Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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E,T

¼ d



D,T

¼ g

E,T

¼ e



D,T

¼ h

ð6:8Þ

The converse-eﬀect coeﬃcients are properly deﬁned similarly:

X,T

¼ d*

X,T

¼ g*



x,T

¼ e*



x,T

¼ h*

ð6:9Þ

It is instructive to outline the thermodynamic argument proving that

d ¼ d*, g ¼ g*, e ¼ e* and h ¼ h*. The First Law states that the addition of an

increment of heat dq to a system produces a change in the internal energy dU and

causes the system to do work dW on its surroundings. Thus

dq ¼ dU þ dW ð6:10Þ

If the changes are performed reversibly, dq ¼ TdS, and

TdS ¼ dU þ dW ð6:11Þ

where S is the entropy. In the case of an isolated piezoelectric material the

mechanical and electrical work is done on the system, and hence

TdS ¼ dU  dW ¼ dU  Xdx  EdD ð6:12Þ

It is possible to deﬁne the thermodynamic state of a system in terms of groups

of certain independent variables, and T, X, x, D and E are available in the

present case. For example, the equilibrium could be expressed in terms of the

342 PIEZOELECTRIC CERAMICS

TEAMFLY

Team-Fly

independent variables (T, x, D) when the appropriate starting point would be the

energy function

A ¼ U TS ð6:13Þ

Now

dA ¼ dU TdS  SdT

which, on substituting for dU  TdS from Eq. (6.12), becomes

dA ¼ Xdx þ EdD  SdT ð6:14Þ

Because dA is a perfect diﬀerential it follows that

x,T

D,T

ð6:15Þ

Similarly, if attention is focused on the independent variables (T, X, E), the

appropriate function is

G ¼ U TS  Xx  ED ð6:16Þ

which leads to

dG ¼SdT xdX  DdE ð6:17Þ

and to the relationship

X,T

E,T

ð6:18Þ

Similarly, the function

¼ U TS  ED ð6:19Þ

leads to

x,T

¼

E,T

ð6:20Þ

and

¼ U  TS  Xx ð6:21Þ

leads to

x,T

¼

D,T

ð6:22Þ

BACKGROUND THEORY 343

Thus the equalities d ¼ d* etc. are proved, and the relationships can be

summarized as follows:

E,T

X,T

¼ d ð6:23Þ



D,T

X,T

¼ g ð6:24Þ

E,T

¼

x,T

¼ e ð6:25Þ

D,T

x,T

¼h ð6:26Þ

6.2 Parameters for Piezoelectric Ceramics and

their Measurement

The discussion is now restricted to matters relevant to the technology of

piezoceramics. If it is assumed that the quantities d, g, e, and s are constants and,

furthermore, the tensor nature of the various parameters (see below) is not taken

into account, then equations of state can be written as follows.

D ¼ dX þe

E ð6:27aÞ

x ¼ s

X þ dE ð6:27bÞ

E ¼gX þD/e

ð6:28aÞ

x ¼ s

X þ gD ð6:28bÞ

Where s

, s

are the elastic compliances and the superscripts denote the

parameters held constant; similarly e

denotes the permittivity measured at

constant stress (the usual condition); to measure the permittivity at constant

strain e

is more diﬃcult to accomplish.

Because ferroelectric ceramics have non-linear characteristics the eﬀects are

more correctly described by the equations,

344 PIEZOELECTRIC CERAMICS

dD ¼ ddX þe

dE ð6:29aÞ

dx ¼ s

dX þ ddE ð6:29bÞ

dE ¼gdX þ dD=e

ð6:30aÞ

dx ¼ s

dX þ gdD ð6:30bÞ

dE ¼hdx þdD=e

ð6:31aÞ

dX ¼ c

dx  hdD ð6:31bÞ

dD ¼ edx þ e

dE ð6:32aÞ

dX ¼ c

dx  edE ð6:32bÞ

in which the coeﬃcients are recognized as being functions of the independent

variables, that is dðX,EÞ, gðX,DÞ, etc.

