theory and design of the surface acoustic wave multistrip...

124 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS VOL SW-20 NO 2 APRIL 1973

Theory and Design of the Surface Acoustic

W a v e Multistrip Coupler

F GRAHAMMARSHALLCLELAND 0 NEWTON AND EDWARD G S PAIGE

InvitedPaper

Abslrucl-The multistrip coupler performs the function of a directional coupler for freely propagating surface acoustic waves on a piezoelectrically active substrate Its operation is analyzed in terms of a transmission line based equivalent circuit Expressions are obtained for the directionality (transmission and reflection) in terms of the number of coupler strips and the acoustic frequency Theory and experiment are shown to be in very good agreement 50-percent metallization is foqnd to give optimum performance progressive in- crease in the proportion of the coupler area covered with metal is shown to involvea progressive change from an in-line field modelto a crossed-field model

Outside a stopband region it is found that a simplified expressionfor the directionality is valid this greatly facilitates component de-sign Design criteria for multistrip components are discussed in terms of substrate and bandwidth requirements An analysis is included of resistive and capacitative effects on coupling and problems associ-ated with coupling between dissimilar materials arediscussed

I NOMENCLATURE

Upper track as defined in Fig 1 Antisymmetric mode Lower track (Fig 1) Capacitance between adjacent strips for individual track aperture Multistrip couplersas listed in TableI Repeat distance of periodic multistrip coupler Geometric filling factor Fraction of energy transferred between tracks Surface-acoustic-wave frequency Normalized frequency ( =ffo) Frequency of fundamentalstructurestopband ( = v2d) Equivalent circuit current Electromechanical coupling constant Couplerlengthforcompleteenergytransferbe-tween tracks Aperture width of individual track Multistrip coupler Number of strips Number of strips for complete energy transfer be-tween tracks Power in individual track Resistance of strip along an individual track aperture Transformer turns ratio in equivalent circuit Surface acoustic wave Symmetric mode Surface-acoustic-wave velocity Antisymmetric mode velocity

c Symmetricmodevelocity z Impedance of individual track l2 Activefraction of periodicrepeatdistance of MSC f fR Energyattenuationcoefficientforantisymmetric

wave Y Parameterforproportion of crossed to in-line field

model A1 Separationdistancebetweencoupledtracks AU Differencebetweenfreeandmetallizedpropaga-

tion velocities 7 Metallizationratiostripwidthequals qd e Phaseelapseanglefortransmissionline x SAWwavelength P Mismatchcoefficientforimpedances PA 0

Massperunittracklength Phaseelapseangleperturbedfrom 0 bycoupling action

11 INTRODUCTION

A The Basic Component

TH E multistripcoupler (MSC) [ l ] is a directional coupler for use with surface acoustic waves I t consists of an array of parallel metallic strips on a piezoelectric

substrate Its specific property is the coupling of nonguided acoustic waves on a single substrate in contrast to the opera-tion of boththewaveguidecoupler [ Z ] andtheadjacent medium coupler [ S ]

T h e periodic MSC is shown schematicallyinFig 1 If a surface acoustic wave is launched from one of the transducers into one half of the structure [trackA for example Fig 1(a)] potential differences are set up between adjacent metal strips becausetheSubstrateispiezoelectricThesepotentialsalso appearintrack B where a secondsurfaceacousticwaveis generated

I n common with other simple coupled mode systems the behavior of thecoupler is best describedinterms of the modes of the system a mode being a field pattern which is not distortedduringpropagation I n theelementarycase of parallelequalaperturetrackson a uniformpiezoelectric medium the modes can be identified(Section 111-A2) a s t he symmetric mode S and the antisymmetric modeU as shown in Fig l(b) Any input pattern that is uniform in the individual tracks can be decomposed into these two modes I n general the symmetric and antisymmetric modes will propagate with different velocities U and v respectively I t is the beating of these two modes which leads to a periodic change of energy between tracks and hence to the coupling action

I n designmanufactureanduse the MSC iscompatibleManuscript received October 4 1972 revised October 23 1972 withtheconventionalinterdigitaltransducer No extraThe authors are with the Royal Radar Establishment Great Malvern

Worcestershire England processingstagesareinvolvedinthephotolithographic

125 MARSHALL et d MULTISTRIP COUPLER

rnsc

Fig 1 (a) Schematic of multistrip directionalcouplerwith interdigital transducers at input and output ports(b) Input and output field die- tributions resolved into symmetric S and antisymmetric a modes for acouplerinwhich100-percenttransferfromtrack A to track B occurs

reproduction the precision required is similar to that for the interdigital transducers used to drive the system No connec-tions internal or external are required by the coupler Thus nopenalty is paid by the introduction of the coupler other than the occupation of space on the substrate and the addi-tionaldelaytimenecessaryforthewave to traversethe structure On material of highelectromechanicalcoupling constant such as lithium niobate the penalty is not severe

Afamily of surface-wavecomponentshasbeenderived from the MSC in addition to its use as a conventional direc-tional coupler The family includes broad-band highly reflect-ingmirrorsbroad-bandunidirectionaltransducersbroad-bandtripletransitsuppressorsbeamwidthchangers (SAW impedance transformers) and redirectors of acoustic beams These components are listed and described in [17] The ob-jective of this paper is to present a theory of theperiodic MSC based on an equivalent circuit model and hence to es-tablisheddesigncriteriawhicharerelevantbothtothe coupler and to components derived from it

B Apflroach t o the Analysis Our primaryconcerninthispaper is to developusable

designcriteria for theMSC-criteriawhichemployreason-ablysimpleanalyticalexpressionsThisisone of themain reasonsforadoptinganelectricalequivalentcircuitmodel fromtheoutsetThemodelwhich we use t o fit the experi-mental data (Section IV) is no more sophisticated than that usedpreviouslytoaccountforthebehavior of interdigital surface-wavetransducers [S] Furthermoretheparameters required to fit a wide range of gap to strip width ratios for the MSC on lithium niobate show a striking resemblance to thoseemployedforinterdigitaltransducers [4] on a poled PZT ceramic [S]

Themainfrequencyrange of interest forprediction of MSC performance lies in a broad band below the frequency of the first stopbandThisstopbandis a specificfeature of periodic couplers and occurs at the main structure resonance frequency fa=v2d where v is the surface-wave velocity and d is the MSC periodicrepeatdistance I n thisfrequency range simple analytical expressions are found which predict performance very satisfactorily these are used extensively in [l71 when analyzing device performance Another virtue of theequivalentcircuitapproach is that factors such asre-

sistivity of metal strips and added capacitance can be readily incorporated (Section V)

