IEEE TRANSACTIONS O N SONICS AX11 OLTR.tSONICS, VOL. SU-14, NO. 4, OCTOBER 1967 167

REFERENCES [I1 A. A. Ballman, “The growth and properties of piezoelectric

bismuth germaniunl oside 13i,2Ge020,” Internat’l J. Crystal Growth, vol. 1, pp. 37-40, Janmry 1967.

bismuth germanium oxide (Bi,zGeOzo),” ibid., vol. 1, pp. 45-46. P I J. L. Bernstein, “The unit cell and space group of piezoelectric

elastic and piezoelectric constants for crystah in class 3m,” J. ($1 A. W. Warner, M . Onor, and G. A. Coquin, “Determination of

Acoust. Soc. Am. (to be putdished).

J. Acoust. Soc. Am., vol. 35, pp. 53-58, January 1963. l 4 1 H. F. Ticrsten, “Thickness vibrations of piezoelectric plates,”

I5l M. Onoe, 11. F. Tiersten, and A. 11. Meitzler, “Shift in the location of resonant frequencies caused by large electronmhanical coupling in thickness-mode resonators,” J . Acoust. Soc. Am., vol. 35, pp. 3 6 3 2 , January 1963.

properties of bismuth germaninm oside,” A p p l . Phys. Lett., vol. 9. 161 E. G. Spencer, P. V. Lenxo, and A. A. Ballman, “Ultrasonic

pp. 290-291, October 1966.

Transmission Parameters of Thickness-Driven Piezoelectric Transducers Arranged in

Multilayer Configurations

Abstract-The individual transducers of an ultrasonic delay line may consist of a multiplicity of piezoelectrically active layers electrically connected in series, parallel, or grouped in series-parallel combinations interspersed with electrically conductive or nonconduc- tive layers of different characteristic acoustic impedances. The stack of transducer layers may be loaded by an absorptive or reactive backing and coupled to the delay medium through bonding and matching layers.

The transmission parameters for such configurations are writ- ten in a form well suited to digital computation. Inspection of numerical results reveals effects which may be qualitatively un- derstood by visualizing separately the effects due to the mechanical resonances of the layer assembly and those due to the arrangement of piezoelectric material with respect to the stress distribution within the stack.

The examples given indicate that transducers consisting of alternately poled stacked h/2 layers of a low coupling factor ma- terial such as CdS give an insertion loss improvement at the cost of bandwidth reduction little different from that obtained with narrow-band tuned terminations. For high coupling factor layers, no significant improvement is obtainable.

I I. INTRODUCTIOX

T HAS BEEN suggested in recent literature, [ 1 1 ’ [ 2 ’

that the “conversion efficiency” of microwave ult.ra- sonic transducers can be improved by arranging them

as multilayer stacks in which either the sign of the driving field or polarity of the layer is inverted every half mave- length, or in which piezoelectrically active layers alternate with inactive ones, both being a half wavelength in thickness. Numerical computations based on simultane- ously fulfilling wave equation solutions at the interfaces between the layers have been obtained and have shown qualitative agreement with the experiment.

hlanwcript received June 5, 1967. The author is with Dell Telephone Laboratories, Inc., Murray

Hill, N. J.

However, all these investigations used cavity resonators as the coupling means between the transducer structure and the electrical circuitry, it mas rarely made clear what the actual circuit parameters were. In particular, the question was left open whether the observed increase in “conversion efficiency” was in fact a simple trade-off of bandwidth versus insertion loss.

For the purpose of de1a.y line design, a gain-bandwidth product is a more appropriate figure of merit, since usually signal bit rate and storage capacity are the variables one wants to maximize simultaneously. It is, therefore, desirable t’o obtain means to assess multilayer systems in terms of circuit parameters which realistically approxi- mate a given design problem by computing a given, easily variable configuration.

To this end, a procedure will be described, based on Mason’s equivalent c i r c~ i t , !~ ’ which is well suited to digital computation while still being a realistic approxi- mation of the actual configuration. Numerical results obtained from it will then be used to obtain a “guided intuit,ive” understanding of the effects due to the stress distributions at the resonance frequencies and the elec- trical output from the piezoelectric sections arranged within the st)ack.

