1 ELECTRICAL MEASUREMENT OF A HIGH FREQUENCY, HIGH ...

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ELECTRICAL MEASUREMENT

OF A HIGH FREQUENCY, HIGH CAPACITANCE

PIEZOCERAMIC RESONATOR WITH RESISTIVE ELECTRODES

Alex V. Mezheritsky, “ELECTRICAL MEASUREMENT OF A HIGH FREQUENCY,

HIGH CAPACITANCE PIEZOCERAMIC RESONATOR WITH RESISTIVE ELECTRODES”,

adapted from the article published in the journal IEEE UFFC, Jan. 2005.

ABSTRACT - In a thin and large area PZT-ceramics piezoresonator (PR) with relatively low resonance

impedance, caused by high frequency resonance and high PR capacitance, the effect of electrode resistivity

and parasitic resistive and inductive elements in the measurement fixture results in significant distortion of the

measured thickness-mode (longitudinal TL, shear TS) resonance response - resonance frequency shifts and

characteristics deformation. This distortion may not allow the precise measurement of the PR characteristic

frequencies, quality factor and electro-mechanical coupling coefficient so essential to a complete PR and

material characterization. A theoretical description of the “energy-trap” phenomena in a thickness-vibrating PR

with resistive electrodes is presented. To interpret electrical measurements, the electro-mechanical model,

including for completeness both the PR with resistive electrodes (as a system with “distributed parameters”)

and the measurement fixture, is developed. The method of two contact points on the electrode provides deep

sharpening and exact determination of the PR resonance. For the optimal disposition of the contact fingers the

resonance bandwidth of a real PR with resistance electrodes is even more pointed than that for the ideal PR.

Keywords : piezoceramics, resonator, resonance frequency, electrode, fixture, energy trap.

Author: A.V. Mezheritsky e- mail: [email protected] t ( CC: [email protected])

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I. INTRODUCTION

Thin piezoplates made from piezoelectric ceramics are widely used as high-frequency resonators,

transducers, and monolithic filters employing thickness modes of vibration. The plate is polarized and covered

with metallic electrodes in order to excite it with an applied RF field by means of the piezoelectric effect.

The complete characterization of piezoelectric thickness-mode resonators [1] includes measurements of the

parameters which indicate the effectiveness of the electromechanical energy conversion. Such a basic parameter,

the coefficient k of electro-mechanical coupling (CEMC), can be defined as the difference between the material

elastic stiffness i jcE,D [2],[3] at constant strength E and induction D of an electric field. In an elementary theory,

the thickness-mode CEMC k is practically determined from resonance frequency data of PR impedance Z [2],

neglecting other modes influence:

( )2

0

tan 211

2KH

Z ki C KHω

= −

, (1)

where 0C Hε= ΦS is the PR capacitance, 2 2k h cε= S D is the CEMC squared, 2 2K cω ρ= D is the

thickness mode wavenumber squared, h is the piezocoefficient, ε S is the dielectric permittivity, H is the PR

thickness, Φ is the electrode area, ρ is the piezomaterial density. The piezomaterial constants [2] set

, , ,c h kε S D for the TS vibrational mode is 11 44 15 15, , ,c h kε S D , and for TL mode is 33 33 33, , , tc h kε S D ,

respectively. The typical frequency responses of a hypothetical PR are shown in Fig. 1. Such characteristics have

both the resonance rf and antiresonance af frequencies [3], defined here as frequency extremes of the output

voltage (absolute value). The IEC Standard [2] determines the CEMC squared according to a practical expression

for the fundamental thickness harmonic as 21 0.405 0.81rk δ= + , where 1r a rf fδ = − is the relative resonant

frequency interval.

In general, thickness vibrations are very complicated, since coupling of different modes may occur in the

vicinity of the fundamental resonance [4]-[7]. So, the basic TL-mode with a coupled TS vibration, or the basic

TS-mode with coupled flexure-twist vibrations, under certain conditions, can provide the “energy-trapping”

effect widely used in frequency selection devices. Application of such simple expressions to piezoelectric

3 ceramic materials is difficult due to the influence of competing modes of vibration and the effects that are

caused by boundary conditions. Typically, these result in response distortions leading to erroneous

characteristic frequency values. Several techniques have been developed to reduce the difficulties that result

from these effects.

Fig 1 (a,b,c). Comparative frequency responses (c) with the resonance rf and antiresonance af frequencies

of a thickness-mode PR for traditional (a) and proposed (“new”) (b) measurement techniques with in series

connected parasitic inductance L and resistor R in the PR fixture.

One of them [8] determines the value of k based on the shift in location of the overtone resonance

frequencies. This technique is well suited for overcoming the problems associated with the use of rδ

determining. The overtone frequencies of the thickness-mode are sufficiently high [1] to reduce the problems

associated with any competing lateral modes. Also, for a thickness-mode PR the influence of the boundary

effects is less for overtones than it is for the fundamental mode. However, the distortions of the frequency

responses of a high capacitance PR at high frequencies are effected in a manner that is not a result of unwanted

mode or boundary effects. Rather, this behavior is related, as to material parameters, to the high dielectric

constant ( 0i jε εS ; 100… 5000, where 0ε is the vacuum dielectric permittivity) of the piezoceramic (PZT)

materials [9],[10]. This so-called “inversion response” effect was firstly reported in [11], and then was

described in details in [12],[13]. The observed resonance frequency shift and inversion effect for higher

harmonics are the result of the relationship between the specimen capacitance and the holding fixture series

4 inductance. The essential shift and distortion of the resonance peak (Fig. 1a) of a system including PR and an

electrical circuit is determined by inductive and resistive elements in the part of the circuit common for the PR

input and output circuitry loops.

Modern practical needs of PR applications at high frequencies 10…30 MHz, and higher, require usage of thin

plates of 200…50 mc thickness or even less. Meantime, large area piezotransducers (up to tens sq. cm) have

some applications to produce a wide wave-front radiating effect (in medicine, etc.), or are used for technology

optimization (as intermediate large area piezoelements for tiny delay line piezotransducers, etc.). All these lead

to high PR capacitance (tens nF), and hence, to low resonance PR resistance (much less 1 Ohm) and low

equivalent PR inductance (tens nH), so the very small inductive and resistive parasitic elements in the

measurement fixture and PR electrode resistance have an increased significance. The thinner piezoplate

thickness, the thinner electrode thickness is required. Particularly for this reason, limited methods can be

applied for electrode deposition. It also leads to worsening of electrical properties of a thin electrode layer [14]

which has lower conductivity than pure metal. The same situation concerns thin and thick piezo-films [15],

and bulk multilayer piezotransducers [16] consisting of multiple thin piezolayers co-fired through thin metal

layers. Determination of such PRs resonance properties is a problem.

