losses in piezocemics e phenomenologiclly consideed to hve thee coupled mechnisms: dielectic mechnicl nd piezoelectic losses. xpessions fo the esonnce nd ntiesonnce qulity fctos which ultimtely detemine the tnsduce efficiency hve been eceived on the bsis of complex mteil constnts fo both stiffened nd unstiffened vibtion modes. Compisons of electic nd mechnicl fields theml nd electicl losses of powe supply nd thei distibution in the tnsduce volume hve then been mde. Fo given constnt mechnicl displcement of the tnsduce top the equied electic voltge pplied to the tnsduce t the ntiesonnce fequency is popotionl to the esonnce qulity fcto while the chnges in the intinsic electic nd mechnicl fields chcteistics in the common cse e not too essentil. The equiements on the piezocemic pmetes types of tnsduce vibtion nd especilly on the fcto of piezoelectic losses in nge of physiclly vlid vlues wee estblished to povide mximl qulity fctos t the ntiesonnce fequency. Autho: A.V. Mezheitsky e-mil: mv5@optonline.net ( CC: mv.five@veizon.net )

2 I. INTROUCTION. Piezocemic mteils (PCM) of PZT system [] hve unique electo-elstic popeties. Piezocemic tnsduces (PT) e used in electonics nd hydo- nd electocoustics fo wide nge of pplictions. High powe mechnicl output PT chcteistics e tditionlly chieved t the esonnce fequency becuse of the eltively low equiement fo electic voltge. Howeve the pocess of PT heting nd s consequence filue of output PT chcteistics in stong electic nd mechnicl fields esticts poweful PT pplictions. It equies optimiztion of egimes of PT excittion. As pedicted in [] the dvntges of excittion of the esonnt poweful PT t n ntiesonnce fequency ginst the tditionl esonnt egime with egd to PT heting nd powe supply losses hve found expeimentl nd pcticl confimtions [3] - [6]. Conclusive specific nd decisive oles of piezoelectic enegy losses wee detemined. The diffeence between the ntiesonnce f nd esonnce f PT fequencies [] [8] depends on degee of PCM poliztion nmely on the vlue of the coefficient of electomechnicl coupling (CMC) k ij nd hmonic numbe n of used type of vibtion. The qulity fcto Q chcteizes esonnt PT behvio. PZT-bsed PCM povide stong coupling of mechnicl nd electic fields (CMC vlue cn ech 0.9 close to theoeticl limit [7]) so the mechnicl nd electicl components of enegy losses s well s the piezoelectic component descibing losses t inteconvesion of the mechnicl nd electicl enegies should be involved in the phenomenologicl desciption of PT behvio. Tditionlly the esonnce qulity fcto t the fundmentl (lowest) hmonic of PT pln mode (disk) [] [8] ws chosen fo chcteizing PCM qulity. lecto-elstic PT behvio cused by nisotopy of polized piezocemics genelly is descibed by ten independent constnts [8] [9]. Becuse it is n electo-mechnicl oscilltoy system PT hs two limit egimes of opetion: shot-cicuit (s.c.) nd open-cicuit (o.c.) t closed nd boken PT electodes espectively. The CMC vlue is detemined by diffeence between el pts of the elstic PCM constnts unde constnt [] electic field stength nd induction coesponding in common cse to the s.c. nd o.c. egimes in pticul k is the CMC s T S = S ( k ) whee k = d S ε. The CMC s el vlue defines she of the totl enegy ccumulted in PT nd conveted fom mechnicl into electicl foms nd vice ves. In tun the vlues of the qulity fctos coesponding to the s.c. (Q s.c. ) nd o.c. (Q o.c. ) egimes in common cse e defined by the complex elstic constnts (e.g. Sˆ nd Sˆ ) whose imginy pts e defined by the mechnicl dielectic nd piezoelectic mechnisms of enegy losses [9]. Tking into

3 ccount the piezoelectic component of losses fo considetion ws offeed fo the fist time in [0]; the pesence of non-zeo imginy pt of the piezocoefficient ws found by diect mesuement []; high ccucy method of complex mteil constnt detemintion ws poposed in []; the physiclly vlid limits on the piezoelectic loss fctos ws estblished in [9] ccoding to the condition of positivity of locl theml losses in piezoelectic medi; the concept of complex mteil constnts ws used in [3] [4] to model dynmic PT esponse. Involving domin mechnism of enegy loss [5] with dmped (time dely) movement of wlls of 90-degee domins the unifom physicl ntue of coupled mechnicl dielectic nd piezoelectic enegy loss components ws shown. In the common cse thee e two bsic types of PT vibtions: stiffened mode (SM) with diection of vibtion long n exciting electic field (e.g. od 5xx mm PT (Fig. 3)) when the electic field induction is homogenous between electodes [8] nd unstiffened mode (UM) with diection of vibtion pependicul to the vecto of n exciting electic field (e.g. b 5xx0.5 mm PT (Fig. 7)) when the electic field stength is homogenous between electodes in the cse of thin PT plte. Ultimtely esech egding the inteeltion of the PT dissiptive chcteistics t the esonnce nd ntiesonnce fequencies the esonnce Q nd ntiesonnce Q qulity fctos is of pcticl nd theoeticl inteest. II. COMPLX LASTIC COMPLIANCS QUALITY FACTORS. The electomechnicl behvio of el PT with enegy losses genelly is descibed [9] by: the complex constnts of the PCM mteil mtix ˆ S S ( iq ) ˆ T T d d ( iγ ) εˆ ε ( iδ ) ; ( ) ij ij ij the line equtions of piezoeffect s i = kl kl kl Ŝij T j + d ˆki k n = mn mn mn T εˆ mn m + d ˆnlT l ; nd the eqution of motion ρ u tt = T t given boundy nd initil conditions whee i Q ij e the qulity fctos of the complex elstic complinces ˆ S ij δ mn nd γ ki e the dielectic nd piezoelectic loss ngles espectively ρ is the PCM density nd s nd T e the electic field induction nd stength mechnicl stin nd stess espectively in vecto o tenso pefomnce is the vecto opeto u is the locl mechnicl displcement. Simil pefomnce is used fo the constnts of the mteil mtixes though C ˆ β [8] [9]. T S ij gˆ kl eˆ kl hˆ ˆ kl mn

