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Elastic, Dielectric, and Piezoelectric Losses

in Piezoceramics – How It Works All Together

Alex V. Mezheritsky, “Elastic, Dielectric, and Piezoelectric Losses in Piezoceramics –

How It Works All Together”,

adapted from the article published in the journal IEEE UFFC, June 2004.

Abstract - The quality factor along with electromechanical coupling coefficient (CEMC) is commonly used

as a measure of the energy efficiency of a piezoelectric transducer (PT) working as an energy converter.

Losses in piezoceramics are phenomenologically considered to have three coupled mechanisms: dielectric,

elastic, and piezoelectric. Their cumulative performance first of all determines the PT quality factor

characterizing the efficiency of vibrational energy accumulation, and related to it dissipative effects.

The extended definition of the PT electro-mechanical quality factor (EMQ) with permanent energy

exchange between electrical source of excitation and PT was proposed. The EMQ analysis has been

conducted on the basis of complex material constants for both stiffened and unstiffened canonical

vibrational modes. The efficiency of mechanically free and electrically excited piezoceramic transducers in

a wide frequency range of PT harmonics, especially between the fundamental resonance and antiresonance

frequencies, was investigated, and the effect of “piezoelectric loss anomaly” with extremely low total

losses was predicted. Particularly, optimization of PT excitation with connected reactive (capacitive)

element was conducted to provide higher PT mechanical vibrational characteristics with less total losses.

The requirements to the piezoceramic material parameters, types of transducer vibrations, and especially to

the piezoelectric loss factor in the range of physically valid values were established to provide maximal

EMQ.

Keywords: Piezoceramic transducer; Quality factor; Energy losses; Series capacitor

Author: A.V. Mezheritsky e- mail: mav5@optonline.net ( CC: amezheritsky@ieee.org)

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I. INTRODUCTION.

Resonators and transducers (PT) made from piezoceramic materials (PCM) of a lead-zirconate-titanate

(PZT) system [1] are used in electronics and hydro- and electroacoustics for various applications. High

power mechanical output PT characteristics [2,3] are traditionally achieved at the resonance frequency

because of the relatively low requirement for electric voltage. The advantages of excitation of the resonant

PT at the antiresonance frequency, against the traditional resonance regime, with regard to PT heating and

power supply losses, were considered in [4,5]. Connected in series to PT an electrical capacitor [6] greatly

increases voltage on PT and its energy efficiency at the frequency close to antiresonance, a decisive role of

the piezoelectric energy loss was established.

PZT-based PCMs provide strong coupling of mechanical and electrical fields (CEMC value can reach

0.9, close to the theoretical limit), so the elastic and dielectric components of energy losses, as well as the

piezoelectric component describing losses at interconversion of the mechanical and electrical energies,

should be involved in the phenomenological PT description [7]. Traditionally [1], the resonance quality

factor mQ at the fundamental (lowest) harmonic of the planar mode of a disk piezoelement, determined by

the use of electric measurements of PT resonance admittance, was chosen for characterizing PCM quality.

The CEMC as a real value [8-10] determines a share of the total energy converted in a single act from

mechanical into electrical forms, or vice versa, and practically is determined directly by the relative

resonance frequency interval rδ (fundamental harmonic) between the antiresonance af and resonance rf

PT frequencies [1,8], whose difference in turn is determined basically by the real part of the elastic

constants (e.g., ,E Di js ). The quality factor characterizes the process of energy accumulation (it includes

multi-cycle energy transformation process with losses) in a PT and can be described in common by the

complex elastic constants (e.g., ,ˆE Di js ), whose imaginary parts depends on elastic , dielectric, and

piezoelectric mechanisms of energy losses [10,11].

A number of high accuracy methods of complex material constants determination was proposed in [10,

12-15]; the physically valid limits on the piezoelectric loss factors were phenomenologically established in

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[7] according to the condition of “positivity” of local thermal losses in a piezoelectric medium; the concept

of complex material constants was used in [16-18] to model PT dynamic responses. Involving the domain

mechanism of energy losses [11] caused by damped (time delay) movement of walls of 90-degree domains,

the uniform physical nature of coupled elastic , dielectric, and piezoelectric energy loss components was

shown. In a phenomenological description the elastic and dielectric losses are considered as independent,

while the piezoelectric loss can not exist alone, but only with connection to the elastic and dielectric losses

together in couple. Probably for this reason, there is an opinion that at present the usefulness of such

approach has not been clearly established.

Fig. 1. Rod, ring and bar PTs with stiffened and unstiffened vibrational modes. Frequency-specifying

dimension L is much larger than other PT dimensions, c denotes the distance between electrodes.

Two basic canonical types of PT vibrations are traditionally considered [8]: stiffened mode (SM) with

direction of vibration along an exciting electric field (e.g., rod 15x2x2 mm PT, Fig.1) when the electric field

induction D is homogenous between electrodes (also known as a thickness-field excited mode [19]), and

unstiffened mode (UM) with direction of vibration perpendicular to the vector of an exciting electric field

(e.g., bar 15x1x0.5 mm PT, Fig.1) when the electric field strength E is homogenous between electrodes in

the case of a thin PT plate (longitudinal-field excited mode [19]). The corresponding PTs all are vibratory

systems with “distributed” parameters. Their characteristics will be compared further to the case of a ring

PT (e.g. ∅15x1x 0.5 mm, Fig.1) with radial vibrations which refers to a vibrator with “concentrated”

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parameters, so that its E and D fields do not depend on coordinate. In an ideal case of a “thin” ring, it does

not have radial harmonics but the fundamental only, and its vibrational characteristics correspond to both

UM and SM.

Study of the loss mechanisms and loss reduction is important for the design of high-efficiency devices

and driving circuits, particularly for the cases of powerful transducers and devices for frequency selection.

The magnitude of the vibration-level in the high-power usage is limited by heat generation as well as a

drastic change in piezoelectric properties, and is restricted by temperature limit. The vibration-level is

represented by the effective vibration velocity ( 0v ), which is a universal, pure mechanical parameter and is

usually determined by the maximum vibration-amplitude of the PT top at the resonance [2]. The EMQ

concept introduced here ultimately characterizes the ratio 20v T∆ % . The temperature T∆ % of a PT increases

with 0v , and sharply goes up when 0v reaches a certain saturated value, while losses increase and quality

factor decreases unlimitedly. The piezoelectric loss, which is a consequence of complex piezoelectric

phenomena, is involved as a high-promising factor of efficiency increasing.

The EMQ of commonly seen one-dimensional vibration piezoelectric elements are formulated and

analyzed using notations defined in [7,8]. We suppose that the vibrations are excited only by an alternating

voltage V applied to the electrodes of a mechanically-free PT. The coordinate system and direction of

polarization for each piezoelectric element is shown in Fig. 1.

II. PT LOCAL AND TOTAL ENERGY STORED AND LOSSES.

For a piezoelectric medium in the case of a relatively low level of excitation the electrical and mechanical

field characteristics [7,20] are related by the linear “equations of piezoeffect”

ˆˆEi i j j mi mS s T d E= + , ˆˆT

l ml m l j jD E d Tε= + , (1)

where , 1...6i j = , , 1...3m l = , ,E Di js is the elastic compliance under constant (usually zero) electric field

strength rE or induction

rD , ,T S

m lε is the dielectric permittivity under constant (usually zero) mechanical

stress jT or strain iS , m id is the piezocoefficient. Similar equivalent performance can be used for the

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material matrix constants through ,ˆE Di jc , ˆˆ ˆ, ,mi m i m ig e h , ,ˆT S

m lβ [8].

Time dependence as ie ωt is used further, z z i z′ ′′= − denotes a complex material constant, z∗ - complex

conjunction. The PT vibrational characteristics typically are presented as complex amplitudes (not time-

current values), generalized field vectors and tensors are expressed by boldface letters.

The electromechanical behavior of a real PT with energy losses is phenomenologically described [7] by

the complex constants of the PCM material matrix:

, , ,ˆ (1 )E D E D E Di j i j i js s i Q≡ − , ˆ (1 )mi mi mid d i γ≡ − , ˆ (1 )T T

m l ml mliε ε δ≡ − , (2)

where 1−≡i , ,E Di jQ are the quality factors of the complex elastic compliances ,ˆE D

i js , m iδ and m lγ are

the dielectric and piezoelectric loss factors, respectively. Further the sign " "≅ is used to express the

approximation on dissipative parameters with relatively high resolution of the order 21, , 1Q δ γ− = when

typically 21 0.5 1 , , 1Q δ γ+ ≅ is taken. In a generalized form (indexes omitted) , for the two vibrational

sets of the complex constants ( 11 31 33 31ˆ ˆˆˆ , , ,E Ts d kε ) and ( 33 33 33 33

ˆ ˆˆˆ , , ,E Ts d kε )with corresponding constants’

indexes (Fig. 1), we can express the real k and complex k CEMCs as:

2 2 T Ek d sε≡ , ( ) 2 2 2ˆ ˆ ˆ ˆ 1 2 1 ET Ek d s k i Qε γ δ≡ ≅ − − − . (3)

From the condition of “positivity” of the local thermal energy losses, the phenomenological limitation on

the value of the relative piezoelectric loss factor t was established [7] :

( ) ( ) ( ) ( ) ( )2 22 2 2 1T E Et d s k Q yε γ δ γ′′ ′′′′≡ = = ≤ , (4)

where 2 Ey k Qδ≡ is the maximal piezoelectric loss factor value (Appendix). Then

( ) ( )2 2

2 2 2 2 22 2

1 1 1ˆ 1 1 1 2(1 )(1 )

E EE E

kk k i G k i k Q t k Q

k Q Q kδ δ

− ≅ − ⋅ = − − − + − − , (5)

where 2 1 EG Qγ δ= − − is the CEMC loss factor. Particularly, the complex compliances ,ˆE Ds with their

relationship 2ˆˆ ˆ (1 )D Es s k= − have the compliances’ quality factors related by:

( )2 2

2 22 2

1 1 1 1 2(1 )1 (1 )

E ED E E

k G k Q t k QQ Q k Q k

δ δ ≅ − = − + − − − , (6)

then ( ) ( )2 2ˆ1 1 1 1 1E Dk k i Q Q − = − + − . Particularly, based on the experiment data and domain model

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results [11], factual values of the piezoelectric loss factor are positive, so that the interval [ ]0, yγ ∈ is of

the main interest.