C Essence of MSC Behavior I t is useful as a preliminarystagetodemonstratethe

behavior of the coupler simply in terms of the electromechan-ical coupling constant K Z T h e model of the MSC employed inthisinstance is t h a t of the perfectlyanisotropic film of MarshallandPaige [ l ] T h e film spans a pair of surface acoustic tracks Its conductivity parallel to the surface-wave vectoriszeroitsconductivityperpendiculartothewave vector is infinite T h e velocity of the symmetric modev is the stiffenedvelocitysince no currentcan flow parallel tothe propagation direction but since current can flow perpendicu-lar to this direction charges in the two tracks associated with theantisymmetricmode will canceleachotheroutandthe velocity of this modev is the unstiffened velocity The length for 100-percent transfer of energy referred to as the transfer length LTis obtained in termsof the acoustic wave frequency f as half the beat wavelength for the modes Thus

where X is the surface acoustic wavelength andK Zis given by [61 as

The number of metal strips involved is obtained as

assuming total activity This demonstrates the close relation-ship between the size of the coupler and the value of KZ

In practicehowevertheperiodicMSCdiffersfromthe perfectly anisotropic film in two main ways 1) stopbands are introducedbytheperiodicity 2 ) coupling efficiency is re-ducedbygeometricfactorswhichdepend on themetalliza-tion

T o gain aninsightintothe significance of thesecond factor consider a sinusoidal signal sampled in one track and regenerated in the second track by a periodic structure (the MSC) Thesamplingfrequencyissignificant if only a few samplesaretakenperwavelengthThesamplingelement widthissignificant if i t is comparable to a wavelengthbe-causeonlytheinstantaneousaverage of signalamplitude across the sampling window can be transmitted to the second track If the phaseshiftinthesamplingwindowis 0 the coupled intensity is reduced by the factor

0 is given by 2nadX where ad is the active length in each repeatdistanceIncorporation of thereducedeffectiveness due to sampling and of the effective repeat distance gives

A sin ( 0 2 ) -2 lsquoVT = -

P a d (-Fgt (5)

Thisexpressionisverysimilartothatwhichresultsfrom

126

simplifyingthedetailedequivalentcircuitapproachdevel-opedinSection 111 I t illustrateshowcoupleractionworks but does not demonstrate a stopband

An alternative approach is to consider a pair of strips to beequivalentto a singlefingerpairtransducerlocatedin each track one feeding the other directly through their elec-trical ports This is the approach taken up in Section 111 I t illustrates the broad bandwidth inherent in the IUSC struc-ture in thatnostripsare interconnectedVerygoodagree-mentbetweenpredictionandexperiment isobtainedusing empiricalparameters

The ultimate approach involvesa self-consistent computa-tion of the piezoelectric properties of the particular material toyieldexactelectric field configurationsandhence all couplingparameters [4] [7] [S] Thisisoutsidethescope of a paperondesigncriteriainview of the success of the equivalentcircuitapproach

111 EQUIVALEKTCIRCUITTHEORYOF THE MSC

A The Equivalent Circuit Model 1) TransducerAssembly T h e MSC spanning a pair of

acoustic tracks may be viewed as a structure whose repeat distance can be represented by a transducer in one track con-nected to a transducer in the other track through their elec-trical ports this concept was presented in the introduction Thus the coupler asa whole can be represented as an assembly of interconnected transducers Smith el al [g] in their anal-ysis of interdigital surface-wave transducers used equivalent circuit models based on those employed for bulk wave trans-ducers In view of the success of their analysis we shall adopt the same bulk wave transducer circuit models These are the so-calledin-lineandcrossed-fieldmodels Thein-linemodel isemployedwhentheelectrical field vector andthewave vectorarecollinear it pertainsprimarily to gapsbetween electrodes T h e crossed-field model is employed when electric field and wave vector are perpendicular it pertains primarily t o regions underelectrodesTwoelaborations of the equiv-alentcircuit of Smith et al [g] areemployedFollowing KraiojanananandRedwood [ lo] weintroduce a passive transmissionlineelementintoeachsectionandfollowing Milsom and Redwood [5]we employ mixed models which can be intermediate between in-line and crossed field

Fig 2 illustrates the equivalent circuit model applied to each MSC repeat distance termed a section Identical acous-tictrackshavebeenassumedforsimplicityEachtrackis regarded as a transmission line and is represented by T net-works Each section comprises a passive element (subscript 1) and an active element (subscript2) of impedance Zipropaga-tionvelocity zlz andunperturbedphaseshift 8 The act ive fraction a of the section is introduced by writing

The ratio of wave velocities in the elements is defined by a parameter p where

Thus p takes the significance of a loading term related to the piezoelectricstiffeningWeconsider

2 t = PAVi (8)

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS APRIL 1973

Fig 2 Transmissionlineequivalentcircuit for onerepeatdistance of MSC showinga pair of coupled lines withpassiveandactiveele-ments

where pA is the substrate mass per unit track length Then the element impedance values for Fig 2 are given [g] by

The value of y associated with the negative capacitor of the active element in Fig 2 determines whether an in-line model (y=0) or a crossed-fieldmodel (y= 1) has been adopted i t permitstheadditionalflexibilitythat a mixedmodel (Olty lt l ) can be investigated [ S ] T h e electrical ports represented bythesecondarycircuits of idealizedtransformers of turns ratiorlareconnectedfromonelinetotheother togive coupling action The transformer turns ratio is obtained as forsurface-wavetransducersfromthebulkwaveresult of Berlincourt et al [l11 as

Here F is a filling or geometric factor which depends on the degree of mentallization The preceding result is obtained for both in-line and crossed-field models and is taken to apply to mixed models also

2 ) NormalModes Weconsidernormalmodes of the coupler at thisstagebecausewiththeiraidanimportant simplification can be made in the equivalent circuit the two coupled transmission lines can be analyzed in terms of wave propagation on a single line

T h e modes of the coupler symmetric and antisymmetric havebeendiscussedintheIntroductionandillustratedin Fig l(b) The modes are now defined by reference to coupled potential (ebectrical potential o n each strip) and coupled flux (chargeflowingfromonelinetotheother)Thesymmetric mode is defined as the condition of zero coupled flux I t is es- sentially a wave launched with the same amplitude and phase in each track Because the velocity is the same in each track phaseandamplitudecorrespondence is maintained during propagationirrespective of whether or not themetal strips are connected a t the line separating the two tracks The po-tentialsoncorrespondingstripsineachtrackareidentical whether joined or not consequently no charge flows between themgivingthecondition of zerocoupledflux Theant i -symmetric mode is defined as the condition of zero coupled potential The mode is composed of waves propagating in the two tracks in antiphase equal and opposite surface charges would be generated in each strip if they were not joined By connecting them charge flows and exact compensation takes

averaging effects have become incorporated

127 MARSHALL et al MULTISTRIP COUPLER

number of strips for 100-percent transfer directly as

The preceding derivation is referred to as the full theory Substantialsimplificationispossible if thefollowingcondi-t ions are met

a) K2ltltl b) p = O c) 8 is not in the region of nA

Conditiona)holdsfor all known piezoelectricsubstrates conditionb) will be showntobeacceptableconditionc) means that the simplified expression is not valid in the region of the stopbands Under these conditions the elapse angles for t h e passive elements of the line can be equated

