IEEE TRANSACTIONS O N SONICS AND ULTRASONICS, VOL. SU-24, NO. 5 , SEPTEMBER 1911 331

ACKNOWLEDGMENT The author wishes to thank his colleagues A. Abusetta and

A. Obada for their highly appreciated discussions. He also ex- presses his gratitude to I. Abutahir and L. Fox of Mineral Re- sources for their great assistance in the computational work.

REFERENCES

Soc. Amer., vol. 48, pp. 135-138, 1969. [31 P. R. Stepanishen, “The time-dependent force and radiation im-

pedance on a piston in a rigid infinite planar baffle,” J. Acoust. Soc. Amer., vol. 49, pp. 841-849, 1970.

[4] D. E. Robinson et al., “Nea field transient radiation patterns for circular pistons,” IEEE Trans. Acoustics, Speech, and Signal Pro- cessing, vol. ASSP-22, pp. 395-403, 1974.

151 G. Goudet, Les Fonctions de Bessel e t Leurs Applications en Physique, 2nd ed. Paris: Masson, 1954.

[6] P. M. Morse, Vibration and Sound, 2nd ed., Asian Students’ Ed. New York: McGraw-Hill, 1948.

[7] L. Camp, Underwater Acoustics. New York: Wiley-Interscience, 1970.

Ahtrocr-Measurementsct-Measuremenb have been made of the acoustic loss factor and piezoelectric coupling coefficient of poled polyvinylidene fluoride (PVF2) plastic polymer fiims which are of interest for high-frequency ultrasonic applications. A theory for piezoelectric resonance in unsup- ported fiims was developed for interpretation of the measurements. Unloaded acoustic Q of approximately 14 has been observed at fre- quencies of 41 and 21.5 MHz for f ‘ i i s of 25 and Sop thickness, respectively.

I I. INTRODUCTION

T HAS BEEN shown by a number of authors [ 1 ] , [2] that polyvinylidene fluoride (PVF,)’ polymer plastic films can

be rendered relatively strongly piezoelectric by a simple poling process, opening the way to new possibilities for the design of inexpensive ultrasonic transducers. The piezoelectric coupling constant and the acoustic loss factor of PVF2 were determined by measurements of the components of the electrical imped- ance about acoustic resonance. Film samples were approxi- mately 25 p and 50 p thick and had evaporated conducting electrodes on the film faces. The thickness dilatational vibra- tion mode was employed, which is of interest for high- frequency transducers.

We have found that the properties of PVF2 lead to piezo- electric resonance curves having different characteristics than those of the more familiar ceramic materials such as PZT. As a result, standard methods for measuring acoustic loss factor Q for the latter materials, involving the simple measurement of series and parallel resonances, are not applicable to PVF, . We

Manuscript received August 16, 1976;revised December 20, 1976. This work was supported by the Office of Naval Research under Con- tract N00014-75-C-0632.

Stanford University, Stanford, CA 94305. The authors are with the W. W. Hansen Laboratories of Physics,

’Sometimes referred to in the literature as (PVDF).

have developed the piezoelectric resonator theory appropriate for the ranges of piezoelectric coupling factor k: and acoustic Q of PVF2, and have used a comparison of theoretical and ex- perimental response curves in obtaining quantitative measure- ments of these qualities.

11. THEORY The theory of thin film microwave acoustic transducers for

volume wave excitation has been covered extensively in the literature [3], [4] . However, in those treatments, the acoustic loss in the transducer is assumed small enough to be neglected, and the transducer element is considered in conjunction with an acoustic substrate into which acoustic energy generated by the transducer is radiated. Here, we formulate a one-dimen- sional model of a transducer including internal acoustic loss. We work with acoustically unloaded transducers, without a substrate. Using this model, quantitative predictions can be made for Q and k: which can be checked experimentally through simple impedance measurements.

We consider a piezoelectric resonator configuration consist- ing of a plate or film of piezoelectric material with conducting electrodes on both faces. It can be made to vibrate in one or more acoustic modes by application of a harmonic voltage across the two electrodes. The lateral dimensions of the reso- nator are assumed to be large compared to both its thickness and the acoustic wavelength. The sample orientation is chosen such that the resonator is excited in the thickness vibrational mode only. We assume that the electrode layers are acousti- cally thin, so as not to load the transducer.

