Analysis of Multiterminal Piezoelectric Plates

5.4, 5.18 Received 29 August 1966

Analysis of Multiterminal Piezoelectric Plates*

RICHARD HOLLAND

Sandia Laboratory, Albuquerque, New Mexico 87115

Mathematical techniques are discussed whereby one may analyze a thin piezoelectric plate with arbitrary electrodes, external electrical loads, and external mechanical tractions. These techniques include tensor Green's-function operations, Fourier-eigenmode expansions, and circuit-theory manipulations. Applica- tion of these techniques leads to admittance matrices interrelating the linear responses of the various mechanical terminals (i.e., the sites of external-force application) and electrical terminals (i.e., electrodes). It is demonstrated that these admittance matrices may be represented by quite simple equivalent circuits. External mechanical forces may be applied either at a point or over a distributed area. These two types are represented best by separate methods, both of which are discussed here. Two specific, nontrivial examples of actual piezoelectric devices are analyzed at the conclusion of this work, and results are compared with experimental observations. These examples are a symmetrically electroded ferroelectric ceramic disk bonded to a substrate and an asymmetrically electroded, mechanically free, ferroelectric ceramic disk. Effects due to loss, nonuniform density, and nonuniform material coefficients (elastic, dielectric, and piezoelectric) have been included in this work. However, finite-plate-thickness effects, such as electrical fringing and particle accelerations perpendicular to the plane of the plate, are not included.

INTRODUCTION

IEZOELECTRIC devices with several electrical- and mechanical-input terminals have been pro- posed or designed for a number of applications. These applications include multistage transducers, •-4 frequency-selective transformers, 5.ø multistate memory elements, 7.8 FM discriminators, ø•ø and electrically tuned filters. ø-n

Various methods have been developed for predicting mathematically the characteristics of these devices. For instance, van der Veen • has solved the problem of a

* Parts of this paper were included in a doctoral thesis sub- mitted to the Massachusetts Institute of Technology (Jan. 1966).

• G. E. Martin, J. Acoust. Soc. Am. 35, 510-520 (1963). • G. E. Martin, J. Acoust. Soc. Am. 36, 1366-1370 (1964). 8 G. E. Martin, J. Acoust. Soc. Am. 36, 1496-1506 (1964). • D. K. Winslow and H. J. Shaw, IEEE Intern. Cony. Record

14, Pt. 5, 26-31 (1966). 5 E. C. Munk, Philips Res. Repts. 20, 170-189 (1965). 60. M. Stuetzer, "Linear Theory of Piezoelectric Power Trans-

formers," Sandia Corp. Res. Rept. SC-RR-66-414 (1966). 7 C. E. Land, G. W. Smith, and I. D. McKinney, IEEE Intern.

Cony. Record 12, Pt. 9, 149-160 (1964). s D. G. Schueler, Wescon Cony. Record 10, Paper 3/4 (1966). 9 C. E. Land and D. G. Schueler, "Ferroelectric Ceramic

Devices for Use in Linear Microcircuits," IEEE Trans. Sonics & Ultrasonics (to be published).

•0 C. E. Land, Wescon Cony. Record 10, Paper 3/5 (1966). n C. E. Land, IEEE Intern. Conv. Record 13, Pt. 7, 51-68

(1965). n B. van der Veen, Philips Res. Repts. 11, 66-79 (1956).

slim piezoelectric bar with two symmetric electrodes. More recently, Munk 5 has analyzed a thin ferroelectric disk with two concentric electrodes. These authors have

taken the approach of solving the elastic-wave equation separately for each of the electrode regions, and then of splicing the solutions at the electrode interfaces. This splicing operation consists mathematically of applying the boundary conditions of continuous stress and displacement across the interfaces. Equivalent approaches have been applied to N-segment tubes by Martin •-a and to bars and disks with two or three electrodes by Land. n The work of Land and Martin is superficially different from that of Munk and van der Veen in that the former

authors chose to represent the mechanical aspects of their problem by electrical-circuit analogs, rather than treat the mechanical variables directly.

The primary difficulty with the splicing approach to complicated piezoelectric structures is its tediousness. If N is the number of electrode interfaces at which solu-

tion splicing is performed, one must solve 2N coupled transcendental equations. Consequently, this technique rapidly becomes impractical with increasing electrode complexity.

For thin piezoelectric plates with complicated electroding, it is more convenient to observe that particle displacement obeys a wave equation that is homogeneous everywhere except at electrode edges. The inhomo-

940 Volume 41 Number 4 Part 2 1967

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17:15:10

ANALYSIS OF PIEZOELECTRIC PLATES

geneous terms that occur at electrode edges are equivalent to concentrated mechanical force or drive. Taking this approach, it is possible to find a single wave equation that is valid over the entire plate, so that no solution splicing is necessary at the electrode edges. This single wave equation may then be integrated by a simple extension of conventional Green's-function techniques or by eigenmode expansion. Jacobsen •a has utilized the Green's-function approach outlined here in studying a fairly simple class of one- or two-interface, straight-rod microwave transducers. The eigenmode-expansion technique has been applied to a very general class of piezoelectric bodies by Lewis) 4 Lewis has, however, re- stricted himself to configurations having only one electrical terminal and no mechanical terminals.

In the present paper, we utilize a combination of Green's-function and eigenmode techniques. The approach of this paper has been previously applied by the present author to solve specific problems, such as N- electrode bars, TM N-electrode symmetric disks, TM and N-layer plates?