Fig 3 Single-line circuits mode (b) For 4 d S ) = +l(4equivalent (a) For symmetric antisymmetric mode y =0 gives the in-line model y = 1 the crossed- field model and Oltyltl the mixed model activeFor theelements

place Consequently no potential appears at any element and 4 4 s ) = cos-rsquo cos (e) + sin (e) the conditionof zero coupled potential applies

(7Exactly the same argument can be applied to the system sinZ( B r 2 ) ) (14b) of Fig 2 withthesubstitution of ldquotransmission linerdquofor ldquoacoustic trackrdquo This establishes a correspondence between coupler and equivalent circuit +(a) = COS-rsquo COS (e) + sin ( 0 2 )

3) Reduction of Equivalent Circuit to a Single Transmission Line Since for thesymmetricmode thewaves on the two lines propagate without interaction propagationis unaffected if the equivalent circuit is open circuitat X and Y the system can be represented by a single line with an open circuit across Then the differenceof the arccosines may be simplified to give the capacitance C of Fig 2 ( a t XY)in every section For the antisymmetric mode the waves propagate so that each ele- e12

= __ ___-ment is at zero potential This is equivalent to a single line FK2 sin2 (e2)with a short circuit across the capacitanceC (a t X Y ) in every section The resulting single-line equivalent circuits are shownwhere in Fig 3 for either modeIn effect Z P fhas become

Z q cosec (e) + -W C Thisexpressionissimilartotheestimate (5) forananiso-

tropic film of sampled and averaged conductivity as consid-ered in Section 11-C

Throughout this papera frequency scale normalized to the representing symmetric mode(S) and antisymmetric mode( a ) firststructurestopbandisused on theassumptionthat

4) Deduction of Coupler Behavior The phase shifts across MSC behaviorscaleswithrepeatdistanceThus f amp = f l f a each section for the symmetric and antisymmetric modes are =eTff given by T h e N T expression for a perfectly anisotropic film may be

obtained from(3) as

On a plot of againstfrequencythisgives a rectangular hyperbolaThesimplifiedexpression (15) differsinthat

and sampling where The result isa U-shaped curve as demonstrated by the dotted

amp(S) = 211 + 2 2 1 + 21 + Z 2 ( S ) (l2lsquo) curve of Fig 4T h e fu l l expression (13) furtherincorporates

Z(a) = Zl + 2 2 + 2 1 2 + amp(U) (12d)stopbandeffectsduetostructureperiodicity5 ) NT as a Function of Frequency NT the number of strips

Since100-percenttransfer of power occurswhenthephasefor100-percenttransferbetweentracks is plotted against shift between the modes is A theserelationsyield N T thenormalized frequency in the regionbelowthesecondstructure

128 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS APRIL 1973

TABLE I

c1c 3 c 2 c4 c 5 C6 Coupler Number

c7 C8 c 9

2020 Stopband frequency 108 86 86 86 86 54 54 54 54 Track aperture 4 3 3 3 3 4 4 4 4

16 20 32 32 32 32Repeat distance 20

Number of strips Metallization ratio

140 140 700 0 45

140 0 450 620 17

I40 0 5

700 0 3 7

700 0 3 4

700 0 6 70 1 2

700

Measurement technique

Effective coupling constant Mismatch factor

Activity factor OE E

0042 0 8 5

0 00430042500345 0 8 50 8 50 8 5

0

E

0

E

0

E

0043 0

085 0

0041 0

0 75 OE

00395 0

0 77 0

0 038 0

0 85 0

00275 0

0 8 0

Model factor 0 45

Note Measurement techniques optical 0electrical E

0 4 50 6 2 0 1 7 05 0 3 7 0 3 4 0 6 70 1 2

-LO

- 30 db

- 20 - theory expt

5

l =theory H expt

0 05 10 15 normailsed frequency ( f f o ]

Fig 4 Number of strips for transfer N T plotted against frequency normalized to the first stopband frequency for coupler C2

Fig 5 CW reflection intensity response of MSC C2 Theory curve refers to coupler of 40 strips Experimental points show reflection intensity maxima far coupler of 140 strips

dUsing t h e simplified expression (15) for NT thesemaybe written

PA = Pi cos2( N F K Zsin2( ~ a j ~ 2 ) ~ O t f ~ )(19a)

Po = Pi sin2(VFK2sin2( x ~ t j ~ 2 ) ~ a ~ ~ ) (19b)

Theprecedingexpressionsrefertopowertransferfor losslesspropagationandidenticaltracksExperimentaland predicted curves are compared in SectionIV 2 ) Rejection I n the region of the stopband reflection may be particularly strong because the reflections from individual stripscanaddconstructivelyAwayfromthestopband though the reflection is weak its estimation is significant in thedesign of structureswherelowtripletransitsignalsare desired

T h e reflection is calculated using the two-element model Steady-stateresponsesforsymmetricandantisymmetric modes are calculated independently from the equivalent cir-cuits A method employing repeated multiplication of section transfermatricescanbeused [12] howeverithasbeen found more convenient to work in terms of an effective char-acteristicimpedanceThecomputedreflectionintensityfor thesymmetricmode is showninFig 5 for a coupler of 40 strips with the parameters of coupler C5 (Table I ) T h e frequency response with its 40 zero reflectances below the stop-band is typical of thesteady-statereflectionresponse of a slightlymismatchedlumpedline 20 resonantwavelengths long The computed reflection intensity for the antisymmetric modeisvirtuallyindistinguishablefromFig 5 T h e significance of thisand of theexperimentalpoints shown for a coupler of 140 strips will be discussed in Section IV The en-velope of the reflection response is not affected by the number

stopband in Fig 4 Curve parameters refer to coupler C2 of TableIasdoexperimentalresultsshownforcomparison Four specific features may be noted from the curves obtaine

a)Thereis a clearlydefinedfrequency j o at which a stopband occurs

b)Disregardingthestopbandeffect(dottedcurve) a frequency is obtained at which the number of strips N T has a minimum

c) A minimum value of NT is obtained d) The stopband has a characteristic shape These features are controlled by separate parameters with

relatively little interaction a)Thestopbandfrequencycomesfromthestructure

repeat distancej=u 2 d b) The frequency of theminimumfor NT iscontrolled

by the length of the active element of the line ad c)Thevalue of the minimumfor NT is determinedby

