Model for s.a.w. scattering by i.d.t. and itsapplication to s.a.w. track changer

P. Sudhakar and A.B. Bhattacharyya

Indexing terms: Surface acoustic waves, Surface-acoustic-wave devices , Ultrasonic transducers

Abstract: A superposition model for surface-acoustic-wave (s.a.w.) scattering by an interdigital transducer(i.d.t.) has been -developed. In essence, it consists of two steps, (a) The incident s.a.w. induces a voltageacross the i.d.t. (b) The bidirectionally radiated s.a.w., due to this induced voltage, is superposed with theincident s.a.w. to yield the scattered s.a.w. field. It is shown to considerably simplify the s.a.w. scatteringparameter calculations for i.d.t. The results agree with earlier observations. Scattering parameter calculationsof nonuniformly illuminated, resonantly loaded hybrid-junction i.d.t.s (r.l.h.i.d.t.) show that (i) s.a.w.s areunidirectionally radiated over the unilluminated part of the h.i.d.t. and (ii) it is described by a (track-changing)transfer function of bandpass type, with flat bandwidths up to 20% of nearly 0 dB loss, under appropriateconditions. The r.l.h.i.d.t. track changer should, therefore, be useful in realising long delay lines. Theattractive feature of the r.l.h.i.d.t. as an s.a.w. track changer is that its implementation is practically feasibleeven on a low piezoelectric-coupling substrate like ST quartz. However, its main disadvantages are the re-quirement of (c) multilevel high-resolution lithography and (6) the additional high-Q tuning inductors forresonant loading.

1 Introduction

When an s.a.w. is incident on an i.d.t, the i.d.t. acts as ascatterer: it delivers only a part of the incident s.a.w. energyto the electrical load across the i.d.t., and, of the remainingpart, a fraction is reflected and the rest is transmitted ass.a.w.s The three-port equivalent circuit of Smith et al}can be used to calculate the scattering parameters. However,it lacks the mathematical as well as the physical simplicityfound in the analysis of the problem by Joshi and Sudhakar.2

The basis of the latter analysis is that the voltage generatedacross the i.d.t. due to the incident s.a.w. causes bidi-rectional s.a.w. radiation by the i.d.t., and the resulting s.a.w.field is obtained by superposing the radiated s.a.w. fieldwith the incident s.a.w. field.* The basic equations obtainedfor the scattering parameters were, however, not selfcontained, because one has to fall back on the relationsbetween (a) the voltage across the i.d.t. and the radiateds.a.w. amplitude and (b) the incident s.a.w. amplitude andthe short circuit current it causes in the i.d.t., as derived byJoshi and White.3 Further, the phase relation between theincident and radiated waves was suggested rather intuitively.

In the present paper, we show that the single-portelectrical equivalent circuit of the i.d.t., along withreciprocity and energy conservation, has all the informationneeded to describe its scattering parameters. The method issimple and can easily explain the interesting track-changingfeatures of nonuniformly illuminated resonantly loadedi.d.t. (r.l.i.d.t.), recently described by Sudhakar et al* Itcan also be easily extended to the scattering parametercalculations of the hybrid junction i.d.t. (h.i.d.t.).s Theresults for the uniformly illuminated h.i.d.t. are in agree-ment with those of Matthaei et al.5 In the case of non-

Paper T342 M, first received 30th August 1978 and in revised form14th February 1979Professor Bhattacharyya and Mr. Suhakar are with the Centre forApplied Research in Electronics, Indian Institute of TechnologyNew Delhi 110 029, India*This is essentially the same as the superposition principle statedearlier by De Vrieser al.10 with the second-order effects, such as s.a.w.reflection by shorted metal fingers, s.a.w. diffraction and bulk-waveradiation by the i.d.t.,7 neglected.

uniformly illuminated h.i.d.t., the method yields thefollowing interesting results:

(i) An s.a.w. is unidirectionally radiated over the un-illuminated part of the h.i.d.t.

