1

A METHOD of “WEAK RESONANCE”

for QUALITY FACTOR and COUPLING COEFFICIENT

MEASUREMENT in PIEZOELECTRICS

Alex V. Mezheritsky, “A METHOD of “WEAK RESONANCE” for QUALITY FACTOR and

COUPLING COEFFICIENT MEASUREMENT in PIEZOELECTRICS”,

adapted from the manuscript submitted to the journal IEEE UFFC, Jan. 2005.

Abstract - In the absolute-immittance spectrum of a piezoelectric resonator (PR), if the relative resonance-

antiresonance frequency interval of a high intensity resonance is basically determined by the coefficient of

electromechanical coupling (CEMC), the relative resonance - antiresonance frequency interval of a low

intensity resonance with the resonance-antiresonance attenuation less than 15 dB, regardless of the reason,

is determined by the quality factor Q of the resonance, and its intensity is proportional to CEMC.

The technique for the quality factor and CEMC determination based on the “weak resonance” (WR) concept

has been formulated and then applied to low-Q and/or low-CEMC piezoelectrics, including the initial stage

of piezoceramics polarization, to piezotransducers under electrical or mechanical loading with maximum

efficiency. The WR - method allows to determine the quality factor on PRs under specific conditions, such as:

arbitrary PR shape resulting to a broken frequency spectrum; PRs with an extremely large or extremely low

electrical capacitance; at high-order PR harmonics; electrodeless piezoelements under mechanically

contactless excitation; determining the local thickness - mode material quality factor value and its distribution

along the surface of a thin electrodeless piezoplate – all where the traditional methods appear a poor

performance or do not work at all.

Keywords: piezoelectrics, piezoceramics, coefficient of electromechanical coupling, quality factor.

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

2 I. INTRODUCTION

The CEMC and quality factor both characterize PR electro-mechanical efficiency, depending on the mode

of vibration and regime of mechanical and/or electrical loading. The difference between the PR frequencies

at resonance ( sf ) and antiresonance ( pf ) [1-3] generally is determined by the CEMC value i jk as follows

(lossless case) [3]:

21 i j rk a bν νδ≅ + , (1)

where 1r p sf fδ = − is the “piezoelectric” relative resonance-atiresonance frequency interval, aν and bν

are the coefficients [3] depending on harmonic number ν of a used type (mode) of vibration. Particularly for

a linear PR vibration 2 2 2( ) 4r i jkνδ π ν; , where ν = 1, 3, 5… is the harmonic number. The quality factor Q

mostly characterizes the resonance PR behavior. Piezoelectric materials (PM) provide strong coupling of

mechanical and electrical fields (CEMC value can reach 0.9, close to theoretical limit), the quality factor

values vary from several units to thousands. There is a number of CEMCs and quality factors, corresponding

to different types of possible vibrations [4,5]. Traditionally, the resonance quality factor and planar CEMC

pk at the fundamental (lowest) harmonic of a PR planar mode (disk) [1] were chosen for characterizing

piezoceramics quality. The standardized method for the PM “mechanical quality factor” determination [1,2]

is based on the expression:

201m r p rQ C k Rω≅ , (2)

where rR is the resonance PR resistance, 0C is the static PR capacitance, 2r rfω π= is the PR resonance

frequency. The standardized accuracy in mQ and CEMC determination is less of the order 15% and 4%,

respectively. Connected to PR reactive elements, such as a series or parallel capacitor, change effective Q

and CEMC parameters, but their influence is supposed not to be worsening the total resonance intensity.

There are six characteristic frequencies of interest [2,6] ( mf , sf , rf ) near resonance and ( nf , pf , af )

near antiresonance for some isolated resonant peak of a lossy PR corresponding to maximum absolute

admittance (impedance), maximum conductance (resistance), and zero susceptance (reactance), respectively

3

(Fig. 1a). The frequencies sf and pf are supposed to be those hypothetical frequencies that would be

measured if there were no losses and the material parameters were otherwise unaltered. The losses at

resonance are directly characterized by the frequencies 1 2sf ± of the PR susceptance extremes.

The methods of PR electro-mechanical parameters measurements are divided into two major groups:

the first one is a lossless case or when energy losses in a PR are too low [1-3,6], and the second one is a lossy

case when energy losses are relatively high or with an arbitrary value of the Q - parameter [6-14].

The specific equipment, measuring and calculating procedures, including phase measurements, are required

for most methods.

Fig. 1 a,b. Schematic diagram (a) showing the definition and relative location of the critical frequencies on

the PR normalized admittance loop (circle) for various degrees of resonance “piezoactivity” : B M>> (loop

1) and B M<< (loop 2), where B and M are the PR resonance and capacitive factors (4), and its AFCh (b).

A comparative analysis of different methods was made in [6] for the specific case of low-Q PM, such as

PVDF, PZT piezocomposite, lead metaniobate. The shift of critical frequencies in respect to the ideal

4 (lossless) characteristic frequencies, caused by k2 Q factor, was calculated and limitations on the standard

methods were established. The original iterative method [8,9] provides high accuracy determination of the

real and imaginary parts of the material constants describing the given (pre-known) type of PR vibration.

In the literature the PR quality factors ( )rQ at resonance and ( )aQ at antiresonance are considered separately

[5,16], their difference typically does not exceed 2 times. The quality factor of a PR under electrical or

mechanical load [15] is used in piezotransducer performance characterization, and its deeply decreased values

reach up to several units under high efficiency regime of loading.