A. Common Description

Peak energy stored ( )w rr

( rr - space coordinate) in a local elementary volume and W in a whole

piezoelectric body of volume Ω are expressed [20] as:

( ) ( )0.5Rew r = ⋅ +r * *E D T : S , ( )0.5 ReW dΩ

= ⋅ + ⋅ Ω∫ * *E D T : S , (7)

then, averaged for a period local ( )h rr

and total H energy losses (heat power) are expressed as:

( ) ( )0.5 Imh r ω= ⋅ +r * *E D T : S , ( )0.5 ImH dωΩ

= ⋅ + ⋅ Ω∫ * *E D T : S . (8)

The mechanical state of a vibrating solid body is described by the equation of motion ρ ′′∇ = ttr

T u for the

elementary unit volume, that can be rewritten in the energy representation as:

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( ) ρ∇ ⋅ = −r r* *T u T : S v , (9)

where ωr r ru, v = i u are the local mechanical displacement and velocity, ρ is the material density, ∇ and :

are the vector and tensor operators. For an elementary volume the difference between potential and kinetic

energies (9) equals the changing (redistribution) of the mechanical energy flux (acoustical Umov-Pointing

vector) flowing through the volume [20]. According to the boundary conditions on the surface of a

mechanically unloaded PT it follows that ( )dΩ

∇ ⋅ Ω =∫r*T u ( ) 0d

ττ⋅ ⋅ =∫

r r*T u n , where τ is the full PT

surface with normal vector rn . Then, the equality of potential and kinetic energies for a whole mechanically

free and electrically excited PT follows from (9):

2

( ) d dρΩ Ω

⋅ Ω = ⋅ Ω∫ ∫r*T : S v , (10)

so that 2

Re( ) d dρΩ Ω

⋅ Ω = ⋅ Ω∫ ∫r*T : S v and ( )Im 0d

Ω⋅ Ω =∫ *T : S , and finally

( ) 2( ) 0.5 Re Re ( )w r vρ = ⋅ + + ∇ ⋅

rr r* *E D T u , ( ) 20.5 ReW d v dρ

Ω Ω

= ⋅ ⋅ Ω + ⋅ Ω ∫ ∫r*E D , (11)

( )( ) 0.5 Im Im ( )h r ω = ⋅ + ∇ ⋅ rr * *E D T u , ( )( ) 0.5 ImH h r d P dω

Ω Ω≡ ⋅ Ω = ≡ ⋅ ⋅ Ω∫ ∫

r *E D , (12)

where P is the total electric field loss, while its local characteristic *( ) 0.5Re( )p r E j≡ ⋅ =r rr

( )0.5 Imω= ⋅ *E D , where j D′= − t

rr is the current density, is related to the local thermal losses as:

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( ) ( ) 0.5 Im ( )p r h r ω = − ∇ ⋅ rr r *T u . (13)

Using the basic relationships for dielectrics φ=−∇E , 0∇⋅ =D , ( )el

elI dτ

τ′= − ⋅∫ trnD , where φ is

the electric field potential, I is the total current through PT, elτ is the electrode surface, we have

( )φ=−∇ ⋅* *E : D D , and according to the electrical boundary conditions for the whole PT volume

2* *( )el

eld d V d iV I i V Yτ τ

φ τ τ ω ωΩ

⋅ ⋅ Ω = − ⋅ ⋅ ⋅ = − ⋅ ⋅ = ⋅ =∫ ∫ ∫r r* * *E D D n D n , (14)

then ( ) 20.5 Im 0.5 Red V Yω

Ω⋅ Ω =∫ *E D , ( ) 2

0.5 Re 0.5 Imd V Y ωΩ

⋅ Ω =∫ *E D , and finally

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0.5 ImW V Y v dω ρΩ

= ⋅ + ⋅ Ω ∫r

, 2

0.5 ReH P V Y= = ⋅ . (15)

The total PT energy stored W consists of two time-averaged parts: convertible 2

0.5 ImconvW V Y ω= ⋅

and unconvertible 2

0.5unconv kinW W v dρΩ

= = ⋅ Ω∫r

(as a sum of equal (10) kinetic and potential energies),

equal to peak kinetic energy. In (15) Im 0Y > (capacitive component) means that the voltage on PT is

created basically by free charges of the electric generator partly compensated by the induced coupled

charges on the PT electroded surface, while Im 0Y < (inductive component) means that the voltage on PT

is basically created by induced coupled charges on the PT electroded surface partly compensated by free

charges supplied by the generator.

Then from the side of the generator loaded by PT with Re ImY Y i Y= + , the total generator power are

described by

2 2*0.5 0.5 Re 0.5 Im Re ImP VI V Y i V Y P i P∑ = = − ⋅ ≡ − , (16)

where the real generator power ReP P= compensates the thermal energy losses inside PT to support

unconvW constant, and the imaginary generator power ImP provides convertible PT energy convW . A

schematic representation of the energy balance of the system “generator-vibrating PT” is shown in Fig. 2.

Note that current energy characteristics change with as much as twice frequency of excitation.

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Fig. 2. Schematic diagram of the energy balance in the system “electrical power supply – PT”.

Ψ is the period of the harmonic excitation voltage.

B. Complex Material Constants and Losses

Using generalized notations , the local energy stored and losses can be expressed through the complex

material constants as:

( ) ( ) ( ) ( )2 2( ) 0.5 ( ) ( ) 2 ( ) ( ) cos ( )T E ETh x E x s T x d E x T x xω ε ϕ ′′ ′′ ′′= + + ⋅ ⋅

, (17)

( ) 2 2( ) 0.5 ( ) ( ) 2 ( ) ( ) cos ( )T E ETw x E x s T x d E x T x xε ϕ= + + ⋅ ⋅ , (18)

where ETϕ is the phase angle between E and T . If [ ]Re ( )E

T x r is the “time projection” of T on E, then

( ) [ ]( ) ( ) cos ( ) ( ) Re ( )ET

EE x T x x E x T xϕ⋅ ⋅ ≡ ⋅ r , (19)

In common case, the electric field intensity and mechanical stress characteristics of PT under electrical

excitation can be presented as ( )( ) ( )E x V c E x= ⋅ % and ( )( ) ( ) ET x V c T x d s= ⋅ % . We have from (17-

19) the following expressions for the total losses and energy stored ( Ei jQ Q≡ ):

( ) 2d e pxP H h x dx H H H= = = + + =∫ (20)

( )2 22 2 1 20.5 ( ) ( ) 2 ( ) Re ( )T

x x x EV c E x dx k Q T x dx k E x T x dxω ε δ γ− = + + ⋅ ∫ ∫ ∫ %

% % % % ,

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( ) 2d e pxW w x dx W W W= = + + =∫ (21)

( )2 22 2 20.5 ( ) ( ) 2 ( ) Re ( )T

x x x EV c E x dx k T x dx k E x T x dxε = + + ⋅ ∫ ∫ ∫ %

% % % % ,

where the sum of the terms reflects relative shares of the dielectric ( ,d dH W ), elastic ( ,e eH W ) and

piezoelectric ( ,p pH W ) losses and stored energy, then the following terminology should be used:

Electric loss - * *0.5Re( ) 0.5 Im( )E j E Dω⋅ = ⋅r r rr

(loss of electric field)

Dielectric loss (Hd) - 2

0.5 ( )T Eω ε ′′ (as in the case of pure dielectric medium)

Mechanical loss - *0.5 Im( : )T Sω (loss of “mechanical” field)

Elastic loss (He) - 2

0.5 ( )Es Tω ′′ (as in the case of a pure solid body)

Piezoelectric loss (Hp) - 0.5 cos( )ETd E Tω ϕ′′⋅ (energy conversion loss).

Factor 2 of pH and pW terms in (20,21) is a result of double energy conversion. According to (20), each

of the material dissipative factors ( , ,Q δ γ ) can be determined separately. For example, “elastic” quality

factor Q ( Ei jQ ) is determined as a ratio of the elastic stored energy to elastic loss, not mechanical or total

losses. The traditional PT resonance quality factor ( rQ ) well coincides with Ei jQ at the fundamental PT

resonance (they are equal for UM) where stored energy and losses are predominantly elastic.

