QUALITY FACTOR OF PIEZOCERAMICS

Transcription

1 UALITY FACTOR OF PIZOCRAMICS Mezheritsky A.V. uality factor of piezoceramics adapted from Ferroelectrics 00 v. 66 pp Abstract. uality factors of a piezoceramic resonator at the resonance and antiresonance frequencies in "weak field" have been investigated. Their interrelation depending on the electroelastic piezoceramic constants type of vibration harmonic number relative size of electrode and degree of polarization in view of the mechanical dielectric and piezoelectric energy losses has been researched. INTROUCTION. Piezoceramic materials (PCM) of a PZT system [1] have unique electro-elastic properties. Piezoceramic resonators (PR) are used in electronics and electroacoustics as filters resonators transducers for various applications. The principle of PR operation is based on direct electromechanical energy conversion from electrical into mechanical and vice versa. A wide range of working frequencies (from statics up to tens megahertz) defines necessity of usage of various types of PR vibrations including high-frequency harmonics. The difference between the resonance f r and antiresonance f a PR frequencies [ 3] depends on a degree of PCM polarization namely on the value of the coefficient of electromechanical coupling (CMC) k ij and harmonic number n of a given type of vibration. The quality factor characterizes the resonant PR behavior. PZT PCM provide strong coupling of mechanical and electrical fields (CMC value can reach 0.9 close to theoretical limit) so the mechanical and electrical

2 components of energy losses as well as piezoelectric component describing losses at interconversion of the mechanical and electrical energies should be involved into phenomenological description of the PR behavior. Traditionally the resonance quality factor at the fundamental (lowest) harmonic of a PR planar mode (disk) [1-3] was chosen for characterizing PCM quality. The electro-elastic PR behavior due to anisotropy of polarized piezoceramics generally is described by ten independent constants [1-4]. Being an electro-mechanical oscillatory system PR has two limit regimes of operation (excitation attenuation) - short circuit (s.c.) and open-circuit (o.c.) - accordingly at closed and broken PR electrodes. The CMC value is determined by the difference between the real parts of the elastic PCM constants at a constant [1] strength and induction of an electric field corresponding to the s.c. and o.c. regimes in particular S = S (1 k ) T k = d Sε where k - CMC. CMC as a real value determines a share of the total energy accumulated in PR and converted from the mechanical into electrical forms and vice versa. In turn the values of the quality factors corresponding to the s.c. ( s.c. ) and o.c. ( o.c. ) regimes are determined by the complex elastic constants for example S ˆ and S ˆ whose imaginary parts are determined by the mechanical dielectric and piezoelectric mechanisms of energy losses [4 5]. The piezoelectric component of losses was offered for consideration for the first time in [5]; the presence of a non-zero imaginary part of the piezocoefficient was found by direct measurements [6]. In particular the research of interrelation of the PR dissipative characteristics at the resonance and antiresonance frequencies - resonance r and antiresonance a quality factors respectively for various types of vibrations including harmonics is of practical interest. In this connection predicted in [7] advantages of excitation of a resonant power PR at the antiresonance frequency against the traditional resonant regime as to PR heating and power supply losses have found experimental and practical confirmations [8 9]. The conclusion on a specific role of the piezoelectric energy losses was made in [10] where the dependence of the quality factor on PR shape at the frequencies much less than the fundamental PR resonance was found. In a common case it was proposed [4] to perform the entire matrix of the electro-elastic PCM constants as complex. Then the quality factors of any PR can be determined by an appropriate combination of the imaginary parts of the matrix elements so the relationships between quality factors of different types of vibrations can be established. It was offered [] and realized [ 1] the original iterative method of experimental determination of the complex elements of the complete matrix. The method provides high accuracy of determination of the real

3 and imaginary parts of the constants describing a given type of PR vibration. However if the real parts of the constants practically do not depend on frequency their imaginary parts show generally a strong frequency dependence [13]. For calculation of the complete set of the complex constants it is needed to collect the values of the partial (base) constants which are determined at strongly differing frequencies because of the strict conditions on relative PR dimensions for realizing the required types of vibrations. For this reason in some cases the final result of calculation contradicts to the physical limitations on the value of the dissipative parameter [4]. Involving the domain mechanism of energy losses [14] with a damped (with delay) movement of walls of 90-degree domains the uniform physical nature of the interconnected mechanical dielectric and piezoelectric energy losses was shown. The traditional PR equivalent circuit (C) representing an electrical analog of the PR electromechanical system with the typical units R L C d Cs ( C 0 = C d + Cs δ r = (ω a ω r ) /ω r ) does not allow to predict a change of the quality factor value in respect to frequency (frequencyspecifying dimension) harmonic number especially to the type of vibration. The resonance resistance according to R r ~ (ω r C 0 δ r r ) -1 contrary to the quality factor has explicit proportional functional dependence on the PR capacitance and piezoactivity. At the same time if a resistive element reflecting the mechanical losses in PR as it is traditionally considered is used the antiresonance quality factor value is more than the resonance one (on the value of the relative resonance interval). Taking into account the dielectric losses as an additional resistive element r connected in parallel to the C ( tanδ = 1/ r ω r C 0 ) the value of the antiresonance quality factor is reduced as a r (1+ δ r ) / ( 1+ δ r r tanδ ). For better performance in a "new" C [15] it is offered to connect in series the additional resistor to the "traditional" C. The presence of the two resistive elements (actually independent for the series (resonant) and parallel (antiresonant) C partial circuits) allows to change r and a of the C independently and over a wide range of values consequently to closer match the C properties with the real PR behavior according to the experimental data. The theoretical explanation of such a practical circuit was presented in [16 17] with using a dominant piezoelectric mechanism of energy losses. The OBJCT of RSARCH: PCM and PR. To conduct research of interrelation of the quality factors at various types of vibrations the following PR types realizing general (base) resonant modes were used: expansion-compression

