Ultrasonic energy measurements using a ceramic crystal

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL M. I. Ialsi* and B. BROWN Department of Pure and Applied Physics, University of Salford, Salford 5, Lanes. (Great Britain) (Received: 9 October, 1972)

SUMMARY In this paper we report on measurements of the energy output of a transducer unit under linear and non-linear conditions of operation, using a ceramic crystal. The theoretical method of analysis of the observations, and their significance, are discussed in some detail.

INTRODUCTION

The phenomenon of reversibility exhibited by piezomagnetic and piezoelectric transducers indicates the ability of the transducer to convert energy at the same efficiency in either direction from one form to the other. These transducers are mainly polarised polycrystalline magnetostrictive alloys and electrostrictive materials, ceramics and piezoelectric single crystals. For convenience, transducers which convert any form of energy into acoustic energy will be referred to as transmitters and those which convert acoustic energy into any other form of energy as receivers. Transmitters and receivers do not differ in construction (shape and geometry) and modes of vibration and, by the reciprocity theorem, their performance may be represented by the same mathematical expressions and physical principles. Piezoelectric and ceramic transducer materials have been studied by numerous workers. 1'2 The reports include those of Langevin 3 and Williams 4 on their use as acoustic receivers in liquid media, and van der Burgt 5 and Onoe and Sawabe 6 * Present address: Joseph Lucas Limited, Group Research Centre, Shirley, Solihull, Warks. (Great Britain). 255 Applied Acoustics (6) (1973)-,-O Applied Science Publishers Ltd, England, 1973--Printed in Great Britain

256

M. I. IBISI, B. B R O W N

on receivers used in solid media, either as motional feedback or vibration pick-up systems. Although the highest efficiency and maximum sensitivity of the receivers are obtained from operations at both mechanical and electrical resonances, in general they are designed for wide-band operation achieved at frequencies well below resonance. Mechanical resonance requires that, at the receiving frequency, the length or thickness of the receiver be a half wavelength, while electrical resonance demands the cancellation of the reactive component of the transducer impedance by tuning with a shunt component. The system of interest here is based on a motional feedback or vibration pick-up unit operated at electrical resonance.

EXPERIMENTAL SET-UP

The ultrasonic transducer unit, whose energy output was observed in the experimental work, is based on a commercial cleaning tank system. It is composed of a mild steel piston, flange-mounted at the bottom of a stainless steel tank and to the lower end of which are attached six 4 per cent C o - N i laminated transducer rods designed to resonate at a b o u t 13.0 kHz. The unit was driven by a variable frequency electronic generator capable of a variable power output of about 2 kW. The transducer unit and electronic generator 7 were described in detail in an earlier paper. 8 To simulate industrial ultrasonic equipment where the transducers may not easily be reached and the resulting ultrasonic field must not be disturbed in any way, a dual purpose ceramic crystal was mounted on the cylindrical surface of the mild steel piston at a distance of 0-057 m from the lower end of the piston and underneath the stainless steel tank. The crystal, a PXE5 ceramic disc (MBI001) of diameter 16-0 x 1 0 - 3 m and thickness 1.1 x 10 -3 m, was supplied by Mullard Equipment Limited. It has a Curie temperature of 280°C, a frequency constant of 2000 Hz m and an overlapping negative electrode, so that connecting leads could be soldered on to one surface (Fig. l(a)). The coupling of the crystal to the mild steel piston was accomplished by the use of a circular groove of similar dimensions to the crystal turned out on the piston. Figure l(b) is an illustration of the cross-section of the piston and the position of the groove. Care was taken to ascertain that the bottom of the groove was fiat and that the groove was of such dimensions that the crystal could easily be lowered into it without strain. The pick-up crystal was cemented into the groove with Araldite and the bond strengthened by curing in an oven at an elevated temperature (150°C). Connecting leads were soldered on to the two electrodes with a low thermal capacity soldering iron, care being taken to minimise the time of contact between the soldering iron and the crystal. The crystal was used both as a motional feedback unit 7 and as a vibration pick-up system for energy measurements.

257

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

"J" V e

E L~-CTRODE

OVER L FIPPINCI

-V~

~'LE CTROD E

(a)

~--.--

EN',~TT=NG;END

._.__~x

J'~ouNTING F'LANC, E

G

_

_

B

GI /

.F.t-- Y = R

I

(b)

(c)

Fig. I. (a) Sectional diagrams of the ceramic disc used, illustrating the design of the electrodes; (b) (left): Cross-section of mild steel piston showing the position of the groove; (c) (right): Plan of the plate element showing the axes of co-ordinates.

