188
MICHAEL
H.
I..
HECKER
ACKNOWLEDGMENT
In order to circumvent this limitation, a control
systemwhichcan provideeachsectionof DAVO with
The author wishesto expresshis gratitudeto Proan individualset of temporalinstructions
is required. fessorKennethN. Stevensfor the helpfulsuggestions
thisresearch.
The cooperation
Each sectionwould undergoa given changein its hehasgiventhroughout
of all members
of the SpeechCommunication
Groupat
cross-sectional area at a different time and with a difthe
Research
Laboratory
of
Electronics
is
also
vet5'
ferent rate, with the result that the vocal-tractanalog
much
appreciated,
particularly
the
kind
assistance
of
would be capable of making almost any transition
Dr. Arthur S. Houseand Miss JaneB. Arnold.
between two configurations.It is impractical to inThis workwassupported
in part by the U.S. Army
corporatesucha controlsysteminto the presentpro- SignalCorps,theAir ForceOfficeof Scientific
Research,
grammingdevice,however,and the problemof con- and the Officeof Naval Research,in part by the Air
trolling DAVO by meansof computermust be even- ForceCambridge
Research
Laboratories,
and by the
National Science Foundation.
tually considered.
THE
JOURNAL
OF THE
ACOUSTICAL
SOCIETY
OF AMERICA
VOLUME
34, NUMBER
FEBRUARY.
1962
Ultrasonic Intensity Gain by CompositeTransducers
Wxr.sz•
J. F•t't* a.•ro F•otra Duss*
Interscience
Research
Institute,Urbarm,Illinois
(ReceivedAugust10, 1961)
The radiationof planeacoustic
wavesinto a mediumby composite
layeredtransducem
composed
o[ a
piezoelectric
elementand two lossless
couplingmediais described
analytically.The increase
in acoustic
intensityin themediumreceiving
theradiationproduced
by thecomposite
structure
overthatproduced
by
directcoupling(piezoelectric
elementin contactwith the medium),for the samevalueof the electricfield
strength,is considered
quantitativelyand the resultsare presented
graphically.Composite
transducers
of
the type described
canbe expectedto achieveconsiderably
higherultrasonicintensities(1000-foldincrease)
in both unfocused and focused beams than have heretofore been realized.
I. INTRODUCTION
particularlylackingin informationregardingoptimum
designconditions,i.e., valuesof the acousticparametersanddimensions
for the couplingmedia,for maximizingthe acousticoutput powerper unit radiating
area for a fixed value of the electricfield strength.6,*
An objectiveof this paperis the presentation
of transducerdesigncriteriafor achievingincreased
power
HEproduction
ofultrasonic
radiation
bycom-
positetransducers
has receivedattentionin the
technicalliteraturea'•and informationon the propaga~
tion characteristicsof sound through multilayered
structures(includingengineering
designcriteria)is to
be found in almostall bookson acoustics.
a A general
treatment of acoustic propagation through lossless outputs.
High-intensity
ultrasound
hasbeenandcontinues
to
structuresof four or more layers has been published
be
useful
in
research
investigations,
medical
applicarecently by Dianov,4 who comparesexperimentalreand it is expectedthat
sults of radiated acousticintensity and bandwidth for tions, and industrialprocesses,
increasing
the
available
intensity
will make possible
the direct radiated case with that incorporating two
a
broader
scope
of
basic
research
studies
and technical
quarter-wavelayersbetweenthe piezoelect'ric
element
Ultrasonicintensitiesgreaterthan have
and the radiation medium. The use of compositeim- applications.
pedancetransformingstructuresto couplevibrating been realized heretofore from unfocnsedsystems can
designedas
piezoelectric
elementsto a mediumhasbeendiscussed be producedby compositetransducers
described
in
this
paper.
In
the
field
of
biophysics,
inbriefly by Fry and Dunn.a The technicalliteratureis
tensities
in therangefromapproximately
(10)2to (10)•
w/cmx are usedto produceuniquechangesin tissue
Urbana, Illinois.
a.s In some
• W. I- Fry, J. M. Taylor, and B. W. Henvis, Designof Crystal structuresof complexbiologicalsystems.