The various coeﬃcients which appear in electromechanical equations of state

such as (6.29a) to (6.32b) are not all independent. For example, from Eqs (6.23)

and (6.24)

d/g ¼ð@x/@EÞ

/ð@x/@DÞ

¼ð@D/@EÞ

¼ e

ð6:33Þ

It is understood that in all cases the temperature is maintained constant.

In a similar way, from Eqs (6.24), (6.25) and (6.26), it follows that

e/h ¼ e

and g/h ¼ s

ð6:34Þ

The electromechanical coupling coeﬃcient (k) is a measure of the ability of a

piezoelectric material to transform mechanical energy into electrical energy, and

vice versa. It is deﬁned [5] by

¼ W

ð6:35Þ

in which W

, W

and W

are respectively the piezoelectric, mechanical and

electrical energy densities. In the piezoceramics’ context an eﬀective coupling

coeﬃcient k

eﬀ

, deﬁned below, is in common usage.

There are important relationships which follow from the above equations of

state. When a piezoelectric material is stressed, electrically or mechanically, the

developed energy densities are, from Eqs (6.29a) and (6.29b).

dD dE ¼

d dXdE þ

ðdEÞ

ð6:36Þ

dx dX ¼

ðdXÞ

d dEdX ð6:37Þ

PARAMETER S FOR PIEZOELECTRIC CERAMICS AND THEIR MEASUREMENT 345

The terms involving (dE)

and (dX)

are the electrical and mechanical energy

densities, and those involving dEdX the piezoelectric energy densities. It follows

from Eq. (6.35) that

¼fd dX dEddE dX/s

ðdXÞ

ðdEÞ

1=2

ð6:38Þ

¼ d/ðs

1=2

ð6:39Þ

The elastic compliance s, (and stiﬀness c) has very diﬀerent values depending

upon whether the electric ﬁeld E within the material is maintained at zero (the

component is ‘short-circuited’), or whether the electric displacement D remains

constant (the component is ‘open-circuited’). The ‘short circuit’ value is speciﬁed

by the superscript E and the ‘open circuit’ condition by the superscript D.

Similarly the permittivity can be measured with the specimen free to deform, that

is at constant stress e

, or in the clamped state, e

Important relationships between s

and s

(and between c

and c

) can be

derived as follows.

Equating Eqs (6.29b) and (6.30b) leads to

dX þ d dE ¼ s

dX þ g dD

After substituting for dD and dE from Eqs (6.29a) and (6.30a) and noting from

Eq. (6.33) that g ¼ d/e

it follows, after a little manipulation that

¼ s

ð1  d

or, from Eq. (6.39)

¼ s

ð1  k

Þð6:40Þ

Also, because c ¼ 1/s

¼ c

ð1  k

Þð6:41Þ

In a similar way, but now equating Eqs (6.30a) and (6.31a) and substituting for

dx from Eq. (6.30b), it follows that

 e

ð1  k

Þð6:42Þ

[the approximation rests on k

1]

Rearranging Eq. (6.42) leads to k

¼ðe

 e

Þ/e

or to

¼ð



Þ/

ð6:43Þ

When an electric ﬁeld is applied to an unconstrained piezoelectric body then both

electrical and mechanical energies are stored. If the body is mechanically

346 PIEZOELECTRIC CERAMICS

clamped then only electrical energy is stored. Eq. (6.43) is the basis of the

deﬁnition of the ‘eﬀective coupling coeﬃcient’ k

eﬀ

, namely

eff

input electrical energy converted into mechanical energy

input electrical energy

ð6:44Þ

Similarly from Eq. (6.40)

eff

input mechanical energy converted into electrical energy

input mechanical energy

ð6:45Þ

In practice the energy transfer electrical-to-mechanical (or vice versa) occurs in a

complex 3-dimensional way. The strains caused by applied electrical or

mechanical stresses have components in three orthogonal directions necessitating

the description of the piezoelectric eﬀect in terms of tensors, as outlined below.