(ii) It is described by a bandpass-type transfer functionwhich under appropriate conditions has flat bandwidths ofup to 20% with nearly 0 dB loss.

This should prove to be useful in realising long delaylines using hj.d.t.s as track changers. The attractive featureof the h.i.d.t. as an s.a.w. track changer is that its implemen-tation is practically feasible even on low piezoelectriccoupling substrates, such as ST quartz. The multistripcoupler track changers are practical only on high-couplingsubstrates like YZ Li Nb 0., .6

In Section 2, we illustrate the method for the problemof s.a.w. scattering by a normal i.d.t. and apply it to that ofthe h.i.d.t. in Section 3.

G, YL =pc 2 B2 G2

Fig. 1 Nonuniformly illuminated i.d.t.

a Nonuniformly illuminated i.d.t.b Single-port electrical equivalent circuit

1280308-6976/79/030128 + 05 tOl-SOfO

MICRO WA VES, OPTICS AND ACOUSTICS, MA Y 1979, Vol. 3, No. 3

2 S.A.W. scattering from normal i.d.t.

To illustrate the method, we first consider the general caseof a nonuniformly illuminated i.d.t. (Fig. la) of which theuniformly illuminated i.d.t. is a special case. Note that theunilluminated part of the i.d.t. (track 2) can be treated as adifferent i.d.t., say i.d.t.2, connected electrically andacoustically in parallel with the illuminated part of thei.d.t., say i.d.t.1, (track 1). The single-port electricalequivalent circuit is shown in Fig. 16, where GX,G2 are theradiation conductances, Blf B2 are the radiation sus-ceptances and Cx, C2 are the electrostatic capacitances ofi.d.t.l and i.d.t.2, respectively. YL is the admittance of theelectrical load across the i.d.t.s with GL and BL as its realand imaginary parts, respectively. The voltage across thei.d.t.s is given, from Fig. lb, by

where

= Y, + n + YT

(1)

(2)

and Yx and Y2 are the admittances of i.d.t.l and i.d.t.2,respectively. 70 is the short-circuit current induced byincident wave of amplitude Ai} on i.d.t.l. The powersdissipated in Gx and G2 (say, Pg and P2, respectively),because of voltage Vt, give the magnitudes of the s.a.w.amplitudes radiated by i.d.t.l and i.d.t.2 in track 1 andtrack 2, respectively. The resultant s.a.w. field is to beobtained by superposing the incident s.a.w. field with theradiated s.a.w. fields. Obviously, in track 2, the only s.a.w.is the one radiated by i.d.t.2, while in track 1 there are boththe incident s.a.w. field and the s.a.w. field radiated byi.d.t.l. Pg gives only the magnitude of the complexamplitude Ag of the s.a.w. radiated by i.d.t.l, bidirectionally,due to the voltage Vt. The phase information, as we shallsee, is not needed for obtaining power-scattering parameters.It is needed only when amplitude-scattering parameters areneeded, and can be readily obtained from the energy-conservation requirement.

Of the incident power Pt in track 1, Pe, say, is absorbedin the electrical load YL, and P2 is the s.a.w. powerscattered to track 2. The remaining power ^ is left over intrack 1, in the form of reflected and transmitted s.a.w.powers. Energy conservation requires

Pi = Pe+P3+Pl

From Fig. \b, we have

Pe = Wt\2GL/2

Pi = Wt\2G2/2

Pg = Wt\2GJ2

(3)

(4)

(5)

(6)

Now, in order to obtain power-scattering parameters, it isonly necessary to express P{ in terms of Vt. It may berecalled that 70, which is related to Vt through eqn. 1, isalso related to P( through reciprocity.1 For the sake ofcompleteness, we briefly outline the relation between 70

and Pi.1

Consider the two cases shown in Figs. 2a and 2b, whereGt is the radiation conductance of the i.d.t. In the firstcase, Fig. 2a, the i.d.t. is excited by an r.f. generator thathas an internal admittance, which is the complex conjugateof the i.d.t. admittance (hence, the net susceptance, beingzero, is not shown in Fig. 2), and s.a.w.s of amplitude Aare radiated out bidirectionally. In the second case, Fig. 2b,