Vibrating Systems(Dover Publications,Inc., New York, 1948). cases,whereit is desiredto producehigh intensities
* Also,BiophysicalResearch
Laboratory,Universityof Illinois,
• W. G. Carly, J. Acoust.Soc.Am. 21, 65 (1949).
aSeefor example,L. E. Kinslerand A. R. Frey, Fundamentals aT. F. Hueter and R. H. Bolt, Sonics(JohnWiley & Sons,Inc.,
New York, 1955).
of Acoustics
(JohnWiley & Sons,Inc., New York, 1950).
• B. Carlin, Ultrasonics(McGraw-Hill Book Company,New
4D. B. Dianov, (translation) Soviet Phys.--Acoustics$, 30
(19so).
• W. J. Fry and F. Dunn in PhysicalTechniques
in Biotogical
Res*arch,edited by W. L. Nastuk (Academic Press Inc., New
York, 1961), Vol. 6.
York, 1949).
sW. J. Fry, in Advances
in Biologicaland MedicalPhysics,
editedby I- H. LawrenceandC. A. Tobias(Academic
PressInc.,
N-ewYork, 1958), Vol. 6.
COMPOSITE
TRANSDUCERS
in relativelysmall tissuevolumesin deep structures
without affectinginterveningtissue,lens systemsare
employedto focusthe acousticenergy.Someof the
presentdesignsemploya layeredstructurecomposed
of a piezoelectricelement,a singlecouplingmedium,
and a lens; the essentialcomponentsof the transducer
which producesand couplesthe acousticenergyto the
transmitting medium. The technical literature con-
g•-•V•
189
%-•nV•
%- •v•
g•v r
Fro. 1. •hematic diagramof compositetransducer.
tainsdesigncriteriafor suchtransducers
whichspecify
that the specificacousticimpedanceof the lens matogetherwith relation(1), constitute
terial be as nearly equal to lhat of the propagating Theseformulas,
analyticexpressions
from whichthe inmediumas possibleand that the thicknessof the single the necessary
systemover that of a
couplingmediumbe as thin as possible.•An objective tensitygain of the composite
of this paper is the demonstrationthat the choiceof systemin which the piezoelectricelementradiates
conditionsusually stated in the literature to realize directlyinto mediumr comparedfor equalvaluesof
"optimum" performancefrom compositetransducers the driving electricfield strength,can be computed.
is neither necessarynor desirable,that is, it is shown For the directradiationcase,the intensityI• is
that other choicesof parametersand dimensionsenI•a=4e•E•/R•.
(5)
joy distinct advantagesover thoseusuallyprescribed.
Ratherthan express
the intensitygainin the general
caseby eliminationof the appropriatevariablesfrom
the aboverelations,it is moreappropriate
to consider
The type of systemconsidered
here is illustratedin
and
interpret
a
series
of
special
c•es.
Fig. 1. Onefaceof a piezoelectric
elemente is coupled
to a mediumr, in which acousticradialion is desired,
Case 1
by the planar layersof spacingmediums and coupling
Let
medium p. The characteristicacousticimpedancesof
(6)
thesemediaare, in general,all different.It is assumed In this case Z, is
that the piezoelectricelementis operatedat its fundamental resonantfrequency,at which it is a half-wave(RvF
wLv-I r /Rv\2-I wLv]
lengththick (or an oddharmonicin whichcaseit is an
I --/1 + ta,f---/+J/1-/--I
I tan--/
odd number of half-wavelengthsthick), that it is
terminatedon the oppositeface by a materialof zero
acoustic
impedance,
andthat all mediaareacoustically
lossless.
The analysisproceeds
by computingthe mottonal part of the electricalinput impedanceat the
electricalterminalsof the piezoelectricelement.This
(7) into (1), the expression
for the
impedanceis dependentupon the acousticinput im- Uponsubstituting
acoustic
intensity
radiated
into
medium
ß
is
pedanceinto mediums, i.e., Z, of the figure.The
acousticintensityI• into mediumß can then be expressed
in termsof the real part of the electricalimpedancefunctionasfollows:
IL
ANALYSIS
Z,=R;I
R•L
I,•=4e,•'E • Re(I/Z,),
vz,
J L \R•/_1 •v•
(1)
The gain,G(X,/4) (thegainfor L,={?,,) in intensity,
wheree•}designates
the appropriatepiezoelectric
stress for equalvaluesof the drivingelectricfield,of the
systemoverthe directdrivesystemis obconstantfor the transducermaterial (en in the caseof composite
tainedby formingtheratioof Eq. (8) to Eq. (5).
an X-cut quartzplate) and E is the amplitudeof the
electricfield strength.