4 along length of a thin bar PR-P1 15x1.0x0.40 mm with polarization along the PR thickness; expansion-compression along length (along vector of polarization) of a long rod PR-B 15x1.5x1.5 mm; radial vibration of a disk PR-P 15x0.4 mm including planar vibration of a square plate PR-T (planar); vibration of shear PR-S1 S and expansion-compression PR-T 6x6x0.45 mm along thickness of a thin plate (figure 1). There are two basic types of vibrations: unstiffened mode (UM) with direction of vibration perpendicular to the vector of an exciting electric field (PR-P1 P S1 T (planar)) and stiffened mode (SM) with direction of vibration along an exciting electric field (PR-B T S) [3]. In particular in the case of linear one-dimensional UM vibrations the resonance frequencies of harmonics (harmonic number n ) are integer multiples of the fundamental resonance while in the case of PR SM vibrations the antiresonance frequencies of harmonics are integer multiples of the fundamental antiresonance. PCM PZT-35Y (0.97(Pb 0.96 Sr 0.04 )(Zr 0.5 Ti 0.48 ) O {Bi Zn Ni Mn}) was taken as a representative of filter and acoustic PCM [3] which has a strongly expressed difference of the quality factor values at appropriate resonant modes because of specific features of structure and used manufacturing techniques. The complete set of the real electro-elastic PCM constants is given in the Appendix. To provide an equal degree of polarization PRs were cut off from a single piezoceramic block polarized in air under pressure at the temperature transition through Curie point. Figure 1. Basic PR types and corresponding resonant modes.

5 MTHO of UALITY FACTOR MASURMNT. A number of methods of measurement of the quality factor of a resonant system are known which are based on measurement of: the rate of attenuation of a resonant vibration after cancellation of the PR excitation; the frequency difference corresponding to the "level 0.7 (3 db)" of a resonance (antiresonance) peak or to the "level 0.5" of the real part for example of the PR admittance or impedance in a stationary regime; the frequencies of extremes of the imaginary part of the resonant PR characteristics. Calculation of the value of the resonance quality factor using the values of the resonance resistance R r static capacitance C 0 resonance f r and antiresonance f a PR frequencies is put in a basis of the standardized method of PCM certification [1-3]. However these methods are applicable only for PR with large enough value of "piezoactivity" defined as a product of the relative resonance frequency interval by the quality factor δ r >> 4 that provides the methodical error less than 10% in the case of 3 db method. This condition corresponds to the intensity (range) of the amplitude-frequency PR characteristics (AFCh) more than 0 db otherwise the error steeply increases up to an invalid value. This problem is important here because the AFCh range at harmonics steeply decreases (as ~ 4 1 n ). For this reason a special "weak resonance" method for the quality factor measurement [18] was developed and used. Its basis is in the following. The common physical principle is known which restricts (low limit) the relative frequency resonance interval of an oscillatory system by the reciprocal value of its quality factor. The reason resulting in lowering of the resonance frequency interval does not matter. It can be caused by a connection to PR of large connected in parallel or small connected in series capacitors [19] (or their combination) a low degree of PR polarization (piezoactivity) (for example at the initial stage of polarization process) a decrease of the relative resonance interval at harmonics etc. If the AFCh range of a resonant peak (between levels of the resonance and antiresonance) does not exceed 8 db in this case the quality factor is equal to the reciprocal value of the relative resonance interval with a methodical error no more than 10 % (figure ). The usage of a capacitive load in the circuit of measurement (for matching with PR) is more preferred at a very weak resonance intensity ( < 1 db ). One of the advantages of the method is its applicability to the quality factor measurement of PR with arbitrary geometrical shape when strong broken AFCh takes place. Taking into account a wide frequency range of measurements that should be done the special attention was given to PR electrodes holder and matching circuit.

6 Figure. Basic elements of the "weak resonance " method for PR quality factor measurement. At low frequencies the main problem concerns measurement of the antiresonance characteristics of PR (PR-B) with low capacitance for which a special balancing unit with the holder was used providing the total parasitic capacitance no more than 1 pf as a result of specific screening. At high frequencies the main negative factors at the measurement of the resonance characteristics of thin plates (PR-T S) are connected with: not ideal plate planes parallelism (the widening of the resonance peak width should not exceed f / f h / h << 1 on the active electrode area); the resistance of the PR electrodes; the inductance of a contacted to PR wire which besides shift of the resonance frequency results in essential decreasing of the effective PR resonance resistance because of large losses brought by the wire inductance. The critical parameter in this case is the intrinsic equivalent PR inductance L which value can reach some nh at high frequencies 3 1 ωr above 50 MHz that is commensurable with the inductance of a wire with the length of about several mm. To exclude the indicated factors a special connection system was used [0].