4 Pticully the complex constnts ˆ with thei inteeltion S ˆ ˆ ˆ S = S( k) whee ˆk is the complex CMC hve elstic constnt qulity fctos detemined by the bse eltion: Q ( k ) + Qδ ( t3) kqδ k = = γ δ Q k Q Q( k). ( ) Fom the condition of positivity of the locl theml enegy losses the limittion on the vlue of the piezoelectic loss ngle ws phenomenologiclly estblished [9] s follows γ y 3 δ k Q o t 3 γ / y 3 [-; ]. ( 3 ) Unde tht condition the theml locl dissiption is lwys positive tht mens heting so integl dissiption fo the whole tnsduce is positive too consequently. Menwhile it does not men tht ech of the components cusing the theml losses such s locl losses of the electic o mechnicl fields must be positive septely in piezoelectic body especilly with stong coupling of the electic nd mechnicl fields. Accoding to () Q cn be genelly gete o less thn Q. The mximum incese of the Q Q is povided by the non-zeo imginy pt γ of the piezocoefficient nd coesponds to the conditions k Q δ nd t 3. In specific cse of bsence of the dielectic (δ = 0) nd piezoelectic (γ = 0) losses: Q Q k Q δ = ( ) ( ) whee influence of the eltive fequency esonnt intevl δ hs opposite sign nd stonge effect in compison to the tditionl PT equivlent cicuit (C) tking into ccount s it is consideed only the mechnicl losses [] (see chpte VI). Futhe the following definition of the qulity fcto of PT esonnce (esonnt pek) nd ccodingly method fo its clcultion e used. Tking the expession descibing the esonnt PT behvio fte decomposition it by the smll dissiptive pmetes ( Q ij δ mn γ kl ) nd eltive fequency displcement χ = f f0 fom the "idel" fequency f 0 of the esonnt pek (loss-fee) B this expession is pefomed [] by fist-ode ppoximtion s ia + + iξ whee A B e coefficients nd the qulity fcto Q of the esonnt pek is defined s coefficient in the genelized fequency displcement ξ = Q χ. The given ppoch methodiclly coesponds in full mesue to the known fequency methods [] [8] of qulity fcto detemintion which e bsed on the mesuement of chcteistic fequencies of the el nd imginy pts of

5 esonnt PT chcteistics (e.g. dmittnce o impednce) nd elted to them methods [3] [4] [6] [7]. III. PT with STIFFN VIBRATION MO. A. Resonnce nd Antiesonnce PT Qulity Fctos Let's futhe conside long od PT epesenting the SM vibtion. The given type of longitudinl PT vibtion long the vecto of poliztion is genelly descibed by the complex electo-elstic PCM constnts [] [8] [9]: The PT complex impednce is the following ˆ (qulity fctos S Q ) T ˆε (δ ) ˆd (γ ). ˆ tn( Kh/) Z = k iω Cˆ ( kˆ ) Kh/ / 0 ( 4 ) T whee Ĉ 0( ˆε ) ˆ K = ω ρs ˆk = ˆd / T ˆε Sˆ e the qusisttic PT cpcitnce nd the expessions fo the complex wvenumbe nd CMC espectively h is the PT length. Using (4) fte decomposition of the expessions fo the PT dmittnce (/Z) nd impednce (Z) by the smll pmetes of the eltive fequency displcement nd dissiptive coefficients in vicinity ccodingly of the esonnce fn = Bn h ρ S nd ntiesonnce fn = n h ρ S fequencies [] whee B n = q π k q n is the n th -oot of the fequency eqution n tn q k q = ( e.g. B = 0.04 t k = 0.5 ) we will eceive in common cse the expessions fo the esonnce Q n nd ntiesonnce Q n qulity fctos of n-hmonic: k = H n in δ Q k Q Q n H ( 3 ) = Hn + k Q δ + t kqδ + kq δin (5) Q k n γ ( ) k = γ δ Q ( ) = kqδ + ( t3) kqδ + kq ( δin + δ out ) (6) n Q Q k Q Q( k) whee H n ( k) =. Note tht the coefficient H n ( k ) hs vlues (t k = 0 0.7) : ( q k + k 4 ) n H = H 3 = H 5 = nd futhe Hn. Fo the fundmentl hmonic (n = ) we will futhe use the following ppoximtion (no moe ( k) 8 0%) fo the coefficient 4 q k + k π nd ntiesonnce PT esistnces e: so tht H 8/ π Thus the esonnce

6 8 R ωc0 k Q π 8k Q R = πω ( ) C0 k. ( 7 ) Hee δ = δ + δ (δ = δ in + δ out ) δ in = δ + δ in γ y 3 δ kq t 3 = γ / y 3 [-; ] δ is the dielectic (domin) component of the dielectic loss ngle δ is the component cused by the conductivity of fee chges (intenl δ in nd extenl δ out ; the lst one cn be PT sufce conductivity o esisto connected in pllel to PT). In ccodnce with (5) nd (6) fo the fundmentl hmonic (n = ) of ou inteest the esonnce qulity fcto Q Q (tking into ccount H << ) nd Q hs wek dependence on the othe dissiptive PCM pmetes. At the sme time the nge of vition of the ntiesonnce qulity fcto Q vlue cn ech in common cse some odes tht depends on combintion of the dissiptive PCM pmetes. The theml (ievesible) PT enegy losses W = 0.5V Re (/ Z ) unde n electic excittion by the voltge V e detemined by the el pt of the PT complex dmittnce. Re(/Z) is the most sensitive to the piezoelectic loss component ( t-pmete) t the fequencies bove the esonnce including ne-ntiesonnce fequency nge (Fig. ). Figue. Clculted fequency dependence of the el pt of the od PT dmittnce fo diffeent vlues of t (3) pmete nd Q =00 k = 0.7 nd δ = Clculted ccoding to (4) the fequency dependence of the el pt of the od PT dmittnce t the fundmentl hmonic is given in Fig. which illusttes descibed PT dissiptive popeties t vition of the piezoelectic loss vlue in the pesence of dielectic loss. The fequency dependence of the dissiption hs minimum (Fig. ) which fequency mtches the ntiesonnce t