As to the piezoelectric loss factor, it follows an understanding of the fact of negativity of the piezoelectric

loss occurring when the piezoelectric (mutual) energy stored is negative as well in the case of positive

piezoelectric loss factor 0γ > in (2) representation. A cross-product power loss (piezoelectric loss)

depends on both E and T, including the phase angle between them. It equals zero when either

E = 0, or T = 0, or both, and additionally when ( )cos 0ETϕ = (at 0, 0≠ ≠E T ).

As easily follows from (20) for a ring PT, the piezoelectric loss

( )cos ETp e dH H H

yγ

ϕ= ⋅ ⋅ , (22)

where 2y k Qδ= and yγ ≤ [7], can not be a dominant loss, its extreme value is

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( )p e dextr H H H= , and the sign depends both on the material parameter of piezoelectric loss factor γ

and on the frequency of PT excitation (phase ETϕ ). The existence of the piezoelectric loss factor itself is

not so important as with connection to the elastic and dielectric losses. Its energy effect is determined only

as a relative value, elastic and dielectric losses must be present both.

In common, the influence of the piezoelectric loss factor (γ ) on the PT characteristics, including

dissipative, is not the same as the piezoelectric loss. They coincide for UM (bar and ring PTs) when E and

T (their resonance factors) do not depend on the piezoelectric loss factor γ at all, and are sufficiently

different for SM (rod PT) under certain frequency conditions as the resonance mechanical stress and

electrical field strength strongly depend on the piezoelectric loss factor. Nevertheless, piezoelectric loss

influence equals zero for both UM and SM at the frequency 1 2 EEf L sρ= (in the case of UM Ef

coincides with the PT resonance frequency where the phase angle 090ETϕ = − ).

Thermal and electric field losses distributions depend on piezoelectric loss factor differently for UM and

SM. So, for a bar PT (UM) a “short-circuited” regime ( 0V = ) causes 3 0E = locally, and an “open-

circuit” regime ( 0I = ) does not lead to 3 0D = locally, but integrally. For a rod PT a “short-circuited”

regime (V = 0) does not lead to E3 = 0 locally, but integrally, and an “open-circuit” regime ( 0I = ) causes

D3 = 0 locally. For a ring PT all these electrical conditions are satisfied locally as it is a vibrator with

concentrated parameters.

C. Frequency behavior of the piezoelectric loss

As the total loss of the source of excitation is determined as 2

0 00.5 Re( )P H C V Y Cω ω= = , after

decomposition with a high resolution 3 1χ << the expression for UM PT ( )0Re Y Cω (see (31) further),

we have the following three characteristic terms:

- term of the dielectric loss : δ ,

- term of the elastic loss : 22 2 2 2

1 1 31 ( 2) ( 2) ...

1 4 1 4k a a

Q Q Qχ

χ χ

⋅ ⋅ + + + + + ,

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- term of the piezoelectric loss : 22 2

22 1 ( 2) ...

1 4k a

Qχ

γχ

− ⋅ ⋅ + + +

.

For a ring PT 1 2a = and for a bar PT 24a π= [8]. It follows that, as to resonance frequency

displacement, the dielectric loss is constant. The elastic loss is described by a constant term, plus a

symmetric bell-like resonance term with its maximum at the resonance frequency, and an asymmetric term

with “left-side decreased” asymmetry. The piezoelectric loss has a constant term, plus an asymmetric term

with the symmetry type depending on the piezoelectric loss factor sign (Fig. 3). As the PT admittance can

be expressed in first-order approximation as

2

2 2 2 20

1 2( 2)

1 4 1 4Y k

a i QC Q Q Q

χω χ χ

⋅ − ⋅ + +

; , (23)

then both the asymmetric terms can be considered as a mixing to ReY (23) some portion of ImY

component. Note particularly that a serious error can occur caused by any phase shifts ( e.g, R-C circuitry

units) when electrical phase-sensitive measurements [13] are provided to determine the piezoelectric loss

factor. The absolute extremes of the piezoelectric loss are at the frequency displacements equal 1 Q± ,

where a half of the elastic loss maximum takes place. The piezoelectric loss is both negative and positive

(the stored piezoelectric energy as well), depending on the resonance frequency displacement and sign of

the piezoelectric loss factor. Losses are pure dielectric at low frequencies because of 0

0f →

→T .

Fig. 3. Qualitative characteristics of the frequency (resonance

frequency displacement) behavior of the dielectric, elastic and

piezoelectric losses. ( )V const f= .

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The characteristic of the piezoelectric loss factor influence ( , ) 1 2( 0)

p

d e

HPM

P H Hχ γ

γ≡ = +

= +for

UM coincides with the relative share of the piezoelectric loss in the total losses, and is a linear function of

the piezoelectric loss factor γ with two extremes at the frequencies 0χ+ > and 0χ− > , which do not

depend on γ . So, for a ring PT (constants’ indexes k31, 11EQ , δ33, γ31, y1 )

1

11 k Q

χδ

± = −∓

, 1

11

kk

χ± = −∓

, 2

11

1r

kδ = −

− , (24)

where kχ± are the frequency displacements of maximal share of the mutual (piezoelectric) energy in

respect to the sum of elastic and dielectric energies (one of the CEMC definitions [10]). Note in this respect

the similarity of the expressions for the CEMC and parameter ( ),M χ γ of piezoelectric loss factor

influence in the case of UM with the replacement k k Qδ→ , that can be used to apply known results

[10] to a new characteristic . The higher-frequency displacement χ+ coincides with antiresonance one

rχ δ+ = at 2 1k Qδ = . The extreme values of the piezoelectric loss factor influence are equal to

( , ) ( ) 1M extr M yχ γ γ± = = ∓ , (25)

where 2y k Qδ= , and absolute extreme values ( , )M yχ γ± = equal 2 and 0, that means zero total

losses at the frequency displacement χ+ for t = +1 and all possible values of 2k Qδ (Fig. 4a). If the basic

material parameter of the piezoelectric loss influence 1k Qµ δ≡ << , then we have

22 3 8 ...χ µ µ± ± + +; . (26)

It can be shown from the approximate approach for a bar PT that :

22 4χ µ π µ± ± +; , 2( , ) 1 8M yχ γ π γ± ⋅; ∓ , (27)

where absolute extreme values ( , )M yχ γ± = equal 1.9 and 0.1, that gives maximum 10 times decreased

total losses at rχ δ+ = for t = +1 and 2 1k Qδ ; (Fig. 4b), so there is no “zero-loss” condition.

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The difference between (25) and (27) for ring and bar PTs is related to the volume characteristics

distribution in the case of a bar PT, when a zero-loss condition takes place at different frequencies for

different PT volume parts. A rod PT (SM) has a similar expression for χ± , and a “zero-loss” condition is

possible only at the antiresonance frequency for 2 1k Qδ ; and t = +1 [4].

Fig. 4 a,b. Calculated comparative frequency dependences of the ring (a) and bar (b) PT normalized

admittances and susceptances for different values of the dielectric δ and piezoelectric t loss factors

at Q =100, k = 0.7 .

At harmonics in the case of a bar PT (UM) the influence of the piezoelectric loss factor sharply decreases

(replacing nπ π→ in (27), where n = 1,3,5… is the harmonic number), while the “zero loss” condition

stays in force for a rod PT (SM) and extremely low losses are possible at high harmonics.

It is proposed to call the described possible effect as “piezoelectric loss anomaly”, when the piezoelectric

loss significantly compensate elastic and dielectric losses at the frequencies ( )f χ± ± .

III. PT ELECTRO - MECHANICAL QUALITY FACTOR.

The quality factor concept usually relates to the resonance regime of PT operation, however PTs are

exploited not only at their own resonance. We will use in this respect the following extended definition: the

quality factor is the energy storedW permanently stored in the vibratory system divided by the energy

lossP (power) dissipated per radian of oscillation as stored lossQ W Pω≡% . As was shown in the previous

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chapter, the total PT energy stored and losses under electrical excitation are expressed by (15), where the

permanently stored PT energy is the mechanical energy equal to the kinetic energy amplitude kinW (due to

equality of the kinetic and potential mechanical energies). Then we define the PT “electro-mechanical

quality factor” Q% at any arbitrary frequency as :

2

2Re

kinv dW

QP V Y

ρω ω Ω

⋅ Ω≡ =

⋅∫

r% , (28)

where P is the effective electrical power of losses time-averaged through a cycle, V is the voltage applied

to PT. Such an extended definition of PT EMQ under permanent energy exchange with electrical source of

excitation is introduced and considered here, and EMQ frequency dependence for SM and UM, expressed

through PT resonance frequency displacement for a given PT, is mainly analyzed (all PCM dissipative

parameters are supposed to be constant on frequency displacement near the resonance). The EMQ coincides

with the resonance (antiresonance) quality factor at any PT resonance, where the PT admittance is active,

without converted energy traveling between the generator and PT, and the stored energy is mechanical.

Out of resonance, some part of stored energy is circling between PT and energy source for a period of

vibration. Note, that it additionally increases total energy consumption inside the generator and is not

desirable. Any connected to PT reactive element (capacitive, inductive) compensates the convertible part of

the stored energy (related to ImY in (15)), so that at a resonance (antiresonance) of the system the total

admittance is active (from the side of a generator), the convertible PT electrical energy is traveling between

PT and the reactive element. In the system, the kinetic (mechanical) energy and energy consumption are

concentrated in a PT body, so the EMQ of the system is determined only by the PT intrinsic parameters

(CEMC, elastic, dielectric, piezoelectric loss factors) at the system’s resonance.