A number of frequency methods is known for the resonant system quality factor determination which

are based on the measurement of the frequency difference (resonance bandwidth F∆ ) corresponding to:

the level 1 2 0.7; (3 dB) of the resonance (antiresonance) peak in the PR admittance (impedance),

or to the level 0.5 (6 dB) of the PR conductance (resistance); the frequency extremes of the PR susceptance

(reactance); the frequencies with the phase shifts 045± in the PR admittance (impedance) – all according to

the expression:

( , ) ( , )r a r aQ F F= ∆ , (3)

where, for example, r sF f= and 1 2 1 2s sF f f+ −∆ = − in the resonance quality factor ( )rQ determination.

A modified method with an arbitrary attenuation level (usually less than 3 dB), known as a perturbation

method, is proposed in [16].

However, all these methods are applicable only to a PR vibration with relatively large value of the

resonance "piezoactivity" defined as a product of the relative resonance-antiresonance frequency interval by

the quality factor r mQδ >> 4, that provides a methodical error less than 10% in the classical “3 dB” method.

This condition corresponds to the intensity of the PR amplitude-frequency characteristics (AFCh), expressed

as a resonance-antiresonance attenuation difference, more than 25 dB – otherwise the error steeply increases

up to invalid values [10], and even the ,r af frequencies do not exist (no zero phase shift) (Fig. 1, curve 2).

The Standard methods do not work at all for the resonance intensities less than ~ 10 dB.

5 II. ELECTRICAL LOSSY PR IMMITTANCE.

A. Common Description of the “Weak Resonance” Method

In a generalized consideration, the admittance of the traditional equivalent electrical circuit (EC) of a PR

(Fig. 2) is described by the following expression:

( )

10 01

01 1

1 12 1

1C

Y i C C i M B Ci y CR i L i C

ω ω ωω ω

−

= + = + = − ⋅ Ψ ++ +

, then

1

21

i pi y

Ψ = ++

, and ( )2

2

2

1 2

1

p y

y

+ −Ψ =

+ (4)

where Ψ is the generalized PR admittance, 0 1C C C= + is the static PR capacitance,

21 0 ( )0.5 rB Q C C a k Q Qν νδ= ⋅ ≅ ≅ and 2

1 01 1M C C b kν= − ≅ − are the parameters of the PR resonance

piezoactivity and permittivity clamping, p B M= is the resonance to capacitive factors ratio , 11Q R Cω=

is the generalized quality factor, 2 2y x Q Qχ= ≅ is the generalized frequency displacement,

( ) ( )2 20.5 1 1 0.5sx ω ω χ χ= − = + , 1sχ ω ω= − is the resonance frequency displacement,

1 11s L Cω = is the EC resonance frequency, 1r p sδ ω ω= − is the relative resonance-antiresonance

frequency interval for the case of lossless PM, depending only on respective CEMC.

Fig. 2. Traditional PR equivalent circuit.

For relatively small frequency displacements 1Qχ −∼ of further interest, where the PR quality factor Q

is considered constant, the extremes of the normalized complex PR admittance (4) 0Y Cω correspond to

2, 1n my p p= ± + . The value of the apparent (actual) relative resonance-antiresonance frequency interval

0( )n mG f f f≡ − , where ( )0 2m nf f f= + , is determined then by

6

2

0

12

n m n mf f y yQ G Q p

f− −

⋅ = = = + and 2 1

r pG p

δ=

+ , (5)

so, the following fundamental relationship takes place:

2 2 2rG Q δ−= + . (6)

Substituting ,n my into (4) for the normalized admittance, we have

( )2 20 2 2 22

0

/ 11 ( )

/ 1m n

rmn

Y C yp p Q G

Y C y

ωδ

ω+

= = + + = ++

, (7)

and the intensity (dynamic range) of the resonance, defined as the attenuation difference between

the maximum (resonance) and minimum (antiresonance) PR normalized absolute-admittances (Fig. 1b),

is expressed as

( ) ( )0 2

0

/20lg 40 lg 1

/m

n

Y CA dB p p

Y C

ω

ω≡ = + + . (8)

As a result, the parameter of the resonance-antiresonance interval QK Q G≡ ⋅ and resonance intensity

( )A dB are related through the parameter of PR piezoactivity rp Q δ= ⋅ .

Fig. 3. Calculated dependences of the

QK , p and GK ( GK ) parameters on the

intensity A (dB) of a resonant peak:

( )r a r rf f fδ = − , 0( )n mG f f f= − ,

( )0 2m nf f f= +

7

Among the “piezoelectric” rδ and actual G parameters, the last one is measured experimentally by the

amplitude method. Functional dependences of the parameters (5,8) are shown in Fig. 3 for low and middle

values of the p - parameter. For a large degree of “piezoactivity” (A >> 25 dB, or 2p >> ), the relative ratio

1G rK Gδ≡ → , that traditionally is used for calculating PM parameters on the basis of expressions (1,2).

However, in the case of low “piezoactivity”, the traditional standard methods [1-3] are not applicable because

of an extremely large methodical error [6,10]. This is the case when a negative feature of the traditional

methods becomes a positive basis for a new approach. Note that 1r Qδ −= at p = 1 with the resonance

intensity A = 15 dB which can be considered as a boundary value for the WR - method.