F P d) The shape of the stopband is controlled by the choice

of parameters p and y Comparisonbetweenpredictionandexperimentisde-

ferred to Section IV

B Predictions f o r Transmission and Rejection I) Transmission ThetransmissionresponseforanMSC

with a fixed number of strips N is deduced from the value of NT For input power Pi in track -4 output powers are the following

for track A

for track B

129 MARSHALL el d MULTISTRIP COUPLER

inwt

L Fig 6 Transmissionandreflectionresponsefor a nlultistrip

coupler showing virtually 100-percent transfer

of stripsfor140strips 140 reflectionpeaks vould appear below the stophand

In pulsed operation transient reflection responses can be observedassociatedwithreflectionsfromtheleadingstrips their amplitude is up to 3 d B higher than the CU amplitudeh analysis can be achieved satisfactorily without using the equivalent circuit modeli t will not be discussed here

I PREDICTION COMPARED W I T H EXPERIMENT A Experimental Methods

Several experimental methods have been used to investi-gate the transfer of energy in the coupler The use of a phase- gratinglaserprobingapparatushasalreadybeendescribed [13]Theapparatususes first or seconddiffractionorders of light reflectedfromthegratingformedby a singlefre-quency surface wave [14] The system is particularly suitable for theexamination of multihtripcouplerbehaviorsince thehurfacewavecanbeprobedwithinthemetallizedarea Couplers of 700stripsgivingmanytransfershavebeen examinedby thih techniqueElectricaldeterminations of thenumber of strips fortransferhavebeenmadeThear-rangement of Fig l(a) was taken with power input to trackA only Frequencies were noted which gave zero power at just one of theoutputtransducers A 2 B 2 thisrepresentsthe situation of an integral number of transfers Ambiguity over the numller of transfers was removed by referring to optical probing results Again couplers having 700 strips were exam-ined I n an alternative method couplers of 140 strips vere employed toshow a single transferstripsyeredeactih-atedby vaporizing the metal at the midpoint of the strips with a laser burner This yielded a series of single transfer points on the NT-f cure Full transmissionandreflectioncurveswere also obtained at outputs 112 B 2 and B1 as a function of frc-quency for selected couplers

All measurementsreportedrefertoMSCstructures of parallel aluminium strips perpentlicular to the Z axis on F c u t Z-propagating12iSl)03Parametersforindividualcouplers are given in Table I The track aperture quoted refers to the width of indix-idual launching-receivingtransducersineach track The metallization ratio q is defined as the ratio of strip width to repeat distance a range of metallization ratios was esamined with the aim o f the following

1) 0t)taininy the minimum pohsible value of N T 2) Obtainingtheflattest posible curve foragainstfre-

cluency to nlavimize the Ilandwidth of the coupler 3) Obtaininginformation on thepropercircuitmodelfor

the f u l l theory

B Coupler Performance Theoperation of the coupler is demonstratedinFig 6

Signalinput is at transducer 111 the three coupler outputs together vith a calibration delay line output are shown i n the four photographs The coupler (device C2 of Table I) was of

-theory + expt

Fig 7 N T - ~ ~ for C7 showingexperimvtalpointscurve coupler aroundandabovestopband 4 trace of bulk waveinteraction a t fn = 136 is visible

+ couplerC3thick strips coupler C4thinstrlps

Fig 8 N T - f n curvesformetallizationratios 062 (coupler C 3 ) and 017 (coupler C4)

140 strips its stopband frequency was 86 M H z operation is shown at 43 MHz The coupler transfer loss was found to be 05 d B the uncoupled surface-wave signal was more than 30 d B down on the coupled signal as was the reflected signal

C ExperimentalVariation of NT with Frequency Compared zi th Theory

An example of the result of fitting theory to experimental results is shown in Fig 4 for coupler C2 structure and fitting parameters are listed in Table I T h e fu l l curve refers to the full theorythedottedcurveshowsthesimplifiedtheory where it is distinct from the full curve Though the fu l l curve is plotted in the stopband region the electrical method used forobtainingtheresultsdoesnotyieldgooddatathere A better comparison for this region is shown in Fig 7 for optical results on coupler C7

Above the stopband thefit between theory and experiment is generally moredoubtfulandthemodesarefoundtobe appreciably attenuated in the coupler especially a t f = 136fo This is attributed to coupling to bulk waves and some evi-dence for this has been previously published [13] We do not considertheseinterestingeffectsfurtherherebecausethey are outside the scope of the theory and because from a de-vice point of view their very existence renders this frequency range of little interest

The results of Fig 4 are for a coupler of metallization ratio q=O45 Resultsandfittedcurvesfortwocouplers of more extreme geometry are shown in Fig8 T h e metallizatioI ratio forthickstrips(coupler C3) was062andforthinstrips

~

130 IEEE TRANSACTIONS ON SONICS A N D ULTRASONICS APRIL 1973

Fig 9 Experimental and theory curves for energy transmission s n two tracks of coupler C2 (a) For 100 strips (h) For 125 strips

(coupler C4) 017 Fitting parameters for the curves are given in Table I Table I also shows results and fitting parameters for further couplersA curious experimental featureof varying

0F 0 5 ~ ~

FK

00 05 10

metallizationratio (L)

Fig 10 Variation with surface metallization ratio of effective coupling constant FK2 and of field model y Equation of fitting parabola is (q-005)~=10(0043-FK2)

Over the MSC working range (06 fo-085 fa) the signal re-flected is shown to be more than 30 dB down on the incident signal Deviation from the theory curve level could possibly be due to mass loading bulk wavesor coupler imperfections Methods for further reduction of unwanted reflected signals are discussed in SectionV I Experimental reflection resultsfor theantisymmetricmodeforcoupler C3 gave a similarre-sponse shape 3 dB stronger in intensity Other couplers with q ~ O 5 howeverhaveshownsimilar intensity levels for the two modes

E Discussion of Comparison of Predict ion and Experiment Since in the next section we shall elaborate the equivalent

circuit to deal with a wider range of experimental situations here we will discuss the parameters which have been used to achieve the very satisfactoryfit shown between prediction and experiment Good fits have been obtained with both simplified and full expressions using the fitting parameters F a yp for NT and transmission curves F and a are common to the two expressions and common values are employed They are the crucial parameters from a device viewpoint because they en-tirely characterize the simplified expression and because they dictate the behavior i n the frequency range in which coupler components are constructed (as discussed i n [ 1 7 ] ) Therefore

the metallization ratio between its extremes was found to be awe consider values of F and a first reversal of the asymptotic shape of the stopband The sig-nificance of this and other parameter variationsis discussed in Section IV-E

D Transmission and Reject ion Compared with Theory The transmission response of the basic MSC is shown i n