11. CHAIN MATRIX DESCRIPTION OF A hlULTILAYER CONFIGURATION

The following assumptions are made at the outset so that simple equivalent circuits will represent the various layers.

1) The transducers have lateral dimensions comprising

2) The transducers, the delay medium, and all backing many wavelengt,hs of sound.

and intermediate layers are loss-free.

168 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, OCTOBER 1967

3) The crystal symmetry of t,he transducers permits description by a single coupling factor.

4) The transducers are thickness-driven. 5) A single strain component suffices t)o describe the

sound propagation.

Under these circumstzlnc*es the complete delay line, seen as a two-port network, can he described as a tandem connection of two-port networks representing the various backing and intermediate layers, the transducers, and the delay medium. The transmission properties of the whole device are expressed in terms of the complex com- ponents of its chain matrix. In the case where each trans- ducer consists of only one piezoelectric layer, t’his chain matxi.: is obtained by simply multiplying all the two-by- two chain matrices describing t,he individual layers.

In the case of the transducers consisting of a multi- plicity of piezoelectric layers interspersed possibly with piezoelectrically inactive ones, the delay line in toto can still be split up into a series of two-ports describing its parts as shown in Fig. 1. However, the boxes labeled input and output transducer may have, in fact, a configuration as shown in Fig. 2. Each transducer may consist of a series of layers with subscript cn ( c for “crystal,” n =

1 . .. N C ) , each described by its acoustic impedance Z,, = pricnAn (A cross-sectional area, p density, c sound velocity), its geometrical thickness I,, its electrical capaci- tance C,, and, if piezoelectrically active, its electrome- chanical coupling factor k,. The transducer stack may be backed by layers subscripted bn ( b for “backing,” n = 1 . . . N B ) , and may be coupled to the transmission medium through intermediate layers subscripted tn ( t for

The additional con~plication over the case of the single active layer transducer arises from the possibility of circulating currents between or unequal voltages across different active layers when these are connected in series, parallel, or groups. I n addition, these modify the effective sound velocity in each active layer and thus affect its mechanical resonance behavior. Finally, elect,rical energy extraction from the electrical terminals of an active layer is tantamount to sound wave absorption between its mechanical terminals. This affects t’he contribution of layers further along the stack, or, in other words, causes a finite penetration depth of the sound wave into the stack.

In order to include all these effects, it is advantageous to visualize each transducer layer as a four-port network with two pairs of electrical and two pairs of mechanical terminals. This representation is shown in Fig. 3 using Mason’s equivalent circuit for a thickness-driven trans- ducer. The force F and displacement velocity U were chosen as the mechanical variables, the voltage E and current i as the electrical variables. The values of the sound velocity c, the coupling factor k , and the clamped capacitance CO may be obtained from the elastic and piezoelectric constants by equations given in Berlincourt et al.

As indicated in Fig. 3, the electrical terminals can

Id transmission,’’ n = 1 . . . N T ) .

ELECTRICAL

TERMINATION TRANSDUCER M E D I U M I N P U T I N P U T T R A N S M I S S I O N

TRANSMISSION OUTPUT E L E C T R I C A L

O U T P U T MEDIUM TRANSDUCER TERMINATION

v c + V w ‘ Z F T ~ X \ E l J 0 I

Fig. 1. Block diagram of an ultrasonic delay line with piezoelectric transducers.

COMMON 1 1 TERMINALS

TRANSDUCER (, TERMINALS y 0 L , ( > 0

zb2 ‘ZNT l t l pc6 IC4 ec3 pc2 ebl ‘zNE

zt2 Z t l zc6 zc5 zc4 zc3 zc2 zcl zbl

BACKING ‘11 ‘c6 ‘c5 ‘c4 ‘c3 ‘c2 ‘c1 ‘bi

c02 c06 c05 c04 c03

k i k 3

I MEDIUM TRANSMISSION

tmn THICKNESS A,, CROSS - SECTIONAL AREA Cmn SOUND VELOCITY Qmn DENSITY Z,, = Amn Q C,,, CHARACT. IMPEDANCE m, n

OF LAYER WITH SUBSCRIPTS

Con CLAMPED CAPACITANCE k n COUPLING FACTOR

Fig. 2. General arrangement of the individual layers o f a multilayer transducer.

be either connected in series into the common feed line (continuous lines) or in parallel (dashed lines). One may then indicate polarity reversal or an inactive layer by giving k a positive, negative or zero value.

