IEEE TRANSACTIONS ON SOMCS AND ULTRASONICS, VOL. su-22, NO. 2, MARCH 1975 105

Simplifications for the Analysis of Interdigital

Abstract-Equivalent circuits for interdigital transducers were previously published which predict identical terminal properties as those obtained using the equivalent circuit methods of Smith, et al., but which are much simpler in that they utilize a maximum of five component parts regardless of how many fingers are used in the transducer. The parameters of these minimk-complexity equiva- lent circuits are restated in a more convenient. way than previously, using surface-wave parameters adapted from those used by Smith, et al. The use of these circuits facilitates analysis yith a minimum of mathematical impedament. Short-ciicuit admittance parameters for the overall properties of apodized or unapodized transducer pairs on a substrate are obtained in relatively simple, compact form. Using these parameters the overall transmission properties for any electrical terminations are easily calculated. The minimum-com- plexity equivalent circuits involve the evaluation of certain Fourier and Hilbert transforms in order to obtain the frequency functions for some of their parts. Closed-form equations are given for these transforms for cases of transducers with or without phase reversale or phase coding. One of the components of the minimum-complexity eqriivalent circuit is a transmission line. I t is &own that at least for transducers with up to 60 fingers a satisfactory approximate method for accounting for transducer finger reflections effects is to simply include small steps in the impedance of the transmission line in order to represent the finger discontinuities.

I. INTRODUCTIOX

I N A PREVIOUS paper simplified equivalent circuits were presented which are suitable for representation of

interdigital transducers having either even or odd sym- metry [l]. Later Krimholtz extended this work to treat transducers having arbitrary symmetry [Z]. These equiva- lent circuits can give considerable simplification over the widely used approach of using a Mason circuit to repre- sent each finger of a transducer [3], [4], an approach which will lead to a very complex overall equivalent circuit if many fingers are involved. It can be shown that with properly chosen parameters the simplified equivalent circuits will predict identical results to those obtained by the methods of Smith, el aE. [3]. However, unlike the methods in .[3], the simplified equivalent circuits always have a maximum of five parts regardless of 71-hether the transducer has 10 or a 100 fingers. The distinction between physically different transducers is carried in the mathe- matical functions which describe the frequency charac- teristics of certain elements in the equivalent circuit. Thus, using these equivalent circuits having minimum complexity, the circuit analysis of interdigital transducer

the National Science Foundation nnder Grant GK-3.5478X. Manuscript received Jnne 1 3 , 1974. This work was supported hy

and Computer Science, University of California, Santa Barbara, The authors are with t,he Department of Electrical Engineering

Calif. 93106.

devices can be accomplished with a minimum of mathe: matical impedament, and the circuits make apparent certain physical insights that are not as readily evident from other points of view.

In [l] and [2], though surface-wave applications were discussed, the mathematical treatment was developed largely in terms of bulk waves and parameters appropriate for bulk waves. This left somewhat unclear how the widely used parameters in p31 should properly be applied in the simplified equivalent circuits. To rectify this sitmuation, in this paper the simplified equivalent circuits are restated in surface-wave terms using parameters consistent with those in [3]. The parameter definitions are not all quite the same as in [3], but their relation to those in [3] will be noted. The circuits will then be used t,o obtain general relations for computing the overall transfer function and input admittance properties for ’two apodized or un- apodized transducers in cascade on the same substrate. New closed-form Fourier and Hilbert, transform relations will be given which are useful in the analysis of various interdigital transducers. Finally, it. will be shown how the simplified equivalent circuits can be modified so as to approximately account for reflection effects due to t.he presence of metal fingers on the piezoelect’ric subst>rate. This approximate approach will be seen to be much simpler than the method presently used for this purpose CFil-C71.