I. DEFIBIITIONS AND SYMBOLS

Consider the thin, piezoelectric plate completely covered by arbitrarily shaped electrodes that is shown in Fig. 1. In the plane-stress and no-fringing approxi- mations, stress, strain, electric field, and electric displacement are related by

T • = • • •S• •- a •Ea, (1 a)

Da = •aktSktq- eaas•ga. (lb)

The Einstein summation convention is applied in Eqs. 1 and • all subsequent equations; subscripts i, j, k, l are understood to range over 1 and 2. The s•bol •q•t• represents the usual two-dimensional stiffness tensor evaluated at constant electric field, •8 while •aq and eaa • are two-dimensional, plane-stress piezoelectric and dielectric coefficients related to more-standard symbols • by

eaasp= eaaY--daiida•t•o•t •, i, j, k, l= 1, 2.

One can show that eaa sp is the dielectric constant evMu- ated with mechanical motion prevented in the plane of the plate but not pe•endicular to the plate.

We assume that all electroelastic variables contain a suppressed e i•* dependence. In order to represent the

•a E. H. Jacobsen, J. Acoust. Soc. Am. 32, 949-953 (1960). •a J. A. Lewis, Bell System Tech. J. 40, 1259-1280 (1961). •* R. Holland, Proc. IEEE 54, 968-975 (1966). •0 R. Holland, Wescon Cony. Record 10, Paper 3/3 (1966). •* R. Holland, IEEE Trans. So'cs & Ultrasonics 14, 21-33

(1967). •sW. Voigt, Lehrb•h der Kristallphysik (B. G. Teubner,

Lepzig, 1910), pp. 679-681. • "Standards on Piezodectric Crystals, 1949," Proc. IRE 37,

1378-1395 (1949).

-V 4 14 +

•j Y

-v• +• v 2

>x

Fro. 1. Piezoelectric plate covered by an arbitrary electrode configuration.

most general phenomenological loss, all the coefficients appearing in Eqs. 1 are assumed to be complex. zø These coefficients are permitted to vary over the plate.

II. WAVE EQUATION AND BOUNDARY CONDITIOBIS

Let us represent the density per unit area of the plate by p and the thickness by r. We assume that r is constant, but p is permitted to vary. Let • represent particle displacement and let a force per unit area •5j be applied in the plane of the plate (see Fig. 1). This force may represent either an external mechanical drive or a mechanical load. The differential equation of motion for the plate then becomes

-- txo•'•i= rr•.•q-Sj, (3)

where the comma represents covariant differentiation. Combining Eqs. 1 and 3 leads to a wave equation that

is valid over the entire plate'

(4)

If the plate is free, O;i is zero, and this equation is homogeneous everywhere except at the electrode edges. This consideration led the authors mentioned in the Introduction to solve it piecewise in the individual electroded regions, and then to splice solutions at the interfaces. However, Eq. 4 clearly demonstrates the equivalence of the piezoelectric driving term - r ([a•Ea) .• to the mechanical driving term •Yi. It is possible to for- mulate procedures to solve the inhomogeneous equation of motion for the entire plate at once, and, thus, to eliminate the need for splicing of solutions.

If a force per unit length F• is applied in an outward direction around the edge of the plate (see Fig. 1), the stress boundary condition there may be expressed as

(s)

In this equation, N• is the outward-pointing unit normal vector.

•o G. E. Martin, USN J. Underwater Acoust. 15, 329-332 (1965).

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III. GREEN'S FUNCTION

Equations 4 and 5 constitute an inhomogeneous bounary-value problem for the displacement Analogous boundary-value problems with scalar un- knowns are referred to as Sturm-Liouville systems.2•. 2•- We now illustrate a method of extending the conventional Green's-function approach to Sturm-Liouville problems so that it is applicable to the present vector differential equation.

We must discuss points on the plate from two view- points: (1) let r represent a point at which drive (either mechanical or piezoelectric) is being applied and (2) let R represent a point at which resulting motion is being observed. We may thus refer to r as the source coordinate and R as the observer coordinate.

It is possible to define vector transformations from the source to the observer system. For example, if we characterize vectors evaluated at r by lower-case indices and vectors evaluated at R by upper-case indices, then /5a represents the identity transformation or uniform translation from r to R:

where Vi/5a = Vz, (6)

V•= V•(r), (7)

Vz= Vr(R), i, I= 1, 2.

We note that this transformation is independent of the points r and R at which it operates. A more general vector transformation may, of course, be defined to depend on both r and R.

The Green's function relevant to the problem at hand may be regarded as such a vector transformation de- pending on r and R. We represent it by G`4(rIR)a, and define it to obey the differential equation

= -/5`4 (r- R)/5jr. (8)

In Eq. 8,/5`4 represents a finite delta function whose total integral is unity'

-•, for [r-R]<(A/,r)i (9)

OTHERWISE.

Equation 8 does not completely specify the Green's function: some stress-type boundary condition such as Eq. 5 must be given as well. The most useful result will subsequently be obtained if we specify that boundary condition to be homogeneous:

(lO)

•' 21 p.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Co., Inc., New York, 1953), Pt. 1, pp. 719- 757.

• F. B. Hildebrand, Methods of Applied Mathematics (Prentice- Hall Publ. Co., Inc., Englewood Cliffs, N.J., 1952), pp. 144-147.

Having defined the Green's function by Eqs. 8 and 10, we now utilize it in the conventional way by defining an interaction integral between •(r) and G`4 (rl R). In this case, this integral is an R-system vector'

-- ['•i•k•//•, z (r)-],•G`4 (rlR)ir}da. (11)

Here, a is the total surface area of the plate and integration is over r. This integral has properties that are very useful in solving Eqs. 4 and 5. Application of the product rule for derivatives and of the divergence theorem enables us to convert this surface integral over the plate to a contour integral around the edge plate s:

J`4•= -• (Fiq-rN,ga,iEa)G`4 (rl R)i,ds. (•2)

Boundary conditions Eqs. 5 and 10 have also been used in obtaining Eq. 12.