As shown in Fig. 3, for small values of the p – parameter ( p << 1 , or B << M) (Fig. 1a, loop 2) the actual

relative resonance-antiresonance frequency interval becomes equal to 1G Q−→ , and we have the following

linear approximations [17]:

( ) ( )2 2( ) 40 lg 1 17.4 1 0.5 ... 17.4A dB p p p p p= ⋅ + + = − + ≅ ,

2 21 1 0.5 ... 1Q G p p⋅ = + = + + ≅ ,

2 21 (1 0.5 ...)r G p p p p pδ = + = − + ≅ , (9)

where 40 ln10 17.4≅ . If the resonance intensity A is less than 10 dB, we have

1

QG

≅ and ( )17.4r

A dBGδ ≅ ⋅ (10)

with the methodical error no more than 15% for the Q and no more than 8% for the rδ determination (for

CEMC no more than 4%). For higher precision, or for a wider range of the ( )A dB values, the following

correction coefficients presented in Fig. 3 can be used:

QK

QG

= and ( )

17.4r G

A dBK Gδ = ⋅ ⋅ , (11)

where QK and ( )17.4G GK K A dB= ⋅ with G rK Gδ= (Fig. 3) are determined from (5,8) as

8

( )2( ) 40 lg 1Q QA dB K K= ⋅ + − , 1

( ) 20 lg1

G

G

KA dB

K +

= ⋅ − (12)

Such a property can be easily understandable from the graphical admittance representation (Fig. 1a),

where 1 2n sf f +→ and 1 2n sf f −→ under the condition B M<< for a circle admittance loop.

Even using formulas (11) with the correction factors QK and GK for the Q and rδ determination,

the WR - method is not recommended for a resonant peak with the intensity ( )A dB larger than ~25 dB

because of appreciable difference between ( )aQ and ( )rQ values under this condition [5].

The present analysis was made for the normalized PR admittance 0Y Cω instead of typical Y .

In the case of relatively high “piezoactivity” ( 2 2 4k Q >> ) there is no difference in the extremes frequencies

in these two cases. However, when the peak amplitude of the PR admittance Y is too small, the measured

relative resonance-antiresonance frequency interval additionally decreases that is caused by a linear frequency

dependence of the capacitive PR admittance component 0Cω (Fig. 1b). Moreover, the difference between

the levels of the “vibrational” resonance and antiresonance admittances (4) equal to (1 2 )rQδ+∼ can be

fully compensated by the difference in the capacitive admittances of the order ~ (1+ G) ≅ ( 1+ 1Q− ) at these

two frequencies. Thus, under the condition 2 1 2rQ δ ≤ , no resonance peaks can be practically found on

the Y characteristic . As it was received by simulations, the error of no more than 10% in determining

the actual relative resonance-antiresonance frequency interval (G) corresponds to the condition 2 4rQ δ ≥ ,

when both the analyzed expressions for the PR admittances Y and 0Y Cω can be successfully used.

In practical terms, a signal proportional to the normalized admittance 0Y Cω can be experimentally

obtained when a capacitive load is used in the circuitry for PR excitation [13] (instead of a traditional active

load [3] ). For Q ≥ 15 and A ≥ 4 dB there is no essential difference in the frequencies determination for either

type of active or capacitive PR loads. Usually the usage of a capacitive load in the measurement circuit is

preferred at very weak resonance intensity less than 1 dB .

9 The influence of the dielectric loss tanδ , represented particularly as a parallel resistive element to the EC

(Fig. 2) , is described by the parameter tanp p δ→ + in (4-8). A numerical estimation of the influence is

of the order 21 (tan )δ+∼ , and hence is not essential for both Q and CEMC determination for at least

tan 0.1δ < much higher typical PM values. Note, that using the practical expressions 11 sQ R Cω= instead

of (4) and ( , )( )n m n mG f f f≡ − instead of (5) provides the difference not more than 1 2Q under the WR -

method conditions, appropriate for practical purposes.

The main idea of the WR - method can be shortly formulated as follows. Traditionally, the CEMC is

determined by the resonance-antiresonance frequency interval rδ (1), and the quality factor is determined

either by the resonance PR resistance (2) or the frequency bandwidth of the resonance peak (3). Under the

WR - conditions, the quality factor is determined by the resonance-antiresonance frequency interval G (10),

while the CEMC is determined by the resonance intensity (10) – it is a kind of exchange between the CEMC

and quality factor as to corresponding parameters required. In practical plane, the WR method is supposed to

provide not rather higher accuracy in Q and CEMC determination, but widen the field where other known

methods do not work or require too complicated measuring and calculating procedures.

Generally, as it follows from the above consideration, the WR - condition is primarily determined by the

relative relationship M B> between the capacitive parameter M and PR resonance term B.

For a PR resonance with strong piezoactivity, on the one hand, it can be provided by sufficiently extreme

reactive load (a respective capacitor connected in parallel or in series to PR) lowering the effective CEMC

of the resonance. On the other hand, an active electrical or mechanical load under maximum efficiency

condition drastically decreases the effective PR quality factor.

III. PIEZOELECTRICS WITH RELATIVELY HIGH 2k Q .

A. A Capacitor Connected to PR in Series, or in Parallel, with Relatively Extreme Capacitance

For the case of “strong resonance” with a relatively high value of k2 Q >> 1, the requirements for the WR

condition can be provided by connecting a capacitor with sufficiently high capacitance in parallel to PR for

10

determining the resonance quality factor ( )rQ , or with sufficiently low capacitance connected in series to PR

for determining the antiresonance quality factor ( )aQ (Fig. 4).