Fig 9 Fig 9(a) refers to 100 coupler strips and Fig 9(b) to 125 strips In both cases the transmitted outputs (in decibels down on the input signal) are plotted against the frequency normalizedtothestopbandfrequencyFullcurvesreferto experimental results for coupler C2 and dashed curves to the fu l l theory The theory shows a signal null in track A and a stopband agreement for100 strips is excellent The simplified theory gives identical curves outside the stopband region and is adequate for most transmission prediction purposes Signal loss a t maximum transfer for coupler C2 at 100 strips was al-most l d B however the normal transfer loss for full length couplers is 05 dB

T h e reflection response of the h4SC is shown in Fig 5 for the symmetric mode for coupler C5 (140strips) A long pulse isemployedtogiveeffectively CW behaviorExperiment gave a closely spaced comb of peaks and troughs as predicted Only the reflection intensity maxima are plotted correspond-ing to the envelope of the theory curve Agreement in shape and level is quite satisfactory on both sides of the stopband

A plot of F K 2 against the metallization ratio 17 is shown in Fig 10 The curve drawn is of parabolic form to demonstrate the symmetry about 7=05 The result shows a remarkable similaritytothetheoreticalbehaviorpredictedbyprevious workers [4] [ 7 ] where F is referred to as a geometrical factor T h e conclusion is t h a t r] =05 gives optimum performance not only in lithium niobate but more generally

Infittingwehavefound c y whichmeasurestheactive length of the section to show only small inconsistent varia-tion between coupler structures At first sight this is surpris-ingInattempting to relatetheequivalentcircuit of the coupler to the structure of the device one might have antici-pated a direct relationship between activity a and metalliza-tion 9Our results show that no such direct relationship exists A similar result has been found by Milsom and Redwood [ 5 ] in comparing an exact theory of the transducer on PZT with the equivalent circuit model

In fitting the full theory we find a direct relation between the field model y and themetallization ratio 77 asshownin Fig 10 though in practice some latitude yinexists in optimiz-ing the stopband shape Thus results show thatfor a structure with ultranarrow strips and wide gaps an in-line field model is appropriate for the active element while for ultrawide strips andnarrowgapsacrossed-fieldmodelisappropriateThis easily explains the reversal of asymptotic shape at the stop-


bandobservedexperimentallyMilsomandRedwoodhave predicted a similarrelationshipbetweenmodelandmetal-lization for PZT [S ]

There has always been some problem in deciding on the type of equivalent circuit model to adopt for a surface wave in-line or crossed field This work strongly suggests that the metallizationindeterminingthe field distribution alsode-terminesthemodelthecrossed-fieldmodelisassociated with a high metallization ratio and the in-line model with a

and (a) refer to the modes In considering energy transfer all energy input occurs at track A the input energy is propor-tional to current squared ie to(iA(S)+iA(a)) At maximum transfer the energy output at track A is a minimum and is proportional to ( ~ A ( s )- iA(a)) Hence the maximum fraction of energy transferred to trackB is obtained as

low metallization ratio The ideaof an active strip or an active gap now becomes untenable rather there is a shift of impor- tance fromone to theotherasmetallization increasesNow referringback to theearlierdiscussion of aweshouldno longeranticipate a monotonicrelationshipbetweenactive length and metallization ratio

T h e full theory has throughout been fitted withp =0 This meansthat no distinctionismadebetweenthevelocityin activeandpassiveregionsotherthanthroughelectrical activity This of course is consistent with the rejection of a fixed identification between strip and gap and active or pas-sive elements Use of p =0 is also attractive in termsof the der-ivation of the simplifiedtheoryHoweveritshouldbe stressed that the parametersp and y as given in TableI do not constitute a unique set of fitting parameters The feature of the set presented is that they were obtained with a conscious attempt to make a physical interpretation possible

V EXTENSION APPROACHOF EQUIVALENT CIRCUIT Theprevioussectionshavedealtwithan MSC which

spanned two identical tracks In this section this restriction is relaxed results are presented for unequal track widths and theextension of the model to tracks on dissimilar materials is also consideredTheequivalentcircuitapproach is also used to derive an expression for attenuation due to strip re-sistance and to consider effects of addition capacitive loading such as occurs i f the strips span inactive regions between two tracks

A NonidentidalTracks I ) UnequalTrackWidths The s tandard hlSC hasparallel

strips on a singlesubstrateforunequaltrackwidthssince the phase velocities in each track are unchanged symmetric andantisymmetricmodesexistaspreviouslydefinedTheir velocities are unaffected but the fraction of energy transfer-able is reduced below 100 percent Thus the number of strips required for maximum transfer is not affected but the maxi-mum fraction of energy transferred FT is changed Analytic-ally FT is calculated as follows

From the definition of the symmetric mode the voltages

lent circuit transformer of Fig 2 the symmetric mode condi-tion is

rai l(s) ~ B ~ B ( s )___--- (20a) C A CB

From the definition of the antisymmetric mode the fluxes coupled between tracks must be equal and opposite Then the antisymmetric mode condition is

Subscripts A and B refer to the two different tracks and (S)

in each track are equal for every strip Referring to the equiva-adjacent the strips have to be of extra length Two device

Usingtheprecedingmodeconditionsthecurrentsmaybe eliminated to give

where LA and le represent track aperture widths 100-percent transfer is thus only obtained for equal track widths the case studied previously

Experimentally the laser probe apparatus has been used to examine the transfer distance aasfunction of relative track width As predicted the transfer distance was found to be un-affectedby a changein therelative trackwidthsAccurate verification of the fraction of power transferred was not pos-sible

2 ) DissimilarPropagationDirectionsandDissimilarMa-terials Dissimilarpropagationdirectionsonthesame sub-strate is a realistic problem to consider because as discussed in [17] various device applications emergeif the strips in one trackareinclinedtothoseinanothertrackTheproblems raised are similar to those which existif the coupler spans two different materials permittivity SAW velocity acoustic im-pedance and K Zare no longer identical There is n o difficulty inincorporatingtheseintotheequivalentcircuitshownin Fig 2 associating one set of parameters with line A and the other with line B However i f the values of K and hence of Avw differ it is no longer a simple coupled mode system I t is not possible simultaneously to set up the two modes defined previously(Section 111-B) Thereforeit is not possible to transformthepair of lines to a singlelineAlternativeap-proachestotheproblembased on theequivalentcircuit model take us beyond the scopeof this paper

B The M S C with Additional Capacitance One of the attractions of the equivalent circuit approach

to the analysis of the multistrip coupleris the ease with which the effects of electrical loading of the strips can be examined Capacitative loading is involved in several applicationsof the couplerWhenthecoupledacoustictracksarenotdirectly

examples are the multistrip reflector and the track separator coupler (see [17]) Both rely on velocity mismatching where strips lie on material outside the track apertures to avoid sur-face-wave generation there The result of extra length is extra capacitance between strips Considering equal track apertures l and a track separation Al the capacitance increases from C to ( l+ampAl l ) C Theconsequence is a changeinthetrans-former turns ratior of Section 111 which may be incorporated into the calculation asa degradation in the coupling constant K 2 Then

