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

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sðlÞ¼e

ðlÞð6:83Þ

The charge Q

on the element dl is

¼ e

ðlÞwdl ð6:84Þ

and the total charge is

dl ð6:85Þ

From Eqs (6.83), (6.85) and (6.86)

dze

ðL  lÞdl ð6:86Þ

so that

dz ð6:87Þ

If C

is the clamped capacitance of one of the bimorph plates, then the voltage

across it is U ¼ Q

and that across two oppositely poled plates connected in

series is U

¼ 2U. Therefore





dz ð6:88Þ

If L ¼ 10 mm, H ¼ 1 mm, dz ¼ 1 mm and h

¼6.2  10

¼ 2.3 V

Although, by the converse eﬀect, a bimorph will bend when a voltage is applied

to it, this will not be according to the inverse of Eq. (6.88) because the

deformation due to an applied ﬁeld is governed by d ¼ð@x=@EÞ

which is not the

inverse of h ¼ð@E=@xÞ

. Furthermore, the applied ﬁeld will produce uniform

strains along the length of the bimorph so that it will be bent in the form of a

circular arc, in contrast with the more complex shape given by Eq. (6.79).

Under an applied voltage the stress-free ﬁlaments in a bent bimorph will be in

two planes, one above and one below the central plane and at equal distances a

from it. The centre plane is strain free because the stresses on opposite sides of

the joining layer are equal and opposed to one another. The steps in the

argument are illustrated in Fig. 6.29 and the stress system is similar to that set up

in a heated bimetallic strip, as analysed by S. Timoshenko [20].

If y in Fig. 6.28 is replaced by a, the strain at the neutral planes is a/R which,

from Fig. 6.21, is Ed

, i.e.

a=R ¼ Ed

ð6:89Þ

392 PIEZOELECTRIC CERAMICS

TEAMFLY

Team-Fly

For a circular arc the depression of the free end of a bimorph relative to the

clamped end is

dz ¼

ð6:90Þ

a can be determined by taking moments in a cross-section of the bimorph and,

provided that the thickness of the material joining the two plates is less than a

ﬁfth of the beam thickness, a  H=3, so that

d ¼





ð6:91Þ

In terms of the applied voltage,

dz ¼





U ð6:92Þ

With the same dimensions as before, and with d

¼80 pC N

, a voltage of

84 V must be applied to produce a deﬂection of 1 mm.

APPLICATIONS 393

Fig. 6.29 A bimorph under an applied ﬁeld: (a) the two free halves of the bimorph; (b) the

free halves in a ﬁeld; (c) behaviour when the two halves shown in (a) are joined to form a

bimorph and the neutral planes have lengths equivalent to those in (b).

It is clear that useful movements for most purposes can only be obtained

for moderate voltages if L/H is made large, but this leads to a fragile

component. One solution is to replace one of the ceramic plates by a metal

strip. This will at least halve the deﬂection per volt, but it can also be

arranged that the bending always compresses the ceramic and the sense of the

applied ﬁeld will then coincide with the poling direction so that depoling

eﬀects are avoided. Higher maximum voltages can be used since the ceramic is

stronger in compression than in tension. Since the ﬁelds must be high to

obtain large movements it is advantageous to use electrostrictive materials,

and this also eliminates the hysteretic behaviour to which piezoelectric

materials are prone.

Very recently the viscous polymer processing route (see Section 3.6.7) has been

exploited by D.H. Pearce and co-workers [21] to fabricate a helical form of the

bimorph, the ‘Helimorph’, illustrated in Fig. 6.30. Both axial and rotational

actuations are developed which have potential for a variety of applications in the

mechanical, optical and acoustical ﬁelds.

394 PIEZOELECTRIC CERAMICS

Fig. 6.30 ‘Helimorph’

actuators. (Courtesy of ‘1Limited’ Cambridge, UK.)