The input terminal value of the force F , the displace- ment velocity U , the voltage E, and current i are related to the corresponding output terminal variables by the following equation

in which ( T ) , is a 4 x 4 matrix whose complex components T;, may be obtained by straightforward circuit analysis

SITTIG: PIEZOELECTRIC TRANSDUCERS

of the network shown in Fig. 3. With the variables in- dicated there, dropping the subscript.s nzn (W being, how- ever, the angular frequency, not unm) and the abbrevia- tions

S = I? sin r/y c = Ic'(1 - cos y)/y, (2)

one obtains for :I parallel connected tmnsducer

169

cosy - S jZ,,(sin y - 2r) (cosy - l)@ l - S 1 --S

-

0 1

or parallel connected so that E,, = EN,+,, i,,,, = 0, one advsntageously reduces a t this point to a two-by-two matrix relating E,, i, and F.vc+l, u ~ ~ + ~ . Writing thus

one finds the components of ( A ) , to be:

1) For parallel connection of the NCth layer

i o 0 0 1 1 A piezoelectrically inert layer or an electrical connection network can be included in this fonnalisnl by a matris

cosy jZ, sin y 0 0

j sill y / Z , cosy 0 0

0 0 8, B,

0 0 DE.

( R = (5)

in which for a purely electrical network y = 0 and A,, B,, CE, D E are the standard two-by-two chain matrix components of the two-port network connecting the elec- trical terminals. A mechanical layer would have y # 0, A , = D E = 1 and eitJher CE = 0, BE = l/jwC,, or B, =

0, C, = jwC, depending on whether its electrical capaci- tance C, appears connected in series or in parallel between the electrical ternlinals.

For a series of N C layers the input variables F,, U,, E,, i,, and t'he output vnriuhlcs F,..,,, E,,+,, z N c + , are again related by

and the matrix describing the transducer with all its intermediate layers from

in which (1') is given by F"-

(T) = Jj (ll)=. n-l

If the stack of transducers described by (2") is loaded by an impedance Z b and if the layer with subscript NC is either series connected so that iNc = iNc+,, ENc+I = 0,

2) For series connection of the NCth layer

However, Z , may in practice be t'he impcdance presented to the back face of the transducer stack by another stack of NB-1 piezoelectrically inactive I:~yers, subscripted by bn, terminated at its far end by an impedance Z b N B . Furthermore, the transducer stack nmy be loaded at i ts transmitting face by :m impedance Z,, seen through NT-1 intermediate layers.

These layers may be described by two-by-two matrices as

The chain matrices describing the layer assemblies ilz toto are given by

w i t h N = N B - l f o r n z = b a n d N = N T - l f f o r m = t . The 2, to be inserted into (9) or (10) may then be ob-

t,ained from

It can be shown that det (A) = 1 so that the transducer assembly with the mechanical port as input and its electrical port as output is described by the transpose

Finally, the input and output electrical terminat,ion networks, described by their respective two-by-two ma-

(A)* of (A) .

170 IEEE TRANSACTIOXS ON SONICS AND ULTRASONICS, OCTOBER 1967

trices ( A ) , and (A)o can be included by performing the multiplications (A) - ( A ) and (A)* (A) , .

I n terms of the components of (14), the elect,rical input impedance 2, seen at the electrical terminal pair when loaded by an impedance Z,, is given by

z, = ( A Z ' V T + B,'(CZ.,T + D) (13

and the voltage transfer rat,io between input voltage and output force

VI , = Z L V T / ( ~ 4 Z . V T + B). (16)

Equivalent equations hold for the reverse transmission direction, if one exchanges A and D and replaces ZdVT by the electrical load impedance R,, yielding a mechanical input impedance 2, and a voltage transfer ratio ITLo.