7 k Q δ (we tke hee δ s δ component only). At 0.5 t 3 the totl losses in the mentioned nge of fequencies e less thn the dielectic losses cused by the specific ole of the piezoelectic component of losses [] [8]. Figue. Clculted fequency dependence of the el pt of the od PT dmittnce fo diffeent vlues of the dielectic loss pmete δ ( k Q δ ): cuve 0 (0) 0.0 (0.49) (0.74) (0.98) (.7) t Q =00 k = 0.7 nd t 3 = (fo ech cuve) line of minim. B. Bsic lectic nd Mechnicl Field Chcteistics The min chcteistics of the od PT vibtion [] [8] e descibed by: V kˆ N( x) h k F 3( ) = ˆ + x V ux ( = h/) = dˆ ( kˆ ) V dˆ N( x) T ( x) = h S k F 3 ˆ ˆ + F + kˆ F V ˆ ˆ Nx ( ) s ( x) = d ( k ) h k F 3 + ˆ ( ) V ( ) 3 x = = const x ( 8 ) iω Z whee tn( Kh / ) F Kh / cos( xkh /) N( x) e esonnt fctos x [0;] is the eltive cos( Kh /) coodinte fom the cente to the PT top U u(x=h/) is the bsolute vlue of the tnsduce top mechnicl displcement. Tking into ccount in common cse tht k Q >> ( stong esonnce ) nd consideing just the fundmentl hmonic (n = ) we eceive mximum vlues of the ppopite tnsduce chcteistics. Fo the tnsduce top mechnicl displcement (x = bsolute vlue) :

8 U( ω ) Vd 8 π Q Vd k Vd U ( ω) = U ( ω 0) = ( 9 ) k then U( ω) V( ω) 8 k Q. U( ω ) V( ω ) π ( k ) Fo the mechnicl stess (t the cente x = 0 of PT bsolute vlue): V d 4 T ( ω) Q h S π V d T π ( ω ) = then h S k T ( ω) V( ω) 8 T ( ω ) V( ω ) π Fo the mechnicl stin (t the cente x = 0 of PT bsolute vlue): k Q. ( 0 ) V 4 s ( ω) d ( 4 k / π ) Q h π V π ( k ) s ( ω) = d then ( ) h k s ( ω) V( ω) ( 4 k / π ) 8 s ( ω ) V( ω ) k π k Q. Fo the mximum vlues of the electic field stength (t the cente x = 0 of PT t the ntiesonnce ω nd t the top x = of PT t the esonnce ω fequency bsolute vlues): V 8 V ( ω ) k Q π ( ω) V( ω)6 ( ω ) = then k 3 Q. ( ) h π h ( ω ) V( ω ) π Tking into ccount tht the electic field induction 3 = const(x) t ny fequency (8) in the cse of od PT we hve ( ω ) V( ω ) V( ω ) 8 ( ω ) V( ω ) ω R V( ω ) π k 4 ω R k QQ =. ( 3 ) When the voltge pplied to the PT does not depend on fequency the mximum (long PT length) vlue of the mechnicl chcteistics (T s u) hs the esonnce the mplitude of which is popotionl to the esonnce qulity fcto Q. At the sme time they do not hve n ntiesonnce pek s in the cse of the electic field induction o totl PT dmittnce. The electic field stength is inhomogeneous long the PT length nd the chcteistics of its distibution depend on the fequency of PT excittion. If t the ntiesonnce fequency the electic field stength hs sinusoidl fom of distibution long PT length while t the esonnce fequency the electic field eches two locl mxim t the cente nd t the top of od PT popotionl to the esonnt fcto k Q (Fig. 3). Tht is the popety inheent to the bsolute vlue of the electic field stength nd the following eltionship must be nywy stisfied 3 xdx = V h ( 4 ) 0 Re ( )

9 Figue 3. istibution of electic field (bsolute vlue) long length of od esonto t the esonnce () nd ntiesonnce () fequencies t equl PT top displcement. Q =00 k = 0.7 δ = 0.05 nd t 3 =. Fo the enegy loss efficiency unde the sme conditions tking into ccount common expession fo the ctive enegy losses of geneto of tnsduce excittion W = 0.5 V Re(/ Z) we hve : V 8 W( ω ) = V C k Q ω 0 R π V π k W( ω) = = V C ( 5 ) R 8 ω 0 kq then 4 V k QQ W( ω ) ( ω ) ω 8 W( ω ) V( ω ) ω π k If we conside the condition U(ω )= U(ω ) (denoted by veticl line) of ou inteest then. V( ω ) 8 k V( ω ) π ( ) k Q (...) k Q ( 6 ) (the lst estimtion is given fo the nge of k = 0 0.7). Fo the othe chcteistics we hve T ( ω) T ( ω ) k... s ( ω ) s ( ω ) 4k π...5 ( 7 ) ( ω) π ( ω ) k ( ω) π ( ω) 8 k Q ( 8 ) W( ω ) Q ω Q ( k ). ( 9 ) W( ω ) Q Q (...5) ω So to povide equl mplitude of the PT top mechnicl displcement t both the ntiesonnce nd esonnce fequencies it is equied tht one pplies voltge to PT t ~ k Q gete