K t Z= (1+ 3AE1)-K2

132

Thus the numberof fingers for transfer is increased by the pro-portional rise in track capacitance I n the particular example of the multistrip reflector the number of strips for operation at SO-Q impedance on YZ lithium niobate is predicted to rise from 50 t o 60 Experimental results for the60 finger structure are presented in [17]

C Str ip Resis tance and At tenuat ion Acoustic attenuation in the MSC can arise from both me-

chanical and electrical sources eg mode conversion to bulk waves and ohmic loss in the metal str ips I t is clear that an appreciableohmic loss couldoccuras a result of currents flowing along the length of the strips this is the particular loss mechanisminvestigatedhereThesestripcurrents flow for the antisymmetric mode but not for the symmetric mode

For identical tracks the distributed resistance of the str ip isreplacedby a resistor RP localizedbetweentheacoustic tracks RF is set equal to the strip resistance for one trackA finitevalue of impedance instead of zero impedance is now transformed into the single-line acoustic circuit for the anti-symmetric mode [Fig 3(b)] This impedance is

Under the condition 2rjCRFltltl which is easily satisfied and using the simplified expression (15) for NT an amplitude at-tenuationcoefficient CYR percouplertransferlengthforthe antisymmetric mode is obtainedas

(YR 2 R 2 f C R ~ ( 2 5 )

Resistance per square for the evaporated aluminium films was found to be within 10 percent of the bulk value over a wide range of film thickness and strip dimensions Thus for a 4-mm-longstrip of thickness 5000 A and width 12 pm the predicted resistance is 18 Q (PA= 28X10-6Qcm) the mea-sured value was 20 Q From transducer measurements a value of order 1 pF per finger was obtained at 54 MHz Thus for coupler C7 the predicted value of CYR is 002 which represents 018 d B per transfer A measurementforthedifferencein propagation loss betweensymmetricandantisymmetric modes which directly estimates resistiveloss for a coupler of the dimensions of C7 (Table I) gave a value of 1dB for seven transfers Thus prediction and experiment are in good agree-ment i t is shown that resistive loss in the multistrip structure is quite insignificant

VI GENERALDESIGNCRITERIAFOR

MULTISTRIP COMPONENTS Based on theexperimentalandtheoreticallypredicted

behavior of thesimplercouplerstructuresomegeneralde-sign criteria are now presented which are of relevance both to the coupler itself and to multistrip components derived from it

A Substrate Two important factorsfollow from the choiceof substrate

thespaceoccupiedbythe MSC andthetimenecessaryto traverse it Taking the 3-dB coupler as an example (it is the componentmostextensivelyused in [17]) the length of a n optimized structure is approximately X K 2 and the traversal t ime is (fK2)--LOn YZ LiNbOt this gives a lengthof about 25 X and a traversal time at 100 M H z of 025 PS The length is


sufficiently small compared to the aperture of a 50-9 trans-ducer on this material of 108 X 1151 to suggest that the in-creased surface area occupied is not a significant factor par-ticularly in structures with delay times exceeding the transit time On the other hand the MSC is not admissible if the de-lay time required is less than (fK2)-This can be an impor-tant factor in rejection of multistrip components at low fre-quencies or alternatively in forcing the operating frequency up if they are to be employedA further alternative is to raise K Z this is discussed in [ 1 7 ] Other materials have a lower K Z and are therefore less favorable ST-cut quartz for example wouldraisethelengthto -lo3 X and the traversal time to -10 p s at 100 MHz These are unacceptably high for a 3-dB coupler but do not exclude the use of the coupler as a t a p on a delayline [I71 wheresignificantlyfewerstripswouldbe required

Apart fromitshigh K Z YZ LiNbOoffersotheradvant-ages as a substrate for multistrip components eg zero beam steeringstrongself-collimationandappreciableanisotropy The latter is an advantage in bent coupler structures where power loss can be avoided on the bends by phase mismatch Finally we point out that the broad bandwidth offered by a high KZmaterial is entirely compatible with the coupler

B Photolithography Multistrip components require no extra processing stages

since the structure can be designed and produced at the same time as the SAW transducers Optimum MSC performance is obtainedfor50-percentmetallization so theresolutionre-quired photolithographically is similar to that for the inter-digitaldrivingtransducerscouplerstripsare on a slightly smaller scale than the transducer fingers hut shorts and breaks arelesscritical I n operationindiLidualbrokenorshorted stripscanbediscountedThecompositionandthickness of the metal film do not appear to be significant factors though through mass loading and topographical effects they can af-fect the degree of bulkmodeconversionandreflectionfrom the structure As illustrated in the previous section resistive losses in the strips are not significant

C Fixing theFrepuency of theStopband and theNumber of strips

Coupler design for a specific application involves choice of the optimum strip repeat distance The choice is significant when bandwidth is important Though theory suggests that there are ranges of effective coupling at frequencies above the structure stopband frequency f o where f o = v 2 d (cl being the strip repeat distance) experiment has shown that attenuation and spurious signals present severe problems in these ranges The fractional stopband width is found to be of order l K2 thus f o (1- 1K2) canberegarded as theupperfrequency limit The lower limit to the bandwidth rahge is imposed by thedecreaseincouplingeffectivenesswithdecreasingfre-quency The lower frequency limit for performance is easily definedfromtheuse of the simplified expressionfor NT

Random fingers may be introduced with the aim of elim- inating the stopband Experimentally the problem is to get a wideenoughrange of strip variationwithouteithersignifi-cantlyincreasingphotolithographicresolutionrequirements or vastly extending the coupler length Small random varia-tionsmerelywidenthestopbandratherthanremoveitIn practice i t is reckoned to be better to havea periodic structure

133 MARSHALL e+dMULTISTRIP COUPLER

with a narrow stopband and to restrict operation to lower fre-to bulk modes which we have observed to be important above quencies than to t ry t o spread the stopband

D Spurious Signal Reduction CW and transient reflected signals on LiNbOs are in gen-

eral 30 d B down on the input signal If this level proves sig-nificant a further 10-dB reduction can be obtained by finger length weighting the leading coupler strips to provide a tap-ered edge Chebyshev weighing on five fingers has been em-ployed Practical SAW systems tend to be limited by other

the stopband To deal adequately withall these effects a more sophosticated approach is required such as t h a t of Joshi and Ulsquohite [l61 employing the full piezoelectric fluxes and poten-tialsSuchanapproachshouldremovetherequirement for the empirical parameters which we introduce into the equiva-lent circuit

ACKNOWLEDGMENT The authors wish to thank J Oliver for laser probe meas-

urements and A S Young for supply of devices and the Di-spurious signal considerations Thusit is an attractive feature rectorRoyalRadarEstablishmentandtheController ofof the klSC on LiNbOI that such techniques are not necessary M Stationery Office for permission to publish this paperEmployment of the MSC to reduce spurious signal responses

in SAW systems is dealt with in[17 ]