Rotary motion

The direct generation of rotary motion can be achieved by the use of surface

wave transducers based on the generation of acoustic waves conﬁned to a thin

layer near to the surface of a solid. The basic theory of surface acoustic waves

(SAWs) was established by Lord Rayleigh in 1885. The waves are due to a

combination of longitudinal and shear motions governed by the stress-free

boundary conditions at the surface. The particles of the surface layer of the solid

describe elliptical motions as indicated in Fig. 6.31(a), with the amplitude

decreasing with distance from the surface until, at a depth of approximately

1 mm, it is negligible. The velocity of SAWs is approximately 300 m s

, a little

lower than the velocity of bulk waves, and so they require times of the order of

microseconds to travel 1 cm.

Because of the horizontal component of the motions of the surface particles

the ‘slider’ in contact with the surface experiences a force tending to move it in a

direction opposite to that of the travelling wave, as indicated in Fig. 6.31(a).

Rotary motion is achieved by propagating a SAW round an elastic ring driving a

contacting ring slider. Correctly phased vibrations are propagated into the elastic

ring from an appropriately poled and energized piezoelectric ring (Fig. 6.31(b)).

The SAW is generated by superimposing two standing waves of equal

amplitude but diﬀering in phase by p/2 with respect to both time and space. The

time phase diﬀerence results from one wave being generated by the voltage

sin(ot) and the other by U

cos(ot); the spatial phase diﬀerence results from

the 3l/4 and l/4 gaps between the two poled segments. The two standing waves

can be written

¼ y

cosðnyÞcosðotÞð6:93Þ

and

¼ y

cos



ny 



cos



o 



in which y

and y

represent the vertical displacements of the ring surface, y is the

angular displacement round the ring and n is the number of wavelengths

accommodated round the ring. The resultant wave is given by

y ¼ y

þ y

¼ y

fcosðnyÞcosðotÞþsinðnyÞsinðotÞg

¼ y

cosðot  nyÞ

ð6:94Þ

which represents a surface wave travelling with velocity o=n.

APPLICATIONS 395

Miscellaneous

There are a variety of actuators which are the products of the ingenuity of the

engineer, such as the ‘Inchworm’, which develops precise linear motion and the

‘walking’ variety. These and others are described by K. Uchino [18].

6.5.3 High frequency applications

These applications can be divided into those which demand high power (e.g.

ultrasonic cleaner) intermediate power (e.g. ‘Tweeter’) and signal power (e.g.

Delay line). Some examples are brieﬂy described below.

396 PIEZOELECTRIC CERAMICS

(a)

(b)

Fig. 6.31 Principle of the rotary actuator: (a) side view; (b) plan view showing poled segments

and how temporal and spatial phase diﬀerences are established.

Ultrasonic cleaner

For high-power ultrasonic applications, such as ultrasonic cleaning, the

transducers are operated at frequencies typically in the range 20–50 kHz. A

PZT disc having its fundamental thickness mode frequency in this range would

be large; for example, the thickness for 20 kHz would be about 12 cm! The

production of such large pieces would present diﬃculties, and so too would

coping with the energy dissipation. The problems are overcome by adopting a

composite structure comprising the piezoceramic together with carefully chosen

metal end-pieces.

The structure of a typical composite power transducer for ultrasonic cleaning

is illustrated in Fig. 6.32. The ‘PZT’ toroids are kept in compression (stress

25 MPa) by the bolt to reduce the risk of their fracturing under the high drive

ﬁelds. The ‘PZT’ will be of the ‘hard’ variety (‘acceptor’-doped; see Section 6.3.2)

so as to minimize the risk of depolarization under the high mechanical stresses

experienced. The associated high Q

value and low electrical tan d ensure that

the losses are kept within acceptable limits. The structure has the added

advantage that any heat developed in the ceramic can be dissipated through the

massive metal end pieces.