11. TRANSDUCER EQUIVALEXT CIRCUITS Let us consider an un,apodized transducer such as the

one sketched in Fig. 1. Fig. 2(a) shows a cross-sectional view of this transducer and defines finger gap capacitances Cj,j+l. This surface-wave transducer can be modeled by the “crossed-field” model [3] shoTvn in Fig. 2(b). In this model the surface-wave device can be t.hought of as repre- sented by a piezoelectric bar carrying bulk waves, while each finger of the actual device is replaced by a pair of electrodes. Each pair of electrodes on the piezoelectric bar in Fig. 2(b) comprises a simple side-electrodcd bulk- wave transducer which in [3] is represented by the Mason equivalent circuit shon-n in Fig. 3 herein. For Fig. 3, in the notation of this paper,

\ Lrn /

106 IEEE TRANSACTIONS ON SONIC8 AND VLTRABONICS, MARCH 1975

obtained from the gap capacitances shown in Fig. 2(a) by the relations

L C - 1

Fig. 1. Simple interdigital surfacewave transducer.

Fig. 2. (a) Croes-sectional view of transducer in Fig. 1. (b)

responding to crossed-field model. (d) Derivative of excitation “Crossed-field” model of transducer. (c) Excitation function cor-

function.

PORT 2 (Acourlic)

D+:‘*’

PORT 3 (Eleciric)

Fig. 3. lMason equivalent circuit for bulk-wave transducer.

where W, and L, are defined in Fig. 2(b) and where W, = L, was assumed in [3]. The parameter k is the phase constant

(3) Note that the magnitude of the capacitances C, will

be proportional to the length of the fingers. The parameter k, is the surface-wave electromechanical coupling coefi- cient. Of course, in the equivalent circuit in Fig. 3 the voltages and currents at the acoustic ports would actually represent forces and velocities, respectively. In order to represent transducers such as that in Fig. 1 by the methods of [3], an equivalent circuit such as that in Fig. 3 is used for each finger, with the acoustic ports of the equivalent circuits connected in cascade and the electrical ports con- nected in parallel.

Though the equivalent-circuit methods of [S] and [4] have afforded great simplification over previous methods, the equivalent circuits involved are still quite complicated if the transducer has many fingers (which is often the case). This many-element circuit has, however, another entirely equivalent form [l], [2], shown in Fig. 4 for the general case. As previously mentioned, this circuit has the same form regardless of the number of fingers in the transducer, and its simplicity can facilitate the design and analysis of transducers. The number of fingers and their spacings determine the frequency functions for 9, J , and B1. These functions are derived from an “excitation func- tion” f ( x ) which, for the case in Fig. 2 (a), is illustrated in Fig. 2(c).’ In the case of a bulk-wave circuit the excita- tion function f (x) can be t,hought of as being proportional to the strength of the electric field a t a given point x , times the piezoelectric constant. Notice that, there is an excita- tion function rectangle for each pair of electrodes in Fig. 2 (b) . Where the field in Fig. 2(b) is pointing dow-nward, the excitation function is negative, and where t,he field in Fig. 2(b) is pointing upward, the excitation funct’ion is positive.

For the case of the crossed-field model for interdigital transducers, it is appr0priat.e to define the amplitude h, for the excihtion function rectangles in Fig. 2(c) by

h, = * k c t2)”’ csc (- r 2 -) L, W m .

Here the csc factor was included in order to provide for the possibilit,y of using a crossed-field model elec- trode 1vidt.h urn smaller than the spacing L,, while still maintaining the same coupling effect at synchronism. This means that though the model used in [3] implies W, = Lm, the same coupling coefficient values can be used in the equations herein even if W, < L,,,. Making W, < L, in the preceding equations will have no effect

where W is the radian frequency, v is the velocity of the acoustic surface wave, and X is its wavelength. The capacitance C, is the total capacitance associated with electrode pair m in Fig. 2(b). This capacit,ance can be operating in their higher passbands.

and Lee [S] have used a more exact, f(z) ft~nrtic~n for the circuit in 1 Actually, more general excitation funrtions ran be used. Bahr