II. ELECTR0-MECHANICAL MODEL DESCRIPTION

A. Basic Estimations

The IEC and IEEE Standards [2],[3] were used as a guideline to conduct the frequency measurements for

this study. The circuit with a “grounded” PR shown in Fig. 1 (a regime of a constant current through PR

( )EI E R const f≅ =% ) was utilized [17] in order to make the measurements. Two designs for the specimen

holding fixture were modeled - traditional “one contact point” and proposed new “two contact points”

measurement techniques. A TS-mode PR type was chosen for experiments as less acoustically sensitive to

lateral boundaries and planar spurious resonances. The PR 20x10x0.2 mm was made from PZT-36 ([9],

Russia) with polarization direction along a 10 mm side, the resonance fr = 4.0 MHz and antiresonance

fa = 5.4 MHz frequencies (CEMC 15k = 0.7, rδ = 0.28), electric capacitance C0 = 10 nF, mechanical quality

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factor ( ) 100mQ Q = . As the intrinsic resonance and antiresonance PR resistances (ideal electrodes) are:

2 2 208r rR C k Qπ ν ω≅ , 2 2 2 2

08 (1 )a aR k Q C kπ ν ω≅ − , (2)

where , ,2r a r afω π= , we have a numerical value estimation for the fundamental harmonic ( 1ν = ):

0.1rR Ohm; , 242aR Ohm; . The fully metallized PR has chemically deposited Ni - electrodes 20x10 mm

with a thickness of 2 mc and electric resistance elR of 3 Ohm between the farthest points on the electrode,

so that the estimated ratios 2 0.13r elR R ; and 2 0.8el rR R Ohm; .

The best results for a linear PR amplitude-frequency characteristics (AFCh) performance were obtained when

using relatively large PR input electrical load ER = 1.5 kOhm >> aR (Fig. 1). From the magnitude display,

the antiresonance frequency can be discerned with ease. The resonance frequency, however, cannot be

determined accurately by traditional inspection. For estimating the influences of the decisive factors of

the fixture wire inductance and resistance, and resistance of the PR electrodes, it is needed to take into account

two basic intrinsic PR parameters of the resonance PR resistance rR and equivalent inductance 1L :

2

20

1( )r

i j

HR

k Qν

ε ε⋅ ⋅

Φ S∼ ,

3

1 20

1( )

( )i j

HL const

kν

ε ε⋅ =

Φ S∼ , (3)

then 2 21 01 0.32rL C k Hω µ≅; . The series inductance in the metal finger that holds the test specimens in

place and its resistance are comparable to the intrinsic PR resonance resistance rR and equivalent inductance

L1 values. By an approximate estimation, a contact wire of 1 cm length with parasitic inductance 0.1L Hµ;

and typical quality factor 10Q ; has effective resistance 0.1R Ohm; at 5 MHz according to R L Qω≈ .

A PR that displayed normal responses when operating at their fundamental frequency also displayed inverted

responses at their overtone frequencies and their shifts are not be related directly to CEMC. If some inductance

L is connected to PR in series, there is a shift νω∆ of the resonance frequency of ν -harmonic

, 10.5 ( )r L L constν νω ω ν∆ − =; . The antiresonance frequency is not affected very much by the series

inductance, while the resonance frequency is lowered. Since the frequency distance between two nearby

harmonics is twice of fundamental resonance , 12 rω; , the “inversion response” effect definitely occurs when

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the frequency shift , 1rνω ω∆ ≥ takes place at the harmonics ( , ,1r rνω ν ω⋅; ) of the following order

3

12

0

2 1 1( )i j

L HL L k

νε ε

≥ ⋅ ⋅Φ S

∼ (4)

The level of the effect increases to the degree that the “resonance frequency” of the higher harmonic appears on

the positive side of the antiresonance peak of the lower frequency harmonic . Note that two times decreased PR

thickness (characteristic frequencies) causes an increase in critical harmonic number of the inversion response

effect by an order.

The presence of fixture wire and PR electrode resistance increases a voltage response mostly at the resonance

frequency and, hence, leads to a resonance bandwidth widening. It can even result in an absence of any well-

defined resonance frequency peak in the expected frequency range. In the case of resistive PR electrodes (non-

equipotential electrode surface), an effectively excited PR region decreases significantly, so only resonance

characteristics of a local PR region can be detected. A new “two contact points” measurement technique was

proposed and analyzed, which fully eliminate described above effects and, even more, transforms a negative

feature into positive one as to electrode resistivity. In the present discussion, the electrodes are assumed to be so

thin that their mechanical effects can be neglected.

B. Introductory Description of the Electrode Resistivity Effect on “Uncoupled”, Infinite Piezoplate

A linear one-dimensional semi-infinite piezoplate (strip) and its equivalent representation (Fig. 2, regions

A-B) are analyzed. Elementary space units are considered here acoustically independent, or “uncoupled”,

in the lateral x-direction. A direct practical analog of such a model can be a 3-1 or 3-2 piezocomposit.

The coupling-effect influence for a regular PR is considered in the next chapter.

An exciting input voltage inU is applied to the strip-points 0 0′− and an output voltage outU is picked-off

from the strip-points n n′− on the electrodes. Such a PR represents a system with “distributed” parameters

[18],[19] consisting of infinitely repeated r z r− − chains with elr R x= ∆ and 11 z Z x−= ∆ , where elR and

1Z − are the linear electrode resistance and PR intrinsic bulk admittance (1) (per unit length), respectively.

According to [18], the effective input PR complex impedance inZ is determined by ( ) 11 1 2in inZ z r Z−− −= + + ,

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then 2 2 2in el el inZ R Z R Z x= − ∆ , and at 0x∆ → we finally have :

2 2 ( ) ( )in el elZ R Z R t Z t= = , (5)

where ( )elR t , Z(t) are the electrode resistance between points x = 0, t and the intrinsic impedance of the PR

region (0, )x t∈ , respectively, t is an arbitrary chosen point on the electrode. According to the equivalent

schema representation of Fig. 2, the voltage distribution is determined as ( )( ) 1 2n

out in inU U x U r Z≡ = + .