7 The method of "weak resonance" can be successfully applied especially to the quality factor measurement of electrodeless thin PR plates at high frequencies where the contactless method with a regulated air split between the electrode and plate can be used. In this case there is minimum influence on the oscillatory process and PCM properties due to limited technological effects. The excitation area of a PR plate (size of an electrode) could easily vary (up to dot) providing in particular mono-mode PR characteristics (Bechman condition [4]) with the effect of "energy trapping". The THORTICAL ANALYSIS and XPRIMNTAL RSULTS. lectromechanical behavior of a real PR with energy losses generally is described by the complex constants of the PCM matrix [4]: ( ) Sˆ S 1 i ˆ T T d d (1 iγ ) εˆ ε (1 iδ ); (1) ij ij ij the linear equations of piezoeffect kl kl kl mn mn mn ˆ s = S T + dˆ ; = εˆt + dt ˆ ; i ij j ki k n mn m nl l and the equation of motion T = ρ U at given boundary and initial conditions tt where i 1 ; quality factors of the complex elastic compliances ij ˆ ij S ; δ mn γ kl tangent of angle of dielectric and piezoelectric losses respectively; ρ PCM density; s T induction and electric field strength mechanical deformation and strength in a vector or tensor performance; vector operator; U local mechanical displacement. Similar performance is used for the constants of the matrix through ˆ ij C ĝ kl ê kl ĥ kl ˆ TS mn β [3]. Particularly the complex constants ˆ S with their relationship ˆ ˆ S = S(1 k ) where k complex CMC have the constants quality factors determined by the basic relationship: ( ) 1 1 k 1 1 = 1 (1 1) γ δ = k δ + t k δ 1 k (1 k). () From the condition of positivity of the thermal energy losses the limitation on the value of tangent of piezoelectric losses γ y 1 δ k or through t 1 γ / y 1 [-1; 1] parameter is phenomenologically established in [4]. Generally can be more or less than. The maximum increase of (at the phenomenological consideration) is determined by the non-zero imaginary part γ of the

8 piezocoefficient and corresponds to the conditions k δ 1 and t 1 1. In the specific case of the absence of the dielectric (δ = 0) and piezoelectric (γ = 0) losses: = (1 k ) (1 δ r ) that has opposite sign and stronger effect in respect to δ r in comparison with the traditional C taking into account as it is considered only the mechanical losses. Further the following definition of the quality factor of a PR resonance (resonant peak) and accordingly a method for its calculation are used. Taking the expression that describes the resonant PR behavior after decomposition it by the small dissipative parameters ( 1/ ij δ mn γ kl ) and relative frequency displacement ξ from the "ideal" frequency of the resonant peak (loss-free) B this expression is performed by first-order approximation as ia + where A B 1 + iy coefficients and the quality factor of the resonant peak is defined as a coefficient in generalized frequency displacement y = ξ. The given approach methodically corresponds in full measure to the known frequency methods of quality factor determination mentioned above. Let us further consider the UM planar vibration of a thin bar and thin disk (PR-P1 P). The given type of bidimentional PR vibration in a plane perpendicular to the vector of polarization is generally described by the complex electro-elastic PCM constants [1-4]: ˆ S ˆ 1 S (quality factors 1 ); ε ˆT (δ ) ; ˆd (γ ). The admittance of a bar PR-P1 [1] is described by: ~ ~ ~ tan( Kb /) Y = / iω C0 1 k + k ; (3) Kb / ~ where C 0( ε ˆT ) ; K = ω ρ ˆ S ; ˆ ˆ T k ˆ = d Sε - complex quasistatic PR capacitance and expressions for the wavenumber and CMC respectively. After decomposition the expressions describing the PR admittance (Y) and impedance (Y -1 ) by the small parameters of the relative frequency displacement and dissipative coefficients in a vicinity of accordingly the resonance frn = n b ρ S and antiresonance fan = n b ρ S frequencies where n = x 1 k π ; x n n-root of the frequency equation n tan x = x(1 k ) k ( 1 1 = 0.0 at k = 0.5); b - PR length we determine expressions for the resonance rn and antiresonance an quality factors ( n = harmonic number): rn = and

9 1 1 k 1 = F n γ δ = an 1 k ( ) 1 F 1 n 1 (1 1 ) = Fn+ k δ + t kδ + k δ 1 k (4) (1 k) where Fn = 4. Note that the coefficient F n F n ( k ) has the values: x (x 1) k + x k n n F 1 = 8/π ( 0.81) 0.75 at k = ; further Fn ~ 1/ n. Thus the resonance and antiresonance resistances: n R 8ω C k = ; π n 1 rn 0 rn R Fk n an an = (1 k ) ωan C0. Here δ = δ + δ (δ = δ in + δ out ) ; γ y 1 δ k ; t1 = γ y1 [-1; 1] ; δ - dielectric (domain) component; δ component caused by the conductivity of free internal and/or external charges. Thus r value both at the fundamental mode and harmonics is determined by the quality factor of the complex elastic compliance S. From (4) follows that the values rn ( ) and an generally most differ at the fundamental harmonic n = 1 while an rn = at n >> 1. Using the method of decomposition the expressions describing the admittance of the bar PR and its C by the small parameters from the condition of their equality the dependence of the equivalent resonance resistance R of the C on the relative resonance frequency displacement ξ = f f r 1 inside the resonance-antiresonance interval is obtained: 3 ξ R( f) = R( fr)1 ξ γ δ δ r ; R( fa) = R( fr)1 δr γ δ ˆ 3. (5) Let us consider the dependence of the resonance quality factor on the relative size ( ) of the PR electrode for the fundamental mode. At an equal polarization of the PR plate (bar) outside of and under the electrode: 1 1 k 1 = M γ δ = r ( ) 1 k ( ) 1 M 1 1 (1 1 ) = M + k δ + t kδ + kδ 1 k (6)