On the other hand, substitution of the differential equations Eqs. 4 and 8 into Eq. 11 yields

f f gs(r)a(r-R)asda

(13)

It is consequently possible to eliminate J•z from Eqs. 12 and 13, and thus to find

f f •i (r)/5`4 (r- R)/5irda

--//

(14)

According to Eq. 8, the second derivative of G`4(r[R)a does not exist for vanishing A. However, we see shortly that the Green's function itself is well defined in this limit. Consequently, Eq. 14 is well defined in the limit as A -• 0. If this limit is taken, the lefthand side goes to •z(R), the particle displacement at the point of observation. The righthand side of Eq. 14 becomes an integral of the source terms weighted by the Green's function evaluated in the limit of vanishing A.



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IV. EVALUATION OF THE GREEN'S FUNCTION

Let us consider now the homogeneous form of Eqs. 4 and 5 for •j. This system will have nontrivial solutions only for certain discrete complex frequencies •o,. These frequencies are often called the eigenfrequencies of the system, and the corresponding displacement functions are called the eigenmodes:

a, and apply Eqs. 15 and 17, we find that

(19)

(15a)

As the •i (") are obtained by setting Ea equal to zero in Eq. 4, they represent the short-circuit eigenmodes. Note that, in the plane-stress, no-fringing approximation, these eigenmodes are independent of the dielectric and piezoelectric properties of the plate?

Since the •(•) are defined with Ea=0, Eq. lb may be used to associate an electric displacement distribution with these eigenmodes'

(16)

It is known that the eigenmodes of a system of equations such as Eq. 15 are orthogonal when weighted by the material density o. (Ref. 24). (If •o, 2 is a degenerate eigenvalue, the usual diagonalization procedure will be necessary to make the eigenmodes corresponding to orthogonal among themselves.) It is convenient to normalize these eigenmodes to unity according to the rule

(17) We now postulate that Ga (rl R)a can be expressed as

a Fourier-eigenmode series in the •i(") (Ref. 25)'

G,• (r[ R)a=Y'. B,•(•) (R)•//,(•) (r). (18)

The Ba(")(R)r are the Fourier coefficients to be determined. This solution satisfies the boundary condition Eq. 10, as a term-by-term comparison with Eq. 15b shows. The Bx(") may be selected so that Eq. 18 satisfies Eq. 8, the differential equation for Ga (rl R)a, as well. In particular, if we substitute Eq. 18 into Eq. 8, multi- ply both sides of the result by •?)(r), integrate over

•.a The solutions to Eqs. 15 describe the actual resonant behavior of a plate of thickness r only for modes for which r is a small fraction of an acoustic wavelength.

•.4 A. E. H. Love, A Treatise on the Mathematical Theory of Elas- tidty (Dover Publications, Inc., New York, 1944), pp. 180-181.

•.sThis summation and all subsequent summations on the eigenmodes are understood to range over the modes for which plate thickness is a small fraction of an acoustic wavelength. For to low enough to satisfy this condition, Eq. 18 has the proper form to satisfy Eqs. 8 and 10 for G•(rlR).

Equation 19 is well behaved in the limit as A • 0, yielding

S0(rl R)a=S(r I R)a

=E J-•/,(")(r)•/zt")(R)/(cøff-cø•)-J ß (20)

V. EVALUATION OF THE PARTICLE DISPLACEMENT

If Eq. 20 for the Green's function is substituted into Eq. 14 with A-0, one obtains the following expression for the particle displacement at the point of observation'

XZ E//s (•) (r)//, © (R)/

X•. J-•i(") (r)•z(") (R)/ (oo•-co•')-lds. (21)

This equation may be simplified considerably by applying the divergence theorem and the product derivative rule to the piezoelectric terms. It is also convenient hereafter to regard the forces on the edge of the plate Fi as a line singularity contribution to the force per unit area 5:s acting on the plate surface. If we designate the total force per unit area (ifs and F•) by ifs', we can obtain the restfit from Eq. 21

(22)

VI. EVALUATION OF THE ELECTRODE CURRENTS

Let us consider a plate having N electrical terminals indexed by p and q. The current at electrode q is the time derivative of the electric displacement integrated over



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R. HOLLAND

electrode q. In view of Eq. lb, this becomes

(23)

and the 8, are weighting factors multiplying those distributions.

The electromechanical portion of the admittance matrix is obtained by differentiating Iq as given in Eq. 24 with respect to 88 as defined by Eq. 27'

where aq is the portion of the plate covered by electrode 0I• q. Substitution of the particle displacement from Eq. 22 08, then yields

aq

-}-jwVgCo(q), (24)

where Vg= rE3 is the voltage applied at electrode q, and where Co(q) is the capacitance at electrode q evaluated with motion prevented in the xy plane'

(25) Co(q) =- e3aS•da. T

aq

VII. EVALUATION OF THE ADMITTANCE MATRIX

The derivative of I• with respect to the voltage applied at electrode p, V•, represents the electrical portion of the admittance matrix of the device shown in Fig. 1. If we observe that differentiation of rE• by V• yields 1 under electrode p and zero elsewhere, we see from Eq. 24 that

(26) x Equation 16 for Da(") has also been used in obtaining this result. The electrical reciprocity of the configuration in question is readily apparent from this relation.