Fig. 2. PR frequency response measuring circuit and its equivalent representation for a semi-infinite piezoplate.

As 0r → at 0x∆ → , we have in the limit of n → ∞ (n x x∆ = ) the exponential voltage decay:

( ) exp( 2 ) exp( ) exp( )in el in RU x U R x Z x i K xµ= − ⋅ = − = − ⋅ , (6)

in inU I Z=

where the basic distribution parameters 2R elK i R Z= − as an analog of the wavenumber, and

1 2R eli K Z Rµ = = as a parameter of relative electrode conductivity, are used further in acoustical and

electrical approaches. If RK is expressed as Re ImR R RK K i K= + , then the absolute value of the voltage

( ) exp(Im ) exp( )in R RU x U K x x x= ⋅ ≡ − , (7)

where 2

1 Im ReR Rx K µ µ= − = is the characteristic dimension of an effectively excited PR region for

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Im 0RK < at 0x > , that corresponds to an e-times voltage magnitude decay along the PR electroded surface.

Further, as the PR admittance can be represented as 11 1 1Re Im iZ Z i Z Z e φ−− − −= + = , where φ is the phase

of the intrinsic PR admittance, then 11 Im cos(0.5 ) 2R R elx K R Zφ

−− = − = ⋅ . For pure capacitive admittance

10Z i Cω− = with the linear PR capacitance 0C (per unit length), we have the decay parameter

01C R elCx x R Cω≡ = , (8)

At the PR resonance with the intrins ic resonance resistance rR (per unit length) we have

20 , ,2 ( ) 2 ( ) 2R r r el r el C rx x R R t R t R t x k Qν≡ = = ≅ ⋅ . (9)

According to (5, 9), an estimation of ,in rZ and 0x at the resonance frequency of a real finite PR can be made

as follows: , 2in r el rZ R R; and 0 2r elx M R R; , where rR is the intrinsic PR resonance resistance

estimated according to (2) using piezoceramic material parameters, elR is the measured electrode resistance

between the farthest points on the electrode, M is the lateral PR dimension. Initial estimation gives

0 2.6x mm; and , 0.8in rZ Ohm; for the experimental specimen.

Taking into account that ( )EI E R const f≅ =% and in inU I Z= , we have the expression for the output voltage

magnitude outU picked-off from the contact separated with the input contact by distance W :

( )1

( ) exp 2 exp cos(0.5 ) 2out in R el elx WU U x I Z W x I R Z R Z Wφ

−

=

≡ = − = ⋅ − ⋅ ⋅

. (10)

Since the antiresonance PR resistance is greater as much as orders the resonance one, minimal size of an

effectively excited PR region corresponds exactly to the resonance thickness frequency. In particular, for a real

PR with finite dimensions at least near the PR antiresonance with el aR R<< we have Rx → ∞ ( Rx M>> ,

where M is the PR length), and then inZ Z= , out inU U= for any x.

Note that the presented linear description can be easily extended to a PR radial configuration with contact

dot-points on the PR electrodes.

9 III. ELECTRODE RESISTIVITY INFLUENCE ON “ENERGY - TRAP” EFFECT

The vibrations of acoustically coupled space units generally lead to a vibrational modes coupling which is

primarily determined by a lateral gradient of the basic mode vibrational characteristics. In the classical “energy-

trap” phenomenon [5], the gradient is created by sufficiently different electrical boundary conditions on the

electroded (short-circuited) and unelectroded (open-circuited) plate surfaces. So, in a TL-mode PR, a

concomitant coupled (TS) mode is mostly localized in a narrow boundary between the electroded (excited)

and unelectroded (unexcited) regions [4],[5].

In the case of resistive electrodes the gradient is wider distributed in the lateral directions and hence is less

localized. In the limit conditions of infinite (ideal) and zero electrode conductivity, the results of the two

approaches trend to equal dispersion relationships. Moreover, if the piezomaterial parameters of elastic

anisotropy ( 33 44 4E Ec c > ) provide sufficiently strong support of the “energy-trap” effect, the only electroded

region in a partly electroded piezoplate vibrates. Then, the piezoplate can be hard-fixed in the surrounding

unelectroded region without influence on the vibrations of the PR active zone, which can be considered as

independent of surroundings. Such an effect is fundamental in high-frequency resonator and monolithic filter

designs based on “energy trapping”. For this reason, it appears to expect that in certain conditions for a PR with

resistive electrodes the vibrational regions can be considered as acoustically uncoupled because of relatively

small planar gradient. Further the theory of the “energy-trap” effect in a TL-mode PR with resistive electrodes

is developed based on the approach presented in [4], whose mathematical procedures and original notations are

generally used.

A. Common Description

A piezoceramic plate polarized and excited in its thickness direction can vibrate in TL as well as TS modes

which are coupled. The dispersion spectrum of a circular disk PR is derived in [4] from the constitutive

equations. The problem is reduced to a two-dimensional one according to the full rotational symmetry of a disk

PR. All variables are radial and axial components only. A trapped energy PR of this kind consists of a thin

ceramic disk with an inner electroded area where propagation of TL-modes is possible, surrounded by a

nonelectroded marginal region where the modes decay outwards. Allowance is made for different electrical

10 boundary conditions such that electroded and nonelectroded regions of the disk may be distinguished. Thus if

energy trapping is to work, the appropriate dispersion curves must allow, at the resonance frequency, a real

planar wavenumber xK in the electroded region to correspond with an imaginary xK in the unelectroded

region. This is only true for frequencies within a certain respective "energy trapping” interval.

If, however, an interchange of modes occurs, there is no trapping interval for the TL-mode and thus energy

trapping does not work.

The faces of the piezoplate are located at 2z H= ± . The rotation axis of the disk coincides with the z-axis,

which is also the direction of polarization. The electric field is determined by its electric potential as

( ) ( )z x z xE ϕ ′= − . A harmonic time dependence exp( t)iω is assumed for all vibration characteristics.