Fig. 4 in order to predict the frequency responqe a b f transducers

YATl'H.4EI et d. : SURFACE-W.kVE DEVICES

ACOUSTIC TIUNSMISSION LINES

+ + Va - vb -

Fig. 4. Equivalent, cirruit for unapodized. unsymmetrical inter- digital transducers. (In the case of unapodized, symmetrical t,ransducers either the box J or the transformer is not present.)

at synchronism but may yield a slightly more accurate frequency response at frequencies off of synchronism since the field patterns about the fingers are modeled somewhat more accurately. In ( 3 ) and (4), kc, C,, U, L,,,, and number of fingers n corrcspond to k , C,/2, U, L/?, and 2 N , respectively, in reference [S].

The excitation function in Fig. 2 (c) can be described mathematically by

where

The excitation functionf(z) can always be broken up into the sum of an even part fe ( x ) and an odd part f o ( Z) . Thus

where

(7?

I n order t'o obtain the functions of frequency 4, J, and B1 in Fig. 4 for a given transducer configuration, we first evaluate the Fourier transform

Since f(z') is real, the real part of F ( k ) is F , ( k ) , the Fourier transform of fe (d) . Also, the imaginary part of

The transformer turns ratio is then

4 = -j(2,>"2kF,(k)

while the parameter J is given by

where

In Fig. 4 and in the preceding equations 2, is the charac- teristic impedance of the surface wave. 2, cancels out in the computation of electrical input impedance and in most transfer functions so i t usually does not need to be evaluated. Note that since Fo(k ) is imaginary, 4 will be real. The box marked J has the unique property of operat- ing like a transmission line of characteristic admittance J with a f90-degree electrical length a t all frequencies. This box has the transmission matrix

In a manner analogous to a quarter-wavelength trans- mission line, if one end of this box is terminat'ed in admit- tance YL, the admittance seen looking into the other end will be given by

Yi. = J 2 / Y ~ . (13)

The susceptance B1 is obtained from

where X is the Hilbert transform

and

x ' ( I F ( x - ) 121 = X I J F ( ~ ) 1 2 ) + -/ I ~ ( k ) 12dIi (IB) 1 "

klr -*

is a modified Hilbert transform. It might first appear that obtaining the various transforms in (g), (1 l ) , and (14) would be a formidable process, but this is not so. Their evaluation nil1 be discussed in Section V.

This equivalent circuit is useful in giving the trans- mission properties between any of the three ports in Fig. 4 (or Fig. l ) , for any given acoustic or electric terminations. The different properties of distinctly different transducers are carried by the funcbions 4, J , B1 which are all functions of k (hence of frequency). The capacitance C in Fig. 4 is the total interelectrode capacitance of the transducer. The length Z for the transmission line corresponds to the length I in Fig. 1, and v for the transmission line in Fig. 4

F ( k ) is F n ( k ) , the Fourier transform of f o ( Z ) . is the surface-wave velocity.

108 IEEE TR.4NSACTIONS ON SONICS .4ND ULTRASONICS, MARCH 1975

It is useful to see what happens to the equivalent circuit in Fig. 4 when the excitation function f ( x ) has even or odd symmetry [l]. The example in Fig. 1 has even symmetric excitation as sketched in Fig. 2(c). Under this condition we find that the transformer in Fig. 4 disappears. The connections from the upper transformer coil to terminal d-d' are completely removed, while the two coils marked +/2 are replaced by zero-impedance wires. Thus for this even symmctry case only the shunt-connected admittance inverter J is connected to the center of the acoustic transmission line. When the electrical circuit port c is driven, identical acoustic waves will propagate out acoustic ports a and b, as would be expected for even excitation.