In the first case with in parallel connected capacitor pC (Fig. 4b), there is an additional capacitive term in

the normalized PR admittance (4) with replacement the parameter M → M + 0pC C . Then, as known,

for relatively small and middle 0pC C values the relative resonance-antiresonance frequency interval,

as a measure of the effective CEMC, decreases according to the expression ( ) ( )p r pG C Cδ≅ =

= 2 20(1 )pa k b k C Cν ν− + due to the antiresonance frequency predominant decreasing [2,3].

For a relatively large pC value ( 0 1pC C << ) [18] in accordance with (9) :

( ) 22( ) ( ) 01 0.5 ...r r pQ G a k Q C Cν⋅ = + + , (13)

and the methodical error as low as 8 % for the resonance quality factor determination with 1( )rQ G −=

is provided by

2

( )0 2

rp

k QC C

ν≥ , (14)

where 2 2( )2 rk νν δ≅ is the “piezoelectric” relative frequency interval (lossless) of the ν-harmonic.

The relevant PR characteristic frequencies are shown in Fig. 5, when the capacitance of a capacitor connected

in parallel increases.

Fig. 4 (a-d). Basic elements of the WR - method

for measuring the quality factor of a PR resonance

with “high piezoactivity” :

a – AFCh and its resonance intensity;

b – connected in parallel capacitor pC for the

resonance quality factor ( )rQ measurement;

c – connected in series capacitor sC for the

antiresonance quality factor ( )aQ measurement;

d – air gap “electrode - piezoplate” as a natural

capacitor connected to PR in series.

11 A similar consideration can be provided for a connected in series capacitor Cs (Fig. 4c) – the antiresonance

PR quality factor is determined by the formula 1( )aQ G −= under the condition of a relatively small value of

the capacitance (Fig. 5):

2 2 2

0 2( )

(1 )s

a

b kC C

k Qνν −

≤ . (15)

The estimation of the sC and pC values (14,15) can be made on the basis of corresponding PM material

parameters mQ and i jk for a given type of PR vibration.

Fig. 5. Dependence of the resonance

and antiresonance PR frequencies

under a capacitive load connected in

series ( sC ) or in parallel ( pC ) to PR.

B. Electrodeless Piezoelements with Mechanically Non-Contact Excitation

The described WR approach can be applied to determining the quality factor of electrodeless

piezoelements even without a mechanical contact to PR (Fig. 4d). When a metal electrode plate is placed near

the surface of a piezoelectric element with some air-gap h, this is equivalent to connecting to the PR-element

in series some natural capacitor, whose relative capacitance can be sufficiently small because of a great

difference between dielectric permittivities of the air a i r oε ε ≈ 1 and piezoelectric, especially such as PZT

ceramics with i j oε ε ≈ 200…4000, where 0ε is the vacuum dielectric permittivity. Comparing the

capacitance of the air gap “metal plate – piezoelement” with the capacitance of an actively excited

12 piezoelement volume (under a patch electrode), the following requirement to the air-gap in respect to

the piezoplate thickness H is stated to realize the WR condition (15):

2

( )2 2 2

0( )(1 )a

i j

k Qh H

b kνν ε ε≥ ⋅

− , (16)

where for a thickness shear mode 15k k→ , 11T

i jε ε→ ; for a length dilation mode 33k k→ , 33T

i jε ε→ ;

for a thickness dilation mode tk k→ , 233(1 ) T

i j pkε ε→ − , etc. For a typical PZT PM with the parameters

2k Q ≈100 and i j oε ε ≈1000, an estimation gives h H > 1/10 for the fundamental harmonic 1ν = , and

h H > 1/100 for the third harmonic 3ν = . Particularly, for a thin piezoplate made from such a PM with

a thickness of H = 300 mc, the gap must be as high as h ≥ 30 mc for 1ν = and h ≥ 3 mc for 3ν = ,

respectively, to satisfy the condition of the WR measurement.

The WR - method can be applied to the quality factor measurement of electrodeless thin PR plates at high

frequencies, when a mechanically non-contact PR excitation with a regulated air-gap between electrode and

piezoplate provides minimum influence on the oscillatory process and PM properties due to lowered

technological effects. The excited area of a PR plate (electrode size) could easily vary (up to dot) providing,

in particular, a mono-mode PR characteristic (Bechmann condition [7]) under the "energy trapping" effect.

When a thin piezoplate with the polarization direction ( Pr

) along PR thickness is partly covered by

superimposed electrodes on the major and side plate surfaces, the longitudinal and transversal (to the direction

of plate polarization) components of an exciting electrical field ( )tEr

and ( )sEr

are present, so that the dilation

and shear modes can be excited and observed simultaneously on a single PR sample. The quality factors of

these modes were determined (Fig. 6) by the WR - method with an appropriate (16) air-gap. As it was

experimentally confirmed by the measurements, the quality factors of the nearby high-order harmonics of the

shear ( ( ) 310sQ = ) and dilation ( ( ) 2160tQ = ) thickness resonances sufficiently differ up to an order [5,20-

22]. The determined quality factors correspond to ( ) 44E

sQ Q≅ and ( )D

t tQ Q≅ the quality factors in the

respective complex material stiffnesses ( )44 441E Ec i Q+ and ( )33 1D Dtc i Q+ , and their difference is related

to the specifics of elastic anisotropy and piezoelectric losses in PM.

13

Fig. 6. Experimental AFCh (total admittance) of a single electrodeless piezoplate in the WR regime of quality

factor measurement with the air-gap for nearby the 7-th shear ( 15k -mode) and 3-rd dilation ( tk -mode)

thickness resonances. PR 6x6x0.3 mm, PM PZT-35Y (Russia) [5,19].