In order to solve the equations of state simultaneously the variables are separated by

( , ) ( ) ( )zu x z F x f z= ⋅ , ( , ) ( ) ( )x xu x z F x h z′= ⋅ , 33( , ) ( ) ( ) ( )x z F x g z h f zϕ = ⋅ − ⋅ , (11)

where ( )z xu are the local displacements, ( ) , ( ), ( )f z g z h z are some z-coordinate functions. Inserting this into

equations of charge and motion yields the differential equation for radial distribution function ( )F x as

2 0x x x xF F x K F′′ ′+ + = , where xK is the separation constant and turns out to be the appropriate “planar”

wavenumber. As can be shown, a similar procedure can be applied to a finite thin piezoplate equally excited

along an infinite line , e.g. 0x = for certainty. Solving the task leads to (11) with the lateral equation

2( ) ( ) 0x x xF x K F x′′ + ⋅ = , (12)

which will be considered in further analysis. For a disk PR the radial dependence as a solution of Bessel’s

differential equation is simply given by zero order Bessel functions 0( )xK x⋅¢ , for a linear PR configuration

the solution of (12) is in the exponential form exp( )xi K x⋅ . The axial variations, however, must be found from

a system of three coupled equations. To discover their exact solutions represents a fairly complicated problem

which, however, can be simplified considerably by imposing the only restrictive condition between

wavenumbers - the planar wavenumber xK is taken to be small compared with the “axial” quasi-wavenumbers

of the TL and TS modes. In this way the analysis is confined to the fundamental thickness vibrations of a thin

11 plate with planar dimensions much larger its thickness.

For z-variables there are ultimately three basic boundary equations to be solved. Two of them are related to

the mechanical boundary conditions - for thickness vibrations no stresses are allowed to be exerted at the major

faces of the plate. They must be traction-free, so that the stresses z zT and x zT must vanish at 2z H= ± ,

where the linear piezoelectric constitutive relations are used.

The traditional electrical boundary conditions on the plate surfaces are a pair of shorted and grounded

electrodes covering the major PR surfaces for which ( )2

0z H

xϕ=±

= (called e-region), or a pair of

unelectroded and charge-free surfaces - no current is allowed to pass the surfaces of the plate - for which

( )2

0z z HD x

=±= (s-region). For an unelectroded surface, the effect of the electric field in the surrounding

space can be considered by requiring that both the electrical potential and the normal component of the

dielectric displacement vector cross the interface continuously.

In the present discussion, the electrodes are assumed to be so thin that their mechanical effects can be

neglected. The above equations and boundary conditions are homogeneous, and constitute an eigenvalue

problem. Both the boundary conditions applied to the local variations of displacements and potential yield

the dispersion relations in both the electroded e-region and the nonelectroded s-region of the plate.

The dimensionless frequency 440.5 EH cω ρΩ = , related to the second TS resonance, is used for the

dispersion relations. Solving the differential equations together, we have the matrix of coefficients in a system

of linear equations for vibrational amplitudes [4]:

, , ,11 12 13

21 22 23

31 32 33

e s e s e sa a aA a a a

a a a= , (13)

with the requirement 0A = for a non-trivial solution. The sets of 2ia and 3ia are initially appear from

the two mechanical boundary conditions, the last set ,1e s

ia corresponds to “electroded” (e – index) and

‘unelectroded” (s – index) major plate surfaces. Then, ( )xK Ω can be found, providing zero-order Bessel

functions 0 ( )xK x⋅¢ for radial symmetry, or exponential functions exp( )xK x⋅ for a linear task with

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the same A-matrix. Usually, Ω is plotted versus the planar propagation constant xK , which is then found to

be an ambiguous function split up into several branches. Each branch represents the dispersion relationship

between Ω and xK for a definite mode of propagation. A real value of xK means propagation of that

respective mode with a planar linear dependence ( )exp Re xi K x⋅ , whereas an imaginary xK indicates

a mode that decays in 0x > direction exponentially with ( )exp Im xK x⋅ for Im 0xK < .

Fig. 3 illustrates the dispersion curves for PCM A [4] with relatively la rge “energy trap” figure-of-merit

33 44 4c c >E E . The TL mode lies well above the thickness-radial-shear mode (TS) in both the electroded e-region

and the unelectroded s-region. The two modes are connected with imaginary continuation loops which,

however, close at values 2 0.4xK H > beyond the scope covered by the approximation made. As was shown

[4], small deviations in material parameters cause a conversion of the spectrum with an apparent interchange

of modes - which differ slightly in their material properties but differ largely in their dispersion. This has its

marked influence on the phase and group velocities of the respective waves and on the frequencies of the

thickness vibrational modes.

B. Piezoplate with Resistive Electrodes

In the case of resistive electrodes each elementary unit volume is acoustically coupled and electrically loaded

by surrounding regions. If alternative voltage 2

0.5 inz HUϕ

=±= ± is applied to some point on the linear PR

electrodes, then

( ) ( )2 t

2

1x zz H x

z Hel

x D x dxh

ϕσ

∞

=±=±

′′ ± = − ∫ , or ( ) ( )2 2

1xx zz H z H

el

x i D xh

ϕ ωσ=± =±

′′± = , (14)

where σ is the unit electrode conductivity, elh is the electrode thickness. It means that for an elementary

electrode unit with length dx the difference of the currents on its left and right sides equals the surface normal

inductance. Then, following the procedures of [4], the equation (14) can be transformed into

( )2 11 33 1( ) 2 0e s

x i el iK H a i H h aωσ ε−⋅ + ⋅ =S , (15)

and then the A-matrix is modified for the case of resistive electrodes as follows:

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2 2 211 11 12 12 13 13

21 22 23

31 32 33

( ) ( ) ( )e s e s e sx x xK H a G a K H a G a K H a G a

A a a aa a a

⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅= , (16)

where ( )1 233 02 2el elG i H h i C R Hωσ ε ω−= = ⋅S . From the condition 0A = for non-trivial solutions the

value of ( )xK Ω can be found. As (16) is a superposition of the e- and s-cases, generally there are two sets

of ( )xK Ω roots depending on electrode resistivity and corresponding to the respective dispersion curve.

Fig. 3. A common view of the dispersion curve ( )2xK H Ω of coupled TL-TS vibrational modes of a PR

with resistive electrodes for the case of 33 44 4E Ec c > ( PCM A [4], 100Q = ): - level of a given frequency, - line of constant electrode conductivity,

- dispersion dependence on electrode conductivity at a given frequency, .