Depending, however, on whether the delay line is operated in pulse or C W transmission, the influence of the transmission medium has to be evaluated different.ly. For pulse transmission, no interference effects exist between the input and output faces of the delay medium, if the pulse is shorter than twice the single trip delay time. At the interface between the delay medium of characteristic impedance Z N T and the receiving transducer with an input impedance Z,, part of the signal is reflected. The amplitude reflection factor is given by

RF = (2, - z.vT),'(z,w + ZNT) (17)

and the signal appears at the input t)erminal of the re- ceiving transducer mult'iplied by a transmission voltage ratio

v,, = 22,/(2, + Z,T) (18) so that the overall voltage transfer ratio through both transducers, neglecting the delay and absorption of the delay medium, is given by

v,, = v,,-v,,*v,,. (19)

For a CTV signal, however, the transmission medium acts simply as an NTt.h layer of half the actual thickness on each transducer, so that (15) and (16) may be used with the components of

( 4 1 0 = ( 4 . (A)* (20) setting N = N T for m = t in (12).

The insertion loss of the delay line is usually defined as the ratio of the power delivered from the source of impedance R; into the load of impedance E , without the delay line, to the power delivered with the delay line inserted. With (19) it, t'urns out to be

I L = 20 log ] V 1 , [ + 20 log R,/(& + R,) (21)

and will be used in the following. This procedure is straightforward and easily imple-

mented as a computer program. It is also possible to derive explicit equations, e.g., for the electrical input impedance, etc., but these are very unwieldy in anything

but the simplest cases. One example is the case of N transducer layers, series connected with alternate polarit'y, each half a wavelength thick a t a resonance frequency fo. Since at the frequency one has ync = P in (4), (7) becomes

((-1)" 0 0 - jNZ02k2/7r@ 1

l o 0 0 1 J and one obtains the following immediately by going through (lo), (15), and (16)

ZBwoCo,'lv = j + NZo4k2/?r(Z.vT + 2,) (22)

and = j'2@iv/z@oc"(l + z , / z N T ) (23)

as the electrical input impedance and the voltage transfer ratio for the whole transducer stack loaded with Z,, on its transmitting and Z b at its back face.

In order to obtain a more dctailed understanding of the performance of multilayer transducers, a computer program was written, implementing the procedure just described. It uses Wacf l , Z,, k, and wo/wn for the transducer layers, and 2, and ofl/wn for the front and backing layers, respectively. Its output consists of 100 values over a selectable frequency range for the following:

1) the electrical conductance G.w,C,, C, being the clamped capacitance of the stack of transducer layers;

2) the electrical capacitance ratio C/C,; 3) the reflection factor magnitude of the receiving face

4) the reflection factor phase of the same; 5) the overall insertion gain between a source of im-

pedance R, and a load of impedance R.; 6) the phase angle between Eo and Ei of Fig. 1. The graphs and results referred to in the following sec-

of the output, transducer;

tion were obtained by means of this program.

111. THE TRANSMISSION PARAMETERS OF A SINGLE LAYER TRANSDUCER

In an ordinary delay line designed to have a large fractional bandwidth, one uses single layer unbacked transducers, approximately matched to the delay medium with respect to their characteristic impedance.

Figs. 4 through 7 show the set of transmission param- eters computed by the program for this case and k = 0.2 which would typically represent CdS transducers in shear mode and fused quartz as a delay medium.