Fig. 7. Experimental dependences of the HF filter insertion loss 0a (dB) on the planar vibration mQ

( pk -mode, standard method) and thickness dilation vibration ( )tQ ( tk -mode, WR - method) quality factors.

PR 6x6x(0.6…1) mm, PM PZT-35Y (Russia) [5,19].

14 The developed WR method of the quality factor measurement on electrodeless piezoplates, and their

correct selection on earlier technological stages, can be practically used in the production of HF resonators

and filters [20-22], improving manufacturing process quality. The experimental dependences of the HF filter

insertion loss 0a , dB (the third harmonic , 10 MHz) on the planar vibration quality factor mQ (standard

method) at the frequency 300 kHz and thickness (dilation) vibration quality factor ( )tQ (WR - method) at

the intermediate frequency 5 MHz of piezoplates are shown in Fig. 7. The initial piezoplates 6x6x1 mm,

made from piezoceramics PZT-35Y under different technologies and regimes, have had equal enough degree

of polarization with planar CEMC pk = 0.48…0.53. They were metallized and polarized for the planar mQ

measurements, then partly grinded for the thickness ( )tQ determination, and finally grinded and put with the

filter electrode topology. As seen from the data presented in Fig. 7, the measured “electrodeless” quality

factor ( )tQ of the working thickness-dilation vibration, contrary to the planar quality factor mQ ,

is a parameter which directly determines quality of the piezoelectric filters, and hence is more informative

and effective for HF applications. The thickness vibration Q – measurement procedure can be, for example,

combined with and integrated to a grinding process, and is easy for automatizing.

C. High-Order PR Overtones and Spurious Resonances

A typical PR, being an electro-mechanical system with “distributed parameters”, has the frequency

spectrum with an infinite set of harmonics (overtones). As the resonance intensity of a harmonic drastically

decreases with harmonic number ν = 1, 3, 5… increasing as 2 2 4( )a r mR R k Q ν≅ , and when it becomes

sufficiently small (9-11) at 2mak Qν ≥ , the actual resonance-antiresonance interval G is determined

by the reciprocal quality factor 1mQ − .

For the radial vibration of the disk PR ∅23x0.8 mm with the parameters pk = 0.58 ( (1)rδ = 17% ) and

mQ = 290, determined by the standard method at the fundamental harmonic, the measured overtone actual

relative frequency interval G (ν) decreases classically as 2( ) 1r νδ ν∼ for several first overtone numbers, and

15

then reaches a constant value matching to the 1mQ − at higher overtones (Fig. 8), as it was predicted by the WR

approach (the consecutive odd numbers were used for the overtone’s numerate order).

Fig. 8. Experimental dependence of the reciprocal actual relative resonance-antiresonance frequency

interval 1

( )G ν−

on the radial vibration overtone number ν = 1, 3, 5 … 21 of the disk PR ∅ 23x0.5 mm made of

PM PZT-23 (Russia, [19]) with the parameters pk = 0.58 and mQ = 290 (fundamental harmonic).

Fig. 9. Planar vibrations AFCh of an

arbitrary-shaped PR with indefinite

vibrational modes and a WR spurious

resonance.

16 Particularly, there is a unique possibility in determining the material quality factor on a PR having

indefinite vibrational modes with an extremely broken spectrum (generally on an arbitrary-shaped piece of

piezoelectrics) – a low intensity isolated spurious “weak“ resonance always can be found into the complicated

PR spectrum (Fig. 9).

IV. PIEZOELECTRICS WITH RELATIVELY LOW 2k Q .

A. Low-Q and/or Low-CEMC Piezoelectrics

The WR - method can be most successfully used for the low-Q and/or low-CEMC PM characterization.

The characteristics of such piezoelectrics as PVDF ( 2 2 0.1t mk Q ; ), P(VDF-TrFE) ( 2 2 0.7t mk Q ; ), PZT

piezocomposite ( 2 2 7t mk Q ; ), lead metaniobate ( 2 2 1t mk Q ; ), where tk is the CEMC of the basic

thickness-extensional vibration mode, are presented in details in [6] with the thickness-vibration admittance

experimental data available for the WR - method to be applied for. High-anisotropy PMs, like modified

PbTiO3, with extremely low planar to thickness CEMCs ratio 0.1p tk k ≈ [23], where pk is the planar

CEMC, provide a low-intensity planar vibration suitable for material quality factor and CEMC measurement

with the WR - method.

B. The initial stage of piezoceramics polarization (Low-CEMC)

The described WR concept was involved to explain the “paradox” experimental data of the CEMC

and related to it piezocoefficient dependences on a degree of piezoceramics polarization.

When the piezoceramics polarization decreases Pr

→ 0, the CEMC and piezocoefficient quantities, calculated

by standard formulas (1) [1-3], behave themselves as if they have some constant finite nonzero limit value

(Fig. 10). According to the WR phenomenon, this limit is not related to the piezoelectric properties and

is determined by the piezoceramics quality factor corresponding to the quasi-non-polarized state.