- energy-trap dispersion curves for an electrodless and fully electroded (infinite conductivity)

piezoplate.

Using , 0( ) 1C E C E el Ex x R Cω ω ω≡ = = as an intermediate parameter, where 33E

E rc Hω π ρ ω= ;

(short- circuited electrodes generally does not mean local 0=E ), then

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2

,33 44

4E E

C E

HG i

xc cπ

Ω=

. (17)

The dispersion relationships ( )xK Ω for a thin plate with resistive electrodes were calculated and plotted (Fig.

3) as a space graph in perpendicular ( )Re 2xK H and ( )Im 2xK H planes for PCM A [4] with basic material

parameters: 33 0 190Sε ε = , 11 0 425Sε ε = , 0.51tk = , 15 0.65k = , 33 44 4.55c c =E E , 10 233 19.4 10Dc N m= ⋅ ,

10 244 5.5 10Dc N m= ⋅ , and then comparisons were conducted for PCMs which differ in their parameters [9].

Internal energy dissipation was taken into account by representing elastic constants ( )ˆ 1i j i jc c i Q= +E E as

complex, where Q is the intrinsic material (resonance) quality factor. Calculated for the traditional lossless

Q = ∞ and lossy 100Q = cases, the basic dispersion e- and s-curves have similar character, but in a lossy

case the dependence generally is not purely real or imaginary, so that it never intersects the Ω -axis near PR’s

resonance and antiresonance. The dispersion ( )Re 2xK H vs. ( )Im 2xK H dependences on electrode

conductivity (as a relative ,Cx E parameter) for a lossy and lossless PCM are shown in Fig. 4 a,b for

antiresonance (a) and resonance (b) frequency regions, represented as frequency “cuts” of Fig. 3.

For the antiresonance region the resistance-dependent dispersion lines go out from the s-branch

corresponding to the extremely high electrode resistivity elR → ∞ , and then come into the Ω-axis with

0xK H → for relatively high electrode conductivity 0elR → . In particular, above the PR antiresonance,

where the PR admittance is predominantly capacitive, ( )Im 2xK H− has a maximum corresponding to the

condition 0 1elC Rω ; of maximum energy dissipation on a resistive electrode. In general, there are two sets of

A-matrix roots for a given frequency, corresponding to the respective branch. In the PR resonance interval near

the resonance frequency, the first root-set going out from the s-branch provides ( )exp 1xK x → , as 0xK H →

at 0elR → , and the second root-set coming into the e-branch is an effective non-zero solution. As was

established by simulations, for relatively high electrode conductivity with , 1CH x <<E , when

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1xK H Q<< , the ( )Im 2xK H of the first root-set is sufficiently linear in respect to the resistive

parameter ,CH x E (Fig. 5), so that this case is under further detailed analysis. The second solution showing

( )Im 2 1xK H Q− > corresponds to a rapidly decaying distribution function with characteristic decay

parameter comparable with the plate thickness. As seen from Fig. 4b, there is a point of singularity near the PR

resonance on the dispersion surfaces (Fig. 3), which is a basic factor of maximum dispersion distortion and

non-linearity.

Fig. 4 a,b. Dispersion curves of TL-TS coupled modes for a PR with resistive electrodes near the PR

antiresonance (a) and resonance (b) frequencies ( Ω values shown): - equal-resistance lines,

- internal lossy case with 100Q = , - internal lossless case with Q = ∞ ,

- limit e- and s-branches with internal losses, - the point of singularity. PCM A [4].

A comparison of two, “uncoupled” and “coupled”, approaches is presented in Fig. 5. Dependences of the

direct decay parameters ( )( )Im 0.5 x RK H− for the both cases on the relative parameter of electrode resistivity

, 1CH x <<E were calculated for several symmetric characteristic frequencies near the piezoplate TL-resonance

(AFCh for a “uncoupled” case is represented inside the figure). First dependence (dotted lines) is based on

expression (7) for an “uncoupled” case and shows a linear character. Second dependence (solid lines) derived

on the base of expression (16) corresponds to a “coupled” case and shows strong non-linearity mostly into a

16 very narrow 3 dB frequency bandwidth near basic TL-resonance. The two dependences coincide for relatively

low electrode resistivity, when the two approaches, “uncoupled” and “coupled”, give similar results for the

( )( )Im 0.5 x RK H decay parameters. At the TL-resonance ( )2( ) ,0.5Im 0.5x R C Er

K H k Q H x− → ⋅ , that

corresponds to the maximum slope of the dispersion curves. Based on dispersion simulations on wide-range

material parameters [9], the following relationship for the point of singularity (Fig. 4b) was empirically

established

( ) 0

0.5Im min 2x x RK H K H Qβ

∗ →− ≅; , (18)

where 2

2 233

44

2 3(1 ) 14 tc

kc

β

− ≈

E

E; is the parameter of piezo-elastic anisotropy influence. Applying a 0.5

coefficient for the 10% difference between Im xK and Im RK , we have a linearity threshold condition at the

TL resonance

0 1Im x r

QxH K H β

= − > (19)

Fig. 5. Dependence of the imaginary part of the planar wavenumber on electrode resistivity near the PR TL-

resonance for PCM A [4]. Solid lines are for Im xK from “energy-trap” equations, and dotted lines are for

Im RK from “uncoupled” approach – all for 100Q = . Dashed lines are for the resonance with Q = ∞ .

17

Note that the Im xK non-linearity, taking place mostly at the TL resonance, is a piezomaterial dissipative effect

low-sensitive to the figure-of-merit 33 44c cE E , so critical for classical “energy trapping”. The “weak electrode

resistivity” approach with characteristic decay distance more as much as an order of the plate thickness is

considered further, providing a simple engineering description of the measurement technique mostly important

for practice.

IV. “WEAK ELECTRODE RESISTIVITY” APPROACH

A. Effective Quality Factor of an Infinite Piezoplate with Resistive Electrodes

Let us further carry out a comparative analysis of precision features for the two systems of resonance

frequency measurements based ultimately on the frequency dependences ( )inU f and ( )outU f for a given

electrode resistivity of an infinite piezoplate under the condition (19) 10Q H x Mβ − < << . The intrinsic PR

admittance (ideal electrodes) in a vicinity of the fundamental resonance peak with 2 1k Q >> is described [20]

by ( ) (1 )rZ f R i y+; , then ( ) 21rZ f R y+; , where 2y Q χ= is the generalized resonance frequency

displacement, 1rf fχ = − is the relative frequency displacement, Q is the intrinsic material quality factor.