Using the piezoelectric equations, equations of motion, elec- trical circuit equations, and boundary conditions, we derive the plane wave piezoelectric impedance matrix for the resona- tor. Internal acoustic loss is accounted for by introducing, in the expression for elastic stress, a term proportion to j S / Q ,

332 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-24, NO. 5 , SEPTEMBER 1977

CB =O.IpF

T ELECTRODES

- - Fig. 1. Charge integration technique used to detect piezoelectricity in

poled PVF2 films.

where S is the strain and Q is the acoustic Q of the piezoelec- tric material defined as the ratio of acoustic energy stored to the acoustic energy lost per radian. We then calculate the elec- trical input impedance of the resonator, expressing the result in terms of an equivalent electrical series circuit. Further de- tails on the derivation are shown in the Appendix.

The electrical impedance curves for the resonator are strong functions of Q in the vicinity of resonance, and curve fitting of experimental impedance values with theoretical curves allows determination of Q and the piezoelectric coupling constant k: for the piezoelectric material, which is defined as the ratio of electrical energy at the output to the total input mechanical energy.

111. EXPERIMENT PVF2 films exhibit relatively strong piezoelectricity when

stretched, polarized under a relatively high dc field Ep at a moderate temperature Tp , and then cooled to room tempera- ture with the dc field on. The piezoelectricity obtained de- pends on E p , T p , and the poling time. The origin of the piezo- electricity is not yet fully understood, although it is thought that the poling process increases true and/or polarization charges in the film through three kinds of process, namely, injection of homocharge from the electrode, dipole orientation in the film, and true charge separation in the film.

For the present experiments, we prepared samples by first evaporating chrome-gold electrodes of -1000 a thickness on both surfaces of PVFz fdms. The masses of the Cr-Au elec- trodes are 4% and 8% of the masses of the polymer films with thicknesses of 50 p and 25 p , respectively. The effects of these mass loadings are to decrease the resonant frequencies slightly; however, in the quantities of primary interest to us, k: and Q, these effects are small compared to the accuracy of measure- ment. The films were used as received from the manufacturer [5], and were not subjected to additional stretching. They were then poled using a dc field Ep > 500 kV/cm for 2 to 3 h in an oil bath at elevated temperature T > 8OoC.

test for the direct piezoelectric effect in the PVF2 film. A stress applied to the film gives rise to an open circuit voltage across the electrodes on the film. Charge developed on the electrodes is integrated by the circuit shown and displayed as a dc output voltage. While the device of Fig. 1 could be used to measure piezoelectric constants quantitatively, we have not carried out the necessary calibration procedures for using it in this way. However, we find it useful as a qualitative check on

After poling, we use the integrating circuit shown in Fig. 1 to

64tI 25pm PVF2 ( I cm x 0 .4cm)

58 I l I I I 1 l

37 38 39 40 41 42 43 44 FREQUENCY (MHz)

Fig. 2 . Acoustically generated absolute electrical impedance of un- loaded PVF2 resonator (-25 pm).

25pm PVF2 I I cm x 0.4cm)

k t 2 = 0.012 0.001 Q = 14?1 C, = 6 0 p F

-80 fo = 40.9 MHz W

1 - THEORY

-65 1 I EXPERIMENT 1

- 60 l l I I 1 I I 37 38 39 40 41 42 43 44

FREQUENCY (MHz 1

Fig. 3 . Phase of electrical impedance of unloaded PVF2 resonator ( ~ 2 5 pm).

the poling process. In all cases, the quantitative evaluation of k; is carried out by means of electrical impedance measure- ments, as described below.

Fig. 2-5 are examples of matched theoretical and experi- mental curves of absolute value and phase of the electrical input impedance as functions of frequency for poled films of 25 p and 50 p thickness.

It is important that the parameters of the electrical circuit, namely, the lead inductance and the conduction loss of the electrodes, be measured carefully and properly accounted for before any comparison of theory and experiment is made. Contact impedance may be larger than the transducer loss im- pedance for resonators of moderate coupling constant. Values of Q and k: obtained from these curves are shown in Table I.