10 sin( x0 ) 1+ x sin( π ) M M( k ) = [01] π 0 1 sin ( x0 ) 1+ x0 k cos [ (1 ) 1 ] (7) where = l / b [01] ; fr = Ln b ρ S ; L n = x 0 π ; x 0 - root of the frequency equation tan 1 ( x ) (1 k ) 1/ tan[ x(1 )(1 k ) 1/ ] = 0. In this case the relative resonance interval δ r a k ; the resonance resistance 1 4 π Rr ( ) akr( ) ωrc0 where a( ) = sin ( ) π C 0 ( ) - static PR capacitance. If only the under-electrode PR area is polarized: ; = M ( ) r 0 where 0 - quality factor of non-polarized PCM M M ( k ) is defined by (7) ( k corresponds to the under-electrode area). So the antiresonance frequency f a1 of a fully metallized bar PR is determined by the elastic constant S with the correction coefficient 1 (in respect to 1) for this reason F 1 < 1. However at an infinitesimal relative size of the electrode f ra ( 0) 1b ρ S [4] and the quality factor ra ( 0) determined by () (or by (4) at F 1 1). Therefore the difference between the resonance quality factor of a completely metallized PR and both the resonance and antiresonance quality factors of a partly metallized PR increases at a rather small size of the electrode. Among the main types of UM vibrations it is necessary especially to emphasize the radial vibration of a thin ring. Such a PR concerns to the type of resonators with concentrated parameters when the only fundamental vibration without harmonics (n = 1 only) takes place and the local resonant characteristics (deformation strength) are homogeneous inside the PR volume. As a result the resonance and antiresonance frequencies are determined by the elastic constants S without correction factors and consequently the quality factors r = and a = are determined by () ( or by (4) at F (n) = 1 ). Generally the antiresonance frequency corresponds to the condition Re dω= 0 where Ω - electrode area. This condition has to be satisfied integrally Ω n on the surface of the electrode in the case of for example a bar PR having inhomogeneous distribution of an electro-mechanical field while in the case of a ring PR the given condition is

11 satisfied for the local value of induction. The latter is a reason of extension of the resonance frequency interval as a result of relative increase of the antiresonance frequency. As a consequence the conditions s.c. and o.c. are realized in accuracy for a ring PR where are determined by (). Let us further consider the radial vibration of a disk PR-P. As a result of decomposition of the expression for its complex admittance [1] by the small dissipative parameters and frequency displacement we receive [4] σ = 1 + σ (1 V ) + xn + σ 1 σ r n 1 (8) where σ = S1 S - Poisson coefficient; V = 1 ; x n n-root of the frequency equation x J 0 (x) (1-σ) J 1 (x) = 0 ; J 01 (x) the first kind Bessel functions of order zero and one. For further analysis of an we will make simplifying assumption that V = 1 whose competency will be considered below. For a disk PR an is determined as well as for a bar PR by the expression (4) under the replacement of the parameters k k p and t 1 t p where k p = k (1 σ ) (.6 3.) k ; t p = γ / y p = t 1 (1 σ ) ( ) t 1 [-1; 1]; y p δ = y 1 ( 1 σ ) ( ) y 1 at typical for PZT PCM σ = k p 0 F n (σ k p ) 1. It follows that both t p 1 and t 1 1 according to the phenomenological limitations [4] whence t Figure 3. Calculated dependence of a on k δ according to (4) with generalized parameters: k = 0.6; F = 1 ( ); 0.8 (- - -) and t = 1 (curve 1); 0.9 (); 0.85 (3); 0.5 (4); 0 (5)

12 According to the particular mechanism of energy losses [14] the damped movement of walls of 90-degree domains provides t 1 - parameter determined by the only an effective angle α 0 (90 180) of domain orientation: t 1 = ( F 1 F 3 ) (1 F )(1 F 4 ) where F m = sin(mα 0 )/mα 0. etermined experimentally min α 0 = ( ) that corresponds to max t 1 = 0.63 (with typical values of t 1 = ) agrees with the above-mentioned phenomenological estimation. Further we will determine the character of changing of the tangent of piezoelectric losses angle γ = d ˆ ˆ d depending on the PCM polarization P : ˆd ~ sin 3 (α 0 ) and ˆd ~ 1+ cos(α 0 ) [14]. It follows that γ 0 for P 0 ( and δ are practically constant ). However the γ value varies within limits of 0% in a wide interval of the saturated polarization (max k p ) i.e. for the mentioned interval of CMC values it is possible to accept γ const ( P ) then t = k γ δ ~ k. (9) 1 Calculated according to (4) the dependence a r (n = 1) versus k δ for various values of F n and t 1 parameters is shown in figure 3. It has a characteristic extremum at k δ 1. Maximum allowed (accessible) for PCM t and t p 1 values determine the features of the quality factors relationship at the fundamental harmonic: for a bar PR a1 does not exceed r1 more than 0 % for a disk PR (t p 1) a1 r1 can reach abnormal high values. The difference between rn of a disk PR and (or the same rn of a bar PR) is determined by an incongruity of the quality factors of the complex elastic constants S ˆ and S ˆ 1 particularly rn = when = 1. According to [4] the value of V is phenomenologically limited by the condition V 1 / σ. The quality factor rn at V 1 of a disk PR (8) contrary to a bar PR should have the additional dependence on the harmonic number. This property allows to determine on the basis of the experimental data the relationship between imaginary parts of the constants S ˆ and ˆ S 1 as well as the own frequency dependence of the quality factor ( f ). The latter is especially important for the comparative analysis of the quality factors of various PRs which should be in equal conditions including frequencies of measurement.