The acousticimpedances
Z. Zv and Z. lookingto
the right at the boundariesindicatedin Fig. 1 are,
respectively,
Z,=R,
Zt,• Rt. - .......
and
'[-R•+jZ• tan(wl.v.
%,)_1
(RqT l+tan2(wLv/vv)
]
G(•X0=\•/L-i+(R,/Rv)•v/vv)i.
(9)
Thebracketed
partof thisformulaisillustrated
graphicallyin Fig. 2. The gainis shownasa functionof the
quantityL•,/Xvwith theparameterfromcurveto curve
impedances
of
(3) beingthe ratio of the characteristic
mediap andr. It isapparent
that thegainof thesystem
(2)
increases
as the characteristic
impedanceof mediump
(the transmission
plate) increases,
that is, as Rv beconms
largerrelativeto R•. The maximum
gainfor a
190
W.J.
FRY'
AND
F.
DUNN
iooo
Ol
o..ol
o
specificchoiceof material is achievedwhen the thicknessLp of the transmission
plateis equalto onequarter-
0.0,5
o.i
with the samepiezoelectric
elementradiatingdirectly
into the medium, r, is achieved as the result of the
0-2
0.2.%
Fro. 3. Gain of compositetransducerfor • wavelengthof
spacingmaterial vs thicknessof transmission
plate and for [
wavelengthof transmission
plate vs thickness
of spacingmaterial,
respectively.
Symmetrical
aboutLp/Xp,Ldh,--0.25 and repeats
everyLp/Xp,LdX,=0.5, respectively.
Then Z, is
wavelengthor an oddmultiplethereof.
A further increasein output intensitycanbe achieved
for a constantvalue of the driving electricfield by
choosinga material for the couplingspacerwith a
characteristicacousticimpedanceR, smallerthan the
characteristicimpedanceof the mediuminto whichthe
acousticenergyradiates.The valuesof the gain from
the graph of Fig. 2 are directly multiplicableby the
ratio (R,/Rs)•-of thesecharacteristic
impedances.
and the radiatedintensityis
It shouldbe notedthat the gain in intensityof the
compositesystemover that characteristicof a system
0.15
(11)
F 1+tan"
(coLv/,:,) (12)
I•= R,LI+(R,/R,)-'
tan2(wL,/vv)l
'
reductionof the electricalinput impedance(the motionalbranch)into the piezoelectric
element.This de- The gainG(«X,)in thiscaseis
creasein electricalinput impedanceresultsfrom the
1+tan2(coLv/vv)
reduced mechanicalimpedanceat the face of the
piezoelectricelementwhich is producedby the compositecouplingassembly,therebyenablingthe element
to draw more electriccurrent and consequentlymore This expression
is represented
in graphicalform in Fig.
powerat the samedriving voltage.
3, where the gain is exhibited as a function of the
parametersLv/X• and the ratio RvR, of the characCase 2
teristicimpedances.
It is seenthat the gainis lessthan
Let
unity for all valuesof the ratio Rv/R•> 1 and greater
œ,=«x,.
(lO) than unity for all valuesof Rv/Rr<I.
G(«Xs)
=1+(Rv/Rr)
•tan'(toL•/,•)'
(13)
COMPOSITE
TRANSDUCEI(S
191
Let
(14)
Then Z• is
(15)
and the radiatedintensityis
The gain,G(«Xv)in this caseis
lq-tan-ø(wL•/v•)
=
4ea:2E2
(16)
R•.
The gain,G(•Xv), in thiscaseis
\R•/
(17)
h-1
L\Rd \• /J
v•
This expression
is illustratedin graphicalformin Fig. 4,
wherethegainisplottedasa functionof theparameter
L•/X• for variousvaluesof the ratio R•/R, for the case
R•/R,=i. Thesecurvesshowthat the gain can be
greaterthan, equal to, or lessthan unity, depending
upon the thicknessof the spacingmediumin wavelengths,for any fixed valuesof the two ratios of characteristicimpedances.