Let us assume that there are M independent force distributions acting on the sample (where M may be infinite). Let these distributions be indexed by s and t. Under this assumption, we can expand ff• in the form

M

•,'= Y'. f •(•) •., (27)

where the f•(•) represent the M distribution patterns

aq •

x (28)

The 8• weighting factors may be regarded as the mechanical voltage analogs in this system. Correspond- ing mechanical-current analogs may be defined by •'6

Ee=jw//fk (t) l&da. a

(29)

In the case of a point-force application, fk(e) is a delta function, and this definition reduces to particle velocity, which is a conventional current analog. In the more general case of distributed force application, this definition leads to a convenient electromechanical reciprocity, which is discussed below.

Using Eq. 29, we may alternatively express the electromechanical portion of the admittance matrix as the derivative of •e with respect to V•. If we substitute Eq. 22, for • into Eq. 29, we obtain

OV•

X (w•'-w•')-•]. (30) Comparison of Eqs. 28 and 30 indicates that the electromechanical portion of the admittance matrix also possesses reciprocity. This reciprocity is a consequence of the definition of •.

Finally, we may obtain the mechanical portion of the admittance by differentiating • with respect to 88'

x •.0 The product f•(o gt has dimensions of force/area in this for-

mulation. Similarly, the product g,•, has dimensions of power. However, the quantities f•(e)_ •e 2- , gt, and are individually determined only within a multiplicative constant by Eqs. 27 and 29. That constant is not necessarily dimensionless, and may even depend on t; its choice is dictated by convenience. As there is no single choice that is universally most convenient, f•(o, g•, and E• may have t-dependent dimensionality in a given problem and different dimensions in different problems.



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VIII. EQUIVALENT CIRCUIT

The piezoelectric plate shown in Fig. 1 represents a distributed medium problem with an infinite set of resonances. As such, it cannot be represented by a finite number of lumped equivalent-circuit elements. How- ever, it will usually happen that one is interested in operating a piezoelectric device near a resonance that is well separated from all other resonances. Under this circumstance, only one term of each admittance-matrix series, Eqs. 26, 28, 30, 31, will contribute significantly. If this term is designated as the nth, the admittance matrix of the plate shown in Fig. 1 may be represented by the equivalent circuit given in Fig. 2. The elements of this circuit are related to the material coefficients and

geometrical parameters of the plate by

and

L,•C,•=co. -•-, (32)

•= (1/oo•c•o f f Da(•)da, (33)

(34)

The nature of the transformer arrangement in this equivalent circuit is such that one of the following-- [lr/,,(s= 1, 2,. --, M), •,v(p = 1, 2,..., N), L,, or C,--- may be chosen arbitrarily. Once this has been done, the other circuit elements are all specified by Eqs. 32-34.

IX. APPLICATIONS

A. Point-Force Application

In many cases of practical interest, external mechanical forces or loads are essentially applied to the plate at a point. This is the case, for example, if the plate is supported by the wires connecting the electrodes to external circuitry. Let us now examine the form taken by our definitions of mechanical terminal variables in this case.

A point force applied at r0 may always be expressed in a form compatible with Eq. 27 by the equation

2

if/= Y'. /•(r-- r0)ui(') 8,, (35)

! Cn L n

'•o(P) P '-/•n s: I

Vq TCo(q) q ,•/nt: I I - i I

ELECTRICAL. TERMINALS

I !

I •t•t ,, ,,

MECHANICAl TERM INALS

Fro. 2. Equivalent circuit of an arbitrarily electroded piezoelectric plate near the nth resonance.

a point in a plane. The mechanical current analogs are obtained by substituting Eq. 36 into Eq. 29'

•,= jcou•,C•) • (r0). (37)

Finally, we obtain the equivalent transformer turns ratios by substituting Eq. 36 into Eq. 34'

(38)

If the point force acting at r0 represents an independent driving source, Eqs. 35-38 completely describe that source in a manner consistent with the previously developed formulation. It is represented by two mechanical terminal pairs having equivalent voltages specified by Eq. 35 and equivalent currents specified by Eq. 37.

However, it will usually happen that a point force is of a loading rather than a driving nature. For example, we may have a mechanical load at r0 representable by a mechanical admittance tensor ykz, where

2

jco•k(r0)=y•t Y'. u•c')5,. (39)

In Eq. 39, y• relates the particle velocity at r0, jco•(r0), to the total force applied by the load at r0, • u• (') 8,.

It is desirable to determine the effect of the admit-

tance y•z on the equivalent circuit of Fig. 2. This effect may be evaluated by multiplying Eq. 39 by uk (t) in order to obtain •, as specified by Eq. 37:

2

Zt= u•C*)y•z Y'. u•C,) $,. (40)

where the u• (') are any two convenient orthogonal unit vectors, such as the unit vectors along the x and y axes. Thus, each point force can be represented by two independent force-distribution patterns, which are given by

f•c,)=/5(r-ro)u• (•), s=l, 2, (36)

and which correspond to the two degrees of freedom of

Differentiating Eq. 40 with respect to $, then yields the admittance matrix of the equivalent two-port load'

(Y•,) eq = u•Cøy•zuzC'). (41)

This equivalent load is connected to the two mechanical terminal pairs associated with r0 to incorporate the effects of y•z into the equivalent circuit (see Fig. 3).

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R. HOLLAND

• Cn I

Vq i:,./•'nq I I

ELECTRICAL TERMINALS

L n

"////n z: !

MECHANICAL LOAD

Fro. 3. Effect of a mechanical point load on the equivalent circuit of a piezoelectric plate.