If the excitation function f (x) has odd symmetry, we find that the admittance inverter box J in Fig. 4 dis- appears, and only the series-connected transformer is con- nected to the cent,er of the transmission line as shown in the example of Fig. 5 . Then an electrical signal applied at port c will result, in signals passing out acoustic ports a and b with identical waveforms, but with opposite polarities, as would be expected for odd excitation. If the excitation function has both even and odd parts, outputs a t a and b may have quite different waveshapes. The example in Fig. 5 has impedance steps in the transmission line in order to account for finger reflections, as discussed in Section VI.

111. SINGLE TIIAXSDUCER IMPEDANCE AND TRANS1;ER PROPERTIES

In many cases the acoustic ports Q and b may be re- garded as having terminations which match the acoustic- lint: impedance Zo, and it is of interest to compute the real part of the input admitt.ance seen looking in at, the elec- trical port under these conditions. From the equivalent circuit in Fig. 4 along with (9)-(11) and (13) it is easily seen that for unapodized transducers (the apodized case will be treated in Section IV)

-+'Yo J 2 ( G i n ) c = - 2 +-

2 Yo

(17)

From (17) and (14) we see that in Fig. 4

R1 = X ( ( G i n ) c ) . (18)

I

+ + V, "b

Fig. 5. Equivalent circuit for transducer having an excitation function with odd symmetry. Impedance steps in transmission line were introduced to account for finger reflections.

It is also of interest to compute the overall transfer function for the circuit in Fig. 4 with acoustic loads Zo a t ports a and b. Let us first assume that t,he electrical port is being driven by a zero-internal-impedance generator with voltage V,. By analysis of the circuit in Fig. 4 or by use of the transmission matrix parameters given by Krim- holtz C23 i t is easy to show that

and

where the asterisk indicates the complex conjugate. Note that for transmission between electrical port c and acoustic port a the transfer function is proportional to j k F ( k ) . But this is the Fourier transform of CV( x ) / d x where df( x) /dx is indicated in Fig. 2(d) for the transducer in Fig. 1. This is a direct result of the fact that acoustic waves emanate from points where there is a change in stress versus dis- tance rather than from regions where the stress is uniform. As can be seen from Fig. 2(d) there is an impulse in the df (x) /dx function for each edge of each excit,ation function rectangle. This point of view is very closely related to the point of view used in [lo] where an approximate method of analysis of interdigital transducers is described using an impulse source function versus distance at each edge of each finger of t,he transducer. It is also of interest to note from (20) and (21) that the transfer function V,/V, is the same as V,/V, except that in (21) F ( k ) is replaced bv its complex conjugate F ( k ) * .

IV. IRIIPEDANCE AND TRANSFER I'ROPERTIES OF TRANSDUCER PAIRS

Since interdigital transducer devices require both an input transducer and an output transducer, let us now

Thus it is evident that B, is the minimum-susceptance consider the analysis of transducer pairs on single sub-

function [g] which goes with the input conductance strates. We will first treat an unapodizcd pair of trans-

( G i n ) c (which is the electrical input conductance due to ducers as shown in Fig. 6. It can be seen t'hat, according

acoustic radiation). With the acoustic ports matched, the to the equivalent circuit in Fig. 4, if the elect'rical port total imaginary part of the electric-port input admittance c is short circuited, the equivalent circuit can be reduced in Fig. 4 is seen to be to a uniform acoustic transmission line of impedance 2 0

between ports a and b. Thus if the output port 2 in Fig. G ( B i n ) c = RI + WC. (19) is short circuited, t,ransducer 1 on the left' will see matched

MATIlIAEl et d.: SURFACE-WAVE DEVICES 109

\----ACOUSTIC L O A D S 1

Fig. 6. Two unapodized interdigital transducers on a piezoelectric substrate.

acoustic loads and its input admittance can be obtained from (17)-( 1 9 j . The short-circuit admittances yll and yZZ for the two-port device between electric ports 1 and 2 in Fig. 6 are therefore given by

where (C) 1 and (C)Z arc the total capacitances of trans- ducers 1 and 2, respectively.