0-58

M . I . IBISI, B. BROWN THEORETICAL CONSIDERATIONS

For the system described above, the voltage across the ceramic disc represents, in amplitude and phase, the response of a motional variable, i.e. the strain in the mild steel piston, the displacement, velocity or acceleration at the position of the disc and within the piston. These motional variables are interrelated and are also related to the stressed state of the piston material by which the ceramic disc was adapted as an energy measuring device. In a theoretical paper, Chaban 9 employed the stressed state of a solid medium into which a spherical ceramic disc was 'frozen' to determine the sensitivity of the disc. He defined the sensitivity as the voltage per unit pressure on the disc and arrived at a relationship wholly dependent on the material properties of the solid medium and the disc. A similar approach is adopted in the analysis of observations made on the six-rod transducer unit described here. Let 21, ~t t be the Lam6 parameters of the ceramic disc of radius r and thickness d frozen into the mild steel piston. Assuming that : (a) the applied force at the lower end of the piston is uniformly distributed over the end face of the piston, (b) that there are only longitudinal vibrations in the piston, i.e. all points in a cross-sectional area perpendicular to the axis of the piston vibrate in unison with the same amplitude and phase, and:

(c) that, relative to the dimensions of the ceramic disc, the piston may be considered to be infinitely large; an axial section of the piston (Fig. l(c)) represented by the z-)' plane (the force is applied in the x-direction) and of thickness d may be regarded as a plate cut in such a manner that it contains the pick-up crystal. In other words, the problem has been reduced to that of an elastic plate of thickness d and length equal to that of the piston and with a circular hole into which an elastic circular disc with an equal radius has been inserted. The axis of polarisation of the pick-up crystal lies in the z-direction. According to Muskhelishvili, ~o the components of stress in the disc may be determined by a consideration of the elastic equilibrium in the circular disc and in the plate with a circular hole. If 22, ~£2 are the Lam6 parameters of the mild steel medium and if it is assumed that there is no friction between the disc and mild steel plate; the components of stress in the disc will be given t° in terms of the pressure, P, along the circular boundary as: X~ = p f l t + 61. ~1 - cSt

r,=PG

= G

= o

2 (t)

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

259

where Xx and Yy are the x and y components of stress and fit, fit are constants dependent on the Lain6 parameters 2 t, Pt, 22, P2. By plane stress theory, the z-component of stress is: z : = v(Xx + r~) (2) where v is Poisson's ratio. The constants fit and 6 t are defined as: ~,1(z2 - 1) ~ t = 2~L~ + ~,2(z~ - - 1)'

#1(z2 + 1) ~

2* + 3p. z = ;, + ~,

=

2* =

~t/,2 --~ t / l •

2

2).~t (~. + 2~L------~

Using the relation between the stress components and the electric field vector in the piezoelectric medium, the potential difference across the pick-up crystal is: V = g33Z._ + g3t(X'x +

ry)

= P#gl(g31 + vg33) V = f l l ( g a l + vg33)

(3)

g33 and ga 1 are the piezoelectric output voltage coefficients relating electric field to mechanical stress, applied parallel, and perpendicular, to the polarising axis respectively. In order to determine the power output of the transmitting transducer unit, the transmission of mechanical waves through the length of the mild steel piston has to be considered. Since the piston is of finite length, a system of standing waves will be set up when a sinusoidal force is applied to the lower end. The system can be equated to a transmission line and may be represented as in Fig. 2(a) (Z). The design criterion of the transducer unit required that the mid-point of the piston, "Us;

7a Zs

J

Z-~

Fig. 2(a). Electrical eq uivalent circuit of a transmission line.

260

M. I. IBISI, B. BROWN

F=~.

X'= if/2. ~ "tL=O 2f_ = ~ 1

, ~ENTRF=

oF

CERRtc~tr.-

I)tSC~

tt='~o,

F=F'o

e

ZK

::x.=O, Z~;--O~'t/,a;=o~ Fig. 2(b) Fig. 2.(b)

The mild steel piston and the conditions assumed in the application of the principle of transmission line.

i.e. x = 1/2, be a modal point of motion (Fig. 2(b)) and the particle velocity (u) be zero. This yields an approximate expression for the force (Fo) at the lower end of the piston:

Fo = F v / (Zk sinh O (l - x~) + Z, cosh O (l - xl)) Z~-s~nl~ ~ + Z, cosh Ol

(4)

where x~ is the distance of the pick-up crystal from the lower end of the piston, Fp is the force detected by the crystal, Z k and Z, are the acoustic impedance of the mild steel piston, and the terminating impedance respectively, I is the length of the piston and 0 is the propagation constant. The output force, Fw, is given by (Z s = 0)

FoZ, F~, = Zk sinh Ol + Z, cosh Ol

(5)

and the power output at the emitting end of the piston by:

nr2po2Z, W = 2(Zk sirth Ol + Z, cosh Ol)

(6)

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

26 l

EXPERIMENTAL OBSERVATIONS AND RESULTS

The experimental set-up for the pick-up crystal consisted of a decade inductor connected across the crystal and an oscilloscope as a voltage detector (Fig. 3). Over a frequency range (12.0 to 15.0 kHz) and for air and water load on the magnetostrictive transducer unit, readings of the voltage across the pick-up crystal tuned for electrical resonance were recorded. The polarisation current to the transducer unit was 3.0 amps and water at a height of 0-17 m above the mild steel piston face was used as load.