The device must be dimensioned so that at resonance a half-wavelength is

accommodated in the distance L. The amplitude of motion at the outer metal

surface will be related to that of the ceramic through the ratio q of their acoustic

impedances:

q ¼

ð6:95Þ

APPLICATIONS 397

Fig. 6.32 The essentials of a power transducer for ultrasonic cleaning (the component is

cylindrically symmetrical about the vertical axis).

where r

, r

, v

, A

and A

are the density, sound velocity and sectional areas

of the metal and the ceramic respectively. By making one end-piece of

magnesium, for which r

¼ 8.4  10

kg m

(8.4 Mrayl) and the other

of steel with r

¼ 41 Mrayl, and suitably adjusting the sectional areas of the

components, about 15 times the amplitude of motion can be obtained at the

magnesium surface than at the steel surface, so that the major part of the

acoustic energy is emitted in only one direction.

A ﬁnal beneﬁt of this type of structure is that the acoustic behaviour is

primarily governed by the properties and dimensions of the metal parts and only

to a minor extent by those of the ceramic, so that small variations in the ceramic

properties become unimportant.

For eﬃcient transfer of power from the generator to the medium, usually

water, the two must be acoustically matched. The discontinuity can be smoothed

by ﬁxing a l/4 thick layer of material having an acoustic impedance intermediate

between that of the radiating surface material and water, and polymers having

impedances of about 3.5 Mrayl are readily available. The velocity of sound

in them is approximately 2500 m s

1

so that the thickness required at 50 kHz

is about 12 mm. In practice the transducer is often bonded to an ultrasonic

cleaning tank and then the tank and water become a complicating part of the

transducer.

‘Tweeter’

Perhaps the best-known example of the use of piezoceramics for the generation

of sonic energy is in the ‘tweeter’. The active element is a lightly supported

circular bimorph. An alternating voltage causes the disc to ﬂex which, in turn,

drives a light-weight cone. Compared with other shapes the circular bender oﬀers

the advantages of a high compliance suited to radiating power into air, a high

capacitance necessary for delivering power to the speaker and a high

electromechanical coupling coeﬃcient. A typical bimorph construction is

shown in Fig. 6.33. The frequency response is nearly ﬂat from about 4 to

about 30 kHz. It is evident from the dimensions of the bimorph that to

manufacture the piezoceramic plates to an acceptably high standard presents a

challenge which has been successfully met.

Simple ‘buzzers’ are made in very large numbers for all types of tone

generators used in phones, smoke alarms, toys etc. The device comprises a poled

PZT disc cemented to a thin metal disc which vibrates at the frequency of the

applied voltage.

398 PIEZOELECTRIC CERAMICS

Wave ¢lters and delay lines

Piezoelectric crystals, notably quartz, are used to control or limit the operating

frequency of electrical circuits. A well-known example is their use in ‘quartz

clocks’. The fact that a dielectric body vibrating at a resonant frequency can

absorb considerably more energy than at other frequencies provides the basis for

piezoelectric wave ﬁlters. The equivalent circuit for a piezoelectric body vibrating

at frequencies close to a natural frequency is given in Fig. 6.3. At resonance the

impedance due to L

and C

falls to zero and, provided that R

is small, the

overall impedance is small.

A ﬁlter is required to pass a certain selected frequency band, or to stop a given

band. The passband for a piezoelectric device is proportional to k

, where k is the

appropriate coupling coeﬃcient. The very low k value of about 0.1 for quartz

only allows it to pass frequency bands of approximately 1% of the resonant

frequency. However, the PZT ceramics, with k values of typically about 0.5, can

readily pass bands up to approximately 10% of the resonant frequency. Quartz

has a very high Q

(about 10

) which results in a sharp cut-oﬀ to the passband.

This, coupled with its very narrow passband, is the reason why the frequency of

quartz oscillators is very well deﬁned. In contrast PZT ceramics have Q

values

in the range 10

–10

and so are unsuited to applications demanding tightly

speciﬁed frequency characteristics.

The simplest of resonators is a thin (about 400 mm) disc electroded on its plane

faces and vibrating radially. For a frequency of 450 kHz the diameter needs to be

APPLICATIONS 399

Fig. 6.33 Hi-ﬁ ‘tweeter’ showing ceramic bimorph plate element. Inset, cross-section of

element showing piezoceramic plates sandwiched between metallized glass ﬁbre mesh.

about 5.6 mm. However, if the required resonant frequency is 10 MHz then the

diameter would need to be reduced to about 250 mm, which is impracticable.