The sharpness of a resonance peak is defined by the frequency bandwidth at a level of 3 dB, or 2 , of the

resonance factor magnitude. For the ideal PR out inU U Z= ∼ , then 2 1 2(1 )outU y+∼ , and the PR resonance

quality factor 3r r dBQ f f Q= ∆ = .

In the case of resistive electrodes, for the traditional schema with a one-point contact (Fig. 1a) in the regime

of a frequency-constant current, according to (6): out in inU U I Z Z= = ∼ , then 2 1 4(1 )outU y+∼ . Then,

the PR input quality factor equals :

3 3r r dBQ f f Q= ∆ =% , (20)

that means a ~1.7 wider resonance peak.

For the proposed schema with two-point contacts (Fig. 1b) separated by distance W, in the regime of

a frequency-constant current, the output voltage frequency dependence near the resonance is determined in

18 accordance with (6,7) as:

( )( ) ( ) 2 1 4

2 1 40

cos(0.5arctan )(1 ) exp 1

(1 )out

out r

U f W yW y

U f x y

≅ + ⋅ − + , (21)

Fig. 6. Effective quality factor dependence vs. 0W x parameter at the resonance frequency for two-point

contacts on the resistive electrode separated by distance W. The solid line is predicted from (23), and the solid

dots are experimental data for the specimen with 0 2.6x mm= and 20M mm= .

Then, we can define the effective PR output quality factor ( )effQ W as a measure of output resonance peak

sharpness as follows:

( ) 3eff r dBQ W f f Q N= ∆ = , (22)

where the N-factor is determined by

( )

[ ]

2

02 1 4

0.5 ln2 0.25 l n 1

cos 0.5 arctan( )1

(1 )

NWNx

N

− +≅

−+

(23)

As seen from Fig. 6, when the distance between contacts 0W x> , the output resonance bandwidth of a PR with

resistive electrodes is more pointed typically as much as 2 – 3 times even of the case of the ideal PR. For a two-

sided infinite piezoplate (“A” plus “B” regions with “C” region on the left, Fig. 2), the exponential voltage

distribution factor does not change, but the effective PR impedance inZ , and hence outU , becomes twice lower.

19 B. Finite Piezoplate Frequency Responses (Electrical Lateral Boundary Effect)

The basic dimension parameters of a finite PR with resistive electrodes are presented in Fig. 7. Lateral PR

dimensions are supposed to be relatively large in respect to the PR thickness, so that the “uncoupled” approach

can be further used (19) for 0 Qx HM Mβ

> ⋅ , or as an estimation for a finite plate 2

22r

el

R H QR M β

> ⋅

.

From the continuity of currents along the electrode, the basic equation for voltage distribution

( ) ( ) ( )2 2z H z H

U x x xϕ ϕ= =−

= − is expressed as:

12 ( )m

x el xU R Z U x dx−′− = ∫ , (24)

which can be further transformed into the differential equation to be solved 2 ( ) 0xxU U xµ ′′ − = , where

2 elZ Rµ = is the voltage distribution parameter, with the common solution in the form

1 2( ) exp( ) exp( )U x P x P xµ µ= − + , where 1 2,P P are coefficients. Applying the electrical boundary

conditions (0) inU U= and 2

0 0( )

m

x xU U x dxµ

→+′− = ∫ , the total current through PR is expressed as:

( ) ( )010 0( ) 0

( ) ( )m

x x elx xM mI Z U x dx U x dx U U R−

→− →+− −′ ′= + = −∫ ∫ . (25)

Finally we have the expressions for the output voltage and effective input PR admittance:

( ) ( , )out inU W W m U= ℑ ⋅ , in inU I Z= ⋅ , ( )1 1,inZ M m Zµ− −= Ψ ⋅ , (26)

exp( ) exp( )( , )

1 exp( 2 ) 1 exp(2 )W WW m

m mµ µ

µ µ−ℑ = +

+ − + , (27)

( ) ( ) ( ) ( ) ( )1 1 1 1,

1 exp 2 1 exp 2 1 exp 2 1 exp 2M m

m m M m M mµ µ µ µ

Ψ = − + − + − + + − − + −

, (28)

where ( , )W mℑ is the voltage transfer function, ( ),M mΨ is the coefficient of the boundary effect influence,

m is the distance between the input contact point and plate corner in the direction of the output contact point on

the electroded PR surface, so that W < m < M , [0;1]W M ∈ , [0;1]m M ∈ ,

1[ ; ]Rx M Q H Mβ −∈ ∞ (7).

20

Fig. 7 (a,b). Basic geometry of a real finite PR plate (a) with resistive electrodes and input and output contacts

(the shadow density reflects the electric field intensity distribution), and lateral voltage distribution (b) along the

electrodes.

Once the PR voltage transfer function is known, it can be incorporated into the electrical measurement

system. Different particular variants of PR connection with corresponding basic expressions are summarized in

Appendix, Fig. 11 (A1-A5). The terms “center” and “corner” mean that the point of the applied input voltage is

located at the plate center with 2m M= , or at the plate corner with m M= , respectively, with “flowing”

output contact.

The traditional technique (Fig. 11 A1, A3) with one contact point is described by (26-28) with

( , ) 1 W mℑ = ( W = 0 ). For this case, comparative AFCh for connected in series inductance L and resistor R

are presented in Fig. 8a for the PR 20x10x0.2 mm with electrical load 1.5ER kOhm= . The influence of the

parasitic fixture elements result in resonance peak shift and distortion - it becomes hardly detectable.

The effect of L and R influence strongly depends on the point of the contact.

The new technique (Fig. 11 A2) with two spaced contact points is described by (26-28). Comparative AFCh

for both techniques are presented in Fig. 8b. For the new technique (Fig. 1b) with two spaced contacts there is

no influence on the resonance response from fixture parasitic inductance and resistance elements connected at

any place in the schema practically up to at least 10L Hµ= and 10R Ohm= , several orders greater than the

21 corresponding PR equivalent parameters. It takes place due to the absence of the part of the circuitry common

for the PR input and output loops. The resonance peak becomes sharp and detectable with ease. Experimental

AFCh fully correspond to those simulated with the “uncoupled” approach (Fig. 8) , as to resonance peak shape,

amplitude, and frequency shifts.