The curves A of the set describe untuned transducers. For maximum power transfer, it is then necessary to provide a source and load impedance Ri and R. so that

Qi = woCoRi = 1 and Q. = w0C,,RR, = 1 (24)

SITTIG PIEZOELECTRIC TRANSDUCERS

l

171

FREPUENCY R A T I O F I F O

Fig. 4

I .06 I I

1.05

g 1.03- 3- o 1.04-

-

:$ 1.02-

!: 1.00- v a iit 1.01 - + U

W U

0.99

0.90-

-

l

0.6 l .o 1.4 1.8 FREQUENCY RATIO F I F O

Fig. 5

- $ - I 2

F-2.0 -

- U

-1.6 - K

2-2.4 - 0

'j-3.2

g-3.6 0.2

_- 0.6 1.- E

1.0 1.4

FREPUENCY RATIO F I F O

Fig. 6

-70- g -eoLL- . - ~ . L ~ ~ ~. .~ 0.2 0.6 1.0 l .4 1.e

FREPUENCY RATIO F I F O

Fig. 7

19.74 0.89 0.63 1.16 0.60 14.69 0.985 0.78 1.20

c 0.43

9.50 1.00 0.84 1.16 0.32 1.00 0.89 1.13 0.24 1.00 0.90 1.10 0. 20

_I 1.8

if the transducers are untuned. As Fig. 8 indicates, the insertion loss at the frequency of maximum response amounts to 20 dB.

A lower loss can be achieved at the cost of bandwidth reduction by tuning out C, at the center frequency f o with appropriate parallel inductors. Curves B, C, D, and E show the transmission parameters with Qi and Q. in (24) set to 1, 2, 3, and 4, respectively. This amounts simply to driving and loading the transducers with connecting circuitry at successively higher impedance levels.

Specifically, one obtains the values listed in Table I for the insertion loss minimum ILmin in decibels, the frequency fm/fo at which this minimum occurs, the fre- quencies f L / f o and fH/fo at which the loss has increased