There is a critical PT characteristic defined [2] as the ratio mv P of effective vibration velocity to

power consumption, whose a dimensionless measure is EMQ. Actually EMQ for a given type of PT

(shape, size, etc.) reflects the characteristic 2

mv T∆ % , where T∆ % is the PT temperature increase

(usually at the vibrational node), and vm corresponds to the PT point of maximal mechanical displacement

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(usually PT top). Such a velocity characteristic on the active transducer boundary determines PT output

characteristics in a surrounding medium, also it is a convenient parameter of the vibration-level for

powerful transducers (piezotransformers, etc.).

Note, that the considered EMQ frequency variation is not a quality factor frequency dependence in a

traditional sense, but ultimately is the PT resonance quality factor when the PT has system’s resonance at a

given frequency (for instance, due to connected non-dissipative non-mechanical elements, such as

capacitor, inductance, etc.).

The following generalized PCM parameters are used for the further analytical analysis , for both UM and

SM : Q ( )Ei jQ = 100, k = 0.7, δ = 0.02 (0.01 and 0.04 for comparison), physically allowed interval of the

relative piezoelectric loss factor [ ]1, 1t ∈ − + . The piezoelectric loss factor γ (or t-parameter) variation is

taken in the simulation discussed below, while the elastic constant’s quality factor Ei jQ and CEMC k are

assumed to be fixed.

A. EMQ of a ring PT

The primary ring PT parameters are expressed as: radial displacement 31 3ˆ0.5u Ld E N= and velocity

v i uω= , tangential stress ( )3 31 11ˆ ˆ1 1 ET E N d sϕ = − ⋅ and total admittance

( )2 233 31 11 31 11

ˆ ˆˆ ˆ ˆT E EelY i d s d s N cω ε τ= − + ⋅ , where L is the ring diameter, elτ is the electrode area, c is the

PT thickness, 2 211ˆ ˆ4 E

r L sω ρ= is the complex resonance frequency squared ( 111 Erf L sπ ρ= ),

2 2ˆ1 rN ω ω≡ − is the resonant factor. Then

16

Fig. 5. Relative quality factor of a ring PT vs. resonance frequency displacement.

Q = 100, k = 0.7 ( 0.40rδ = ), δ = 0.01 ( 2 0.5k Qδ = ), 0.02 ( 2 0.98k Qδ = ), 0.04 ( 2 2k Qδ = ),

t-parameter varies (step 0.05) for δ = 0.02 .

( ) ( )22 1 2 1N i Qχ χ χ= − + + + , ( ) ( )2 42 2 24 1 2 1N Qχ χ χ= + + + . (29)

Using the expression for kinetic energy 220.5kinW m uω= , we find EMQ as

( )

( )( )

22 2 23 2

2 2 220 0

1

Re Re Rekin

r

m uW kQ kP V Y N Y C N Y C

χωω ωω ω ω

+≡ = ⋅ ≅ =% , (30)

1rf fχ = − is the resonance frequency displacement, k is the CEMC ( 31k ), m is the PT mass.

Then, out of the resonance region for 1Qχ −>> : ( )

( )2

20

1 ...Re 4

kQ

Y Cχ

ω χ+ +

≈ ⋅% .

The relative ring PT EMQ frequency behavior is presented in Fig. 5, including resonance frequency

interval between the resonance and antiresonance frequencies. To simplify the expression (30), we will use

the decomposition process on small parameters of dissipation and resonance frequency displacement:

( ) ( ) ( ) ( )( ) ( )

22 2

2 42 20

1 2 1 2 2 1Re2 1

4 1 2 1

Q QYk Q k

C Q

χ χ χ γδ γ

ω χ χ χ

+ − + −= − − +

+ + + , (31)

( )

( ) ( ) ( ) ( )

2

2 2 2 2

1

1 2 2 1 1 2 2 1

QQ Q Q Q N Q Q N k

χ

χ χ γ χ γ δ

+=

+ − − + − − +

% . (32)

Then with a total resolution not worse than 5 %

( )( )( ) ( )1 1.51 1 1

2 2 1 21 1.5

rr

r rQ QQδχ χ

δ γ δ χδ χ δχ

+ − ⋅ − − ⋅ ⋅ + = + ;% (33)

( )( )

( )( )

2

2

1 1.5 1 0.51 12

1 1 1.5 1 0.5r

r r r

kQ k Q

δ χχ χγ δ

δ χ δ δ

+ + = − − − ⋅ ⋅ − + + .

The last relationship coincides with 231

31 33211 11 31 11

1 1 12

1D E E

kQ Q k Q

γ δ

= − − − − at rχ δ= , relating the ring

PT resonance 11E

rQ Q Q= ≡ and antiresonance 11D

aQ Q= quality factors (6).

We have the following approximate formula structure, whose components are shown in Fig. 6:

17

( )

21 1 12 2 2

rQ QQχ

χ γ δδχ

≈ − ⋅ − + ⋅

% , (34)

A negative share of the “piezo-elastic” loss c0mponent ( 1 2Qγ > ) increases linearly in magnitude with

frequency, the last term in (34) reflects the dielectric loss presented by a partly short-circuited equivalent

dielectric loss resistor.

Fig. 6. Dissipation factors in the quality factor expression (34) vs. relative frequency displacement for

Q = 100, k = 0.7 ( 0.40rδ = ),δ = 0.02 ( 2 0.98k Qδ = ), t = 1.

B. EMQ of a bar and rod PT

In the case of one-dimensional vibration with the velocity distribution ( )sin 2mv v xKL= ⋅ (Appendix) ,

where [ ]1,1 +−∈x is the relative coordinate, mv is the velocity space amplitude (its location can be at the

PT top, inside PT, or outside PT), the total PT kinetic energy is 2 20.5 0.5 mv d m v BρΩ

Ω = ⋅∫ , where

( )0.5 1 sinB KL KL = − (35)

is the “velocity distribution” parameter, presented in Fig. 7 and reflecting the non-coincidence of the

maximum velocity vm location with the PT top (Fig. 2) depending on frequency. The B-factor depends on

PT shape and vibration mode. It can be estimated as ( )0.5 1 1B χ χ + + ; in the vicinity of the bar PT

resonance, and as ( )0.5 1 1B χ χ + + ; in the vicinity of the rod PT antiresonance (B = 1 for a ring PT).

18

Fig. 7. “Velocity distribution” parameter B (35) and its approximation for the fundamental harmonic,

,1 1rf fχ = − ( ,1 1af fχ = − ).

According to the EMQ definition 2

32

Remkin

m u BWQ

P V Yω ω

⋅≡ =

⋅% with ( ) ( )2 cos 2N KL KL≡ ⋅

for a bar PT ( )

( )( )

222 2 231 2

312 22, 1 0 0

14 4Re Rer

Bk BQ k

N Y C N Y C

χω π πω ω ω

+≅ =% , (36)

and then ( ) ( )

231 2

0

1cos 2 Re

BQ k

KL Y Cω≅ ⋅ ⋅% with ( ) 1

2 1 12 2 EKL i

Qπ

χ

≅ + −

. (37)

For a rod PT ( )

( )( )

222 2 233 2

332 2 22 2,1 0 33 0 33

14 4Re 1 Re 1a

Bk BQ k

N Z C k N Z C k

χω π πω ω ω

+≅ =

⋅ − ⋅ −

% , (38)

and then ( ) ( )

233 2 2

0 33

1cos 2 Re 1

BQ k

KL Z C kω≅ ⋅ ⋅

⋅ −

% with ( ) 12 1 1

2 2 DKL iQ

πχ

≅ + −

. (39)

The EMQ frequency dependences for bar (UM) and rod (SM) PTs are presented in Fig. 8, 9. There is a

characteristic point on the graphs with no influence of the piezoelectric loss factor on the PT quality factor.

For both UM and SM cases it takes place on the characteristic frequency 1 2 EEf L sρ= determined by

the corresponding elastic compliance at constant electric field E. Only in the case of UM that frequency

coincides with the PT resonance frequency (including multiple to integer harmonics). For SM , 1E rf f< ,

19

and there is no such effect at harmonics. For UM the resonance quality factor at harmonics ,r nQ equals Ei jQ

and does not depend on dielectric and piezoelectric loss factors, while for SM a strong effect from these

factors takes place.

Then, in the case of UM and SM for a relatively high frequency displacements ( ) 1 Qχ χ >>

( )( )2

312 2

0

1 121Re kYC Q

χ χ

ω π χχ

+ +⋅ ⋅; % and ( ) ( )

( )22 33

0 33 2 2

1 121Re 1

kZ C k

Qχ χ

ωπ χχ

+ + ⋅ − ⋅ ⋅ ; % . (40)

Fig. 8 a,b. Relative quality factor vs. frequency displacement in a wide frequency range for a bar (a) and

rod (b) PT. Q = 100, k = 0.7, δ = 0.02 ( 2 0.98k Qδ = ), t-parameter varies.

20

Fig. 9 a,b. Relative quality factor vs. resonance frequency displacement in the vicinity of the fundamental

PT resonance for a bar (a) and rod (b) PT. Q =100, k =0.7, δ =0.01 ( 2 0.5k Qδ = ), 0.02 ( 2 0.98k Qδ = ),

0.04 ( 2 2k Qδ = ), t-parameter varies (step 0.05) for δ = 0.02 .