Fig. 8 (a,b). Simulated AFCh for the traditional schema A3 (Appendix) (a) with fixture inductance L and

resistance R for different locations of the contacts on the plate, and input and output AFCh for schemas A1, A2

(Appendix) (b) with different connections. PR has two resistive electrodes with 3elR Ohm= , M = 20 mm, W =

20(10) mm. Dashed line is for intrinsic PR AFCh ( 0elR = ).

Generalized dependences of relative PR resonance responses on the parameter 0x M of electrode

conductivity in comparison to the ideal PR ( outU I Z= with 0elR = ) and their distribution along the PR

electroded surface are shown in Fig. 9 for a finite PR plate. There is a maximum of output voltage outU

magnitude under 0x (electrode conductivity) variation at the PR resonance corresponding to 0x W= for

0.4W M< in a “corner” configuration. The resonance peak amplitude for a real PR is less than that for the

ideal PR when the condition 0.4W M> is satisfied [21] in a wide range of electrode conductivity. Note that

the input voltage 0in out W

U U=

= amplitude at the PR resonance is inversely proportional to the product of

effective quality factor (22) and capacitance of the effectively excited PR zone near the input point.

22

Fig. 9. Relative PR responses vs. parameter 0x M of electrode conductivity at the resonance frequency for a

set of input-output distances W ( ( )I const f= ). Finite PR plate.

Since the voltage distribution factor for a finite PR in comparison to the case of infinite PR (6) can be

expressed as ( )exp( ) ,out inU U W W mµ= − ⋅ ℜ , then

( , ) exp( ) ( , )W m W W mµℜ = ⋅ ℑ , (29)

and similar for the output voltage ( )2 exp( ) , ,out elU I R Z W Dim W M mµ= − ⋅ with

( ), , exp( ) ( , ) ( , )Dim W M m W W m M mµ= ⋅ ℑ Ψ , (30)

where ( , )W mℜ and ( ), ,Dim W M m are the boundary “correction” coefficients to the case of a finite plate.

Taking place an increase of the output response (resonance peak widening) is caused by the influence of the

boundary effect (Fig. 10). The two cases of finite and infinite plates can be considered approximately equal for

the condition 0 0.5x M< and 02m x> at the resonance under “uncoupled” approach restrictions.

In the limit of relatively high electrode conductivity with 1Rx M << it follows from (26-28) that:

inZ Z µ= and ( ) exp( )W W µℑ = − , then ( )expoutU I Z W µ µ= − when the input contact is placed at

23

the corner of the plate, and 2inZ Z µ= and ( ) exp( )W W µℑ = − , then ( )exp 2outU I Z W µ µ= −

when the input contact is placed at the center of the plate. These results are similar to the case of infinite PR.

Fig. 10. Distribution of relative, in respect to the ideal PR, resonance output responses along the PR electrode

for different points ( x ) the input voltage applied to, Wx x W− = , 0x M = 0.26. Solid lines are predicted from

(26-28), and the solid dots are experimental data for the specimen 20x10x0.2 mm.

V. CONCLUSION

In thin (200…50 mc) and large area (tens sq. cm) PZT-ceramics PRs with low resonance impedance (much

less 1 Ohm), caused by high frequency (tens MHz) and high PR capacitance (tens nF), the effect of electrode

resistivity and parasitic resistive and inductive elements in the measurement fixture results in significant

distortion of the measured thickness-mode resonance frequency response, including high-order PR harmonics.

The theory of the coupled-modes “energy-trap” effect in a PR with resistive electrodes was developed.

As the effect of electrode resistivity influence is dissipative, the lateral gradient of vibrational characteristic s,

being proportional to Q , allows to apply an “uncoupled” approach to the PR resonance behavior description

for typical piezoceramic materials.

24 A simple fixture principle with two contact points on the electrode was proposed providing deep sharpening

and exact determination of the resonance frequencies even in the presence of the parasitic factors due to absence

of common with the PR input and output circuitry loops. As a result, for optimal disposition of the contact

fingers on the finite piezoplate the resonance bandwidth of a real PR with resistive electrodes is more pointed

than even in the case of the ideal PR. There is no need any additional compensative elements or to make very

accurate measurements of the parasitic elements that are associated with a specimen holding fixture to

algebraically subtract their influence.

The proposed analysis and method can be also applied to bulk multilayer piezo-transducers consisting of

multiple thin piezo-layers co-fired through thin metal layers, and can be used particularly for investigating of

resonance properties distribution (inhomogeneity) along the surface of a thin piezoplate (film).

REFERENCES

[1] T. R. Meeker, "Thickness mode piezoelectric transducers,” Ultrasonics, vol. 10-1, pp. 26-36, 1972.

[2] IEC Standard. Guide to Dynamic Measurements of Piezoelectric Ceramics with High Electromechanical

Coupling, Publication 483, 1976.

[3] IRE Standards on Piezoelectric Crystals - The Piezoelectric Vibrator: Definitions and Methods of

Measurement, Proc. IRE, vol. 45, no. 3, pp. 353-358, 1957.

[4] P. Schnabel, “Dispersion of thickness vibration of piezoceramic disk resonators,” IEEE Trans. Ultrason.,

Ferroelect., Freq. Contr., vol. SU-25, no. 1, pp. 16-24, 1978.

[5] R. Holland and E.P. EerNisse, Design of resonant piezoelectric devices, Cambridge: MIT Press, 1968.

[6] A.A. Comparini and J.J. Hannon, “Flexural, Width-Shear, and Width-Twist vibrations of thin rectangular

crystal plates,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. SU-21, no. 2, pp. 130-135, 1974.

[7] D.C. Gazis and R.D. Mindlin, “Extensional vibrations and waves in a circular disk and a semi-infinite

plate,” J. App. Mech., vol. 27, pp. 541-547, 1960.

25 [8] M. Onoe, H.F. Tiersten, and A.H. Meitzler, "Shift in the location of resonant frequencies caused by large

electromechanical coupling in thickness-mode resonators," JASA, vol. 35, no. 1, pp. 36-42, 1963.