VIICONCLUSIONS I n this paper our objective has been to develop a method

of analyzing the behavior of the coupler which leads to useful design criteria We have shown that in the frequency range where the coupler and its derivatives are used a simple two-parameterexpression suffices to predictthebehaviorThis statement assumes FK2 is known for the particular material underinvestigation at SO-percentmetallization a situation whichcommonlyexistsfromtransducermeasurementsand from theoretical prediction

The equivalent circuit we have usedis very similar to that employed by Milsom and Redwood [S] for transducer struc-ture A feature of our equivalent circuit-fitting parameters on LiKbOswhich we regard assignificant is that theybear a striking similarity to those of Milsom and Redwood on poled PZT Furthermore from the predicted behavior of transducer structures on quartz [7] it is apparent that the same trendsin parameter would be required Therefore we are led to suggest that though differences in detail may exist between the equiv-alentcircuitparametersforcouplers on differentmaterials the parameters found for LiNbO have general validity

Using the equivalent circuit model inclusion of some im-portantrefinements is possible lXrsquoe have demonstratedthe inclusion of finger resistance and additional capacitance and have touched on coupling between dissimilar materials Fur-thermore as discussed in Section IV the variation of param-eters with metallization ratio leads to an interesting identifi-cation of model (in-line or crossed field) not with material or itsorientationbutwiththedegreeofmetallizationThe crosscorrelationherebetweenthepropertiesofthernulti-stripcouplerandtheinterdigitaltransducer is particularly interesting

Various aspects of MSC operation have not been fully in-corporatedintotheequivalentcircuitTheseincludemass loading and topographical effects arising from the presence of the metal strips these would become importantat high ratios of film thickness to acoustic wavelength and for low KZthey can have a major influence on the behavior in the vicinity of the stopband and on the level of reflection from the coupler

REFERENCES F G Marshalland E G S PaigeldquoNovelacoustic-surface-wave directional coupler with diverse applicationsrsquolsquo Electron Left vol 7 pp 460-462 Aug 1971 E A 4sh and D P Morgan ldquoRealization of microwave circuit func-tions using acoustic surface wavesrdquo Electron Lett vol 3 pp 462-463 a c t 1967 W L Bond J H Collins H M Gerard T 1 2 Reeder and H J ShawldquoAcousticsurfacewavecouplingacross a n air gaprdquo A p p l P h y s L e f t vol 14 pp 122--124 Feb 1969 SG Joshi and R M White ldquoExcitation and detection of surfacr elastic waves in piezoelectric crystalsldquo J Acousl Soc Amer vol 46 pp 17-27 July 1969 RF Milsom and M Redwood ldquolnterdigital piezoelectric Rayleigh wave transducer An improvedequivalentcircuitrdquo Eleclron Lett vol 7 pp 217-218 May 1971 J J Campbell and W R Jones ldquoA method for estimating optimal crystal cuts and propagation directions for excitation of piezoelectric surface wavesrdquo I E E E Trans Sonics Ultrason vol SU-15 pp 209-217 Oct 1968 G A Coquin and H F Tiersten ldquoAnalysis of the excitation and detection of piezoelectric surface wavm in quartz by means of sur-faceelectrodesrdquo J Acousl Soc Amer vol 41 pp 921-939 Apr 1967 RF Milsom and M Redwood ldquoPiezoelectric generation of surface waves by interdigitalarrayrdquo Proc Inst ElecEng vol 118 pp 831-840 July 19i1 W R SmithH M Gerard J H Collins T M Reeder and H J Shaw ldquo4nalysis of interdigital surface wave transducers by use of an equivalent circuit modelrdquoIEEE Trans Microwow Theory T e c h vol MTT-I 7 pp 856-864 Nov 1969 T KraiojanananandMRedwoodldquoEquivalentelectricalcircuits of interdigital transducers for piezoelectric generation and detection of RayIeighwavesrdquo Proc Inrf Elec Eng vol 118 pp 305-310 Feb 1971

[ I l l D A Berlincourt D R Curranand H JaffeldquoPiezoelectricand piezomagnetic materials and their function in transducersrdquo inPhys-ical Acousfics W P MasonEdvol 1A New York Academic Press 1964 pp 233-242

[l21 E K Sittig and G A Coquin ldquoFilters and dispersive delay lines using repetitively mismatched ultrasonic transmission linesrdquo I E E E Trans Sonics Ultrason vol SU-15 pp 111-119 4pr 1968

[l31 ID Maines F G Marshall J F C Oliver and E G S Paige Frequency dependent behaviour of an acoustic-surface-wave multi-

strip couplerrdquo Electron L e f t vol 8 pp 81-82 Feb 1972 (141 R I Whitman andA Korpel ldquoProbing of acousticsurface perturba-

tionsbycoherentlightrdquoApplOptvol 8 pp 1567-1576 Aug 1969 I151 W R Smith H M Gerard J H Collins T M Reeder and H J

Shaw ldquoDesign of surface wave delay lines with interdigital trans-ducersrdquo I E E E Trans Microwave Theory Tech vol MTT-17 PD 865-873 Nov 1969

[ l a ] S ti Joshi and R M White ldquoDispersion of surface elastic waves producedby a conductinggratingon a piezoelectric crystalrdquo J A p p l Phys vol 39 pp 5819-5827 Dec 1968

[l71 F G Marshall C 0Newton and E G S Paige ldquoSurface acoustic wave multistrip components and their applicationsrdquo this issue pp

The equivalent circuit ignores surface-wave energy conversion 216-225

125 MARSHALL et d MULTISTRIP COUPLER

rnsc

Fig 1 (a) Schematic of multistrip directionalcouplerwith interdigital transducers at input and output ports(b) Input and output field die- tributions resolved into symmetric S and antisymmetric a modes for acouplerinwhich100-percenttransferfromtrack A to track B occurs

reproduction the precision required is similar to that for the interdigital transducers used to drive the system No connec-tions internal or external are required by the coupler Thus nopenalty is paid by the introduction of the coupler other than the occupation of space on the substrate and the addi-tionaldelaytimenecessaryforthewave to traversethe structure On material of highelectromechanicalcoupling constant such as lithium niobate the penalty is not severe

Afamily of surface-wavecomponentshasbeenderived from the MSC in addition to its use as a conventional direc-tional coupler The family includes broad-band highly reflect-ingmirrorsbroad-bandunidirectionaltransducersbroad-bandtripletransitsuppressorsbeamwidthchangers (SAW impedance transformers) and redirectors of acoustic beams These components are listed and described in [17] The ob-jective of this paper is to present a theory of theperiodic MSC based on an equivalent circuit model and hence to es-tablisheddesigncriteriawhicharerelevantbothtothe coupler and to components derived from it