10 (popotionl to the esonnce qulity fcto). At the sme time the mximum vlues (with espect to PT volume) of the mechnicl stin nd stess e no moe thn.3 nd times gete espectively t the ntiesonnce fequency nd these diffeences depend only on the CMC vlue but not dissiptive chcteistics (qulity fcto etc.). Note tht even if the voltge pplied to od PT t the ntiesonnce fequency is too high (~ k Q 00 t k = 0.7 nd Q =00) the mximum stength of the electic field (eltive vlue) does not exceed 3 times nd this tio does not depend on qulity fctos. Fo this eson non-line effects t stong level of PT excittion must be commensuble fo the esonnt nd ntiesonnt egimes of PT excittion t equl PT top mechnicl displcement tht llows to extend pesented esults to the pcticlly impotnt cse of high powe loding. At the sme time chcte of field distibution in the PT volume cn diffe significntly. The piezoelectic loss fcto cuses chnges not only in totl PT enegy losses but lso influences the distibution of the theml losses in the PT volume nd cn povide specific PT egions with n extemely low o high heting (Fig. 4). Figue 4. istibution of theml losses long length of od PT t the esonnce nd ntiesonnce fequencies t equl PT top displcement fo diffeent vlues (- 0 +) of t (3) - pmete of piezoelectic losses. Q =00 k = 0.7 δ = Thee e no piezoelectic losses t t 3 = 0. The pmete (9) of eltive enegy losses of the electic souce of PT excittion (totl theml losses in the whole PT volume) ultimtely defining eltive efficiency of PT excittion t the ntiesonnce fequency e popotionl to the tio of the ntiesonnce qulity fcto to the esonnce one with some coefficient depending only on the CMC (eltive esonnt fequency intevl). The expession nd vlue of this fcto depends on initil tsk in espect to the egime of

11 PT excittion. We hve consideed hee the chcteistics of the sme PT t the esonnce nd ntiesonnce fequencies whose eltive diffeence depends only on the CMC vlue nd cn exceed bout 5 40% (fo k cse). Fo this eson the vibtionl velocities hve n dditionl diffeence by the sme fcto t equl mplitude of PT top mechnicl displcement t the espective fequencies. Fo pcticl puposes the compison of two diffeent PTs with equl esonnce nd ntiesonnce fequencies espectively esily cn be mde on the bsis of the given eltionships. Figue 5. Clculted dependence of totl loss enegy efficiency fo the esonnce nd ntiesonnce fequencies on t (3) pmete t equl top displcement of od PT. Q = 00 nd δ ( cuve 0 (0) 0.0 (0.49) (0.74) (0.98) (.7) ll t k = (0.98) t k = (0.98) t k = k Q δ ): ithe wy fo the totl losses t the ntiesonnce fequency to be less thn losses t the esonnce fequency the tio of the espective qulity fctos must be t lest Q Q >.5 nd fo highe efficiency should essentilly exceed this vlue (Fig. 5). Functionl dependences of the esonnce Q nd ntiesonnce Q qulity fctos on the mechnicl dielectic nd piezoelectic components of losses e descibed by (5) nd (6). Tking into ccount tht fo the SM vibtion H << t the fundmentl mode (n = ) we hve Q Q nd Q = Q. Fom (5) nd (6) follows tht mximum vlues of the qulity fctos Q nd Q coespond to the conditions k Q δ nd t. ( 0 ) ( ) 3 The esonnce qulity fcto mx Q = Q ( H) Q π 8.5 Q. Theoeticlly mx Q is unlimited nd the diffeence between Q nd Q cn ech some odes in n idel cse. When consideed tditionlly only the mechnicl nd dielectic mechnisms of enegy losses cuses decese (moe thn times in espect to Q ) in the ntiesonnce qulity fcto.

12 The possibility of n incese of the Q qulity fcto vlue is diect consequence of the pesence of the non-zeo imginy pt (t t 3 > 0) of the ˆd piezocoefficient. An incese of the Q qulity fcto vlue is definitely impossible t δ < 0. k Q nd δ > 3.5 k Q (Fig. 6). Figue 6. Clculted dependence Q () / Q vesus kqδ ccoding to (5 6) nd (3 4) using genelized pmetes fo: k = 0.6 H n (F n ) = ( ) 0.8 (- - -) nd t = (cuve ) 0.9 () 0.85 (3) 0.5 (4) 0 (5). Fom the expeimentl dt fo PZT-bsed PCM it follows tht the eceived vlues of t 3 = coespond to mximum degee of poliztion. In common cse the pmete t 3 is popotionl to the CMC k. Accoding to the pticul mechnism of enegy losses [5] the dmped movement of wlls of 90-degee domins povides the t 3 - pmete tht is detemined by only n effective ngle α 0 (90 80) of domin oienttion s t 3 = A A ) ( A )( ) ( 3 A4 whee A m = sin(mα 0 )/mα 0. We cn estimte the chcte of chnge of the piezoelectic loss ngle γ = d ˆ ˆ d depending on PCM poliztion ( P ): ˆd ~ sin 3 (α 0 ) nd ˆd ~ + cos(α 0 ). Fo P 0 it follows tht γ 0 ( Q nd δ e supposed to be pcticlly constnt). Howeve the γ vlue vies within limits of 0 % in wide intevl ( 0.) of stuted poliztion (mx k ) i.e. fo the mentioned intevl of the CMC vlues it is possible to ccept tht γ const ( P ) then t 3 = k δ Q γ ~ k. ( )