Therefore other modes of vibration, e.g. the thickness compression mode, are

exploited for the higher-frequency applications. Unfortunately the overtone

frequencies are not suﬃciently removed from the fundamental for ﬁlter

applications, but the problem is solved by exploiting the ‘trapped-energy’

principle to suppress the overtones.

The plate is partly covered with electrodes of a speciﬁc thickness. Because of

the extra inertia of the electrodes the fundamental frequency of the thickness

modes beneath the electrodes is less than that of the unelectroded thickness.

Therefore the longer wave characteristic of the electroded region cannot

propagate in the unelectroded region; it is said to be trapped. In contrast, the

higher-frequency overtones can propagate away into the unelectroded region

where their energy is dissipated. If the top electrode is split, coupling between the

two parts will only be eﬃcient at resonance. A trapped-energy ﬁlter of this type is

illustrated in Fig. 6.34(a).

Filters based on this principle made from PZT ceramic have been widely used in

the intermediate frequency stages of FM radio receivers. More stable units suitable

for telecommunication ﬁlters are made from single-crystal LiTaO

and quartz.

Filters and delay lines of the form shown in Fig. 6.34(b) are made which

exploit the surface acoustic wave (see Section 6.5.2 above). The SAW is

propagated in the direction normal to the overlap of the interdigitated electrodes,

the wavelength launched being related to the electrode spacing and width. For a

combined space and half-width of 15 mm(l

surface

¼ 30 mm) the structure will

propagate a centre frequency of 100 MHz. That is it operates as a ﬁlter.

The transit time of the SAW between the launch and receiving electrodes can

be tailored to introduce a speciﬁc time delay into a microwave circuit.

A delay line can also be formed from a slice of a special glass designed so

that the velocity of sound is as nearly as possible independent of temperature.

(The term ‘isopaustic’ has been coined to describe such a material.) PZT

ceramic transducers are soldered on two 458 metallized edges of the slice. The

input transducer converts the electrical signal to a transverse (‘shear’) acoustic

400 PIEZOELECTRIC CERAMICS

Fig. 6.34 (a) Trapped-energy ﬁlter. (b) Surface acoustic wave (SAW) ﬁlter.

wave which travels through the slice and is reﬂected at the edges as shown in

Fig. 6.35. At the output transducer the signal is reconverted into an electrical

signal delayed by the length of time taken to travel around the slice. Unwanted

reﬂections inside the slice are suppressed by the damping spots which are

screen-printed epoxy resin loaded with tungsten powder. By grinding the slice

to precise dimensions the delay time can be controlled with an accuracy to

better than 1 in 10

Such delay lines are used in colour television sets to introduce a delay of

approximately 64 ms, the time taken for the electron beam to travel once across

the screen. The delayed signal can be used to average out variations in the colour

signal that occur during transmission and so improve picture quality. Similar

delay lines are used in videotape recorders.

The intravascular catheter

The intravascular catheter, used on a routine basis to image arterial tissue, is an

essential medical tool enabling the surgeon to identify disease type and to take

appropriate steps to combat it, and the demand is for very large numbers

(currently approximately 175 000 p.a.).

In a typical design the transducer head, illustrated in Fig. 6.36(b), comprises

an array of 64 ‘PZT’ bars (approximate dimensions 5003070 mm) arranged

around the catheter tip.

To form the array a 70 mm thick poled sheet (50.6 mm) of PZT is bonded

between two metallized polymer sheets. The PZT is sliced into the 64 elements by

diamond-cutting, the cut just penetrating the bottom polymer layer so allowing

the whole to be wrapped around the catheter tip. The outer polymer ﬁlm, about

20 mm thick, serves as a l/4 acoustic matching layers and also carries the thin

copper tracks to electrically address each element.

APPLICATIONS 401

Fig. 6.35 A signal delay device.