13 Figure 4. xperimental frequency dependence of the resonance quality factor of the bar PR-P1 at the fundamental (g) and higher (+) harmonics of the disk (n) and square plates (1) PR-P T (planar ) at harmonics in a range of frequencies MHz: interpolation of a set of the experimental data on average and limiting values accordingly. The resonance quality factor of a bar PR both at the fundamental mode and harmonics is equal to and for a disk PR - / K(σV n) where K(σV n) - coefficient (8) depending on harmonic number n only if V 1. It was experimentally established that the statistically average value of the ratio of the resonance quality factor of the disk PR (PZT-35Y) to the resonance quality factor of the bar PR cut off from the disk is at the frequencies of the measurement khz that corresponds to V = according to (8). However the obtained small difference of V in respect to 1 can be caused by the influence of sawing procedure. It proves to be true by comparison of the frequency dependences of the resonance quality factors of bar and disk PRs both at fundamental harmonic when the frequency-specifying PR dimension changes and at frequencies of high harmonics when the PR dimensions are fixed (figure 4). From the presented data follows that possible at V 1 distinction of dependences of the resonance quality factors of disk and bar PRs on harmonic number for the researched PCM was not found that allows to consider V 1. Thus the resonance quality factor ( ) depends on frequency as r ( f) = Af λ where λ = ( 3.7 ± 0.8 ) -1. This dependence corresponds to the PZT PCM system that is conformed also by researches of a tetragonal PbTiO 3 composition (factor λ = 1/4 ) at frequencies up to 5 MHz [1]. The values V 1 are probably possible at the presence in PCM a structural anisotropy. For further consideration it is necessary to emphasize that according to [1] at frequencies above 0 MHz the lowering of the quality factor as ( f ) ~ 1 / f was observed.

14 According to (4) the character of the difference between r and a is determined by the piezoelectric dissipative effect and it increases with growth of CMC. The experimental dependence of the quality factors of PR-P on a degree of polarization ( k p (δ r ) ) is shown in figure 5. The strongest dependence of a (δ r ) takes place which increases with growth of a degree of polarization. For maximum achieved k p = 0.55 (δ r = 0.15) the value of a1 r1 increases more than times and at k p 0 a r 0. The curve of the r (δ r ) dependence (figure 5) reflects the known fact [1] of insignificant decrease of r at increase of PCM polarization. From comparison of the experimental results (figure 5) with theoretical calculation (4) follows that the increase of a1 r1 is a result of the presence of a non-zero imaginary part of the piezocoefficient with t p 1 for the greatest possible for a given PCM piezoactivity. The calculated dependence a1 r1 (4) for a planar mode of vibration in a view of the conditions γ = const (δ p ) and t p ~ k p (9) in the researched range of polarization has shown satisfactory agreement of theoretical and experimental data and t p = 0.90 was observed for the maximum achieved value of the relative resonance interval. The absence of the imaginary part of piezocoefficient ( γ t p = 0 ) should result in the converse effect decreasing of a1 r1 more than times. Figure 5. Influence of the relative resonance interval δ r of a disk PR-P with planar mode of vibration on the resonance r1 and antiresonance a1 quality factors and their ratio a1 r1 : lines 1 - interpolation of the experimental values of r1 (1) and a1 (); lines 3 4 calculated dependence (δ r ) according to (4) of a1 r1 at 0 = 600 δ = and accordingly: 3 - max t p = 1 (at k p = 0.63); 4 - t p = 0 (γ = 0)

15 Rhombohedral morphotropic and tetragonal PCM compositions were researched including PbTiO 3 [1]. The common features of the experimental results for a r dependence for planar UM of vibration are the following. Maximum a r value corresponds to the fundamental harmonic (n = 1). At higher harmonics the ratio of a r steeply decreases that is in agreement with (4) for the parameters F n k /(1 k ) δ rn and F 1 1 F n 0 at n = In view of the mentioned weak increase of r with increase of frequency a steeply decreases at harmonics following right after the fundamental resonance and further smoothly rises and comes nearer to r (f) (figure 8). For PR-P with a radial mode of vibration the ratio a r (n = 1) reaches while for a bar PR this ratio does not exceed 14 % that is explained by the difference of foremost valid values of the t-parameter: accordingly t p 1 and t 1 < 0.6. According to (4) for max F the ratio a r (n = 1) at t p = 0.95 does not exceed 3.0. In the case of a partly metallized PR the resonance and antiresonance quality factors increase with decrease of electrode relative dimension. It was found that maximum r ( ) / r a ( ) / r = 3 ( n = 1 ) at = From the analysis of the experimental data for UM of vibration it follows that a r < 1 at kpr1δ < 0.15 that corresponds to the character of the curves given in figure 3. With increase of k δ the value the ratio a r (n = 1) increases and further at p r1 kpr1δ > 1 decreases. It is necessary to take into account that the tangent of dielectric losses angle consists of two components δ = δ + δ determined accordingly by the domain δ mechanism of energy losses (that determines t p (t 1 ) value) and free charges δ. The last component adds additional term to the expressions (4 6) resulting in decreasing of a r. Furthermore frequency dependence of δ ( f ) (Appendix) whose variation in the range of 1 khz 1 MHz can be essential (up to 3 times) strongly depends on PCM composition. One of the possible UM of vibrations is a shear mode along the thickness (PR-S1). Interrelation of its shear quality factors are described also by the expression (4) at replacement of the parameters k k 15 ; S S ; 44 ; b h ; δ δ ; γ γ 15 t 1 t 4 with the same common properties. On the basis of the model of domain mechanism of energy losses offered in [14] the relationships coupling the imaginary parts of appropriate constants of longitudinal and shear modes of vibrations were obtained: 44