The state of strain in a body is fully described by a second-rank tensor, a

‘strain tensor’, and the state of stress by a stress tensor, again of second rank.

Therefore the relationships between the stress and strain tensors, i.e. the Young

modulus or the compliance, are fourth-rank tensors. The relationship between

the electric ﬁeld and electric displacement, i.e. the permittivity, is a second-rank

tensor. In general, a vector (formally regarded as a ﬁrst-rank tensor) has three

components, a second-rank tensor has nine components, a third-rank tensor has

27 components and a fourth-rank tensor has 81 components.

Not all the tensor components are independent. Between Eqs (6.29a) and

(6.29b) there are 45 independent tensor components, 21 for the elastic

compliance s

, six for the permittivity e

and 18 for the piezoelectric coeﬃcient

d. Fortunately crystal symmetry and the choice of reference axes reduces the

number even further. Here the discussion is restricted to poled polycrystalline

ceramics, which have 1-fold symmetry in a plane normal to the poling direction.

The symmetry of a poled ceramic is therefore described as 1mm, which is

equivalent to 6mm in the hexagonal symmetry system.

The convention is to deﬁne the poling direction as the 3-axis, as illustrated in

Fig. 6.2. The shear planes are indicated by the subscripts 4, 5 and 6 and are

perpendicular to directions 1, 2 and 3 respectively. For example, d

is the

coeﬃcient relating the ﬁeld along the polar axis to the strain perpendicular to it,

whilst d

is the corresponding coeﬃcient for both strain and ﬁeld along the polar

axis. Shear can only occur when a ﬁeld is applied at right angles to the polar axis

so that there is only one coeﬃcient, d

. There are also piezoelectric coeﬃcients

corresponding to hydrostatic stress, e.g. d

¼ d

þ 2d

(see Section 6.4.6).

Since the elastic compliance (s) and stiﬀness (c) are tensor properties, in

general c

6¼ 1/s

; the exact relations are given in Eq. (6.54) and in the Appendix.

Since s

¼ s

and c

¼ c

, only six terms (s

, s

or c

, c

) are needed for poled ceramics. Of these, s

and c

are irrelevant

because shear in the plane perpendicular to the polar axis produces no

PARAMETER S FOR PIEZOELECTRIC CERAMICS AND THEIR MEASUREMENT 347

piezoelectric response. Two sets of elastic constants are needed, one for short-

circuit and the other for open-circuit conditions, as indicated by superscripts E or

D respectively.

The Poisson ratios n are given

¼s

ð6:46Þ

Following the above conventions, Eqs (6.33) and (6.34) become

X,T

¼ e

ð6:47Þ

For example,

¼ e

ð6:48Þ

Similarly

¼ e

ð6:49Þ

Also, Eq. (6.29a) is expanded to

¼ d

þ e

ð6:50aÞ

¼ d

þ e

ð6:50bÞ

¼ d

ðdX

þ dX

Þþd

þ e

ð6:50cÞ

The coeﬃcients e

and e

can be expressed in terms of d

and d

from

Eq. (6.50c):

348 PIEZOELECTRIC CERAMICS

Fig. 6.2 Labelling of reference axes and planes for piezoceramics.

¼ 2d

þ d

ð6:51Þ

¼ d

ðc

þ c

Þþd

ð6:52Þ

A combination of Eqs (6.48), (6.49), (6.51) and (6.52) enables the coeﬃcients e

and h to be calculated from the more usually quoted coeﬃcients d and g.