PXE5 ])ISC O

am~

-_c

O Fig. 3.

Experimental set-up of the pick-up crystal.

Next, keeping the magnetisation current constant at 3.0 amps, observations of the voltage across the crystal were made for: (a) constant excitation current (0.2 amps) into the transducer unit, and : (b) constant voltage (50.0 volts) across the transducer. In both cases the load on the transducer was water, the frequency range was 11"5 to 14-0 kHz and, for every frequency setting, the pick-up crystal was tuned by adjusting the decade inductor.

262

M. I. IBISI, B. BROWN

7(

i~Air

6O

load

30 ~ 4o ~

/_

wo;~jo.,:%o

3o

i ' \/

2o

\ -,,.

\

!

125

I

I

133

130 I

L

I25

140 I

130

'

135

140

Frequency K HZ Fig. 4.

Pick-up crystal method--calculated power output as a function of frequency for air and water loads on the six-rod transducer unit.

7C

Constant .- J J

current

6¢ P~ Constant voltage eye,e,)

=

l.J,~p,a~ed 200

•

j\/!

2C

/;\\

qC

I;)'3

130 125

14'0

135

130

13"5

140

Frequency K Hz Fig. 5.

Power output of the six-rod transducer unit calculated from observations on the pick-up crystal as a function of frequency for constant current and constant voltage.

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

263

The final set of observations made with pick-up crystal was at the resonance frequency (13.13 kHz) of the magnetostrictive transducer unit. Several values of the de polarising current were chosen and, at each of these values, the voltages across the pick-up crystal and the transducer unit were observed for varied values of excitation current. Here again, the transducer unit was loaded with water and tuning of the pick-up crystal was conducted for each observation. The basic relationships derived earlier were eqn. (3) for the sensitivity of the meter, and eqns. (4), (5) and (6) for the initial force (Fo) at the lower end of the piston (x = 0), the force (Fw) at the emitting end (x = I) of the piston and the power output respectively. The sensitivity was obtained by multiplying eqn. (3) by the thickness of the disc and averaging over its surface area: V --- -- = 0.02995 volts/Newton/m 2 P The calculated acoustic power output values are plotted as functions of frequency in Figs. 4 and 5 and as functions of power input to the transducer unit in Figs. 6 and 7, Fig. 6 for magnetisation current 2.0 to 3.5 amps and Fig. 7 for 4.0 to 5.5 amps in steps of 0.5 amps. In both cases, only one complete curve has been drawn (3.0 amps for Fig. 6 and 5-0 amps for Fig. 7), the distribution of the other observations (for 2.0, 2.5, 3.5, 4.0, 4.5 and 5.5 amps) being merely indicated. There are three sections in the curve of Fig. 6: (i) The low ultrasonic intensity section with a very high gradient. This is indicated in all observations (Figs. 6 and 7) by the value obtained for an excitation current of 0.5 amps and a curve in the lower end of the solid line. Below this curved section is the region of linear response of the transducer unit and, when produced, the solid line should pass through the origin. (ii) The straight section of the graph corresponding to observations made for excitation currents of 1.0 to 2.25 amps and with a slightly lower gradient than (i) above. This is the region of medium and high ultrasonic intensities indicative of non-linear response of the transducer unit and the presence of cavitation in the liquid medium. (iii) The flat top of the graph corresponding to observations made for excitation currents of 2-5 to 3"0 amps. This region expresses the limits of electrical input power that the transducer can accept and the possibility of a strain reversal process in the transducer material. In this region, the peak amplitude of the excitation current must have exceeded the value of the magnetisation current (3.0 amps) or the power input (Wr) to the transducer unit is greater than one half the product of the square of the polarising current (Ip) and the total resistive component (R) of the driving point impedance of the transducer unit corresponding to the polarisation current ~1 (i.e. Wr > Ip ~ R/2). This factor is borne out by the curve of Fig. 7 where the fiat top is absent due to the peak amplitude of the maximum excitation current (3-0 amps ac) being less than the polarising current (5.0 amps dc).

264

T

M. I. IBISI, B. BROWN

160

x

AcousTiC OUTPUT

(WATTS)

(9 •

~2-0 a 2.5 R ~ .o n

+

3-5

/~a~NF'I'IZ~TION CUR'RE~T

R

+

J2~

96 X

/ --I-

]

Fig. 6.