MICRO WA VES, OPTICS AND A COUSTICS, MA Y1979, Vol. 3, No. 3

identical s.a.w.s of amplitude A are incident, from both, thesides, on the i.d.t. and the r.f generator is replaced by itsinternal admittance. By reciprocity, the currents Ia and Io

shown in the Figs. 2a and 2b should be equal. This meansthat the incident s.a.w.s are equivalent to an electricalgenerator that can give a short-circuit current of 2/a acrossthe i.d.t. Clearly, this should be independent of theelectrical load across the i.d.t. From the linear nature of theproblem, the s.a.w. of amplitude A incident on the i.d.t.from only one side, should be equivalent to an electricalgenerator, capable of delivering a short-circuit current ofonly Ia. In this case, the incident s.a.w. power is only

2

This formally completes the power-scattering-parametercalculations. We now consider the two cases, when (i) thes.a.w. is incident on the i.d.t.l from one side and (ii)identical s.a.w.s are incident on i.d.t.l from both sides.

G, =

f21, A|

L.

Fig. 2 S.A.W. excitation by i.d.t.

a Radiated s.a.w. power Pa = \Ia\7/(2Gt)

A = amplitude of radiated s.a.w.b Reciprocity theorem /„ = Ia

Incident s.a.w. power P,- = PaA — incident s.a.w. amplitude

2.1 S.A.W. incident from one side

The resonant power in track 1, Pl} is the sum of the reflectedand transmitted powers in track 1 (say Pj:1^ and /f1*, respect-ively). Since P^is simply the power radiated by i.d.t.l inthe backward direction, from the bidirectional nature ofPg we have

= (P8/2)

The transmitted power in track 1, 7J/1) is given by

By reciprocity,Io andPt are related as

Pi - l/al2/(4G,)

(7)

(8)

(9a)

(9b)

The scattering parameters can be readily obtained fromeqns. 4, 5, 7, 8 and 9 and the results are identical to thosereported by Joshi and Sudhakar2 for nonuniformly illumi-nated i.d.t. and by Smith et al.l for uniformly illuminatedi.d.t.

129

If the amplitude-scattering coefficients are needed wefirst note that the reflected and transmitted wave complexamplitudes in the track 1 are Ae and (At + Ag), respectively.We then have

= \(At+Ag)lAt\:

(10s)

and

= \AjAtf

Using these and the others in the energy-conservationequation (eqn. 3), we get

cos0 = ~(GT/\YT\) (11)

where GT is the real part of YT and 6 is given by

(Ag/Ai) = pexp(±/0) (12) —

with p as the magnitude of (Ag/Ai). Also, we have

YT = | y T | exp( /0) (13) _

where 0 is given by

cos0 = G T / | 7 r | (14)

From eqns. 10-14, we get, with the help of eqns. 1, 2, 7 ~~and 9,

= -(GJYT) 05)This is again identical to the result of Joshi and Sudhakar,2

wherein the negative sign was used intuitively.

2.2 Identical s.a.w.s incident from both sides on i.d.t.1

The analysis is same as the above with the followingchanges:

(a) The remnant s.a.w. power in track I, Pu is of theoppositely propagating s.a.w.s of amplitude (At +Ag)

(b) 1Q and Pt are related as

/o = 2/fl

Pi = Ma\2l{2Gx)

(16a)-

(16ft)

The power-scattering coefficients follow easily from eqns. 3to 5 with the help of eqns. 1, 2 and 16. The phase angle 0is again given by eqn. 11, but (Ag/Ai) is now given by

^ = ~(2G1)/YT (17)

The power-scattering coefficient p, defined as (P2/Pi), canbe interpreted as the transfer function of the nonuniformlyilluminated i.d.t. as an s.a.w. track changer. It is given, indB.by