The indirect eﬀect given in Eq. (6.29b) expands to

¼ s

þ s

þ d

ð6:53aÞ

¼ s

þ s

þ d

ð6:53bÞ

¼ s

ðdX

þ dX

Þþs

þ d

ð6:53cÞ

¼ s

þ d

ð6:53dÞ

¼ 2ðs

 s

ÞdX

ð6:53eÞ

The stiﬀness coeﬃcients can be derived from the compliances using the

following relations which hold for either open- or closed-circuit conditions and

also with the symbols c and s interchanged:

 s

fðsÞ

¼

 s

f ðsÞ

¼

ðs

 s

f ðsÞ

 s

f ðsÞ

ð6:54Þ

where f ðsÞ¼ðs

 s

Þfs

ðs

þ s

Þ2s

For convenience the various property relationships are summarized in an

appendix to this chapter.

The values of the piezoelectric properties of a material can be derived from the

resonance behaviour of suitably shaped specimens subjected to a sinusoidally

varying electric ﬁeld. To a good approximation the behaviour of the piezoelectric

PARAMETER S FOR PIEZOELECTRIC CERAMICS AND THEIR MEASUREMENT 349

specimen close to its fundamental resonance can be represented by the equivalent

circuit shown in Fig. 6.3(a) [6].

There is an equivalence between the diﬀerential equations describing a

mechanical system which oscillates with damped simple harmonic motion and

driven by a sinusoidal force, and the series L

, C

, R

arm of the circuit driven by

a sinusoidal e.m.f. The inductance L

is equivalent to the mass (inertia) of the

mechanical system, the capacitance C

to the mechanical stiﬀness and the

resistance R

accounts for the energy losses; C

is the electrical capacitance of the

specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of

the parallel circuit.

350 PIEZOELECTRIC CERAMICS

Fig. 6.3 (a) Equivalent circuit for a piezoelectric specimen vibrating close to resonance; (b)

the equivalent series components of the impedance of (a).

Fig. 6.4 The characteristic frequencies of the equivalent circuit exaggerating the diﬀerences

between f

, f

and f

and between f

, f

and f

The form of the frequency response of the electrical circuit, shown in Fig. 6.4,

has the following characteristic frequencies:

1. f

and f

when the impedance is, respectively, a minimum and a maximum,

and

2. the resonance frequency f

and antiresonance frequency f

when the reactance

of the circuit is zero.

The reactance of the mechanical arm is zero at the ‘series’ resonance frequency f

when joL

¼ 1/joC

; the parallel resonance f

occurs when the currents ﬂowing

in the two arms are in antiphase, that is when jðoL

 1/oC

1

¼1/joC

,or

when o ¼fðC

þ C

Þ/L

1=2

It follows from Eq. (6.44) and the equivalent circuit that

eff

¼ C

/ðC

þ C

Þð6:55Þ

and it can be shown that

eff

¼ C

/ðC

þ C

Þ¼ðf

 f

Þ/ f

ðf

 f

Þ/ f

ðf

 f

Þ/ f

ð6:56Þ

Provided that the mechanical quality factor (see Fig. 5.34 for analogy) for the

resonator is suﬃciently high (Q

5  100) the approximations apply.

Values for f

and f

are readily measured using an impedance analyser.

Speciﬁc (‘33’) coupling coeﬃcient, compliance and permittivity values can be

measured on appropriately dimensioned specimens from which the correspond-

ing d and g coeﬃcients can be calculated; the following examples illustrate how

this is accomplished.

In the case of a piezoelectric ceramic rod ( 5 mm diam. and 15 mm long)

electroded on its end faces and poled along its length, for the fundamental

resonance condition when there is a standing elastic longitudinal wave along the

length with antinodes at the ends and the centre a node, it can be shown that

tan



 f



(6:57)

The appropriate permittivity e

is easily determined from the capacitance C of

the specimen at a frequency well below resonance:

ð6:58Þ

where l and A are the length and cross-sectional area of the rod respectively. The

elastic compliance s

is related to the parallel resonance frequency by

PARAMETER S FOR PIEZOELECTRIC CERAMICS AND THEIR MEASUREMENT 351