,

I

I

1

O b s e r v e d o u t p u t p o w e r by the pick-up m e t h o d as a function o f the power i n p u t to the six-rod t r a n s d u c e r unit for varied values o f rnagnetisation current.

265

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

ACOUSTIC T OUTPUT

~I~

/I.-@ R ~ q N C T I Z R T I O N ~..S ~ CURRENT

• +

$,o R S.S

VFILU(~S

96

x

+

+

/

52

o Fig. 7.

x

at

I

I

I

I

2t, o

~eo

7ao

9,1o

Observed output power by the pick-up method as a function of the power input to the six-rod transducer unit for varied values of magnetisation current.

266

Xt. I. IBISI, B. B R O W N

The acoustic impedance of PXE5 is about 3.15 x i0 v k g m - - ' s--' and there should be some degree of matching between the mild steel piston of impedance 4.6 x 107 kg m -2 s-~ and the crystal. Applying the principle of transmission line to a line (mild steel piston) 0.057 m long, the impedance presented by the pickup crystal to the transducer unit is about 3.9 x 107 k g m -2 s -~. For sound travelling in a semi-infinite medium (1) and incident on the boundary between (l) and semi-infinite medium (2), the intensity transmitted is given by i _,: I2 =

Il

(ir +4r1) 2)

where I is the intensity, r = Z 2 / Z t and ZL, Z2 are the impedance of the media.

30

20

I0

,

Fig. 8.

I 2

i

I 4

i

l

Efficiency of transducer indicated by pick-up crystal as a function of polarising currem.

ULTRASONIC ENERGY MEASUREMENTS USING A CERAMIC CRYSTAL

267

If this expression is applied to the pick-up crystal set-up, 34.5 per cent of the output intensity of the transducer unit is seen to be transmitted into the medium of the crystal. The crystal indicated a transducer efficiency of 22"0 per cent for a de polarising current of 3-0 amps through the transducer unit (Fig. 6). The experimental results indicate a variation of transducer efficiency with the magnitude of the polarising current (Figs. 6 and 7). Values of transducer efficiency

E-

J

0

2

,

|

PoL.mZ,N¢ c u ~ t n r

Fig. 9.

~

I

4

6 (t~l, 0

Apparent efficiency of the six-rod transducer unit as a function of polarising current.

calculated from these results are plotted in Fig. 8. They indicate a rise in efficiency with increasing polarising current to a peak value (22.9 per cent) at 3.5 amps, followed by a decrease in efficiency with further increases in polarising current. This may be explained by a consideration of the relative deformation of the piston and the corresponding mechanical forces, and losses due to eddy current and mechanical hysteresis. While increases in relative deformation and mechanical

268

M.I. IBISI. B. BROWN

force cease at s a t u r a t i o n i n d u c t i o n , the losses increase with further increases in both the polarising and excitation c u r r e n t values. Thus, the power o u t p u t of the t r a n s d u c e r u n i t will increase with increasing polarising c u r r e n t until saturation i n d u c t i o n is attained, a n d will decrease with further increases in i n d u c t i o n due to increases in losses. Further, at m e d i u m a n d high ultrasonic intensities, cavitation in the liquid m e d i u m will tend to load the t r a n s d u c e r unit with further decreases in energy output.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

W. G. CAD'C,Piezoelectricity, McGraw-Hill Book Co. Inc., N.Y., 1964. W. P. MASON,Physical acoustics, Academic Press, London and New York, 1964. R. A. LANGEVtN,J. Aeoust. Soc. Am., 26 (1954) p. 421. O. A. WILLIAMS,J. Acoust. Soc. Am., 16 (1945) p. 231; 17 (1946) p. 219. C. M. VAN DER BURGT, J. Acoust. Soc. Am., 28 (1956) p. 1020; IEEE Trans. PGUE, U-10, 2 1963. M. ONOE, and M. SAWAaE,Proc. IRE, 50 (1962) p. 1967. M. I. Imsr, PhD Thesis, University of Salford, 1971. M. I. Ialsl and B. BROWN,The efficiency of a magnetostrictive tranducer unit : Part 2, Applied Acoustics, 6 (1973) pp. 171-191. I. A. CHABAN,Societ Phy. Acoust., 11 (1966) p. 344. N. I. MtJSKHELISHVILt,Some basic problems of the mathematical theo~3' of elasticity, 3rd Ed., P. Noordhoff Ltd, Groningen-Holland, 1953. E. G. RICHARDS3N, Technical Aspects of Sound, Volume 2, Elsevier Pub. Co., Amsterdam, 1957. R. G, GOLDMAN,Ultrasonic Technology, Reinhold Pub. Corp., Chapman & Hall Ltd, London and New York, 1962.