B. Distributed-Force Application

If force is applied over a distributed area instead of at discrete points, the previously outlined approach is obviously inapplicable. For continuous force distributions, one possibility is to expand ffk' in terms of the mechanical eigenmodes ß

ffk'= Z p• (•) g,. (42) $

Other definitions of the mechanical analogs are, of course, possible, and may be more efficient in cases of specific force distribution. However, expansion Eq. 42 is applicable to an arbitrary force distribution, pro- vided that the usual assumptions regarding the com- pleteness of the •(•) are valid.

There are now an infinite number of independent force-distribution patterns f•cs) (see Eq. 27):

f•(8) = p•('). (43)

However, at frequencies low enough to make our plane- stress approximation valid, it is permissable to ignore the force-distribution patterns that are associated with modes high enough to be excluded by Ref. 25. The equivalent mechanical currents may be obtained by substituting Eqs. 22 and 43 into Eq. 29'

(44)

The orthogonality relation, Eq. 17, is used several times in deriving this result. It may be observed that, if the mechanical-terminal variables 8, and F•, are defined in the manner of Eq. 42, there is no transfer admittance between the various mechanical ports'

Y,,= ;-- o9. (4s)

Alternatively, substitution of Eq. 43 for fk (') into Eq. 34 for •)r•,, the mechanical-transformer-turns ratio, yields

•.,=&.,/co.C,,i. (46)

Consequently, while a large number of mechanical terminals are required by expansion Eq. 42, only one of

these terminals (the nth) will have a nonzero transformer coupling ratio to the external mechanical forces.

Equations 42-46 are sufficient to characterize an independent distributed mechanical drive. However, as in the case of the point force, the characterization of a mechanical load is somewhat more difficult.

Let us assume that a distributed mechanical load is

present that is characterizable by the admittance tensor r•k•, so that

(47)

In other words, let the particle velocity be locally related to the applied force per unit area by •. We now desire to relate this r• to some equivalent mechanical load in our circuit representation. To do this, we multi- ply Eq. 47 by p•k (t) and integrate over a, thus obtaining F•t, as defined in Eq. 29'

(48)

Differentiation with respect to & yields the admittance matrix of a multiterminal mechanical load.

(49) Connecting this load to our equivalent circuit then characterizes pictorially the effect of r•t on the plate (see Fig. 4).

However, only the nth terminal pair of this load is coupled into the mechanical system in the vicinity of the nth isolated resonance. By virtue of Eq. 46, all the other terminals of this equivalent mechanical load are connected to the "1" side of a 0:1 transformer, and thus in effect are open-circuited. Consequently, the effect of this mechanical load is to place an impedance of value

C n L n

(o:•)

d{n n :1 ( !: t.o' n Cn I/z)

•/'{'n,n +1 ' I (o:1)

(Yts)eq

ELECTRICAL TERMINALS MECHANICAL LOAD

FzG. 4. Effect of a mechanical distributed load on the equivalent circuit of a piezoelectric plate.

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] Cn

V_p ?o(P)

1


Ln

,.///{n n; I (1' (unCnl/z)

Znn

MECHANICAL LOAD

Fro. 5. Reduced equivalent circuit of a mechanically loaded piezoelectric plate.

Z,• across the nth mechanical terminal pair of the circuit of Fig. 4, where •7

As the "0" side of a 0:1 transformer is, by definition, a short circuit, all the mechanical transformers except the nth look like short circuits when considered from the

side opposite the mechanical load. This is illustrated in Fig. 5.

C. incompletely Electroded Plates

We have so far dealt only with plates in which the unelectroded areas were negligible. The chief simplify- ing effect of this assumption is that short-circuiting all the electrodes then guarantees that Ea will everywhere be zero.

However, there is a fairly easy technique for extending the complete electroding results to the partial-electroding case. In particular, let us imagine the unelectroded areas to be completely covered by infinitesimal, nontouching, open-circuited electrodes. Excepting fringing effects between the infinitesimal electrodes, this imaginary arrangement is physically equivalent to the unelectroded situation. However, as the piezoelectrically induced voltages on adjacent electrodes will differ only by infinitesimal amounts, fringing effects in the imaginary arrangement will be on the order of second-order differentials and, thus, negligible. Conse- quently, the effects of the unelectroded areas may be represented in the equivalent circuit by an infinite set of electrical terminals, each open-circuited except for infinitesimal capacitors representing clamped capacitance effects [see Fig. 6(a)•. We index these terminals byr.

Upon reflection through the associated transformers [Fig. 6(b)•, the capacitors representing damped dielectric effects will individually present impedances of the form

•'* The (Yts)eq matrix to be inverted in Eq. 50 is truncated according to the same rule as that set forth in Ref. 25 for the trun- cation of series: matrix elements corresponding to modes that violate the plane-stress assumption are excluded.

where, according to Eqs. 25 and 33,

Co(r)- eooS*a,/r, (52)

•,,= Dat")a,/o•,C, «. (53)

It should be observed that the area elements a, are infinitesimal.

Upon summing these infinitesimal impedances and converting the summation to an integral, we obtain an over-all equivalent capacitance for the unelectroded area [-Fig. 6(c)].

•.•'C. (Do t*))•' 1 -• Here, au is the total unelectroded area. This equivalent capacitance enters the equivalent circuit in series with the motional capacitance of mode n, C, [Fig. 6(c)•. Consequently, unelectroded areas have the effect of raising resonant frequencies.