13 IV. PT with UNSTIFFN VIBRATION MO. A. Resonnce nd Antiesonnce PT Qulity Fctos Let's futhe conside long b PT epesenting the UM vibtion. The given type of PT vibtion in plne pependicul to the vecto of poliztion is genelly descibed by the complex electoelstic PCM mteil constnts [] [8] [9]: ˆ (qulity fctos S The complex dynmic dmittnce of the b PT is descibed by: Q ) T ˆε (δ ) ˆd 3(γ 3 ). T whee Ĉ 0( ˆε ) = / ˆ ˆ + ˆ tn( Kb / ) Y iω C0 k3 k3 ( ) Kb / K = ω ρ Sˆ ˆk 3 = ˆd 3/ T ˆε Sˆ e the qusisttic PT cpcitnce nd the expessions fo the complex wvenumbe nd CMC espectively b is the PT length. Afte decomposition the expessions fo the PT dmittnce (Y) nd impednce (/Y) by the smll pmetes of the eltive fequency displcement nd dissiptive coefficients in vicinity of ccodingly the esonnce fn = n b ρ S nd ntiesonnce fn = Gn b ρ S fequencies [] whee G = q k π q n is the n th -oot of the fequency eqution n n 3 tn q = q ( k ) 3 k 3 ( e.g. G = 0.03 t k 3 = 0.5) we will get the expessions fo the esonnce Q n nd ntiesonnce Q n qulity fctos (n = hmonic numbe): k3 = n F 3 δ Q k3 Q Q n Q = Q ( 3 ) n Fn = 3 ( ) Fn + k Q δ + t k3qδ + k3qδ (4) Q k 3 γ ( ) whee ( k ) Fn = q q k q k 3 4 n ( n ) 3+ n 3. Note tht the coefficient F n F n ( k 3 ) hs vlues: F = 8/π ( 0.8) 0.75 t k 3 = nd futhe Fn ~ n. Hee γ 3 y δ k3q t = γ 3 / y [-; ]. Fo the fundmentl hmonic (n = ) the esonnce R nd ntiesonnce R PT esistnces e descibed by (7) with eplcement of k k 3 pmetes. Thus the esonnce qulity fcto Q vlue t the fundmentl mode is detemined exctly by the qulity fcto Q of the complex elstic complince ˆ : Q = Q. Tking into ccount tht fo the UM vibtion F (n = ) we hve Q Q. Fom (4) it follows tht mximum vlue of the ntiesonnce qulity fcto Q t the fundmentl mode coesponds to the conditions S

14 k Q δ = nd t = thus mx Q = Q ( F) 5.3 Q. Theoeticlly in n idel cse the 3 ( ) diffeence between Q nd Q cn be extemely ppecible. B. Comptive Anlysis of the Loss istibution in PT Volume fo the Stiffened nd Unstiffened Modes of Vibtion. The esonnce qulity fcto Q (n = ) of the UM vibtion is only detemined by the mechnicl loss component ( Q fo b PT) but fo the SM vibtion it depends on both Q (od PT) nd to some degee the dielectic nd piezoelectic loss components. The influence of the lst components tkes plce becuse in the cse of SM vibtion the locl electic field is inhomogeneous between the PT electodes nd hs esonnt chcte (similly to cuent defomtion etc.) unde the condition (4). The locl vlue of the field stength t the esonnce inceses by odes tht s consequence inceses she of the dielectic (nd piezoelectic) loss component in the totl esonnce enegy loss. In pticul fo this eson the contibution of conductivity of fee volume chges into the vlue of the esonnce qulity fcto inceses (minly in the PT volume egions with lge esonnt vlues of the field ) but thee is no contibution of sufce o extenl conductivity whee field hs noml non-esonnt vlue. At the sme time in the cse of UM vibtion the electic field stength hs eltively low vlue due to its non-esonnt behvio. Figue 7. Clculted distibution of the locl electicl losses long length of b PT t the esonnce () nd ntiesonnce () fequencies unde the condition of constnt PT top displcement (U) fo: Q = 00 k 3 = 0.7 δ = 0.0 nd t () -pmete in the llowed nge [- ] (shown in pentheses).

15 Reltive popeties of the mechnicl field (stin stess) in the cse of b PT (UM) e descibed by the eltionships simil to (7) fo the od PT cse (SM). Howeve the distibutions of the electic field stength nd induction diffe essentilly. The electic field stength 3 = const(f) does not depend on fequency nd hs homogeneous distibution in b PT volume. The electic field induction hs inhomogeneous distibution long b PT length (fequency-specifying size) hs lso esonnt (mx ~ k3 Q ) nd ntiesonnt (min ~ 3 k Q ) peks nd povides specific volume distibution of both locl losses of electic souce of excittion nd locl theml losses (Fig. 7 8). * * The locl losses of the electic field e descibed by p( x) = 0.5 Re( j ( x)) = 0.5ω Im( ( )) x whee j(x) is the cuent density (* - complex conjunction) then the totl in PT volume Ω enegy loss of powe supply is ( ) 0.5 Re(/ ) nd equls the totl PT theml loss. When Ω P = pxdω= V Z the vlue of t -pmete of the piezoelectic loss lies in the intevl t = 0.8 the enegy losses of the electic field in some fequency nge including the ntiesonnce fequency e negtive t the cente egion of the b PT (Fig. 7). It mens tht the cuent t the PT cente is opposite to the pplied voltge i.e. the locl negtive dynmic conductivity [] [8] of the centl egion of the b PT tkes plce. At the sme time the locl theml dissiption is lwys positive t ny point of the PT volume (Fig. 8). Figue 8. Clculted distibution of locl theml losses long length of b PT t the esonnce () nd ntiesonnce () fequencies unde the condition of constnt PT top displcement (U) fo: Q =00 k 3 = 0.7 δ = 0.0 nd t () - pmete in the llowed nge [- ] (shown in pentheses). The eson of such diffeence in distibution of the locl theml nd electic field losses depends on edistibution (enegy flux) of coupled electo-mechnicl enegy becuse of the concept of the Umov-Pointing vecto [9]. Pticully in the cse of b PT