B Apflroach t o the Analysis Our primaryconcerninthispaper is to developusable

designcriteria for theMSC-criteriawhichemployreason-ablysimpleanalyticalexpressionsThisisone of themain reasonsforadoptinganelectricalequivalentcircuitmodel fromtheoutsetThemodelwhich we use t o fit the experi-mental data (Section IV) is no more sophisticated than that usedpreviouslytoaccountforthebehavior of interdigital surface-wavetransducers [S] Furthermoretheparameters required to fit a wide range of gap to strip width ratios for the MSC on lithium niobate show a striking resemblance to thoseemployedforinterdigitaltransducers [4] on a poled PZT ceramic [S]

Themainfrequencyrange of interest forprediction of MSC performance lies in a broad band below the frequency of the first stopbandThisstopbandis a specificfeature of periodic couplers and occurs at the main structure resonance frequency fa=v2d where v is the surface-wave velocity and d is the MSC periodicrepeatdistance I n thisfrequency range simple analytical expressions are found which predict performance very satisfactorily these are used extensively in [l71 when analyzing device performance Another virtue of theequivalentcircuitapproach is that factors such asre-

sistivity of metal strips and added capacitance can be readily incorporated (Section V)

C Essence of MSC Behavior I t is useful as a preliminarystagetodemonstratethe

behavior of the coupler simply in terms of the electromechan-ical coupling constant K Z T h e model of the MSC employed inthisinstance is t h a t of the perfectlyanisotropic film of MarshallandPaige [ l ] T h e film spans a pair of surface acoustic tracks Its conductivity parallel to the surface-wave vectoriszeroitsconductivityperpendiculartothewave vector is infinite T h e velocity of the symmetric modev is the stiffenedvelocitysince no currentcan flow parallel tothe propagation direction but since current can flow perpendicu-lar to this direction charges in the two tracks associated with theantisymmetricmode will canceleachotheroutandthe velocity of this modev is the unstiffened velocity The length for 100-percent transfer of energy referred to as the transfer length LTis obtained in termsof the acoustic wave frequency f as half the beat wavelength for the modes Thus

where X is the surface acoustic wavelength andK Zis given by [61 as

The number of metal strips involved is obtained as

assuming total activity This demonstrates the close relation-ship between the size of the coupler and the value of KZ

In practicehowevertheperiodicMSCdiffersfromthe perfectly anisotropic film in two main ways 1) stopbands are introducedbytheperiodicity 2 ) coupling efficiency is re-ducedbygeometricfactorswhichdepend on themetalliza-tion

T o gain aninsightintothe significance of thesecond factor consider a sinusoidal signal sampled in one track and regenerated in the second track by a periodic structure (the MSC) Thesamplingfrequencyissignificant if only a few samplesaretakenperwavelengthThesamplingelement widthissignificant if i t is comparable to a wavelengthbe-causeonlytheinstantaneousaverage of signalamplitude across the sampling window can be transmitted to the second track If the phaseshiftinthesamplingwindowis 0 the coupled intensity is reduced by the factor

0 is given by 2nadX where ad is the active length in each repeatdistanceIncorporation of thereducedeffectiveness due to sampling and of the effective repeat distance gives

A sin ( 0 2 ) -2 lsquoVT = -

P a d (-Fgt (5)

Thisexpressionisverysimilartothatwhichresultsfrom

126










2 t = PAVi (8)























obtained from(3) as







TABLE I


c7 C8 c 9




140 140 700 0 45

140 0 450 620 17

I40 0 5

700 0 3 7

700 0 3 4

700 0 6 70 1 2

700




0042 0 8 5

0 00430042500345 0 8 50 8 50 8 5

0

E

0

E

0

E

0043 0

085 0

0041 0

0 75 OE

00395 0

0 77 0

0 038 0

0 85 0

00275 0

0 8 0

Model factor 0 45


0 4 50 6 2 0 1 7 05 0 3 7 0 3 4 0 6 70 1 2

-LO

- 30 db

- 20 - theory expt

5

l =theory H expt





















for track A

for track B


inwt










the f u l l theory



-theory + expt









~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )






































126










2 t = PAVi (8)























obtained from(3) as







TABLE I


c7 C8 c 9




140 140 700 0 45

140 0 450 620 17

I40 0 5

700 0 3 7

700 0 3 4

700 0 6 70 1 2

700




0042 0 8 5

0 00430042500345 0 8 50 8 50 8 5

0

E

0

E

0

E

0043 0

085 0

0041 0

0 75 OE

00395 0

0 77 0

0 038 0

0 85 0

00275 0

0 8 0

Model factor 0 45


0 4 50 6 2 0 1 7 05 0 3 7 0 3 4 0 6 70 1 2

-LO

- 30 db

- 20 - theory expt

5

l =theory H expt





















for track A

for track B


inwt










the f u l l theory



-theory + expt









~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )





















































obtained from(3) as







TABLE I


c7 C8 c 9




140 140 700 0 45

140 0 450 620 17

I40 0 5

700 0 3 7

700 0 3 4

700 0 6 70 1 2

700




0042 0 8 5

0 00430042500345 0 8 50 8 50 8 5

0

E

0

E

0

E

0043 0

085 0

0041 0

0 75 OE

00395 0

0 77 0

0 038 0

0 85 0

00275 0

0 8 0

Model factor 0 45


0 4 50 6 2 0 1 7 05 0 3 7 0 3 4 0 6 70 1 2

-LO

- 30 db

- 20 - theory expt

5

l =theory H expt





















for track A

for track B


inwt










the f u l l theory



-theory + expt









~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )







































TABLE I


c7 C8 c 9




140 140 700 0 45

140 0 450 620 17

I40 0 5

700 0 3 7

700 0 3 4

700 0 6 70 1 2

700




0042 0 8 5

0 00430042500345 0 8 50 8 50 8 5

0

E

0

E

0

E

0043 0

085 0

0041 0

0 75 OE

00395 0

0 77 0

0 038 0

0 85 0

00275 0

0 8 0

Model factor 0 45


0 4 50 6 2 0 1 7 05 0 3 7 0 3 4 0 6 70 1 2

-LO

- 30 db

- 20 - theory expt

5

l =theory H expt





















for track A

for track B


inwt










the f u l l theory



-theory + expt









~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )







































inwt










the f u l l theory



-theory + expt









~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )






































~




0F 0 5 ~ ~

FK

00 05 10

























rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )


















































rai l(s) ~ B ~ B ( s )___--- (20a) C A CB











K t Z= (1+ 3AE1)-K2

132






(YR 2 R 2 f C R ~ ( 2 5 )






































132






(YR 2 R 2 f C R ~ ( 2 5 )






































theory and design of the surface acoustic wave multistrip...

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