After the decomposit ion process the best approximation for a bar PT (UM) is as follows:

( )

( )( ) ( )1 1.51 1 1

2 2 1 21 1.5

rr

r rQ QQδχ χ

δ γ δ χδ χ δχ

+ − ⋅ − − ⋅ ⋅ ⋅ + +

;% , (41)

which coincides with similar expression (33) for a ring PT, and for a rod PT (SM) we have

( ) ( ) ( ) ( )( )

2 1 1.21 1 12 1 1 1 2 2 1

8 1 1.2r

r r rr rQ QQ

δπ χ χδ δ δ γ δ χ δ

δ δ χχ

+ − − ⋅ − + + ⋅ − − − − ⋅ ⋅ +

;% . (42)

IV. EFFECT OF PIEZOELECTRIC LOSS FACTOR ON THE GAIN OF PT OUTCOME

WITH A CONNECTED CAPACITIVE ELEMENT.

Near the resonance frequency, the required power is large and the voltage required is small. If the driving

frequency approaches the antiresonance frequency, the input power can become low for the same vibration

level. However, the very high voltage required may cause a problem in practical applications. To realize a

low-loss generation condition without a high voltage source [6], an external connected in-series capacitor C

is used (Fig. 10). The “resonance of voltages” takes place at the frequency s rf , where the capacitive from

C and inductive from PT components compensate each other:

1 1

Im Re1 1outV Y Y Y

iU i C C Cω ω ω

− −

= + = + −

, then 0 0

1Re( )

out

s r s r

VV CU U C Y Cω

≡ = , (43)

where Re ImY Y i Y= + , U is the generator voltage, out s rV V= is the voltage on PT at the system’s

resonance. The condition for the resonance of the system 1 Im 0Y Cω+ = gives the resonance frequency

displacement ( ) [ ]1 1 0;s r s r r r rf f Aχ δ δ≡ − = + ∈ and its relative value

( ) [ ]11 0;1s r rq Aχ δ

−= = + ∈ , where ( ) [ ]2

0 1 0;A C C b k= − ∈ ∞ is the ratio of the capacitance C to

21

the partly-clamped PT capacitance, ,r af f are the intrinsic PT resonance and antiresonance frequencies

(b = 0.75, 21 1 1 1r a rf f kδ = − = − − for a ring PT). Using (30) for EMQ, we have for 1s r Qχ −>> :

( ) ( ) ( ) ( )

2

2 220

2 2( ) ( ) (1 ) ( )

1 111

s r r rs r s r s r

s r s rs r

NV C AQ Q q q Q

U C Ak

δ δχ χ χ

χ χχ= ≅ ⋅ ⋅ = − ⋅ ⋅

+ +++% % % , (44)

Note, that when the PT and generator are connected directly (or at 0C C>> ), the resonance frequency of

the system coincides with the own PT resonance frequency, the voltages 1V U → , and the expression (44)

must be corrected by 1 for very small 1s r Qχ −<< (that is not critical for further consideration).

Fig. 10. Schema of the PT excitation with an in-series

connected capacitor C.

As A becomes smaller, the resonance frequency of the system becomes higher, while the antiresonance

frequency does not change. A high voltage V appears at the PT when A is small. This high voltage is

induced by the system’s resonance composed of the PT equivalent inductance and capacitances C0 and C .

The source voltage U is amplified by the effective quality factor, the induced voltage is divided by C0 and

C into voltage V on PT. The calculated relative voltage V U , related strongly to EMQ, is shown in

Fig 11a for a ring PT. The “capacitive” coefficient in the V U expression (44) has the maximum

approximately equal to 0.25 at A = 1 (or q = 0.5), when ( )20 1C C b k= − , and the curves are

approximately symmetric in respect to the vertical line 0.5 q .

The vibration-level is represented by the effective vibration velocity, whose relative value can be

expressed using the ring redial velocity expression ( ) ( ) 31ˆ2v L i V c d Nω= and its resonance value

( ) ( ) 312 rrv L U c d Qω= , then for 1

s r Qχ −>>

22

( ) ( )(1 ) ( ) 11

1 2sr s r s r

s rr sr

v QV qv U Q N Q

χ χ

χ

+= − ⋅ ⋅

+

%; . (45)

The relative velocity decreases approximately linear with q if the quality factor does not change. For the

typical piezoelectric loss factor values 0.5...1t ≈ the relative reduction in peak values of vibration velocity

is much smaller than the increase in the losses, when A becomes small, the voltage on PT becomes high,

and the resonance frequency of the system moves to the PT antiresonance (Fig. 11b). This means that we

can have the same vibration velocity with lower losses (temperature increase). The input power can be

lowered, as EMQ increases when the capacitance ratio A becomes smaller.

Note that the system of PT with an in-series connected capacitor at its resonance frequency has the only

unconvertible energy as a whole system in respect to the power supply, and the PT convertible energy is

circling between PT and the capacitor. EMQ is a quality factor of the system, but is determined by the PT

parameters only because the capacitor C is a lossless and non-mechanical element (without kinetic energy).

To cover all basic vibration types, calculated voltage gain and relative PT velocity characteristics for rod

and bar PTs in the resonance frequency interval are presented in Fig. 11c,d and Fig. 11e,f , respectively. In

the vicinity of the system’s resonance (resonant voltages on the capacitor C and PT are equal) we have:

( ) [ ]1 1

1 1Im Re1 1 1 1 Im ReoutV Y Y Y

i i C Z C Z i C ZU i C C C

ω ω ωω ω ω

− −− −

= + = + − ≅ + = − +

, (46)

where Re ImY Y i Y= + and Re ImZ Z i Z= + , with the conditions for the system resonance in a

convenient form for the bar and rod PT characteristics estimates, respectively:

1 Im 0 s rY Cω χ+ = → , 1 Im 0 srC Zω χ− = →r

. (47)

Executing similar transformations (Appendix), the bar ( ( 0)r s rQ Q Qχ≡ = =% % ) and rod ( rQ Q≠% ) PTs

voltage gain and velocity ratios are expressed as:

( )

(1 ) 2 ( )1

rs r

s r

Vq q Q

Uδ

χχ

≅ − ⋅ ⋅+

% , (48)

( ) ( )(1 ) ( ) 11

1m sr sr sr

m r s rr sr

v QV qv U Q N Q

χ χ

χ

+= ≅ − ⋅ ⋅

+

%% . (49)

23

24

Fig. 11 a - f. Calculated comparative dependences of the voltage gain and relative PT velocity for a ring

(a, b), rod (c, d) and bar (e, f) PT in the resonance frequency interval for generalized parameters Q = 100,

k = 0.7, δ = 0.01 ( 2 0.5k Qδ = ), 0.02 ( 2 0.98k Qδ = ), 0.04 ( 2 2k Qδ = ), t-parameter varies for δ = 0.02

(step 0.05). Bold dot line “a” corresponds to the case of constant on frequency EMQ Ei jQ Q=% .

According to the presented in Fig. 11 data, all velocity characteristics are approximately similar for the

middle values of t-parameter (t < 0.8), while for 1t → + there is an essential difference between them

depending on the type of vibration – for a rod PT (UM) and ring PT the voltage gain and velocity

characteristics can reach extremely high values. Note only, that similar results can be received for the case

of an in-parallel connected capacitor at constant on frequency current of PT excitation [21].

V. CONCLUSIONS

Local and total energy balance for a vibrating piezoelectric transducer was analyzed. Losses in

piezoceramics are phenomenologically considered to have three coupled mechanisms: dielectric, elastic,

and piezoelectric. Piezoelectric loss, which is a consequence of complex piezoelectric phenomena, is

involved as a high-promising factor of PT efficiency increasing. The efficiency of mechanically free and

electrically excited piezoceramic transducers in a wide frequency range of PT harmonics, especially

between the fundamental resonance and antiresonance frequencies, was investigated, and the effect of

“piezoelectric loss anomaly” with extremely low total losses and highest PT effic iency was predicted.

The extended definition of the electro-mechanical quality factor of a PT with permanent energy

exchange with electrical source of excitation was formulated and analyzed. Expressions for EMQ have

been received on the basis of complex material constants for both stiffened and unstiffened canonical

vibrational modes of commonly seen piezoelectric elements in a wide frequency range. The results are in

good agreement with those found for traditional quality factors at the resonance and antiresonance

frequencies. As was shown, the maximum EMQ increasing in the PT resonance interval corresponds to the

maximal normalized PCM piezoelectric loss factor t → +1 and for vibrational modes with 2 1k Qδ ; . The

EMQ expressions are finally presented in an easy form for practical use.

25

Optimization of the PT excitation with a connected reactive (capacitive) element was conducted to

provide higher PT mechanical characteristics with less total losses and without a large increase of applied

voltage required. Using the same method, it’s possible to calculate the EMQ for PTs with different

boundary conditions under a combined electrical and mechanical excitation.

REFERENCES

[1] B. Jaffe, W.R. Cook, and H. Jaffe, Piezoelectric ceramics. London, Academic Press, 1971.

[2] S. Takahashi, S. Hirose, K. Uchino, and K.Y. Oh, “Electro-Mechanical Characteristics of Lead-

Zirconate-Titanate Ceramics Under Vibration-Level Change,” Proc. IEEE, pp. 377-382, 1995.