It is interesting to compare Ceq of Eq. 54 with the equivalent capacitance in series with C, that represents the effect of an open-circuited, continuous electrode covering au. From Eqs. 25 and•33,_iwe•ifind that

f f oaa/If f Do(n)aa] •U

(55)

Consequently, we see that open-circuited electrodes

I

I

V._p Co(p) ß


(a)

Cn Ln I

i:,./•n p ,.//in s: I

I: ,Y['nq ,_ffff'nt i ft X

_• • MECHANICAL -- ••J• _ •--i TERMINALS Co(r-I) . Co(r) Co(r+l)

INFINESITIMAL CAPACITORS EFFECT OF UNELECTRODED AREA

(b)

x x

I ! --- • •/•- -- "-' t J EFFECT OF UNELECTRODED AREA

(c)

x x

•l Ceq - -- l •,•. !__ ................. EFFECT OF ELEGTRODED AREA

FIG. 6. Effect of unelectroded areas on the equivalent circuit of a piezoelectric plate.

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R. HOLLAND

a2• •N-I VN-I

SUBSTRATE

FIG. 7. Symmetrically electroded ferroelectric ceramic disk bonded to a substrate.

determines f•(") completely'

f k ('ø = •Sr•5 (r-- a) / 2,ra. (58)

For simplicity, we assume that 7, p, and the elastic coefficients are uniform. The piezoelectric coefficients e-a•j are permitted to vary from electrode to electrode, but are assumed to be uniform in a given electroded region. It is known 17 that the eigenmode solutions to Eqs. 15 for this disk are

•?n) _ Jx(rKn/a)K,,/{ aJx(K•)•p,r(K,, 2-- 1-3-e2)3•}, (59) co • = • nn ½K •r/pa •,

where e is Poisson's ratio and the K, are the roots of

(60)

Solutions of Eq. 60 are tabulated in Ref. 17. We now illustrate the derivation of the equivalent

circuit for this configuration. Recall that one of the quantities appearing on the lefthand of Eqs. 132-34 may be selected arbitrarily. We choose to set

also raise the resonant frequencies, although not by so much as unelectroded regions of equal area.

(61)

Equation 34 then specifies C•'

X. EXAMPLES

A. Bonded-Down, Symmetrically Electroded, Ferroelectric Ceramic Disk

As a specific example of the techniques presented in this paper, let us discuss the symmetrically electroded, axially poled ferroelectric ceramic disk shown in Fig. 7. We assume that the disk is bonded to the substrate by an idealized bond representable by an admittance tensor (see Eq. 47),

r/k• = c•r/. (56)

Cn = E7r•,1111E (Xn •- 1 +o'2) T3 --1. (62)

In evaluating this quantity, Eqs. 59 for •i(•) and 58 for f•(• have been substituted. The equivalent motional inductance is then specified by Eqs. 32 and 59'

L•= ,ra•(K,, •-- 1 +v•)oK• -•, (63)

while Eqs. 16, 33, 59, and 62 serve to determine the •,

•, = 2ragm (p)Bn,. (64)

In addition to the external mechanical forces asso-

ciated with the bond, let us permit a uniform radial force to act around the edge of the disk. We regard the edge of the disk as a mechanical terminal, and designate it by the subscript a (see Fig. 7). The arbitrary multiplicative constant discussed in Ref. 26 may be selected so that the mechanical analog of current at this terminal is particle velocity-

(57)

The independent-force distribution f•(•) associated with this mechanical terminal is to be applied uni- formly around the edge of the disk in a radial direction. As the arbitrary multiplicative factor for this terminal has already been selected, consistency of Eqs. 29 and 57

In evaluating •v, use is made of the fact that the only independent component of ga•$ is gan for a ferroelectric ceramic. The symbol B,v appearing in Eq. 64 is defined as

av"Jx (%"K./a) -- av'Jx (av'Kda) B,•p - ; (65)

aJx (K,O

e%n(p) represents •'an evaluated at electrode p. The equivalent-circuit quantities represented by

Eqs. 57-65 agree with known results for a ferroelectric ceramic disk mechanically free except at the cir- cumference?

However, the methods of our Application B are ideal for extending these results to the present case of a bonded-down disk. Let us consider the equivalent ad-



17:15:10


mittance matrix of the bond as specified by Eq. 49. Substituting Eq. 56 into Eq. 49, factoring p• outside the integral, and applying the orthogonaltry relation, Eq. 17, the lead to the result

(66)

This matrix may be readily inverted to give

(67)

Consequently, the bond may be represented by in- serting an impedance of value

ELECTRICAL TERMINALS J•a• Ea [

+ -

MECH AN ICAL TERMINAL,

BOND

Fro. 8. Equivalent circuit of a bonded-down, symmetrically electroded, ferroelectric ceramic disk.

(68)

in series with the mottonal capacitance C, (see Fig. 8). Application of Eq. 46 for •Jl•,, then yields

Z= (w,?C,•p•)-L (69)

This impedance is equivalent to a complex capacitance of value

C= --jconCnprl, (70)

so that the resultant capacitance of C• and C is

(71)

If one assttrnes that • is pure real, •7 the quality factor of the nth symmetric contour-extensional resonance Q• of a free disk is just the reciprocal loss tangent of C,, as L, is ideal (see Eq. 63). Moreover, the loss tangent of C, and 3, m• • are the same, so that

Q,= R•,mx•/z•,mf (72)

for a free disk.

However, if the bonding admittance • is real (i.e., re- sistive), this quality factor will be reduced to

Q.-' . (73) (a/O'

for the case of the bonded disk. Two interesting trends are apparent from Eq. 73. It may be seen that the Q-

reducing effects of the bond decrease with increasing resonance mtrnber and with increasing thickness-to- diameter ratio.