H = 10 log 10 (p) = (18)

If the incident identical s.a.w.s illuminate the lower half ofthe i.d.t., which is resonantly loaded (r.l.i.d.t.), then theabove result is the same as given by Sudhakar et al.* Itsinteresting feature, demonstrated experimentally inReference 4 is now obvious from the equivalent circuitof Fig. \b: over the frequency range the imaginary part ofYT remains zero (or much smaller than GT), the unillumi-nated half of the rj.i.d.t. is a perfectly matched load to theilluminated half of the r.l.i.d.t. and, hence, the s.a.w. energytransfer from track 1 to track 2 is complete.

Incidentally, the above condition is same as the one that

minimises the phase dispersion under the i.d.t. and givesoptimum insertion-loss fractional-bandwidth combinationfor the i.d.t., as pointed out by Smith et a/.8 The numberof finger pairs needed for the i.d.t. for the above conditionis referred to as the optimum number of finger pairs. It isabout 4 on YZLiNbO3 and 19 on YX quartz.8 However,the energy in track 2 is of bidirectional s.a.w.s, and hencesuch a structure is not suitable for multiple track changing,though it is useful in the improved design of bandpassfilters.4

Ttrack 2

_]

track 1

^P^

ajVt

J J

k $... w y o « f̂ jJ

-fc

B2 u

Fig. 3 Nonuniformly illuminated h.i.d. t.

a Nonuniformly illuminated h.i.d.t.b Electrical equivalent circuit

3 Scattering parameters of hybrid junction i.d.t.(h.i.d.t.)

The hi.d.t.5 is a combination of two identical i.d.t.sdisplaced by X/4 from each other, where X is the wave-length of the s.a.w. at the i.d.t. centre frequency (Fig. 3a).If the two i.d.t .s are excited electrically in phasequadrature, each one radiates bidirectionally, say withamplitude A, but the resulting wave is unidirectional, withamplitude 2A, over a frequency range much larger than thenormal bandwidth of the i.d.t.s.7

If the electrical loads, across the two i.d.t.s of thehi.d.t., are identical, then an incident s.a.w. generatesvoltages that are in phase quadrature across the two i.d.t.s.Hence, the radiated s.a.w. is unidirectional, and is in thesame direction as the incident wave. Hence the hi.d.t., withidentical loads at its two electrical ports, is reflectionlessand with two identical reactive loads at its two electricalports; the incident wave passes through the h.i.d.t. withoutany attenuation. Both these features were earlier observedby Matthaei et al.5 In fact, the complete scatteringparameters of the hi.d.t., nonuniformly illuminated with

130 MICROWA VES, OPTICS AND ACOUSTICS, MA Y 1979, Vol. 3, No. 3

s.a.w. can be obtained using the equivalent circuit of Fig.3b, in exactly the same way as before.

However, the scattering from each i.d.t. of the hj.d.t.has already been solved in Section 2. A superposition of thewaves radiated by the two i.d.t.s that constitute the hj.d.t.should then lead to the wave radiated by the hj.d.t. as awhole. The overall radiated wave is unidirectional, in thesame direction as the incident wave, and its amplitude isgiven by

A'g/Ai = lAglAi (19)

where Ag is the amplitude of the bidirectionally radiatedwave by each i.d.t. of the hj.d.t. and is given by eqn. 15.Thus, we have

= -2GJYT (20)

YT is now the total admittance seen by either of the twocurrent generators in the equivalent circuit of Fig. 3b.

The complex transmission coefficient in track 1 is givenby

= (Ai+Ag)IAt (21)

= (G'T+jBT)/YT (22)

whereG'T — GT — (23)

The propagation phase shift is omitted in the aboveequation. For the case of uniformly illuminated hj.d.t.with identical loads at its two electrical ports, eqn. 22becomes identical to the result given by Matthaei et al.5

The wave in track 2 is the one radiated by the hj.d.t.,i.e. its amplitude A'e is given by eqn. 20. We note that thepower in an s.a.w. beam is proportional to the square of itsamplitude and the beam width, and Gi} G2 are pro-portional to the widths of tracks 1 and 2, respectively.