Since both Q and k: are unknown quantities to start with, the direct curve fitting of experimental impedance values with theoretical curves requires that one deals simultaneously with two variables. This lengthy process of quantitatively determin- ing Q and k: is greatly facilitated if we use an approximation which allows us to obtain one unknown independently of the other. Our one-dimensional model of a lossy resonator shows that near resonance the ratio of the acoustic reactance X, to

BUI et al.: RESONANCE IN PIEZOELECTRIC FILM 333

50pm PVF2 (I. lcm I( 0.35cm) kt = 0.0185 0.001

I80 I I l l I I

18 19 2 0 21 22 23 24 FREQUENCY (MHz)

Fig. 4. Acoustically generated absolute electrical impedance of un- loaded PVF2 resonator (250 pm).

50pm PVF2 (1.lcrn X 0 . 3 5 ~ 1 ) k t = 0.0185 * 0.001

Q = 13?1 C, = 32.5 pF 1, = 21.5MHz

-80

- 6 5 -

-60 I I I

1 EXPERIMENT

I I I

18 19 2 0 21 22 23 2 4 FREQUENCY (MHz)

Fig. 5. Phase angle of electrical impedance of unloaded PVF2 resonator (=S0 pm).

the acoustic resistance R,, plotted as a function of frequency normalized to the thickness resonance frequency is a straight line. The acoustic Q can be readily determined from the slope of the straight line with Q being equal to half the slope. Figs. 6 and 7 show that this method predicts a Q of 13.8, whereas all the theoretical curves in Figs. 2-5 were computed with a Q of 14 for 25 p film and 13 for 50 p film. The discrepancy be- tween the two methods is less than six percent.

IV. DISCUSSION An absolute-admittance method, sometimes called the two-

frequency method, has been widely used for determining the properties of piezoelectric materials. This method uses a paral- lel equivalent circuit for a piezoelectric resonator described by Van Dyke [6 ] which consists of the series connection of an inductance L , a capacitance C, and a resistance R , in parallel with a second capacitance Co. The series resonance of the resonator, which corresponds to maximum admittance, is modeled by the resonance of the LC circuit, while the parallel resonance, which corresponds to minimum admittance, is mod- eled by the LCo circuit. The popularity of the method stems from its simplicity, where a complete evaluation of the equiva-

1.00 -

2 2 075-

Q = 1/2 SLOPE = 13.8 25pm PVF2 ( I cm x 0.4cm)

kt = 0.012 C 0.001 V-. z x"

2

5 0.50 - CO = 6 0 p F

5 2 fo = 40.9MHz

2 5 0.25 2 G l - W I I I

3 v) U 0.0

2 a -0.25 - W

LLV

-

I

0 4 0.96 0.97 0.98 0.99 1.01 1.02 1.03 1.04 1.05

J -0.50 - $ 2 E K -0.75 -

1.00 -

2 2 075-

Q = 1/2 SLOPE = 13.8 25pm PVF2 ( I cm x 0.4cm)

kt = 0.012 C 0.001 V-. z x"

2

5 0.50 - CO = 6 0 p F

5 2 fo = 40.9MHz

2 5 0.25 2 G l - W I I I I

3 v) U 0.0

2 a -0.25 -

-

I

0 4 0.96 0.97 0.98 0.99 1.01 1.02 1.03 1.04 1.05

W

-1.00 l Fig. 6. Determination of acoustic Q, independently of kf .

1.00 - - a a = 112 SLOPE N 13.8 n a 5 0 p m PVF2 ( I . l c m x 0.35cm)

3 5 k 1 2 = 0.0185 f 0.001 W > 0.75- U

n

> a

* z u w

3 0 1.01 1.02 1.03 1.04 1.05

E 0.50 - CO E 02 .5PF z 0.25-

1, = 21.5MHz

a O'O 0.196 0.197 0.$8 0$9

' 1 l

o z v * a -0.25 -

- 1.00 l n Fig. 7 . Determination of acoustic Qa independently of k f .

lent circuit constants requires only the measurement of the maximum absolute admittance, the minimum absolute admit- tance, the frequency of maximum absolute admittance, and the frequency of minimum absolute admittance. This method is highly accurate for resonators of high Q (Q > 1000) and rel- atively high Qk: product (50 > Qk: > 10) such as quartz. Martin [7] has extended the range of this method to include resonators of moderately low Q (Q > 100) and moderately low Qk: product, (5 > Qkf > 1) such as barium titanate, by de- riving a new set of equations that improve the accuracy of determining resonator parameters from the same four mea-

334 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SW-24, NO. 5, SEPTEMBER 1977

TABLE I 1

Error 1

I I I l

sured quantities. However, PVF2 has a lower Qkf product (Qkf < 0.25), and for this case we have found that the above methods result in substantial errors in determining k: and Q.