16 p ( x ) = w ( x * ) 0.5 ω Im ( T s ) ( 5 ) whee w(x) is the locl theml loss. As PT is supposed to be electiclly excited only ccoding to the boundy conditions the totl in PT volume ievesible flux of the mechnicl enegy x= * Im( Ts ) dω= 0. Note tht the function of distibution of the theml losses t the x= 0 ntiesonnce fequency hs two mxim t the cente nd top of the b PT t positive vlues of t -pmete. The lst fetue is vey impotnt fo comptive expeiment on the detemintion of the efficiency of two egimes of PT excittion nd must be tken into ccount. If tempetue (het) detecto is locted only t the PT cente [3] the eo in totl enegy heting mesuement cn be double nd incoect conclusion on the enegy efficiency could be mde. The specific fetues of the consideed b PT behvio nd the bsic qulity fctos eltionships (3) nd (4) e simil to disk PT of dil (pln) vibtion (UM). Tking into ccount tht CMC k p >> k 3 nd tht the t (p) -pmete of piezoelectic losses is popotionl to CMC descibed effects hve to tke plce to gete degee in the cse of disk PT. Clculted ccoding to (3) nd (4) the dependence Q Q (n = ) vesus k Q δ fo vious vlues of the ij genelized F nd t pmetes is shown in Fig. 6 nd is pplied both to the b nd disk PT. Commonly the qulity fctos such s Q of the esonnce PT pek Q of ppopite elstic constnt nd Q s.c. coesponding to the shot-cicuit egime e ppoximtely equl to ech othe t lest fo the fundmentl hmonic (the sme popety tkes plce fo the set of the qulity fctos Q Q nd Q o.c. t ntiesonnce). The diffeence between them inside ech set depends on the type of vibtion hmonic numbe etc. They e exctly equl just fo PT with concentted pmetes such s thin ing esonto when ll locl esonnt mechnicl nd electicl chcteistics (defomtion stength etc.) e homogeneous in the PT volume. V. XPRIMNTAL RSULTS on QUALITY FACTOR RLATIONSHIPS. Accoding to (3) nd (4) the chcte of the diffeence between Q nd Q is stipulted by the piezoelectic dissiptive effect cused by the non-zeo imginy pt of the piezocoefficient the diffeence inceses with CMC gowth. The expeimentl dependence of the disk PT qulity fctos on degee of poliztion ( k p (δ ) ) is shown in Fig. 9. The ntiesonnce Q (δ ) eltionship hs the stongest dependence which inceses with gowth of poliztion. Fo the mximum chieved k p = 0.55 (δ = 0.5) the tio Q Q eches moe thn t k p 0 Q Q Q 0 to

17 the qulity fcto of non-polized PCM. Menwhile the Q (δ ) cuve eflects the known fct [] tht Q deceses insignificntly s PCM poliztion inceses. Figue 9. Influence of the eltive esonnt intevl δ of disk PT with pln mode of vibtion on the esonnce Q ntiesonnce Q qulity fctos nd thei tio Q Q : lines - intepoltion of the expeimentl vlues of Q () nd Q () lines 3 4 clculted dependences Q Q (δ ) ccoding to (4) t 0 Q = 600 δ = nd ccodingly: 3 - mx t p = (t k p = 0.63) 4 - t p = 0 (γ 3 = 0). PCM PZT-35Y. Clculted dependence Q Q fo pln mode of vibtion in view of the conditions γ 3 = const (δ p ) nd t p ~ k p () in the eseched nge of poliztion hs shown stisfctoy geement of theoeticl nd expeimentl dt nd t p 0.90 ws obseved t mximum chieved CMC vlue. The bsence of the imginy pt of the piezocoefficient ( γ 3 t p = 0 ) should hve esulted in the convese effect decesing of Q Q moe thn times. Rhombohedl mophotopic nd tetgonl [] PCM compositions wee involved into the expeimentl esech. Fo the disk PT with dil mode of vibtion the mximum Q Q vlue coesponds to the fundmentl hmonic (n = ) nd the tio Q Q (n = ) eches.8.4 while the tio Q Q (n = ) cn not exceed 3.0 fo mx F 0.8 nd t p = 0.95 ccoding to (4). It is necessy to tke into ccount tht the dielectic loss ngle consists of two components δ = δ + δ defined ccodingly by the domin mechnism of enegy losses δ (which detemines t p ( t t 3 ) vlue) nd by the conductivity of fee chges δ. The lst component dds dditionl tem to the expessions (5) (6) nd (4) esulting in decesing of Q Q.

18 Figue 0. xpeimentl dependence of the esonnce Q nd ntiesonnce Q pln qulity fctos t the fundmentl hmonic of disk PT on the fcto of dielectic losses tn δ () ( khz). PCM PZT-35Y. The expeimentl dt pesented in Fig. 0 show the dependence of the esonnce Q nd ntiesonnce Q pln qulity fctos t the fundmentl mode of the disk PT on the fcto of dielectic losses tn δ ( khz). The PT smples wee mde of single block 60x60x8 mm of PCM PZT-35Y (Russi) s epesenttive of filte nd coustic PCM sinteed t non-optiml egime with lge gdient of popeties. The smples wee polized in i unde pessue t tempetue tnsition though Cuie point. At enough equl degee of poliztion of the smples (pln δ = 0 4 %) the loweing of density steep incese of the sttic conductivity nd incesed poosity took plce with incese of tn δ (). Unde the effect of the indicted fctos the esonnce qulity fcto of pln vibtion is educed little bit the pln ntiesonnce qulity fcto deceses steeply tht coesponds to the theoeticl nlysis. Thee is one impotnt pcticl conclusion tht the chnge of the ntiesonnce qulity fcto Q vlue is sensitive indicto of the intenl ctive conductivity connected with the intenl mico-defectiveness of PCM stuctue such s poosity not ected metl components o inclusions etc. In most cses the dditionl fiing ( C) of smples with incesed sttic conductivity is enough fo essentil incesing of the ntiesonnce qulity fcto when "heling" of mico-ccks nd ptil buning out of conducting inclusions t incesed poosity tke plce.