[3] Y. Sasaki, S. Takahashi, and S. Hirose, “Relationship between mechanical loss and phases of

physical constants in lead-zirconate-titanate ceramics,” Jpn. J. Appl. Phys., vol. 36, part 1, no. 9B,

pp. 6058-6061, 1997.

[4] A.V. Mezheritsky, “Efficiency of excitation of piezoceramic transducer at antiresonance frequency”,

IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, no. 4, pp. 484-494, 2002.

[5] S. Hirose, M. Aoyagi, Y. Tomikawa, S. Takahashi, and K. Uchino, “High power characteristics

at antiresonance frequency of piezoelectric transducers”, Ultrasonics, vol. 34, pp. 213-217, 1996.

[6] M. Umeda, K. Nakamura, and S. Ueha, “Effects of a series capacitor on the energy consumption in

piezoelectric transducers at high vibration amplitude level,” Jpn. J. Appl. Phys., vol. 38,

pp. 3327-3330, 1999.

[7] R. Holland and E.P. Eer Nisse, Design of resonant piezoelectric devices. Cambridge, M.I.T. Press,

1969.

[8] IEC Standard, Guide to dynamic measurements of piezoelectric ceramics with high electro-

mechanical coupling, Publication 483, 1976.

[9] M. Brissaud, “Characterization of Piezoceramics,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr.,

26

vol. 38, no. 6, pp. 603-617, 1991.

[10] J.G. Smits, Eigenstates of coupling factor and loss factor of piezoelectric ceramics.

Enschede (Netherlands): Technical University of Twente , 1978.

[11] A. Arlt and H. Dederichs, “Complex elastic, dielectric and piezoelectric constants produced by

domain wall damping in ferroelectric ceramics”, Ferroelectrics, vol. 29, pp. 47-50, 1980.

[12] K. Uchino and S. Hirose, “Loss mechanisms in piezoelectrics: how to measure different losses

separately”, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 48, no. 1, pp. 307- 321, 2001.

[13] Xiao-Hong Du, Qing-Ming Wang, and K. Uchino, “Accurate determination of complex materials

coefficients of piezoelectric resonators”, IEEE Trans. Ultrason., Ferroelect., and Freq. Cont.,

vol. 50, no. 3, pp. 312-320, 2003.

[14] H. Wang, Q. Zhang, and L.E. Cross, “A high sensitivity, phase sensitive d33 meter for complex

piezoelectric constant measurement”, Jap. J. Appl. Phys., vol. 32, pp. L1281-L1283, 1993.

[15] A.V. Mezheritsky, “Quality factor of piezoceramics,” Ferroelectrics, vol. 266, pp. 277-304, 2002.

[16] D. Guyomar, N. Aurelle, and L. Eyrand, “Simulation of transducer behavior as a function of the

frequency and the mechanical, dielectric and piezoelectric losses”, Proc. 10 IEEE Symposium on

Applications of Ferroelectrics, pp. 365-372, 1996.

[17] S. Sherrit and B.K. Mukherjee, ”The use of complex material constants to model the dynamical

response of piezoelectric materials”, IEEE Ultrasonics Symposium, pp. 633-640, 1998.

[18] J. Kelly, A. Ballato, and A. Safari. “The effect of a complex piezoelectric coupling coefficient on the

resonance and antiresonance frequencies of piezoelectric ceramics”, Proc. 10 IEEE Symposium on

Applications of Ferroelectrics, pp. 825-828, 1996.

[19] A. Ballato, “Modeling piezoelectric and piezomagnetic devices and structures via equivalent

networks,” IEEE Trans. Ultrason., Ferroelect., and Freq. Cont., vol. 48, no. 5, pp. 1189-1240, 2001.

[20] B.A. Auld, Acoustic fields and waves in solids. Stanford University, vol. 1, 1973.

[21] A.V. Mezheritsky, “A method for manufacturing of a piezoceramic element for frequency

generator”, RU Patent 1823763, 1992.

27

Appendix. Table. Basic bar and rod PT parameters.

Parameter Bar PT (UM) Rod PT (SM)

ix 1x x= 3x x= ,E D

i js 211 11 31

ˆˆ ˆ (1 )D Es s k= − 233 33 33

ˆˆ ˆ (1 )D Es s k= − Q

11EQ Q= , ( )11 11 11ˆ ˆ 1E E E Es s s i Q= ≡ − 33

EQ Q= , ( )33 33 33ˆ ˆ 1E E E Es s s i Q= ≡ −

δ 33δ δ= , ( )33 33 33ˆ ˆ 1T T T iε ε ε δ= ≡ − 33δ δ= , ( )33 33 33ˆ ˆ 1T T T iε ε ε δ= ≡ −

γ 31γ γ= , ( )31 31 31

ˆ ˆ 1d d d iγ= ≡ − 33γ γ= , ( )33 33 33ˆ ˆ 1d d d iγ= ≡ −

2k 231k = 2

31d / T33ε 11ˆEs , 2

31k = 231d / 33

Tε 11Es 2

33k = 233d / T

33ε 33ˆEs , 233k = 2

33d / 33Tε 33

Es y

|γ31 | ≤ y1 ≡ 2

33 31 11Ek Qδ |γ33

| ≤ y3 ≡ 233 33 33

Ek Qδ

t 1 31 1t yγ= 3 33 3t yγ=

Y or Z (Y=1/Z)

( )2 20 31 31

ˆ ˆˆ 1Y i C k k Fω= − + ⋅ 233

20 33

ˆ11ˆ ˆ1

k FZ

i C kω

−= ⋅ −

K 2 211ˆEK sω ρ= 2 2

33ˆEK sω ρ=

,r nf , 112 E

r nf n L sρ= , , 332 Er n nf L sρ= Φ , 2

332 1n n kλ πΦ = −

,a nf , 112 D

a n nf L sρ= Φ , 2312 1n n kλ πΦ = − , 332 D

a nf n L sρ= Freq. eq. 2 2

31 31tan (1 )k kλ λ= − − 233tan kλ λ=

( )T x stress [ ]31

111

ˆ( ) ( ) 1

ˆ E

dVT x N x

c s = ⋅ ⋅ −

[ ]333 2

33 33

ˆ 1( ) ( ) 1 ˆˆ 1E

dVT x N x

c s k F

= ⋅ ⋅ − ⋅ −

( )2

u xL

( )131

sin 2( ) ˆ2

xKLu x Vd

L c N = ⋅ ⋅

( ) 23 33

33 233

ˆsin 2( ) 1ˆˆ2 1

xKLu x kV dL c N k F

− = ⋅ ⋅ ⋅ −

( )E x el. field

3 ( ) ( )E x V c const x= = 233

3 233

ˆ1 ( )( ) ˆ1

k N xVE x

c k F

− = −

, [ ]1

1

Re ( )V E x dx−

= ∫

( )D x el. field ( )

231

3 3311

ˆˆ( ) ( ) 1

ˆT

E

dVD x N x

c sε

= + −

233

3 33 233

ˆ1ˆ( ) ˆ1

T kV VD x

c i Z AAk Fε

ω

− = ⋅ ⋅ = = ⋅− ( )const x=

* tan( /2)

/ 2KL

FKL

≡ , ( )/ 2 cos( /2)N KL KL≡ ⋅ , cos( /2)

( )cos( /2)

xKLN x

KL≡ are “resonant” factors.

Glossary Qm – standardized PCM quality factor

,E Di jQ – PCM quality factors of the complex elastic compliances ,ˆE D

i js (generalized Ei jQ Q≡ )

, ,,r n a nQ Q – resonance and antiresonance PT quality factors of n-harmonic

Q% – electro-mechanical PT quality factor (EMQ)

28

,ˆE Di js ( ,E D

i js ), kld ( kld ), Tmnε ( T

mnε ) – piezomaterial constants (complex and real values)

i jk ( i jk ) – coefficient of electro-mechanical coupling (complex and real values) a, b – coefficients for CEMC determination according to IEC Standard [9] G – CEMC loss factor

nλ , K – root of a frequency equation and complex acoustic wavenumber

, ,( )r n r nf ω , , ,( )a n a nf ω – intrinsic resonance (r) and antiresonance (a) PT frequencies of n-harmonic (real)

rδ – relative PT resonance frequency interval ( rδ = fa / fr – 1 ) [9]

( )χ χ – relative resonance (antiresonance) frequency displacement

,Y Z – PT admittance and impedance

0 0ˆ ( )C C – PT capacitance (complex and real values)

mnδ – dielectric loss factor (generalized δ )

k lγ – piezoelectric loss factor (generalized γ )

1y , 3y – maximal piezoelectric loss factor values related to a rod (ring) and bar PT, respectively

1t , 3t – normalized piezoelectric loss factors (e.g., t1 ≡ γ31 / y1 ) (generalized t ) M , µ – piezoelectric loss influence, and its material parameter L , Ω , elτ – frequency-specifying PT dimension, PT volume and electrode area m, x – PT mass, and relative local coordinate [-1;1] B – “velocity distribution” parameter F, F(x), N, N(x) – “resonant” factors

, ,U V φ – power supply voltage, electric voltage applied directly to PT, and electric field potential ( ),j x I – current density, and total current through PT

(3)D , (3)E , (1,3)T , (1,3)S – electric field induction and strength, mechanical stress and strain

0 0, , ( , , )m mu u u v v v – local , PT top and maximal displacements (velocities) TEϕ – phase angle between electric field and mechanical strength vibrational characteristics

w(x), W – local and total energy stored in PT h(x), H – local and total PT thermal losses p(x), P – local and total losses of electric field in PT

,kin potW W – kinetic and potential PT mechanical energies

, ,d e pH H H – dielectric, elastic and piezoelectric thermal loss components

Re ImP P i P∑ ≡ − – total power of a generator

Ef – frequency of a zero piezoelectric loss factor influence

( ), kχ χ± ± – frequency displacements of maximal influence of piezoelectric loss factor (mutual

energy) in total losses (stored energy) ( )s r s rf χ – system “series capacitor – PT” resonance frequency (frequency displacement)

A , q – capacitance ratio and parameter of a relative location of the “C – PT” resonance T∆ % – temperature increase

" "≅ , " "; – approximations on relatively low losses as 21 0.5 1 , , 1Q δ γ+ ≅ and

frequency displacement as 31 1χ+ ; , respectively

29

ESTIMATION OF THE PIEZOELECTRIC LOSS FACTOR.