When first working with PVF?, with its Q unknown, we at- tempted to use the two-frequency methods for initial estimates and encountered inconsistencies. After applying the curve fit- ting process described above, it became apparent that the Qkf range for PVF2 lies outside of the applicable ranges of the two- frequency methods. The quantitative errors which result from applying the two-frequency method to materials of low Qk: product are shown in Table 11.

Since completing our work, we find that Ohigashi [8] has concurrently and independently made accurate measurements of Q and kf for PVF2. He has used the parallel equivalent circuit of the resonator [ 6 ] , but in contrast with the two- frequency method, he carried out the measurment over a large frequency range. He then varied the parameters R , L , C, and CO of the equivalent circuit to obtain the best theoretical fit of the admittance curve to the experimental data points. The Q and kf were then obtained from the parameters of the equivalent circuit.

Comparing the parallel equivalent circuit with our series cir- cuit (described in the Appendix), they are found to give the same theoretical fit to the experimental PVF2 curves. How- ever, we believed the series model to be more convenient to use because it offers the possibility of determining Q indepen- dently of k f , which is of value if these two quantities happen to be unknown, as is usually the case with newly discovered materials.

Our measured values for the coupling coefficient kt are less than the largest value reported [8] for PVF2, which is 20 per- cent. We believe this difference results from using lower poling voltage and low poling temperature. This is consistent with the fact that our kt for 50 pm film is slightly greater than that obtained by Ohigashi, where our poling voltage was slightly higher.

We have consistently obtained larger values of Q than those obtained by Ohigashi (14 as compared to 9). These values of Q are constant for different samples and different thicknesses.

I L

Modified two-frequency method7

< 4

-2

22

-21

-31

Q

One explanation is concerned with possible structural differ- ences in the films. We used oriented films directly as received from Kureha Co., while Ohigashi started with unoriented sheets and uniaxially stretched them, and the stretching condi- tions may have been different.

APPENDIX ANALYSIS OF AN UNLOADED AND LOSSY

PIEZOELECTRIC RESONATOR We give here a derivation of the series equivalent circuit for a

lossy piezoelectric free resonator. We assume that the lateral dimensions of the resonator are large compared to its thickness so that a one-dimensional analysis can be made, and that the electrode layers are acoustically thin, so as not to load the res- onator. We show in Fig. 8 the one-dimensional resonator with its associated electrical and acoustic variables upon which our derivation of the electrical series equivalent circuit are based [9]. The equations necessary for the derivation of the equiva- lent circuit are given below.

Piezoelectric Equations

Boundary Conditions

F , = -A TI (0)

F2 = - A T2 ( l )

U1 = v(0)

02 = -v( l ) .

Acoustic Equations

dT dz - _ - jwpu

du - =id . dz

BUI et al. : RESONANCE IN PIEZOELECTRIC FILM 335

2 = o 2 ' 1 Assuming small acoustic losses

we find

~~ 1 F = FORCE Y = PARTICLE VELOCITY

v3 A * ELECTRODE AREA t i FILM THICKNESS

Fig. 8. Electrical and acoustic terminal variables of thin transducer.

The total current through the transducer is

I3 = juAD. (9)

Since current is conserved, D must be uniform with z(aD/az = 0). The voltage across the transducer is

2

V , =/ E d z

We derive the plane wave equation for the lossy resonator by modifying one of the two standard linear piezoelectric equa- tions that relate stress T, displacement current D, strain S, and electric field E. To account for internal acoustic losses, a new term is added to the stress equation (l), which then becomes

T = cES t j q S - eE (1 1)

where jwq represents acoustic damping [ 9 ] . From (2) we ex- press E in terms of D and S. We substitute that result into (1 l), which then becomes

T = (cD + juq )S - - eD € 8

where

c D = c E (l +-&)

where cD is the stiffened elastic constant and e is the piezo- electric coefficient. Next using the two acoustic equations, (7) and (8), and (1 2), the following wave equation for the medium is obtained:

where p is the density and U is the particle velocity. The above equation has a complex wave propagation constant y = a + j p as expected for plane wave propagation in a lossy medium. Solutions to the wave equation are of the form

v = A cos yz + B sin yz (15)

where

y* = ( p -jay = tCD W 2 P

+ j w

where p, is the wavenumber in the lossless case. The acoustic Q of the lossy resonator can be conveniently expressed in terms terms of the attenuation constant (Y and the propagation con- stant 0 as