19 VI. QUALITY FACTOR PROPRTIS of PT QUIVALNT CIRCUIT. The tditionl PT equivlent cicuit [] [0] [] epesenting n electicl nlog of the PT electomechnicl system with the typicl elements R L C d Cs (C 0 = C d + Cs) does not llow one to pedict el diffeence in the qulity fcto vlues t the esonnce nd ntiesonnce fequencies. When single esistive element is used (eflecting the mechnicl losses in PT s it is tditionlly consideed) the ntiesonnce qulity fcto vlue is gete thn the esonnce one (on the vlue of eltive esonnt intevl only). Tking into ccount the dielectic losses s n dditionl esistive element d connected in pllel to the C ( tnδ = d ω C0 ) the vlue of the ntiesonnce qulity fcto is educed s Q = Q ( + δ )(+ δ Q tn δ) whee δ = (ω ω )/ω. Fo bette pefomnce in "new" C [0] it is offeed to connect the dditionl esisto s in seies to the "tditionl" C. Pesence of the two esistive elements R nd s (ctully independent fo seies (esonnt) nd pllel (ntiesonnt) C ptil cicuits) llows to chnge the C Q nd Q independently nd ove wide nge of vlues i.e. to close mtch the C behvio with the el PT popeties ccoding to the expeimentl dt. The theoeticl explntion of such pcticl cicuit ws pesented in [] whee the dominnt piezoelectic mechnism of enegy losses ws involved. Using the method of decomposition by the smll pmetes of the expessions descibing the dmittnce of the b PT nd its "tditionl" C fom the condition of thei equlity the dependence of the C equivlent esistnce R on the eltive esonnt fequency displcement χ = f f inside the esonnce-ntiesonnce intevl ws obtined: 3 χ R( f ) = R( f ) Q χ γ δ 3 Q δ R( f) = R( f) δq γ3 δ Q tht llows to detemine the C dissiptive elements with piezoelectic fcto included. Then fo the new C with fixed esistive elements R nd s thei vlues e defined s s ( 3 ) γ Q δ ω C nd ( 6 ) R ω C k Q. ( 7 ) s VII. CONCLUSION. Comptive nlysis of the enegy efficiency of PT excittion t both the ntiesonnce nd esonnce fequencies ws mde tking into ccount the bsic mechnicl dielectic nd piezoelectic mechnisms of enegy losses. It ws estblished tht the chcte of the inteeltion of the esonnce nd ntiesonnce PT qulity fctos is detemined minly by the imginy pt of

20 the piezocoefficient whose influence is the most essentil t the fundmentl ntiesonnce fequency fo both SM nd UM vibtions. Gete vlue of the ntiesonnce qulity fcto in espect to the esonnce qulity fcto is povided unde the following conditions: fo the PCM nd PT-type with the getest CMC vlue k (such s k nd k p fo od nd disk PT) the combintion of the electo-elstic pmetes hs to be ne k δ with mximl vlue up to + of the eltive t-pmete of the piezoelectic losses miniml conductivity of fee volume chges. To povide equl PT top mechnicl displcement mplitude t the ntiesonnce nd esonnce Q( ) ( ) fequencies it is equied tht one pplies voltge to the PT ~ k Q times gete (popotionl only to the esonnce qulity fcto) t the ntiesonnce fequency thn t the esonnce fequency. At the sme time mximum vlues of the mechnicl stin nd stess in the PT volume e no moe thn.3 nd times gete espectively t the ntiesonnce fequency nd these diffeences depend only on CMC but not dissiptive chcteistics (qulity fcto etc.). ven if the voltge pplied to the PT with SM vibtion (od PT) t the ntiesonnce fequency is too high nd typiclly exceed some odes the mximum electic field stength (eltive vlue fo two egimes consideed) does not exceed 3 times nd this tio does not depend on qulity fcto. The piezoelectic loss fcto cuses chnges not only in the totl PT enegy losses but lso influences on the distibution of the theml losses in the PT volume nd cn povide specific PT egions with n extemely low o high heting. Fo the physiclly vlid positive vlues of t-pmete of the piezoelectic losses the locl electicl enegy losses of the exciting electic field t the ntiesonnce fequency cn be negtive t the cente egion of thin PT of UM vibtion when locl cuent t the PT cente is opposite to the pplied voltge ( negtive locl dynmic conductivity ). At the sme time locl theml dissiption is lwys positive t ny point of the PT volume. Pesented esults will be useful fo the optimiztion of egimes of PT excittion to povide highe efficiency. escibed ppoch cn be successfully pplied to the nlysis of the PCM qulity fctos fo othe PT pplictions whee enegy dissiption is the most citicl nd cn be used in the methods of pediction of the PT popeties in diffeent conditions []. Tking into ccount wide nge of vlues (up to 3 times) of qulity fctos fo t lest min types of PT vibtions specifiction of the concept of PCM "mechnicl qulity fcto" is obviously necessy. Since the influence of the piezoelectic component of enegy losses inceses with gowth of the CMC vlue tht cn be essentil fcto in desciption of the popeties of new clss of stong piezoelectic mteils ( k 0.95 ) such s PMN-PT [7].