The effectof varying the value
of the ratio RdRr for a fixedvalue of the ratio R,/Rr
is to alter drasticallythe gain for valuesof L• near
¬Xsandnot to producesignificant
alterations
for values
deviating significantlyfrom L•=¬M. Equation (17)
showsthat the gain is increasedfor R•/R,<I and decreasedfor R•/R,> 1.
tan-ø(oLd,)
'
(21)
This expression
is identicalin form to Eq. (13) and is
alsoillustratedgraphicallyin Fig. 3 wherethe gain,
G(X•/2),isplottedasa functionof theparameter
LdX•
and the ratio, R•/R, of characteristic
impedances.
III.
1+ tan2(wLs/v•)
O(•Xv)=
1+
DISCUSSION
It is apparent from the analysisand graphsof the
precedingsectionthat the radiatedacousticpowerper
unit area of a compositeassembly,consistingof a
piezoelectricelement coupledthrough "spacing"and
io
Case 4
Let
L•=«X•.
(18)
Then Z• is
Z•-
/Rs\/
,•L•\
Rr
v• /
F/RA'-' q
\Rd
-11
_1
(Rd R,)2+ tan2(wL,/v•)
and the radiatedintensityis
0.1
v•
, 09)
0.01
0
0.05
0.1
0.15
0.2
0.25
Ls/•s
I.=4e.,a.'-'h'2]l•(wL.,/t•)
1' (20)
" Rrkl+(&/Rr)tan2(wL•/v•)_]
Fro. 4. Gain of compositetransducerfor • wavelengthof transmissionplate vs thicknessof spacingmaterialfor caseR•/R,= 1.
SymmetricalaboutL•/X•=0.25 and repeatseveryL•/X,=0.5.
192
W.
J.
FRY
AND
F.
DUNN
mustbe designedto com"transmission"
media(seeFig. 1), canbe eithergreater meansthat the transducers
or less than that from a transducerwith the piezo- pensatefor the effectsof temperaturechangesin the
electricelementradiatingdirectly into the radiation materials when the devicesoperate "continuously"
medium,for equalvaluesof the electricfield strength. under high-powerconditions,i.e., the dimensionsof
The ratio of theseintensities,which is definedin this the couplinglayers,in wavelengths,mustremainwithin
paper as the gain of the compositetransducer,can be the necessarytolerances.The gain is, of course,a
i.e., odd mulvaried over a wide range of values by appropriate periodicfunctionof the layer thicknesses,
choiceof the dimensions
and characteristicimpedances tiplesof a quarter-wavelengththicknessyield identical
of the spacingand transmittingmedia.Thus, the maxi- valuesof the gain. However,if temperatureeffectsare
mum intensitiesavailableinto liquidsat the transducer to be minimized,the choiceof thicknessfor many apface at maximumdriving electricfield strengthare plicationswould be one-quarterwavelengthsincethe
considerably
increased.This is to be comparedwith
the gain obtainedby insertinga singlequarter-wave
layer having the root-mean-square
characteristic
impedanceof the piezoelectricelementand the radiation
mediumbetweenthesetwo media.This gain, which is
(Rr/R,) 2,dependsuponthe characteristic
impedanceof
the radiation medium, whereas in the case of two
thinner the couplinglayers, the smallerwill be the
effectsproduced
by temperaturechanges
in the system.
From an examinationof the graphs,it is apparent
that intensitygainsin water (or liquidsin general)in
the range from 100 to 1000 can be achieved.For example, if a material with a characteristicimpedance
approximatelyequal to that of water is chosenfor the
spacingelement and a material such as stainlesssteel
with a characteristic impedance approximately 30
times that of water is usedfor the transmissionplate,
then for one-quarterwavelengththicknesses
of these
materials the gain of the compositesystem compared
quarter-wavelayers,the gain is (R•/R,) 2 and depends
uponthe characteristic
impedanceratio of the coupling
materialto that of the spacingmaterial.It is of interest
to comparethe gainrealizedby employingtwo quarterwave layersover a singlequarter-wavelayer, which is
(R•,/R•)2. The analyticmethodusedin thispapercan with the caseof direct radiation of the piezoelectric
be extendedto any numberof layers,from which it elementinto water is nearly 1000.
Compositetransducersof the type consideredhere,
will be seenthat further increasein the intensitygain,
for equal values of the applied electricfield strength, when combinedwith appropriate lens systems,will
can be obtained. 4 The dimensions of the elements of a
make possiblethe attainment of ultra-high intensity
compositetransducerare of critical importanceespe- acoustic
beams,i.e., intensities
asgreatas (10)*to (10)8
cially if high intensitygainsare to be achieved.This w/cm2.