We thus obtain, for the power transfer function of anonuniformly illuminated resonantly loaded hj.d.t. asan s.a.w. track changer, in dB, from eqn. 20

H(cS) = 101o g l 0 (4G 1 G 2 / | r T | 2 ) (24)

This is identical to eqn. 18, with all its interesting featuresexplained earlier. Here,P2 is the power of the unidirectionals.a.w. in track 2, propagating in the direction of theincident wave in. track 1. Notice that the illuminated andunilluminated parts of the hj.d.t, hj.d.t.1 and h.i.d.t.2,respectively, in fact, can be entirely different h.i.d.t.s. Bychanging the connections between the two h.i.d.t.s fromthose shown by double solid lines to those shown by thedotted lines in Fig. 3b, the direction of the wave radiatedby the h.i.d.t.2 in track 2 gets reversed while the situationin track 1 remains unchanged. The transfer functiondescribing the track changing is still given by eqn. 24.Because this can have near OdB loss, flat passband widthsof up to 20%, it should be possible to realise long delaylines with multiple track changings on the same substrate,using hj.d.t.s. The attractive feature of the scheme is thatit is practically feasible even on low-coupling substrates.As an example, the overall transfer function after (a) one,(b) two and (c) ten track changings, is shown in Fig. 4,using resonantly loaded hj.d.t. (r.l.h.i.d.t.), the two i.d.t.swhich have only twenty-four finger pairs each, as thes.a.w. track changer on ST— x quartz. Also shown in thesame figure is the transfer function of an s.a.w. delay lineon ST quartz with identical unweighted input outputi.d.t.s of twenty finger pairs each, which is the optimum

number for maximum bandwidth and minimum insertionloss for S T - x quartz.8 Clearly, the flat bandwidth afterten successive track changings is about the same as the 3 dBbandwidth of the delay line. Incidentally, the near-rectangular passband, after multiple track changings, may bean added bonus.

In the above calculations, the electrical quality factor, Q,of either of the i.d.t.s (defined as C J Q Q / G , at the i.d.t.centre frequency co0) of the equal finger-equal gap (width)hj.d.t. is taken to be 1-2 times that of an equal finger-equal gap (width) normal i.d.t. This corresponds to assumingthat the two i.d.t.s of the hj.d.t. can be treated as if theyare isolated,3 which is only approximately true. However,the exact value of Q does not significantly affect.the resultspresented here.7

The radiation conductance and radiation susceptance ofthe i.d.t. are, respectively, taken to be the following:8

G(u>) = G

B(oS) = G(

where

G = G(co0) =

sin X

sin 2 ^ -2X2

4k2N

X = Nn(f-fo)/fo

(25)

(26)

(27)

(28)

k2 is the square of the electromechanical coupling constantfor s.a.w.s, and N and C are the number of the finger pairsand the capacitance of the i.d.t., respectively.

4 Conclusions

The scattering parameters of the i.d.t. are obtained using itssingle-port electrical equivalent circuit with the help ofthe superposition principle. The calculations are simple andare in agreement with earlier results. The electrodereflections and bulk-mode conversion are not taken intoaccount. Apart from simplifying the problem of s.a.w.scattering by the i.d.t., the important outcome of this