16 (S ˆ ˆ 44) (S ) = 4 (1+F 4 ) / (1 F 4 ) ; ˆd 15 / ˆd = (F 1 +F 3 ) / (F 1 F 3 ) ; F m = sin(mv 0 )/ mv 0 ; (10) T T ( εˆ ˆ ) ( ε ) = (1+F ) / (1 F ); ( t 4 ) = ( t 1 ) ( ˆd ) / ( ε ˆT ) ( S ˆ ) = (F 1 F 3 ) / (1 F )(1 F 4 ). According to that relationships ˆ ˆ ( S44) ( S) = for a wide range of PCM polarization up to its maximum level. In view of S44 S = 3.0±0.4 for PCM of PZT system ( for a nonpolarized condition S44 S = ( 1+σ 0 ).6 ) the estimated ratio 44 according to the model lies in the boundaries of 1.6 to.0. The given estimation agrees with the experimental data presented in figure 8. The relationships stated in this section are also applicable to other possible types of UM vibrations for example vibration along width of a bar PR etc. The rod PR-B impedance with SM of vibration is expressed as ~ tan( /) = ~ 1 Kl Z ~ 1 k ; () / iωc (1 ) Kl / 0 k ~ where C 0(εˆ T ) ; K = ω ρ ˆ S ; ˆ ˆ ˆ T k ˆ = d Sε - complex quasistatic PR capacitance and the expressions for wave number and CMC respectively; Sˆ = Sˆ (1 kˆ ). After decomposition of the expressions for the PR admittance (1 / Z) and impedance (Z) by the small parameters of the relative frequency displacement and dissipative coefficients in a vicinity accordingly of the antiresonance f a n n b ρs = and resonance f r n Bn b ρs = frequencies where B n = x π 1 k ; x n - n-root of the frequency equation n tan x k x = ( B 1 1 = 0.04 at k = 0.5 ); l - PR length we determine the expressions for the resonance rn and antiresonance an quality factors of n-harmonic: 1 1 k 1 = H n γ δ in = rn 1 k ( ) 1 H n = 1 1 (1 3) Hn + k δ + t kδ + k δin 1 k (1) k 1 = γ δ = an 1 k ( ) 1 = 1 kδ + (1 t3) kδ + k ( δin + δout ) (1 k ) (13)

17 where H n (1 k) = 1. Note that the coefficient H 4 n (k ) has values: at k = ( x k + k ) n H 1 = ; H 3 = ; H 5 = ; further antiresonance PR resistances are: R = ω C (1 H ) k ; 1 rn rn 0 n rn R Hn 1. Thus the resonance and 8k an an = π n ωan C0 k (1 ). Here δ = δ + δ ( δ = δ in + δ out ); δ in = δ + δ in ; γ y 3 δ k ; t 3 = γ / y 3 [-1; 1]; δ - dielectric (domain) component; δ component caused by the conductivity of free charges ( internal δ in and external δ out - the last one can be caused by the surface conductivity or a resistor connected in parallel to PR ). Taking into account that for SM of vibration H 1 << 1 and H n 1 at n 3 in comparison to UM the situation is similar at the fundamental mode (n = 1): a1 = ; r1 however a converse situation is observed at harmonics ( n 3 ): rn an = at n. From (1 13) it follows that maximum values of the quality factors rn and an at the fundamental mode and its harmonics correspond to the conditions k δ = 1 and t 3 = 1. Thus a maximum value of the resonance quality factor max rn = (1 Hn) and in an ideal case the difference can reach some orders. A specific feature of the frequency dependence of rn is its steep increase up to an at the third harmonic ( n = 3 ) and further practically constant value at higher harmonics (figure 8). Traditional consideration of the mechanical and dielectric mechanisms of energy losses only provides decreasing (more than times in respect to ) of the quality factor at harmonics. The possibility of increasing of the quality factor value at higher harmonics is a consequence of the presence of the non-zero imaginary part (at t 3 > 0) of ˆd piezocoefficient. From the experimental data for PZT PCM follows that t 3 = corresponding to a maximum degree of polarization. As well as in the case of UM of vibration for the same conditions (9) the parameter t 3 ~ k. Increasing of the quality factor value at harmonics is definitely impossible in both intervals δ < 0.1 ( k ) and δ > 3.5 ( k ) (figure 3). There is quite a different situation for UM and SM of vibrations in respect to the internal δ in and external δ out dielectric losses caused by free charges. On an example of the bar PR-P1 and

18 rod PR-B in the case of UM of vibration the value of rn = does not depend on dielectric and piezoelectric dissipative parameters while in the case of SM of vibration the value of rn = Ψ ( γ δ δ in ) especially for n 3 in respect to dielectric losses depends along with the dielectric "domain" component only on the bulk conductivity of internal free charges and does not depend on surface or external ones. Figures 6 ab. Calculated AFCh (normalized admittance) of a rod PR-B of SM vibration for the fundamental (a) and 3-rd harmonics (b) at a various combinations of the values of the dielectric ( δ = δ + δ in + δ out ) and piezoelectric losses ( γ = t 3 y 3 ). PZT-35Y type PCM at = 100 k = 0.7 and: line 1 - δ = t 3 = 1; - δ = t 3 = 0; 3 - δ = t 3 = 0; 4 - δ = t 3 = 1; 5 - δ = t 3 = 1. Zero-point on the x-axis is arbitrary shifted.