A B C

E D

3.18 0.73 1.96

0.46 1.12 0.90 0.91

0.38 0.57 1.27 0.774

1.06 0.01

0.69 1.33 0.601 1.10

0.10 0.78

1.12 1.31

0.86 1.30 0.48 0.39

~~~

by 3 dB, and the fractional bandwidth (fH - f L ) / f m . I n this case, the mechanical impedance loading the

transducers is sufficiently closely coupled to the electrical sides of the circuit in Fig. 1 to maintain a low Q of the termination circuits. For a termination condition near that of case D, the transducers are impedance matched a t all terminal pairs and have 0 dB insertion loss at f = f m . Increasing R i and R, further causes the insertion to rise again due to increasing mismatch while the band- width still decreases. Thus little improvement of the already low insertion loss results from tuning and in- creasing t'he Q of the electrical termination circuits.

For the same conditions, but with IC = 0.6 one obtains Table 11.

172 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, OCTOBER 1967

IV. SINGLE LAYER TRANSDUCER WITH REACTIVE BACKIIVG

Fig. 8 shows the insertion loss of a single layer trans- ducer pair backed with 0, 1, 2, 3 inactive layers of a characteristic impedance equaling that of the transducer and delay medium. Since all backing layers equal X/2 at the frequency lo, no insertion loss variation ensues a t f o as layers are added; however, the bandwidth is in- creasingly curtailed.

The mechanism causing this may be visualized from Fig. D which shows a stack of four transducer layers A , B, C, and D, each Xi2 thick at the frequency j o , with layer D abutting on the delay medium DM. ,4 sound wave arriving from DM will produce output volt,ages E , through E, from each layer which are proportional to the strain integral tjal;en bctween its fares. This, in turn, depends on the position of the faces with respect to t'he standing strain wave ensuing from the reflection of the sound wave at the free left-hand facc of layer ti. If one indicates the strain distribution by the drawn sine wave and its integral between the laycr faces by the shaded bars, t'he latter are seen to be equal a t fo for the layer polarities in- dicated by the arrows. Thus, for example, the voltage E , across layer R would equal the voltage E , across layer C. If the frequency is varied to, say, 0.S3 f o as shown, E, has hardly varied, but E, virtually disappears, because layer C appears to be symmetrically disposed across the strain node, so that the strain integral vanishes. I t is, of course, immaterial for this effect if the other layers are trnnsducers or inactive as long as they are left unconnected.

The progressive phase shift caused by ArB layers to the left of the one considered makes the response of that layer exhibit

minima for f,/fo = 2(p- 1)/(2NB + 1)p = 1, 2 ,

maxima for f,/'f0 = ( 2 p - 1)/(2NB + 1 ) p = 1, 2 , . . . so that the response has a multiplicity of lobes. An in- teresting consequence results from (2.5). Xeglect.ing the skewing effect of the transducer response itself, the lowest response lobe ( p = l) will have the same fractional bandwidth as one transducer larger. In fact, as is evident from Fig. 8, the lowest lobe has an only insignificantly higher insertion loss than the lobe a t f = fo. The insertion loss can indeed be further improved by realizing that, for proper impedance matching, Ri and R, ought to be multiplied by a factor, f o / f P , for operation in the pth lobe. This is shown in Fig. 10 for N B = 2, p = 1, 2, 3, and Ri = R,, = 1, 5, and 5, respectively.

Therefore, the frequency of a transducer can be lowered by a considerable amount through the use of inactive backing layers without a significant impairment of frac- tional bandwidth or insertion loss. In fact, by replacing the transmission line backing by mass loading, one arrives a t the sandwich transducer design widely used in low- frequency work.

(25)

FREQUENCY RATIO F I F O

Fig. S. Insertion gain of a single layer transducer as above, un- tllncd and terminated with It',w0CO = IZ,woC:o = 1. Curves A throl~gh I ) indicate hacking with 0, 1, 2, 3 inactive X/2 layers, matched acoustically to thc transducer impedance.

Fig. 9. Signal addition in a four-layer transducer, all series con- nected, X/2 thick a t f o affixed to a delay medium D M of matching acoustic impedance. The sine wave indicates the standing wave stress distribution, and the shaded bars the individual transducer responses if a plane sound wave impingcs on the stack from the right.

O 7 - - - - 7

FREQUENCY RATIO F I F O

Fig. 10. Insertion gain of a single layer transducer backed by two X/2 layers, untuned with IZpoCO = R.wOCo = 1, 1.67, and 5 for the curves A through C.

V. THE SERIES CONKECTED MULTILAYER TRANSDUCER If a number of equal transducer layers, each X/2 thick

a t a frequency f = fo , are connected in series electrically, all having a characteristic impedance equal to that of the delay medium, the behavior of the transmission parameter may be visualized from Figs. 11 through 14, showing

SITTIG PIEZOELECTRIC TRANSDUCERS

10-01 1 b I W

0 z 10-1

- IO*

- 1 6 ~

- 1 1 6 ~

- 8 10-2

- a

a 3 t

c 3

2 n

a 0 a

W

W J

0.2

C l

l

\ / L*

1.8 FREQUENCY RATIO F/Fo

Fig. 11.

\ / l 0.2 0.6 1.0 14 l e

L . . L - I

FREQUENCY RATIO F/Fo

Fig. 12.

I FREQUENCY RATIO F i F o

Fig. 14.