To demonstrate the phenomenon, a set of rod PRs 6x6x15 mm, made from a single block of a soft

piezoceramic, with electrodes on their opposite tops were polarized by relatively small electric field varying

slightly from sample to sample. Two different kinds of the effective CEMC were experimentally determined:

17 the first one by the formal resonance-antiresonance method (1) on the basis of the G actual frequency interval,

and the second quasi-static CEMC according to 2 233( ) 33( ) 33 33

T Est stk d sε= by determining the quasi-static

piezocoefficient 33( )std with the quasi-static method [24] at a 85 Hz axial mechanical load in the range from

12min 15 10 C N−⋅ and up to saturation. As seen from the dependences of Fig. 10, when the quasi-static

CEMC 33( ) 0stk → , the actual resonance-antiresonance frequency interval 01G Q→ , where 0Q is the

quality factor of non-polarized piezoceramics, which can be hence easily determined by the WR - method.

Fig. 10. Dependence of the actual resonance-antiresonance frequency interval G on a degree of

piezoceramic element polarization from the zero-stage and up for the rod 33k -mode PR 6x6x15 mm

at the fundamental resonance. PM PZT-19 (Russia, [19]), mQ = 90, 33k = 0.65, 33 0Tε ε =2000).

V. PRs in PIEZOTRANSDUCERS UNDER “HIGH EFFICIENCY” LOADING

The transducer loading process is changing its electrical impedance on the way that minimal series

impedance under loading is increasing, and that maximal parallel impedance is decreasing. Both quality

factors of series and parallel resonance are also dropping down coincidently following electrical (mechanical)

18

load increase. It will be shown that the quality-factors ratio, between the state of non-loaded ( mQ ) and loaded

( Q% as a variable under load variation) transducers, is one of the best measure of transducer’s dynamic

(loading) performances, particularly its energetic efficiency can be expressed as 1 mQ Qη ≅ − % ;

( )1 1 1m m n m mG Q f f f Q− ⋅ − −; ; under high efficiency regime of excitation. We consider two types of

loading: piezotransformers with electrical load and acoustical piezotransducers under mechanical loading

when system’s quality factor is sharply lowering due to, for example, ultrasonic energy radiated into

surrounding medium. The purpose of this chapter is to demonstrate the WR - method possibilities under such

conditions , so that simplified models and descriptions are further used with linear PM characteristics taken

into account.

A. Eelectrical loading – piezotransformers (PTF)

A simplified 1-D model of the classical Rosen-type PTF (Fig. 11a) with 2λ - mode of operation and

input/output sections with a “node” point in the center [25] of a bar plate is considered. The PTF consists of

a thin piezoplate with a length L and thickness H, divided on two equal parts of input and output sections with

thickness and longitudinal polarization, and input inC (section 1) and output outC (section 2) capacitance,

respectively. The input voltage V forces the piezoplate to vibrate, so that the output voltage U is created on

the load LR .

The system’s quality factor concept in the PTF behavior description is used further. According to the

vibrational system’s quality factor definition [26] kinQ W Pω=% , where kinW is the PTF vibrational kinetic

energy, h RP P P= + is the total power loss (power supply loss) which is divided between heat power hP in

PTF and output power RP , while the material quality factor m kin hQ W Pω= . The output PTF power RP on

the active load LR can be derived considering the electrical equivalent circuit of the PTF output section (Fig.

11b):

{ }2 20.5 0.5 max1 ( ) 1

L outR out el

L out

R C tP U J E C P

R C tω

ωω

∗= = ⋅ =+ +

% , (17)

19

where E% is the “mechanical” electromotive force (EMF), J is the output PTF current, L outt R Cω= is

the dimensionless load parameter, { }max 0.5el outP E Cω= ⋅% is the averaged for a period of vibration

maximum electric power created by the generator (corresponds to the short-circuited PTF regime), then

{ } { }max maxel elW P ω= is the averaged for a period maximum (possible) electric energy created by

the generator E% .

Fig. 11 a,b. Schematic

representation of a PTF (a) and

the equivalent circuit of the loaded

output PTF section (b).

Considering the PTF as a mechanically free-vibrating element with equal input and output parts as to

dimensions and masses, it can be concluded from the momentum conservation principle that under any

electric load (1) (2) 0.5kin kin kinW W W≅ ≅ , and there is no other kinetic energy.

The electromechanical coupling factor (CEMC) relates the amount of electrical energy that is transformed to

mechanical energy, or mechanical energy transformed to electrical energy. Particularly applied to the PTF

output section under direct piezoelectric effect { }233 (2)max el kink W W= , then

{ } 2

332 2(2)

max0.5 0.5

1 1elR

kin kin

WP t tk

W W t tω= ⋅ = ⋅

+ + , then

233 2

1 10.5

1h R

kin m

P P tk

W Q tQ ω+

= = + ⋅+% , (18)

where 2 233 33 33 33

T Ek d sε= is the longitudinal CEMC squared.

20 As follows from the mechanical “stress - strain” relationship applied to a two - sections PTF plate with

arbitrary electrical loading of the output section, the effective (integral) dynamic PTF elastic compliance [29]

is determined as [ ]1 20.5 ( )s s s t= + by the sum of the PTF input compliance 1 11Es s→ and the dynamic PTF

output compliance 2

22 33 33 21

1E t

s s kt

→ − +

, the last one in the limits of 33Es under a short-circuit and 33

Ds

under an open-circuit loading regime. Then, their relative component

2

2233 2

1

11

s tk

s tα β

≡ = − +

, (19)

where 33 11

EEs sβ = is the material elastic anisotropy parameter. The resonance frequency of a free-vibrating

PTF plate under electrical load is then 01 2 0.5(1 )sf L s fρ α= = + , where 0 111 2 Ef L sρ=

is the reference resonance frequency corresponding to a homogeneous bar plate with thickness polarization.