19 The reason of such phenomenon is the following. In the case of UM of vibration when directions of vibration and an exciting electric field are perpendicular the electric field is homogeneous between PR electrodes (to be more specific it is in the case of a thin plate PR with equidistant electrodes). In the case of SM of vibration the directions of vibration and an exciting electric field are collinear and for this reason a local field is inhomogeneous between PR electrodes and has a resonant character (similarly to current deformation etc.) under the condition of equality of an integral of along frequency-specifying dimension to the electric voltage applied to the electrodes. The local value of field strength at the resonance can be increased by orders that as a consequence increases a share of a dielectric component in total energy losses [7]. In particular for this reason the contribution of the bulk conductivity of free charges into the value of the resonance quality factor increases (in the PR region of resonant values of the electric field ) however there is no contribution of the surface or external conductivity where the electric field has normal non-resonant value. Calculated AFCh of admittance of a rod PR-B of SM of vibration at the fundamental and 3-rd harmonics according to () are given in figure 6 which illustrates the described property of SM at variation of the value and type of the dielectric losses at the presence or absence of the piezoelectric losses. Figure 7. xperimental dependence of the resonance r1 and antiresonance a1 planar quality factors of the fundamental harmonic and the resonance quality factor r3 of the 3-rd thickness harmonic on the factor of dielectric losses factor tan δ (1 khz) for PR-T. PCM PZT-35Y. There is one important practical conclusion can be made that the change of the resonance rn value of SM of vibration at higher harmonics ( n 3) is a sensitive indicator of the internal active conductivity connected with internal micro-defectiveness of the PCM structure such as porosity not reacted metal components or inclusions etc. Sometimes the additional calcination

20 ( o C) of the samples with an increased static conductivity is enough for an essential increasing of the resonance SM quality factor at harmonics when healing of micro-cracks and partial burning out of conducting inclusions at increased porosity take place. The antiresonance quality factors an both for UM and SM of vibrations are determined by the total value of δ (δ δ ) ( in respect to δ as an effect of usual shunting of a large antiresonance PR resistance) irrespectively to the character and nature of the active conductivity. The experimental data in figure 7 are showing the dependence of the resonance r1 and antiresonance a1 planar quality factors of the fundamental mode of vibration and the resonance quality factor r3 of the thickness 3-rd harmonic on the factor of the dielectric losses tanδ (1 khz) of PR-T. The researched PR samples were made of a single block 60x60x8 mm of PCM sintered at a non-optimal regime with large gradient of properties. While the PR samples have sufficiently equal degree of polarization (planar δ r = %) lowering of density steep increase of the static conductivity and increased porosity took place with increase of tanδ. Under the effect of the indicated factors the resonance quality factor of planar vibration is a little reduced the planar antiresonance quality factor reduces steeply and the resonance quality factor of the 3-rd harmonic of thickness vibration is even more steeply reduced that corresponds to the theoretical analysis. On the basis of the model of domain mechanism of energy losses [14] the following relationships were obtained coupling the imaginary parts of appropriate constants describing longitudinal (SM) and transversal (UM) (in respect to the direction of polarization) modes of vibrations: ( ˆ S ) = ( S ˆ ) ; ˆd = ˆd ; ( t 3 ) ( ˆd ) / ( ε ˆT ) ( ˆ S ) = ( t 1 ). (14) One of the possible SM is a shear vibration along the plate thickness (PR-S). Interrelation of its shear quality factors are described by the expressions (113) as well at an appropriate replacement of the parameters: k k 15 ; S S ; 44 ; l h ; δ δ ; γ γ 15 ; 44 t 1 t 4 with similar common properties. These relationships together with (10 14) were used for analyzing of the experimental results reflecting interrelation of the quality factors of various modes of vibrations (figure 8). It is necessary especially to consider the case of SM of expansion - compression vibration along the thickness of a thin plate. The oscillatory process is described by the electro-elastic constants ˆ C ε ˆT S ˆ ˆ ˆ ˆ eˆ = dc13 + dc. Taking into account that d > 0 and d < 0 the piezocoefficient e value is determined by the difference between d and d. This effect results in

21 mutual partial compensation of the contribution of dissipative parameters γ and γ in the values of the resonance thickness quality factors rn at higher harmonics. Taking into account ˆ ˆ (1 ) C = C k t ˆ C = C (1 + i t ) where k e Cˆ ε defines "thickness" CMC S ˆ ˆ t = from the analysis of the expression for complete admittance of such PR [1] we determine the expressions for the resonance rn and antiresonance an "thickness" quality factors: ( δ ) kpt (1 tp) kpt k δ p = 1 Hn rn t 1 ( 1 kt δ) + (1 t3) kt δ 1 k and an = t (15) the last quantity is determined by (15) at H (n) = 1. Thus the resonance resistance is determined by R = ω C (1 k )(1 H ) k where H n (k t ) - the parameter to the expression (1) at 1 rn rn 0 p n t rn replacement k k t. Further the resonance and antiresonance frequencies are determined by the formulas to the expression () at replacement S ( C ) 1 b h. Figure 8. xperimental dependence of the resonance and antiresonance quality factors of a set of the researched types of UM and SM of vibrations on frequency (harmonic number).

22 The expression (15) is obtained at the simplifying assumption of equality of the quality factors t = = 13 of appropriate elastic constants and at satisfying the relationship k = ( k k ) (1 k ) [1] that does not change the character of contribution of considered t p p components of energy losses. Particularly at the absence of planar piezoactivity ( k p = 0 ) the expressions (15) and (1) provide the identical results for appropriate PR. It is necessary to emphasize especially the effect of energy trapping [4] at the fundamental harmonic of thickness vibration of a thin plate. The desirable mono-mode resonance is provided in particular by an appropriate choice of electrode size however in the case of thickness vibration it is possible only for PCM with the energy-trapped parameter value C C 44 > 5. The summarized experimental data are given in figure 8 describing the resonance and antiresonance quality factors of a set of considered PR of UM and SM vibrations at the fundamental resonance and its harmonics. ach PR type occupies the frequency range providing a required type of vibration. The collection of experimental dissipative data reflects the main properties considered above. It is necessary to emphasize that no changes of the real parts of the elastic constants in the researched range of frequencies were found their variations lie at a level of the measurement error. Figure 9. xperimental AFCh (admittance) of a single electrodeless PR-T plate in a regime of the measurement of the quality factor using the method of "weak resonance" for nearby the 7-th shear and 3-rd expansion - compression thickness resonances. Overall quality factors values (PR-T S1 vibration types for the researched PCM) from all the possible values in the range of (figure 8) can be observed simultaneously on a single sample PR-T during measurement of the quality factor by the weak resonance method that is illustrated in figure 9. When an electrodeless plate is used with displaced superimposed electrodes