Figs. 11-14. The transmission parameters of a multilayer trans- ducer with the X/2 layers connected with alternate polarity in series, untuned, terminated with woCtR; = LOOC~RQ = 1. Ct =

and D. C Q / N . The number of layers N = 1,2,3, and 4 for curves A , B, C,

TABLE IV ( k = 0.6)

ILrnill f m / f Q fL l fo f r r l fo a f / f m -4 3.18 0.73 0.46 1.12 0.90 B 0.56 0. 82 C 0.004

0.68 1.07 0.47 0.865 0.77

D 1.06

0.26 0.34

0.89 0.82 1.04 0.25

__

in accordance with the computations. As Fig. 12 shows, an insertion loss improvement is obtained for f = f o at the cost of decreased bandwidth. Specifically, the minimum insertion loss ILmi,, in decibels, the frequency fm/fo at which it occurs, and the frequencies f J f o and fH/fo delimiting the 3-dB bandwidth are given in the case of Ri = R, = NCjw,Co (C, capacitance of an in- dividual layer) and no tuning, by Table I11 for k = 0.2 and by Table IV for k = 0.6.

Again, in this case, condition C comes close to complete impedance matching at all terminals for f = fm,

hence zero insertion loss and little improvement of the already low insertion loss can be obtained with more multiple layers.

-4 B

19.74 0.89 0.63 1.16 14.42

0.60 0.955 0.80

C 11.38 0.97 1.12

9.32 0.97 0.89 D

0.33 1.08 0.23 0.86 1.06 0.17

computations fork = 0.2 and alternately reversed polarity of successive layers. A , B, C, and D, respectively, indicat'e the data for 1, 2, 3, and 4 layers. Again Fig. 9 and its explanation in Section IV give a key to predicting the location of the ensuing response lobes: since all layers are series connected, the total voltage E developed across all the layers is simply the sum of the individual layer voltages E, through E4. Zeros of this sum, and thus response minima are therefore encountered for NC layers at

f d f o = p - p 1 , 2 , ... but p # NC (26)

mm TRANSACTIONS ON SONICS AND ULTRASONICS, OCTOBER 1967

part a resonance effect to the stack resulting in shape variations of the individual lobes of the frequency re- sponse, without affecting t’he basic picture of multiple lobes. Some exanlples were computed to verify this, but are not, included i n this paper for brevit’y’s sake.

It is, however, known that the bandwidth reduction, caused by what amounts to a cross-correlation effect between the strain pattern in a stack of active layers and its extraction electrode pattern, can be removed by traveling wave insertion and extraction. By providing a matched absorptive backing to suppress t’he standing wave, and by inserting the signal through a delay line with multiple taps providing appropriate phase matching, one arrives a t structures similar t’o those described by various authors. [‘l * “ l rhese do not seem to have found extensive application, probably due to the inherent com- plexity of the electrical networks involved.

174

VI. CONCLUSIOKS

I n summarizing the results of the the following conclusions emerge.

preceding sections,

1) Maximum bandwidth is always obtained with a single layer transducer with untuned terminations match- ing the impedance of its clamped capacitance.

2) Only in t’he case of low coupling factor can band- width be traded off for an insertion loss reduction. This can be done either electrically by using t’uned terminations with higher impedances resulting in an increased electrical Q or by using alternately poled multilayer transducers having an increased mechanical Q. A comparison of Tables I and I11 shows that the former choice is more advantage- ous in terms of larger bandwidth for equal loss.

3) In the case of high coupling factor layers, no ap- preciable inherent insertion loss improvement can be obtained from multilayer arrangemenk Hon-ever, if the access driving circuit is current limited, more power may be transferred to a series connected multilayer transducer because of its higher inqedance compared with the single layer.

4) The use of inactive backing layers and operation on the lowest lobe of the frequency response results in little degradation of fractional bandwidth and loss, compared with a single layer operating at this lower frequency as its fundament,al. The trade-off consists in operation a t a lower impedance level which may be desirable in special

Although the foregoing remarks were only justified by computations of acoustically matched arrangements, they are not substantially modified in more general cases. Inserted inactive lavers and mismat’ched lavers will im-

cases.

~LCKNOWLEDGMENT

The aut’hor is indebted to G. A. Coquin for helpful discussion and advice, particularly in the writing of the computer program.

REFERENCES J. de Klerk, P. G. Klemens, .and.E. F. Kelly,

enhancement of microwave piezoelectric conversion layers,” A p p l . Phys. Letl., vol. 7, pp. 264-265, 1965.

Microwave Lab., Stanford University, Stanford, Calif., [? I 13. J. Shaw, “Selected studies in microwave

Contract AFlY(CiZY)-4788, November 1965. [31 D. A. Berlincourt. D. R. Curran, and H. Jaffe,

Acoustics. W . P. h Iason’. Ed.. vol. 1, pt. A. New York

“Multilayer in CdS-Si0

acoustics,” Rept. 1382,

in Physical :: Academic,

pp. 233-242, 196-1. I .

J . A C O U ~ . SOC. Am., vol. 30, pp. 528-532, 1955. 1‘1 M. Greenspan and R. hl. Wilmotte, “Distributed transducer,”

coupling t,ransducer,” J . dcoust. Soc. Am., vol. 34 pp. 333-337, 1962. [;l W. J. Trot,t, “Theory of a passive, reversible, distributed-