The system’s PTF vibrational quality factor Q% under electrical load determines all the basic PTF

characteristics at the resonance and near-resonance frequency region. Based on the traditional bar-type PR

vibration description and considering the vibrating input section as a 2λ - mode bar PR with the electrical

induction 3 33 31ED V H d Tε= + , the mechanical stress at the PT node-point is supposed to be

31ˆ4

(0)1

d V QT i

s h iyπ≅ − ⋅

+

% , where 2y Qχ= % is the generalized frequency displacement, 1sf fχ = −

is the resonance frequency displacement, and then the input PR admittance can be expressed as

( )2 2 11 2

1in inY C i bk ak Qiy

ω

→ − + ⋅ + % % % , (20)

where 24a π= , 1b a= − are transverse coefficients ( 1ν = ) [3], 2 2 231 33 312 (1 )Tk d s kε α= = +% is the

effective CEMC of the input PTF section squared, 2 231 31 33 11

T Ek d sε= is the transverse material CEMC

squared.

21

Fig. 12 a,b . Typical normalized input PTF admittance and conductance (power) frequency characteristics (a)

and calculated dependences of the PTF input actual relative frequency interval 1n mG f f= − and reciprocal

system’s quality factor 1Q −% variation on the input resonance intensity A(dB) (b), as a consequence of

loading (t – parameter shown). Material parameters used: 31 0.32k = , 33 0.65k = , 1.30β = , 500mQ = .

22

For a fully-loaded regime mQ Q<<% of PTF excitation, when the t load parameter is in the limits

( )min max,t t according to 1 2max min 330.5 mt t k Q−; ; , the PTF system’s quality factor becomes equal

( )1233

2Q t t

k−≅ +% . According to the initial expressions for the WR - method it follows that the input actual

resonance - antiresonance frequency interval 233 20.5

1t

G kt

≅ ⋅+

. The resonance intensity

( )2 2

131 312 231 33

1( ) 17.4 17.4

1k k

A dB a t tbk G k

−⋅ ⋅ +−

; ; is dependable on the piezoelectric anisotropy parameter

31 33k k , so that for a typical 31 33 0.5k k ; the input resonance intensity 8A dB; for the maximum

efficiency regime ( 1t = ) of PTF electrical loading. Consecutive input section AFCh transformations for a

PTF under load are shown in Fig. 12 a. Note that the frequency characteristics of the input PTF losses, being

proportional to the input PTF conductance Re inY , correspond to the PTF output characteristics. PTF loading

is changing PTF’s input electrical impedance on the way that (under loading) maximal resonance admittance

is decreasing, and minimal antiresonance admittance is increasing coincidently (Fig. 12a), following the PTF

vibrational quality factor Q% drastically dropping down at 1t → (18) (Fig.12b). The limit resonance intervals

for short-circuited ,0rδ and open-circuited ,rδ ∞ regimes of PTF excitation are determined by (1) with the

corresponding CEMCs 231 (1 )k β+ and 2 2

31 33[1 (1 )]k kβ+ − , while the actual resonance interval effG for

maximum efficiency regime ( 1t = ) corresponds to the minimum Q% as 2331 min 4effG Q k= ≅% (Fig. 12 b).

B. Mechanical loading – piezotransducer (PTD)

In a process of immersing, the contacting and pressing transducer’s front emitting face to some other

material (until it properly operates full-power) is required to provide maximal contact between PTD and its

acoustical load. It is supposed a low-signal impedance measurement to be made. For a bar type PTD,

symmetrically loaded on its both tops, with an unstiffened [3] vibrational mode for simplicity (to separate

the directions of the acoustical stress and electrical field), from the boundary conditions for velocit ies and

23 mechanical stress, it easily can be shown [15,27] that its electrical admittance is as follows:

[ ]

( ) [ ]2 2

00

tan 2 11

2 1 tan 2L

KLY i C k k

KL i Z Z KLω

= − + ⋅

+ , (21)

where 2 211ˆEK sω ρ= is the wave number squared, ( )11 11ˆ 1E E

ms s i Q= − is the material complex

compliance, 2 231k k= is the longitudinal CEMC squared, 0C is the static PR capacitance, L is the bar length,

0 1 11 2E EsZ v s L fρ ρ ρ= = = and L L LZ vρ= are the acoustic impedances of piezoceramics and

the medium loading PTD, respectively , 1, Evρ and ,L Lvρ are the density and sound velocity of

piezoceramics and surrounding medium, respectively. Then, after decomposition the expression (21) on small

parameters of frequency displacement 1sf fχ = − and dissipation, it can be expressed in the vicinity of

the fundamental harmonic ( 1ν = ) as

( )2 20 31 31

11 2

1Y C i b k ak Q

iyω

≅ − +

+ % , (22)

where 2y Qχ= % is the generalized frequency displacement, 24a π= , 1b a= − , and the system’s quality

factor Q% is determined by

0

1 1 4 L

m

ZQ ZQ π

= + ⋅% , (23)

which monotonically depends on the relative load parameter 0LZ Z .

Maximum output PTD power corresponds to the equality of the internal (thermal) and output losses, when

the material and “acoustical” quality factors are equal [28], so that 104 L mZ Z Qπ −= . For higher loading,

when the “acoustical” and “electrical” quality factors are equal [15] and the primary loss mechanism is

radiation of ultrasonic energy, it can be shown that the system’s quality factor is determined by

0 4 L mQ Z Z Qπ→ <<% , so that the maximum efficiency regime of excitation corresponds to

2 2 2...3Q kπ ≈% ; for typical piezoceramic CEMCs. Under such conditions according to the WR -

method, the medium acoustic impedance is 2LZ f m Sπ≅ ∆ ⋅ , where n mf f f∆ = − is the PTD actual

24 resonance - antiresonance frequency interval, and m, S are the PTD mass and effective radiating surface.