0 9 3 10normalised frequency

107

Fig. 4 R.L.H.I.D.T. track-changer transfer function on a lowcoupling substrate such as ST — x quartz

a After one transferb After two transfersc After ten transfersd Delay line transfer function with identical input output un-

weighted i.d.t.s each with twenty finger pairsQo = 20; QN = Ql/N,Q= 1-2 QN

MICROWA VES, OPTICS AND ACOUSTICS, MA Y1979, Vol. 3, No. 3 131

analysis is the following: when an s.a.w. illuminates half ofthe resonantly loaded hj.d.t. from one side, a unidirectionals.a.w. is radiated over the unilluminated half of the hj.d.t.and it can be described by a bandpass type transferfunction with up to 20% flat passband widths of nearly 0 dBloss. By appropriate design of the hj.d.t., the s.a.w. trackchanging along with a reversal of the direction of propa-gation is feasible. Long delay paths using multiple trackchangings on a single substrate are thus realisable, with nearrectangular passbands. The attractive feature of therl .hj.d.t. track changer is that it is a short structure tofabricate even on low piezoelectric coupling substrates.Calculated results for the transfer function of rj.hj.d.t.track changer which is only twenty-four wavelengths long(at the i.d.t. centre frequency) on ST— x quartz are pre-sented: they show respectable flat passband widths evenafter ten successive track changings. Using shorter r.l.h.i.d.t.and tuning its two i.d.t.s over a broad bandwidth usingslightly more complex, but well developed, networks9

wider flat bandwidths are feasible. Since electrodereflections are anyway absent in h.i.d.t., only bulk modeconversion is neglected in the analysis.

The anticipated problems in the realisation of hj.d.t.track changers are the following: (a) multilevel, highresolution lithography is needed and (b) high-Q inductorsare needed for resonant loading.

5 Acknowledgments

The authors gratefully acknowledge the financial supportlent by the Department of Science and Technology, NewDelhi, India, for carrying out the present work. The authors

are also grateful to the referees for their useful suggestionson the manuscript.

6 References

1 SMITH,W.R., GERARD,H.M.)COLLINS)J.H.,REEDER,T.M.,and SHAW, H.J.: 'Analysis of interdigital surface wave trans-ducers by use of an equivalent circuit model', IEEE Trans.,1969, MTT-17, pp. 856-864

2 JOSHI, S.G., and SUDHAKAR, P.: 'Scattering parameters ofinterdigital surface acoustic wave transducers', ibid., 1977,SU-24, pp. 201-205

3 JOSHI, S.G., and WHITE, R.M.: 'Excitation and detection ofsurface elastic waves in piezoelectric crystals', /. Accoust. Soc.Am., 1969,46, pp. 17-27

4 SUDHAKAR, P., BHATTACHARYYA, A.B., and MATHUR,BIMAL: 'An s.a.w. bandpass filter with — 50 dB sidelobes usingunweighted i.d.t.s', Electron. Lett, 1978, 14, pp. 437-439

5 MATTHAEI, G.L., HOIEM, B.E., KRIMHOLTZ, R.S., andHANSON, B.A.: 'Non-reflecting surface acoustic wave filtersand switching devices'. Proceedings of the IEEE UltrasonicsSymposium, 1972, pp. 381-383

6 MARSHALL, F.G., NEWTON, CO., and PAIGE,E.C.S.: 'Surfaceacoustic wave multistrip components and their applications',IEEE Trans., 1973, MTT-21, pp. 216-225

7 SUDHAKAR, PUTTAGUNTA: 'Physical modelling of surfaceacoustic wave (s.a.w.) interdigital transducers and its applicationto the design of s.a.w. track changers and bandpass filters'. Ph.D.thesis, submitted to Indian Institute of Technology, Delhi, India,January 1979

8 SMITH, W.R., GERARD, H.M., COLLINS, J.H., REEDER T.M.,and SHAW, H.J.: 'Design of surface wave delay lines with inter-digital transducers', IEEE Trans., 1969, MTT-17, 865-873

9 REEDER, T.M., and SPERRY, W.R.: 'Broad-band coupling tohigh-Q resonant loads', ibid., 1972, MTT-20, pp. 453-458

10 DE VRIES, A.J., MILLER, R.L., and WOJCIK, T.J.: 'Reflectionof a surface wave from three types of ID transducers', Proceedingsof the IEEE Ultrasonics Symposium, 1972, pp. 343-345

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