A. Domain Mechanism of Dissipation.

According to the model [14], involving the domain mechanism of energy losses with damped (time delay)

movement of walls of 90-degree domains, the uniform physical nature of coupled elastic, dielectric, and

piezoelectric energy losses was shown. In a slightly modified model a simple polarization procedure is

assumed – a single domain with spontaneous polarization 0Pr

is reoriented when the component of the

applied electric field Er

along 0Pr

is greater the coercive field strength cE . The domains angle distribution

for the model is presented in Fig 12 with a characteristic angle 0 00 [90 ...180 ]α ∈ of reoriented domains,

so that the total polarization of a sample is expressed as 0 02 sin( )P N P ϑ= . Being dynamically excited by

electrical and mechanical forces, the ferroelectric body has the following dissipative elastic, dielectric and

piezoelectric components of the material constants:

211 00.5 ( )s S Nϑ ω′′∆ = , 2

33 0 ( )P Nε ϑ ω′′∆ = , ( ) 331 0 0 04 2 3 ( ) sin ( )d S P Nπ ϑ ω α′′∆ = − , (52)

where ( ) 2 2( ) 4 1A cϑ ω ωτ ω τ= + , c andτ are the domain elastic and phase-delay (dissipative)

constants, A is the area of a domain, N is the total domain number per volume, S0 is the domain

spontaneous deformation.

Fig. 12. Angle distribution for the domain model.

Then according to the model, we can estimate the relative piezoelectric loss factor value:

( )2

312 6 61 0 02

11 33

64sin ( ) 0.72 sin ( ) 0.72

9d

tS

α αε π

′′∆≅ = <

′′ ′′∆ ∆; , or 1 0.85t < . (53)

30

Then 30sin ( )t α∼ , 0sin( )d k α′ ∼ ∼ , 2

0sin ( )γ α∼ , 10sin ( )y α−∼ ,

so that the piezoelectric loss factors are proportional to the degree of polarization as

2rkγ δ∼ ∼ and 3 1.5

rt k δ∼ ∼ , (54)

where rδ is the relative resonance frequency interval for a given mode of vibration. Moreover, it follows

from the one-dimensional model /14/ that 33 31d d′′ ′′∆ =−∆ and 33 11s s′′ ′′∆ = ∆ , then 3 1t t= .

According to /6/ there are common restrictions on the piezoelectric loss factor values:

231 1 33 31 11

Ey k Qγ δ≤ ≡ (corresponds to ring and bar PT characterization),

231 33 11

Ep py k Qγ δ≤ ≡ (disk PT), assuming 11 12

E EQ Q= [25],

and 233 3 33 33 33

Ey k Qγ δ≤ ≡ (rod PT).

The first two are related as follows 1 1(1 ) 2 (0.55...0.63)py y yσ= − ⋅; at typical for PZT PCM planar

Poisson coefficient σ = 0.2…0.4. Then 31 1 2 (1 )p pt y tγ σ≡ = − ≅ 1(1.6...1.8) [ 1, 1]t⋅ ∈ − + . So,

maximal va lue can not exceed 1 0.6t < when 1pt → . The dissipative characteristics of a disk PT with the

relative piezoelectric loss factor pt are described similar to (34, 43) [24,25] for ring and bar PTs, however

the relative effect of piezoelectric losses is sufficiently stronger, that makes a disk PT more desirable for

experiments [6,7]. It can be seen from two points of view: according to (34), a disk PT has the same

dissipative parameters 11EQ , 33δ , and 31γ with much higher CEMC 2 2

312 (1 )pk k σ= − ≅ 231(2.6...3.2) k ,

or in the expression (6) with generalized parameters pt is greater than 1t , and what is more important pt can

reach 1. The influence of free-charges conductivity is considered in details in [5,25], its effect on PR

intrinsic dissipative parameters can be essential.

31

B. Experimental data.

As the effect of piezoelectric losses depends on 2k Q δ factor, the basic standardized [9,10] dissipative

parameters of sufficiently different (soft, hard, etc.) PZT PCMs were analyzed and are presented in Table .

If the resonance quality factor ( mQ ) is determined at the resonance frequency, the dielectric loss factor

should be measured at the frequency close to the resonance as well. Presented in Table the dielectric loss

factor values, measured at 1 MHz on unpolarized samples, can sufficiently differ up to 2-3 times from

traditionally measured values at low frequency (1 kHz) because of different mechanisms of low-frequency

dielectric relaxation. Taking into account the corrected values, the parameter with maximal CEMC

233 33mk Q δ (as well as 2

33p mk Q δ ) for the longitudinal mode typically is near 1 on magnitude, while for the

transverse mode 231 33 1mk Q δ << .

Table. Dissipative and piezoelectric parameters of commercial PCMs (Russia) [10].

Parameter PCM

Qm tanδ (1 kHz)

tan (1 )tan (1 )

MHzkHz

δδ

Q δ⋅

1Q δ⋅

33k

pk

31k

PZT-1 42 0.025 2.1 2.2 0.67 0.68 0.52 0.29 PZT-19 85 0.015 1.7 2.2 0.68 0.79 0.67 0.40 PZT-36 90 0.020 1.0 1.8 0.75 0.74 0.57 0.34 PZT-24 280 0.004 1.5 1.7 0.77 0.67 0.51 0.30

PZT-35Y 700 0.014 0.3 2.9 0.58 0.66 0.50 0.29 PZT-22 650 0.015 0.25 2.4 0.64 0.4 0.22 0.13

PZT-3 780 0.006 0.8 3.7 0.52 0.70 0.54 0.32

*Average: 2.4 0.7Qδ = ± , 1 0.65 0.08Qδ = ±

The experimental dependences of the resonance rQ , antiresonance aQ quality factors [27], and their ratio

a rQ Q , on the degree of polarization of the disk PT ∅10x0.5 mm made from PZT-19 are presented in

Fig. 13. According to the results of theoretical description, taking into account the influence of free-charges

conductivity which decreases the effect of piezoelectric losses [5, 25], the ratio 2.3a rQ Q ; at maximal

degree of polarization corresponds to 0.9 0.05pt ≅ ± , the most close to 1 .

32

Fig. 13. Dependences of the resonance rQ , antiresonance aQ quality factors, and their ratio a rQ Q ,

on the degree of polarization (planar resonant interval rδ ) for the disk PT ∅10x0.5 mm with planar UM of

vibration: line 1 - interpolation of experimental rQ values; lines 2, 3 – calculated dependences of

( )a r rQ Q δ according to (34,54) at 0Q = 100, 33δ = 0.02 and accordingly: 2 – max pt = 1

(at pk = 0.60); 3 - pt = 0 ( 31γ = 0). PZT-19.

However, as a disk PT refers to UM, there is no extreme “zero-loss” effect. Meantime,

the maximal voltage gain V U on the disk PT was about 23 with in series connected capacitor 0C C; ,

that is near twice higher than expected at constant quality factor (under the condition rQ Q=% ) , and 3 times

higher than the calculated value without taking into account the piezoelectric loss 31 0γ = (Fig. 11e).

So, for planar UM vibrations the maximum allowed (accessible) for ??? values 1t < 0.6 and pt ≈ 1

define the features of quality factors relationship at the fundamental harmonic: for a bar PT aQ does not

exceed rQ more than 20 % , for a disk PT ( pt →1) a rQ Q can reach more then 2 time increasing.

33

The basic expressions for energy components:

( ) ( ) ( )2Re ( ) ( ) ( ) cos ( ) ( ) sinET ETE x d E x T x d E x T xε ϕ ϕ′′⋅ = + ⋅ ⋅ + ⋅ ⋅*E D

( ) ( ) ( )2Im ( ) ( ) ( ) cos ( ) ( ) sinET ETE x d E x T x d E x T xε ϕ ϕ′′ ′′⋅ = + ⋅ ⋅ − ⋅ ⋅*E D

( ) ( ) ( )2Re ( ) ( ) ( ) cos ( ) ( ) sinET ETs T x d E x T x d E x T xϕ ϕ′′= + ⋅ ⋅ − ⋅ ⋅*T : S

( ) ( ) ( )2Im ( ) ( ) ( ) cos ( ) ( ) sinET ETs T x d E x T x d E x T xϕ ϕ′′ ′′= + ⋅ ⋅ + ⋅ ⋅*T : S

( ) ( ) ( ) 22 2 20.5 ( ) ( ) ( ) cos sinET ETkin x x

W v d V c k T x dx E x T x dxρ ε ϕ γ ϕΩ

≡ ⋅ Ω ≅ + ⋅ − ⋅ ∫ ∫ ∫r % % % .