The resonance intensity (4-8) for this case is 0( ) 17.4 17.4 4r r LA dB Q Z Zδ δ π⋅ ⋅%; ; , the actual relative

resonance-antiresonance frequency interval 104 LG Q Z Zπ−%; ; , while “piezoelectric” frequency interval

( ) 17.4r G A dBδ ⋅; remains constant (Fig. 13 a,b). Under fully-loaded transducer’s variable quality factor

3Q →% at typical 0.4k = with 4A dB≈ , the transducer is being still well operational and able to deliver

near 100% of its maximal operating power.

Fig. 13 a,b. Dependences of the system’s quality factor (a) and actual relative frequency interval (b) of an

acoustically loaded PTD on the resonance intensity for the material parameters 31 0.4k = ( 7.2%rδ = ) and

100mQ = .

CONCLUSION

When the intensity of some particular PR resonance, defined as the attenuation difference between the

resonance and antiresonance PR peak impedances, is less than 15 dB, its actual relative resonance -

antiresonance frequency interval is determined by the quality factor, regardless of the reason causing such

a “weak resonance”, and the resonance intensity is proportional to CEMC. It corresponds to the fundamental

physical principle that the generalized characteristic frequency bandwidth of an oscillatory system can not be

25 less than the reciprocal value of its quality factor under any circumstances. Developed on the basis of this

effect, the very effective measurement WR - method can be used for the PM and PR characterization under

the following conditions of a natural weak resonance: high order overtones and harmonics; low - Q and/or

low - CEMC piezomaterials; at the initial stage of piezoceramic element polarization (quasi non-polarized

ceramics); at the fundamental PR harmonic under high level of electrical (piezotransformer) or mechanical

(piezotransducer) loading. An artificially induced “weak” PR resonance can be created by an external

capacitor with extreme capacitance connected to PR in parallel for ( )rQ , or in series for ( )aQ , quality factors

determination. The last condition is naturally realized on an electrodeless piezoplate with the air-gap under

mechanically contactless PR excitation. The simple and universal WR - method corresponds to the standard

measuring procedure that requires determination only extremes of the total absolute-immittance.

It is fast and easy, and is not dependable on computer and phase-sensitive measurements.

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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

28 Glossary ( mf , sf , rf ), ( nf , pf , af ) – frequencies of PR maximum absolute admittance (impedance), maximum

conductance (resistance), and zero susceptance (reactance), respectively

1 12 2

,s s

f f+ −

– frequencies of the PR susceptance extremes

( , ) ,r aF F∆ – generalized PR resonance (antiresonance) frequency and bandwidth of the peak

0 ( , ) ( , )( ) , ( , )m r a t sQ Q Q Q Q – standardized piezomaterial quality factor and generalized PR quality factor

Q% – system’s (total) electro-mechanical PT quality factor

tanδ – dielectric loss factors

31 33( , , , , ) ,i j p tk k k k k k k% – coefficient of electro-mechanical coupling and its generalized notation

,r Gδ – “piezoelectric“ and actual PR relative resonance-antiresonance frequency intervals

( 1, 1r p s n mf f G f fδ = − = − )

( ) ,x yχ – relative and generalized resonance frequency displacements

,a bν ν and ν – coefficients for CEMC determination [IEC] and harmonic (overtone) number

1 1, , ,C C L R and 0C – parameters of PR EC and PR capacitance

, , ,r aY A R R – PR admittance (impedance), resonance intensity, and resonance and antiresonance resistances

, ( )Q G GK K K – correction coefficients

, ,B M p – parameters of PR resonance characterization

( ), ,E D E Di j i js c , kld , T

mnε – piezomaterial constants

1 2, ,s s s – dynamic PT compliance of input section, output section and total PT under electric load

,β α – parameters of PM elastic anysotropy

,p sC C – in parallel and in series connected to PR capacitors

, ,L H h – PR plate length and thickness, thickness of a gap “plate – electrode”

,kin elW W – kinetic PT mechanical energy stored, and electrical energy stored

, ,R hP P P , η – input, output and PT thermal (heat) loss power, and PT efficiency

E% – “electro-mechanical” electromotive force , , ,LU J R t – PT output voltage, load current, load resistance and relative load resistance

,in outC C – PT capacitance (real values)

0 , LZ Z – acoustical impedances of PR and surrounding medium

1, Evρ and ,L Lvρ – density and sound velocity of piezoceramics and surrounding medium, respectively

m , S – PTD mass and effective radiating surface Abbreviations

PR, PM – piezoelectric resonator and material PZT – lead zirconate-titanate CEMC – coefficient of electro-mechanical coupling EC – equivalent circuit PTF, PTD – piezoelectric transformer and transducer WR – weak resonance AFCh – amplitude – frequency characteristics HF – high frequency

29

Fig. …. Qualitative dependence of

the PR AFCh and its resonance

interval on a degree of piezoceramic

element polarization from the zero-

stage to up.

Fig. …. Qualitative dependence of the

PR AFCh on the capacitance of a

capacitor connected to PR in parallel :

, 0 0pC = , 2

, 1 0p rC C k Q< ,

2,2...5 0p rC C k Q≥ .