21 RFRNCS [] B. Jffe W.R. Cook nd H. Jffe Piezoelectic cemics. London Acdemic Pess 97. [] A.V. Mezheitsky nd P.. Kndib Locl enegy losses in n electiclly excited cemic piezoelectic esonto lecticity (Russi) vol. 4 pp [3] S. Hiose S. Tkhshi M. Aoygi nd Y. Tomikw High powe chcteistics of piezoelectic mteils Poc. 9 I Symposium on Applictions of Feoelectics pp [4] S. Hiose M. Aoygi Y. Tomikw S. Tkhshi nd K. Uchino High powe chcteistics t ntiesonnce fequency of piezoelectic tnsduces Ultsonics vol. 34 pp [5] K. Uchino nd S. Hiose Loss mechnisms in piezoelectics: how to mesue diffeent losses septely I Tns. Ultson. Feoelect. Feq. Cont. vol. 48 N pp [6]. Guyom N. Auelle nd L. ynd Simultion of tnsduce behvio s function of the fequency nd the mechnicl dielectic nd piezoelectic losses Poc. 0 I Symposium on Applictions of Feoelectics pp [7] J. Li nd. Viehlnd The complex piezoelectic esponse of elxo feoelectics Poc. 0 I Symposium on Applictions of Feoelectics pp [8] I Stndd on piezoelecticity. Std [9] R. Hollnd nd.p. e Nisse esign of esonnt piezoelectic devices. Cmbidge M.I.T. Pess 969. [0] G.. Mtin ielectic piezoelectic nd elstic losses in longitudinlly polized cemic tubes US Nvy J. Undewte Acoustics vol. 5 N 4 pp [] H. Wng Q. Zhng nd L.. Coss A high sensitivity phse sensitive d mete fo complex piezoelectic constnt mesuement Jp. J. Appl. Phys. vol. 3 pp. L8-L [] J.G. Smits High ccucy detemintion of the el nd imginy pts of the elstic piezoelectic nd dielectic constnts of feoelectic PLZT (/55/45) cemics with itetive method Feoelectics vol. 64 pp [3] S. Sheit nd B.K. Mukhejee The use of complex mteil constnts to model the dynmicl esponse of piezoelectic mteils I Ultsonics Symposium pp [4] J. Kelly A. Bllto nd A. Sfi. The effect of complex piezoelectic coupling coefficient on the esonnce nd ntiesonnce fequencies of piezoelectic cemics Poc. 0 I Symposium on Applictions of Feoelectics pp

22 [5] A. Alt nd H. edeichs Complex elstic dielect?ic nd piezoelect?ic constnts poduced by domin wll dmping in fe?oelectic cemics Feoelectics vol. 9 pp [6] A.V. Mezheitsky A method fo mesuing the qulity fcto of piezocemic element Russi Ptent 7398 Pt. Bull [7] A.V. Mezheitsky A method fo mesuing pmetes of piezocemic esonto Russi Ptent 6546 Pt. Bull [8] A.V. Mezheitsky negy losses in piezoelectic cemics unde electicl excittion lecticity (Russi) vol. 0 pp [9] B.A. Auld Acoustic fields nd wves in solids. Stnfod Univesity vol [0] M. Toki Y. Tsuzuki nd O. Kwno A new equivlent cicuit fo piezoelectic cemic disk esonto Poc. I vol. 68 N 8 pp [] A.V. Mezheitsky nd P.. Kndib negy losses in piezocemics nd thei consideing in equivlent cicuit of esonto Telecom. nd Rdio ngineeing (Russi) vol. 0 pp [] M.J. Zippo K.K. Shung nd T.R. Shout High fequency popeties of piezocemic esontos Poc. 0 I Symposium on Applictions of Feoelectics pp [3] M. Umed K. Nkmu nd S. Ueh ffects of seies cpcito on the enegy consumption in piezoelectic tnsduces t high vibtion mplitude level Jpn.J.Appl.Phys. vol. 38 pp [4] A.V. Mezheitsky Qulity fcto of piezocemics Feoelectics vol. 66 pp Glossy Q Q genelized qulity fcto nd qulity fctos of the complex elstic complinces Ŝ ij ij Q s.c. Q o.c. PT qulity fcto coesponding to the s.c. nd o.c. egimes of PT excittion Q n Q n esonnce nd ntiesonnce PT qulity fctos of n-hmonic ˆ ( S ij S ij ) kl dˆ ( d kl ) T εˆ mn ( T ε mn ) piezomteil constnts (complex nd el vlues) k ˆij ( k ij ) coefficient of electo-mechnicl coupling (complex nd el vlues) χ ξ eltive nd genelized fequency displcement ( χ = f f ξ = Q χ ) q n K oot of fequency eqution nd complex coustic wvenumbe f n (ω n ) f n (ω n ) esonnce () nd ntiesonnce () fequencies of n-hmonic (el vlues) δ eltive esonnce fequency intevl ( δ = f / f ) Y Z PT dmittnce nd impednce R R esonnce nd ntiesonnce PT esistnces

23 Ĉ 0(C 0 ) PT cpcitnce (complex nd el vlues) δ mn ( δ δ δ tnδ ) dielectic loss ngle fcto γ kl piezoelectic loss ngle y y 3 limit vlues of piezoelectic loss ngle t t 3 nomlized piezoelectic loss fctos (e.g. t γ 3 / y ) h b fequency-specifying PT dimension Ω PT volume nd electode e H n F n piezoelectic pmetes of the PT qulity fctos eltionships B n G n fequency coection coefficients x eltive locl coodinte [-;] A m α 0 pmete nd effective ngle of domin oienttion R L C d C S (C 0 ) tditionl pmetes of the PT equivlent cicuit V electic voltge pplied to PT (3) nd (3) T (3) nd s (3) electic field induction nd stength mechnicl stess nd eltive defomtion u(x) U locl nd PT top displcement w(x) W locl nd totl PT theml losses p(x) P locl nd totl PT losses of electic field (powe supply) M pmete of specific dielectic loss Field o dissiptive pmete with point on the top (e.g. T ) - pmetes mximum bsolute vlue in the PT volume Abbevitions PCM PT piezocemic mteil nd piezocemic tnsduce SM UM stiffened nd unstiffened mode of vibtion CMC coefficient of electo-mechnicl coupling C PT equivlent cicuit s.c. o.s. sot-cicuit nd open-cicuit egime of PT excittion The utho A.V. Mezheitsky: 753 Ocen Ave. F Booklyn NY 9 (mv.five@veizon.net) Ph.. in Physics (985 MIPT Moscow Russi) I Membe This wok ws done t the Cemics eptment of PHONON Co. Moscow Russi.