We solve the wave equation in terms of the particle velocities U , and u2 using the boundary conditions of continuity of ve- locity (5) and (6) and obtain

U = u1 sin y( t - 2) - uz sin yz

sin yc

We substitute (20) back into (g), (12), and the two boundary conditions of continuity of force (3), (4). One can character- ize the acoustic resonator as a three-port network by express- ing the two acoustic port variables F 1 , F 2 , and the electrical voltage V , in terms of u l , u 2 , and 13.

can then be described by the following 3 X 3 matrix: The electroacoustic properties of the piezoelectric transducer

- B coth y t B csch y t

B csch y t B coth y t

h h - W

- W -

where B = AZo(l t j [ l/(2Q)] ), h = e / e S is the piezoelectric constant, CO = eSA/r is the clamped capacitance of the piezo- electric transducer, and Zo = (cDP)/w is the acoustic imped- ance. The impedance matrix of (21) reduces to the more fa- miliar lossless case if we set a equal to zero. The above impedance matrix enables us to determine the electrical equiv- alent circuit of any lossy transducer by solving for V3/Z3 with the appropriate acoustic terminations F1 and F z . I f the trans- ducer is loaded on both sides, then F, and Fz take on finite values that are proportional to the acoustic impedances of their respective loads [3]. In our case where we are working with unloaded transducers, we set F , = F2 = 0 and solve for the electrical impedance Z 3 = V3/Z3. We express coth yc and csch y t in terms of cosh at, cos fit, sinh CUT and cos fit, with CY

related to 6 and Q through (19). We expand cosh at and sin sinh at to the first order of ac. The results show that the elec- trical behavior of a free piezoelectric resonator can be charac- terized by a circuit consisting of the total electrode capaci- tance CO in series with an acoustic loss resistance R, and an

336 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-24, NO. 5 , SEPTEMBER 1977

acoustic reactance X,. The expressions for R, and X, are

where k: = (e:3)/(cDe33), B = a(w/wo) is the normalized fre- quency, and wo = (n / t ) (cD/~)"* is the half-wave resonant fre- quency. Both R, and X , are functions of frequency and of the acoustic Q of the piezoelectric material. R, attains its maximum value near the center frequency f o , while X, be- comes negative for a frequency greater than fo and positive for a frequency less than fo .

All of our impedance measurements were done at room tem- perature using a Hewlett-Packard model 4815-A RF vector

impedance meter. The total electrode capacitance CO was determined from measurements at frequencies far from reso- nance. Next, the coupling constant k: and the quality factor Q were used as adjustable parameters to get the best theoreti- cal curve fit to the experimental data points. This process of choosing two variables at the same time is tedious. Examina- tion of the behavior of the resonator near resonance offers us a new method for determining Q independently of k:. In (26), .9/(4QZ) is negligible with respect to sin 0 . We take the ratio of X , to R, and expand it around resonance:

where 8 = n(1 t S). Thus Q can be conveniently determined from the slope of the ratio of X , to R , near resonance.

REFERENCES [ l ] H. Kawai, Japan J. Appl. Phys., 8, 975 (1969). [2] N. Murayama, K. Nakamura, H. Obara, and M. Segawa, Ulfrason-

ics, 14, 1 , 15 (1976). [ 3 ] T. M. Reeder and D. K. Winslow,IEEE Trans. MTT-I 7 , 11, 927

(1 969). [ 4 ] D. A. Berlincourt, D. R . Curran, and H. Jaffe, Physical Acoustics,

W. P. Mason, Ed., Vol. IA, Chapter 3, Academic Press, New York (1964).

NY 10017. 151 Kreha Corporation of America, 420 Lexington Ave., New York,

[6] K. S . Van Dyke,Phys. Rev., 25, 895 (1925). [ 7 ] G . E. Martin,J. Acousf. Soc. Am., 26, 3,413 (1954). [ 8 ] H. Ohigashi, J. Appl. Phys., 47, 3, 949 (1976). [9] B. A. Auld, Acoustic Fields and Waves in Solids, Volume l , Chap-

ter 3, Wiley-lnterscience, New York, 1973.