Appendix B. B1. EMQ decomposition procedure for a bar PR (UM) with the initial expressions

(indexes k31, 11EQ , δ33, γ31, y1 ) :

2 22 2

0 31 31

(1 ) (1 ) tan( /2)1

(1 ) (1 ) / 2i i KL

Y i C i k ki Q i Q KLγ γ

ω δ − −

= − − + − − ,

0C ( 33Tε ) is the PR capacitance (real), 2

31k = 231d / 33

Tε 11ES , ( )2 0.5 (1 ) 1 2KL i Qπ χ≅ + − ,

1rf fχ ≡ − is the resonance frequency displacement. Then

( )2 20 31 31

tan 0.5 (1 ) 1 221 (1 2 ) (1 2 )

(1 2 ) 1

i QY i C i k i i Q k i i Q

i Q

π χω δ γ γ

π χ

+ − ≅ − − − + + − + ⋅ − + ,

where 2 2

22

(1 ) (1 )tan (1 ) (1 ) 1 ...

2 2 2 2 12 4i i

iQ Q Q Q

π π π χ χ χχ χ χ χ χ

+ + − + = − + + − − +

.

With a precision less than 2 2, Qχ − (no more 10 % ), we have the expressions:

[ ]2 2 2

0 31 31 31 2

4 1(1 ) ( 2 1 ) (1 2 ) ,

(1 2 ) (1 ) (1 ) 2i

Y C i k k Q k i i Qi Q i Q

ω δ γ γπ χ χ χ

≅ − + − − − − + ⋅ − + − +

( )( )

22 31

0 31 2 22 2

1 2 2 3 2 (1 )2Re( ) ( 2 1 )

1 4

Q QkY C k Q

Q Q

γ χ χω δ γ

π χ χ

− − + ≅ − − + + +

with ( )( )

2231

231

411r r

kk

π

δ δ−

=+

,

then ( )2

2 310 31 2 2

1 (1 ) 2 2 1 (1 )2Re( ) (2 1 )

Q QkY C k Q

Q

χ χ γ χ χω δ γ

π χ

+ + − − + ≅ − − + =

( )( )22 2

310 2 2 2 2

31

1 (1 ) 4 12 1 (1 ) 11 2 2

1 2 4 1 (1 ) 4k

C QQ Q k

χ π χ χ χ χχ χ πω γ δ

π χ χ π χ χ

+ + ++ + = − − − + + +

.

34

According to the EMQ definition (38) ( ) ( )231

0 2 2

1 121Re

kY C

Q

χ χω

π χ

+ +≅ ⋅ ⋅% and 2

0 312

8r rR C k Qω

π≅ .

Then we have the approximation (43) for EMQ parameter of a bar PT as follows

( )

( )( )

( )( )

231

2 231

1 1.5 1 0.51 1 8 12

1 1 1.5 1r

r r r

kQ k QQ

δ χχ χγ δ

π δ χ δ δχ

+ + ≅ − − − ⋅ ⋅ − + + %

B2. EMQ decomposition procedure for a rod PR (SM) with initial expressions (indexes k33, 33EQ , δ33, γ33, y3 ):

2332 2 2

0 33 33 33

1 tan( /2)1 (1 )

/ 2(1 )(1 ) 1 (1 )KL

Z k iGKLi C k i iGk kω δ

≅ − − − − + − ,

where 0C ( 33Tε ) is the PR capacitance (real), 12G Qγ δ−≡ − − is the CEMC loss factor, 2

33k = 233d / 33

Tε 33ES

is the CEMC, ( )2 0.5 (1 ) 1 2 DKL i Qπ χ≅ + − , 1af fχ = − is the antiresonance frequency

displacement. As in the case of B1, with a precision less than 2 2, Qχ − (no more 10 % ), we have:

233 22 2 2

0 33 33 33

1 4 1(1 )(1 2 ) (1 )(1 )(1 ) 1 (1 ) (1 ) 2D D

iZ i k iGi QC k i iGk k i Qπ χω δ χ χ

− = + − ⋅ − + − − + − − +

,

233

2 20 33

41 1Im( ) 1 0

(1 ) (1 )k

ZC kω π χ χ

−≅ + = − +

with 2332

4 11 0

(1 )r r

kπ δ δ

+ ≅+

.

Since ( )2 2

2 23333 332 2

33 33

1 1 11 1 2(1 )1 (1 )D

kQG k Q t k Q

Q Q k Q kδ δ

= − = − + − − − , then

2 2 22 33 33 33

0 33 2 2 233 33

1 1Re( ) (1 )

1 2 (1 ) 2 1D D

k a k kZ C k G G G

k Q Q kχ

ω δ δχ χ

⋅ − ≅ − − + − − + − − + −

According to the EMQ definition (40) ( )( )

( )2332 22

0 33

1 121

Re 1

kQ

Z C k

χ χ

π χω

+ +≅ ⋅ ⋅

⋅ −% ,

2 2233 33

2 2 233 33 33

1 1 2 2(1 ) (1 ) 1 1D

k kG G G

Q k a k kQχ χ χδ δ

χ χ

≅ + − + − − − + + − − % , then

233

2 2 233 33 33

1 1 2 12

1 (1 ) (1 ) (1 )k G

GQ k k a kQ

χ χχ δ

χ χ

≅ − − + + − − + + % .

Going to the resonance frequency displacement through 1

r

r

χ δχ

δ−

=+

r,

1r

rr

δδ

δ= −

+

r,

233

2331 2r

a kak

δ =−

,

we have the approximation (44) for EMQ parameter of a rod PT as follows

35

( ) ( ) ( ) ( )( )

2 1 1.21 1 12 1 1 1 2 1 2

8 1 1.2r

r r rr rQ QQ

δπ χ χδ δ δ γ δ χ δ

δ δ χχ

+ − ⋅ − ⋅ − + + ⋅ − − − − ⋅ ⋅ +

;% .

Appendix C. The basic expressions for PT EMQ and relative velocity parameters. Parameter Bar PR (UM) Rod PR (SM)

( )χ χ 1 1rf fχ = − 1 1af fχ = −

( )2mu

L 31

ˆ

2mu dV

L c N = ⋅

2

33 33233

ˆ ˆ1ˆ2 1

mu d kVL c N k F

− = ⋅ ⋅ −

B ( )0.5 1 sinB KL KL= −

N ( ) ( )2 cos 2N KL KL= ⋅

kinW 2 20.5kin mW m u Bω= ⋅

Q%

( ) ( )231 2

0

1cos 2 Re

Bk

KL Y Cω⋅ ⋅

( ) ( )( )233 2 2

0 33

1cos 2 Re 1

Bk

KL Z C kω⋅ ⋅

⋅ −

2F For 1Qχ −>> and 2 1χ <<

( )

22

2

4 11

Fπ χ χ

⋅ +

; , 2

8r

QF

π=

For 1Qχ −>>r

and 2 1χ <<r

( )

22

2

4 11

Fπ χ χ

⋅ +

; r r , 2

8 D

a

QF

π=

rδ 21

211r

a kb k

δ =−

, 1 2

4a

π= , 1 11b a= −

21

211r

a kb k

δ =−

, 1 2

4a

π= , 1 12b a=

Resonance (f0) 01 Im( ) 0Y Cω χ+ = → 01 Im( ) 0C Zω χ− = →

0

out

in

UU

( )0 0 0

1Re( )

CC Y Cω

≅

( ) ( ) ( )0 01 2 1rp p Q χ δ χ≅ − ⋅ ⋅ +%

( )( )( )

20 33

20 33 0

1 1

Re( ) 1

C k

C Z C kω

−≅

−

( ) ( ) ( )0 01 2 1rp p Q χ δ χ≅ − ⋅ ⋅ +% p 0 1

1r

pA

χδ

= =+

, ( )2

0 331C

AC b k−

; , 1b a= − , 0 0 1rf fχ ≡ −

m rv

31 2

82

rr

QU Ld

cω

π≅

%, rQ Q=% 31 2

82

rr

QU Ld

cω

π≅

%

0m

m r

v

v ( ) ( )

2

00

1(1 ) 1

8 1FV Q

pU Q Q

πχ

χ+ ≅ − ⋅ ⋅

+

% ( ) ( )

( )0

0

11

1r

Qp

Qχ

χ≅ − ⋅ ⋅

+

%%

36

Abbreviations

PCM, PT – piezoceramic material and piezoceramic transducer

SM, UM – stiffened and unstiffened mode of vibration

EMQ – electro-mechanical quality factor

CEMC – coefficient of electro-mechanical coupling

s.c., o.s. – sort-circuit and open-circuit regime of PR excitation

a – complex material parameters

Sings ≅ , ; – for the approximations on relatively low losses as 2 2 21 0.5 1 , , 1Q δ γ+ ≅ and

frequency displacement 31 1χ+ ; , respectively

, , :∇ ⋅ – vector and tensor operators

The author A.V. Mezheritsky : 525 Ocean Pkwy, # 3 J, Brooklyn, NY 11218 mav5@optonline.net Ph.D. in Physics (1985, MIPT, Moscow, Russia), IEEE Member