1. Experimental Results

Equation 73 has been used to determine the admittance • characterizing sylastic bonds. This has been done by measuring the Q's of the first four resonances of symmetrically electroded ferroelectric ceramic disks, first when mechanically free, and second when bonded to a substrate with sylastic. The disks used were fabri- cated from a lead-zirconate-titanate composition known as PZT 65/35. Values of p/r, (r, and R•1111 E are given in Table I for this composition. Dimensions of the disks used here are 4.0 mil for r and 62.5 mil for a.

Values for z3, m• • may be determined by substituting •3, xm B and the measured free Q• into Eq. 73 with • = •o. Then, in the bonded case, all quantities appearing in Eq. 73 except • are known. Measured values for Q• (free and bonded) are given in Table II, along with the subsequently obtained values for 7.

These results, excluding the free Q•, were found tO vary by as much as 30% as amplitude of excitation was varied. The free Q•, on the other hand, was essentially amplitude-independent over the range studied. It may consequently be deduced that nonlinear loss effects were present in the sylastic but not in the ceramic.

Table II indicates that some frequency dependence in the free Q,, %,mx•/z•,xm, • is present, and that the sylastic admittance is fairly strongly frequency- dependent.

It should probably b•pointed out that these nonlinear and frequency-dependent characteristics of sylas-

TA•.E I. Material properties of PZT 65/35 (a lead-zirconate- TA•.g II. Free- and bonded-disk quality factors and correspond- titanate composition) in mks units. ing sylastic admittance.

Symbol Value n Free Q,, Bonded Q,, n

Density (kg/m a) p/r 7.95X 10 a (mks units) Planar stiffness 1•r1111 H 1.19X 1011 1 140 40 0.109 Poisson's ratio a 0.294 2 128 59 0.081 Quality factor Q,•, ,, 160 3 116 54 0.047 Piezoelectric coeff. •$11 6.57 4 98 52 0.038



17:15:10

R. HOLLAND

8 ELECTRODES

J • r •" / NOT SHOWN

t

FIO. 9. N-electrode asymmetric disk.

tic make it impossible to predict the Q-reducing effects of a syslastic bond with precision greater than 30%. (Perhaps other bonding materials will be found that are simpler to characterize.) However, for disks with only the inner third of the surface area bonded, the Q-reducing effects are on the order of 20% or less, as contrasted with a maximum of 70% reported here for a completely bonded sample. Consequently, in the partial-bonding case, which is actually the more attractive from the practical viewpoint of attaining high Q's, it is possible to predict the resultant Q to within 30% of 20%--i.e., to 6%.

B. Asymmetrically Electroded, Free, Ferro- electric Ceramic Disk

Recently, Onoe and Kurachi have determined the equivalent circuit of a one-electrode, axially poled, mechanically free, ferroelectric ceramic disk for cases in which the electrode shape is 0-dependent. •'s Let us now apply the results of the present paper to the same problem, but relax the restriction that only one electrode be present. In particular, we consider a ferroelectric ceramic disk having N electrodes, the pth bounded by 0v'<0<0v", av'<r<av" (see Fig. 9). The linear terminal behavior of this disk is completely specified by the electrode admittance matrix of Eq. 26.

If we assume that ga•j does not vary under an individual electrode, the two integrals appearing in Eq. 26

may be more conveniently evaluated in an alternative form'

f/Da(n)da-•p•aij•i(n)Ni(P)ds, (74) ap

where sv is the edge of electrode p and _/•j(v) is the outward-pointing unit vector normal to sv. Equation 16 and an integration by parts are required to establish this identity.

It should be noted that the summation of Eq. 26 extends over all sufficiently-low-contour extensional modes of the disk. This includes modes lacking axial symmetry as well as the axially symmetric modes utilized in Example A. This complete set of modes com- poses a two-dimensional series, so that it is convenient to replace the single modal index n with a double index (m, n) in the following discussion. In the double-index convention, the first index is related to radial dependence and the second index is related to angular dependence. With this convention, Eq. 26 becomes

Yv•=jw•]•]I••Eagy•i(m'n)Nj(v)ds

+jwCo(q)Svq. (75)

The modes having zero angular dependence are specified by Eq. 59 with

•0 (m'O) :0. (76)

These displacement functions and the form of g3o relevant to a ferroelectric ceramic lead to the result

•• Km,oga• (p) Bmp (Op t '-- Op t) gaiy•i(m'ø)NJ(P)ds: •wp(Km,o2__ lq-o-2)• « (77)

The-Km,0:appearing ]here r •are the 'Km of Eq. 59. The modes having nonzero angular dependence are

considerably more complicated. They are given by aø

(rn,n,ch [ dJn(rm,nda) •r \m'n's/ : A m,n dr

nBm n "] 1 / cosn0\

•. ' Jn(gm,nO•'/a) •.]•__sii.lI•O) ,

(m,n,c h r•lAm,n dJn(Km,n•)•'/a)•_(sin•lO• •o\m'n's/----!_ •' Jn(Km'nrf C)'AFBr•'n •-; -Ix/p\cos•tO/'

(78)

m,n=l, 2, -.-.

as M. Onoe and T. Kurachi, J. Inst. Elec. Commun. Engrs. Japan 49, 104-110 (1966) (in Japanese). 20 R. Holland, J. Acoust. Soc. Am. 40, 1051-1057 (1966).