[9] OCT 11 0444-87. Piezoceramic Materials. Specifications, Moscow: ElectronStandard, 1988.

[10] D.A. Berlincourt, D.R. Curran, and H. Jaffe, "Piezoelectric and piezomagnetic materials and their function

in transducers", Physical Acoustics, vol. 1, part A, Editor - W.P.Mason, New York: Academic Press, 1964.

[11] A.V. Mezheritsky, Theoretical and Experimental Investigation of Complex Electro-Elastic Constants of

Piezoceramics, Moscow Institute of Physics and Technology (MIPT), Ph.D. Dissertation, 1985.

[12] S.N. Wickstrom and T.R. Meeker, "Electrical measurement of high frequency ceramic resonators,"

Proc. IEEE Ultrasonics Symposium, pp. 717 – 722, 1989.

[13] A.V. Mezheritsky, “Measurement of resonance characteristics of high frequency piezoceramic

resonators with high electric capacitance,” Measurement techniques (Russian), no. 9, pp. 53 – 55, 1990.

[14] S. Sherrit, N. Gauthier, E.D. Wiederick, B.K. Mukherjee, and S.E. Prasad, “The effect electrode

materials on measured piezoelectric properties of ceramics and ceramic-polymer composites,”

Proc. IEEE Ultrasonics Symposium, pp. 346 – 349, 1991.

[15] R.A. Dorey and R.W. Whatmore, “Apparent reduction in the value of the d33 piezoelectric coefficient in

PZT thick films,” Integrated Ferroelectrics, vol. 50, pp. 111 – 119, 2002

[16] Y. Sasaki, S. Takahashi, and M. Yamamoto, “Mechanical quality factor of multilayer piezoelectric

ceramic transducers,” Jpn. J. Appl. Phys., vol. 40, part 1, no. 5B, pp. 3549-3551, 2001.

[17] G.L. Ragen, Microwave Transmission Circuits, New York: Dover Publications, 1965.

[18] R.P. Feynman, R.B. Leighton, M.L. Sands, The Feynman Lectures on Physics. Electrodynamics,

Massachusets: Addison Wesley, 1963.

[19] O.P. Gribovsky, Ceramic Solid Circuits, Moscow: Energy, 1971.

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

[21] A.V. Mezheritsky, “Method of measurement of the resonance frequencies of piezoceramic resonator,”

Russia patent 1273840, Patent Bulletin no. 44, 1986

26

APPENDIX

There are a number of different variants of PR connection, which are presented in Fig. 11 (A1-A5).

Traditional schema

with parasitic elements in the input circuitry loop only.

out in

E in

U ZE R i L R Zω

=+ + +% A 1 )

( , )out in

E in

U Z D mE R i L R Zω

⋅ ℑ=

+ + +% A 2 )

Traditional schema with parasitic elements in the part of the circuit common for input and output loops.

out in

E in

U Z i L RE R i L R Z

ωω+ +

=+ + +% A 3 )

0.5 1 ( , )inout

E in

Z D m i L RUE R i L R Z

ωω

+ ℑ + +=

+ + +% A 4 )

(both resistive electrodes)

( , )out in

E in

U Z D m i L RE R i L R Z

ωω

⋅ ℑ + +=

+ + +% A 5 )

(the only upper resistive electrode)

Fig. 11 (A1-A5). Variants of PR connection and corresponding basic expressions for PR responses.

An increased number of electrical contacts in the proposed PR fixing system definitely increases probability

of a contact failure in one of them. Such situations were modeled for the researched PR 20x10x0.2 mm (M = 20

mm) with input electric load 1.5ER kOhm= (Fig. 12).

27

Fig. 12 (a,b,c). Simulated AFCh for schema A4 (Appendix) under variation of:

a - inductance L (at R = 0) for W = 8 mm, b - inductance L (at R = 0) for W = 20 mm,

c - resistance R (at L = 0) for W = 20 and 10 mm. Dashed line is for intrinsic PR AFCh ( 0elR = ).

The schema A4 (Appendix) with “accidentally” disconnected output “ground” contact was considered. In this

case the effect of the fixture parasitic inductance and resistance takes place again, and the character of the

influence depends greatly on the point of the output contact. Characteristic features of the dependences are

connected to a specific voltage phase rotation along the electrode, and further voltages superposit ion.

Experimental AFCh, corresponding to schema A4 conditions, show satisfactory coincidence with those

presented in Fig. 12, as to resonance peak shape and frequency shifts.

28 Glossary

( ), , ( ) ,r r aZ Z R R Rφ – PT impedance, its phase, resonance and antiresonance resistances (linear values)

,r z , inZ – elementary electrode resistance and PR impedance in the model, effective input PR impedance

i jcE,D, i jh , i jε S

, i jk (k ) – piezomaterial constants (generalized notation)

, , ,D E T S – electric field induction and strength, mechanical stress and strain

, ( ) ,m r effQ Q Q Q – standardized PCM , PR resonance, and effective quality factors

0 0 1( ),C C L – PT capacitance (linear value), and its equivalent inductance

, ,f ω Ω – frequency, angular and dimensionless frequencies

, , ,r a rf f δ ν – PR resonance and antiresonance frequencies, relative resonance frequency interval, and overtone

number , yχ – frequency displacement and generalized frequency displacement

Eω – TL-mode characteristic frequency ( E rω ω; )

,x z – space coordinates

( ), ,el el elR R h σ – electrode resistance (linear value), its thickness and unit resistance

, ,in outU Uϕ – electric potential, PT input and output voltages

, , , ,EE I R R L% – schema electromotive force, total current through PR, load, parasitic resistance and

inductance

K and ,x RK K – complex thickness and lateral wavenumbers

0, , ,R Cx x x µ – characteristic decay parameters

, ,M H Φ – PR length, thickness, and electrode area

, , , ,WW m x x t – parameters of PR measurement configuration

, , , ,Dim Fℑ Ψ ℜ – voltage distribution parameters ,, ,e s

i jA a G – matrix, its elements, and parameter of electrode resistivity influence

,x zu u – local PT displacements

( ) , ( ) , ( )f z g z h z – z-coordinate vibrational functions

29

30

The author A.V. Mezheritsky : 525 Ocean Pkwy, # 3 J, Brooklyn, NY 11218 [email protected] Ph.D. in Physics (1985, MIPT, Moscow, Russia), IEEE Member