23 there are the longitudinal and transversal (to the direction of plate polarization) components of an exciting electric field caused by in particular the fringe effect. As a result of such conditions the thickness SM and shear UM vibrations take place. In a common case the quality factor of an arbitrary PR is determined by some combination of the elements of the matrix of the complex electro-elastic PCM constants. The carried out analysis can be applied to other varieties of types of PR vibrations that in a common case are described by 1. five complex elastic PCM constants Sˆ = Sˆ dˆ εˆ T ˆ ij S with respective quality factors 3. Sˆ = Sˆ dˆ εˆ T [4]: ij 1 1 k 1 = γ δ 1 k. 1 Sˆ = Sˆ dˆ εˆ T 1 k 1 = γ δ 1 k 1 1 k 15 1 = γ 15 δ 44 1 k Sˆ = Sˆ dˆ εˆ T k / σ 1 = + γ δ k/ σ 1 Sˆ = Sˆ dˆ dˆ εˆ 5. T k k / µ 1 = γ + γ δ 13 1 kk / µ where complex ˆd ˆd ˆd 15 - piezocoefficients with appropriate (usually positive) tangents of piezoelectric losses angles γ γ γ 15 (); ε ˆT ε ˆT - dielectric permittivity with appropriate positive tangents of dielectric losses angles δ δ ; k k k 15 > 0 - CMCs [1-3]; σ = S1 S µ = > 0 - Poisson coefficients. S13 SS It follows that determined by the non-zero imaginary parts of the piezocoefficients in view of practical values of the constants 1 < 1 and 13 > 13 even more 13 can have a negative value. There is no physical inconsistency in it because the elastic compliance ˆ S 13 is not a diagonal matrix element and it can not be a single frequency-specifying constant of any type of vibrations. In this connection it is necessary to specify the valid range of the values of the CMC imaginary part [] with a point of view of "passivity" of PCM. On an example of the complex CMC (1 ) k = k iβ the expression for β parameter is similar to (). From the energy dissipation limitations [4]

24 ˆ ( ˆ )( ˆ T ) and ˆ ( ε ˆ)( S) 0 ) follows that β ( - ; M ) where T ( d ε S max M = (1 k ) k (positive) corresponds to the k δ = 1 t 1 = 1 and at t 1 ( 1 ; k ) the value of M < 0 and β parameter has only negative values. It provides positive heat losses in any local PR volume but in the meantime the negative (180 0 voltage-currant phase shift) local dynamic conductivity for example of a thin PR with UM of vibration [17] is possible and takes place. Conclusions. It was established that the character of interrelation of the resonance and antiresonance PR quality factors is determined mainly by an imaginary part of the piezocoefficient whose influence is the most essential at high-order SM harmonics. Specific types of PR vibrations and PCM electro-elastic parameters were found providing essential increasing of the antiresonance and resonance quality factors at higher harmonics. The present results will be useful for determination of a comple te set of the electroelastic complex PCM constants operative estimation of reliability of the obtained data. The described approach can be successfully applied to the analysis of the high frequency PR with vibration along thickness in a wide frequency range up to 100 MHz [3] at higher harmonics [4] in particular. The obtained results and used approach can be put in the basis of methods of prediction of the PR properties in different conditions control their parameters [5] optimization regimes of the transducer operation [8]. Taking into account a wide range of values (up to 3 times) of the quality factors of at least main types of PR vibrations a specification of the concept of PCM "mechanical quality factor" is obviously necessary. Since the influence of the piezoelectric component of energy losses increases with growth of CMC that can be essential factor in description of the properties of a new class of strong piezoelectric materials ( k 0.95 ) such as PMN-PT [6].

25 Glossary generalized quality factor 0 quality factor of non-polarized PCM m standard PCM quality factor determined at the fundamental resonance of a disk PR sc.. oc.. PR quality factor corresponding to the s.c. and o.c. regimes of PR exitation PCM quality factors of the complex elastic compliances ij ˆ ij S rn an ˆ ij ˆkl S resonance and antiresonance PR quality factors of n-harmonic (n = ) d ε ˆT piezomaterial constants (complex values) nm S d kl ij T ε mn piezomaterial constants (real values) k ij k ij coefficient of electro-mechanical coupling (real and complex values) Y Z PR admittance and impedance ξ y frequency displacement and generalized frequency displacement ( y = ξ ) K complex wavenumber f rn ( rn ω ) f an ( ω an ) generalized resonance and antiresonance frequencies of n-harmonic (real values) δ r relative resonance frequency interval R rn an R resonance and antiresonance PR resistances of n-harmonic ~ C 0 C 0 complex and real PR capacitance σ µ Poisson coefficients (real values) V relative dissipative parameter for a planar vibrational type δ nm δ δ δ tan δ tangent of dielectric losses angle γ kl tangent of piezoelectric losses angle t 1 t p t 3 t 4 normalized piezoelectric losses parameters (for instance t 1 γ / y 1 ) y 1 y p y 3 y 4 limit values of tangent of piezoelectric losses angle x n n-root of a frequency equation h b PR dimensions Ω l PR electrode area and dimensions P parameter of a degree of PCM polarization F m parameter of domain orientation n L n B n piezoelectric parameters of the PR quality factors relationships F n M H n frequency correction coefficients

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