17:15:10


The derivation of this result has been discussed by Onoe aø and Love. a• In Eq. 78, {3 is the ratio of dilational to shear acoustic velocity,

O=[2/(1-a)• ,•. (79)

The third index, c or s, indicates whether the radial displacement component has sine or cosine angular dependence. The c and s modes are frequency-degenerate, but orthogonal. The summation of Eq. 75 is understood to extend over both c and s modes. A m., and Bm., are normalization coefficients, which are discussed and tabulated in Ref. 29. The Km,, are so-called eigenvalues of the problem, and are analogous to the Km.0 of the symmetric modes.These Kin., were first computed by Onoe, aø although a more extensive listing of their values is available in Ref. 29. The Km,, are related to the resonant frequency of mode (m,n) by

COm, ff -- 3, zm •Km, ff r/pa •'. (8O)

Substitution of Eq. 78 into the integrals of Eq. 75 yields

(8•)

It should be observed that the Bm.• coefficients of Eq. 78 drop out of Eq. 81. These coefficients are related to shear strains, which are piezoinactive for an axially poled ferroelectric ceramic. On the other hand, the A m., coefficients are related to dilational strains, which are not piezoinactive.

If Eqs. 77 and 81 are substituted into Eq. 75, a bit of algebra enables one to obtain the following formula for Yvq:

ZC=, orS., •Z•,• (0 •"-- 0/) (0 •"-- 0 •')

ß

+5.

4 cos-(Ov"--O "+0 '--0•') sin-(Ov"--Ov') sin-(Oq"--O•') q io

2 2 2

m m•On 0 P (•m,n 2-- 60 2)

' I(m,n,p)I(m,n,q) (82)

In Eq. 82, I(m,n,p) represents the integral

(av"Km,./a)

I(m,n,p)= f tJ,(t)dt. ll

Values for this integral have been tabulated?'

(83)

resonant frequency of mode (m,n)•, the real part of the admittance as predicted by Eq. 82 becomes

RYvq(RO0m.,) = (Qm,,,a/Km,,,or) X (or/Rq, mz•) «' X 4 cosn/2 (Ov"+O•,'- 0•"- 0•') Xsinn/2 (O•,"-O•,') sinn/2 (Oq"-O•')

XI(m,n,p)I(m,n,q). (84)

1. Experimental Verifications

Equation 82 has been subjected to a number of ex- 'perimental verifications. Eight-electrode ferroelectric ceramic disks such as the one illustrated in Fig. 10 were used for this purpose.

If an asymmetrically'•'telectroded disk is excited at o0=•o0•.• [where COrn,n'-'RCOra,n'-•jlcora,n is the complex

•o M. Onoe, J. Acoust. Soc. Am. 28, 1158-1162 (1956). ox Ref. 24, pp. 497-498. •" R. Holland, "Applications of Green's Functions and Eigen-

modes in the Design of Piezoelectric Ceramic Devices," doctoral thesis, Mass. Inst. Technol. (Jan. 1966), pp. 124-126.

Fro. 10. Asymmetric disk electroding and wiring arrangement used to excite n=4 modes experimentally.

.818 •/4 RADI,••

' //_/. MILS / / III /

/ • 150.2 MILS



17:15:10

R. HOLLAND

TABLE III. Comparison of theoretical and experimental ad- mittances and eigenvalues for asymmetric disks.

Mode r K,•,,, K,•,,, t•y(t•com, n) t•y(t%m,n) theoret exptl theoret exptl

(mil) (mrS) a (mrS) a

(1,1) 2.0 1.621 1.658 6.3 5.1 (1,1) 2.9 1.621 1.654 4.4 4.3 (1,1) 2.9 1.621 1.655 4.4 4.2 (2,1) 2.9 3.530 3.583 19.9 18.9 (1,2) 2.0 1.391 1.430 0.186 0.15 (1,2) 3.0 1.391 1.425 0.126 0.13 (2,2) 2.0 2.516 2.574 11.2 9.8 (2,2) 3.0 2.516 2.567 7.7 7.2 (1,4) 2.9 2.783 2.854 1.55 1.65 (2,4) 2.9 4.410 4.503 11.5 12.4

a mG --10 -a ohm -1.

In this equation, Q,•,n is the quality factor of mode (m,n), and is R-•111x•/x-•1n1 • for real cr (see Eq. 72).

For the experiments conducted during this research, all the electrodes were driven in parallel, as shown in Fig. 10. This leads to a total admittance at the driving terminals having a real part

8

Z; (8s) Iv, q----1

at RWm.n. Using this procedure, Eq. 84 was checked experimentally at the resonances of modes (1,1), (1,2), (1,4), (2,1), (2,2), and (2,4). In Fig. 10, q- and -- signs on the electrodes indicate the ceramic polarization di- rections for excitation of modes having angular index 4. Note that the sequence is q- ---[- ---[- ---[- --. For modes of n= 2, +q-----q-+---- is required, and for n= 1. +q-q-+ .... is required.

Table III summarizes the theoretical and experimental results of this comparison of •Y(•oom,•). Also shown in Table III are theoretical and experimental values for Km,•. Experimental values for Km,• were computed using Eq. 80 and measured values of w•,•.

The ceramic material composing these samples was a lead-zirconate-titanate composition known at PZT 65/35. Material coefficients for this ceramic that are necessary for the theoretical evaluation of •Y(•w,•,n) are summarized in Table I.

ACKNOWLEDGMENTS

The author wishes to thank Hans Baerwald, Otmar Stuetzer, Cecil Land, and Errol Eer Nisse, all of Sandia Corporation, for criticizing the manuscript. The ceramic disks used in the experiments were prepared by Cecil Land and Ira McKinney. This work was supported by the U.S. Atomic Energy Commission.

952• Volume 41 Number 4 Part 2 1967


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Analysis of Multiterminal Piezoelectric Plates