Similarly if we rcplacc rach of transducers 1 and 2 in Fig. 6 by its equivalent circuit in Fig. 4, and insert the appropriate connecting transmission line and acoustic loads, we can derive for the short-circuit transfer ad- mittance y21

where OL is the attenuation per unit length for propagation on the piezoelectric substrate and d is the spacing be- tween thc reference centers of the two transducers. Notice in Fig. 6 that distance x is defined separately for each transducer, and the positive direction for x for com- puting t h r transforms F l ( k ) and F,(k) is to the right for both transducers. By reciprocity 2/21 = yl,.

Let us now consider the case of a pair of apodized transducers as indicated in Fig. 7. As was first made clear by Tancrell and Holland [lo], in order to correctly analyze such cases i t is necessary to subdivide the sub- strratc into subchannels such as the subchannel As wide indicated by the dashed lines in Fig. 7. Thus the region where the transducers exist might be divided into Q subchannrls, and cach subchannel can be viewed as con- taining an unapodizod pair of transducers. Then the over- all response can be computed as the response due to Q channels operating in parallel between electrical port 1 and

i Trr 2

Fig. 7. Example of two apodized transducers on substrat,e.

electrical port 2. For transducer 1 in Fig. 7 the short- circuit input admittance would become

m-l m-1 L '

where (yll), would be the short-circuit admittance at port 1 due to the ?nth subdivision of the transducer. Here kFlm(k) is the Fourier transform of a sequence of impulses as shown in Fig. 2 (d) with magnitude coefficients Ah which are computed using (4) , but with the capacitance C, computed for fingers As long. Parameters ( C ) 1 is again the total capacitance of transducer 1. An analogous equation for yn would apply for transducer 2 using trans- forms kFzm ( k ) for each subchannel and using capacitance (C)z. Since the short-circuit transfer admittances of t'wo- po;t networks connected in parallel are additive, by (23) the short-circuit transfer admittances are given by

m-l L

Tancrell and Holland [lo] have shown that if at least one of the transducers is not apodized it is not necessary to subdivide the transducers. Let us see how this fact can be seen from the present point of view. Supposing the number 1 transducer is not apodized, all of the transforms /cF1,(k) * will be the same and thus can be factored out and placed to the left of the summation sign in (25). As a result of the linearity of Fourier transforms one finds that for the case where, say, only the number 2 transducer is apodized, (25) can be replaced by (23) with kFn(k) computed from an array of impulses weighted in propor- tion to the amount of finger overlap. For example, let us suppose the finger overlaps for the unapodized transducer number 1 are S and that the maximum overlap for the apodized transducer 2 is also S. Then the amplitude fac- tors for the delta funct,ions in Fig. 2(d) for the unapod- ized transducer 1 would be computed in thr normal way for a transducer with fingers of uniform length S. In the case of the apodized tmnsducer 2, the magnitude factor for the delta function for fingers with the full overlap S

110 IEEE TRAXSACTIONS ON SONICS AND ULTRASONICS, MARCH 1975

would be j hmax 1, but the amplitude factor for the delta functions associated lvith the apodized fingers would be given by

(overlap to adjacent finger) (max overlap S )

1 h /scaled = I L a x 1 . (26)

In an apodized transducer the overlap with the adjacent, finger on one side of a finger may be different than the overlap with the finger on the other side. Thus in Fig. 2 (a) if finger 1 overlaps finger 2 for a dist'ance equal to one quarter of the maximum amount' S and finger 2 overlaps finger 3 by an amount equal to one half of the maximum amount S, the delta functions in Fig. 2 (d) marked -hl and h2 should be scaled to a magnitude of I h,,, 114, while the negat'ive delta functions marked --hz and h3 should be scaled in magnitude to 1 h,,, ( j2 .

Unfortunately no such simplification can be used for computing the input admit'tance to an apodized transducer even if the other transducer is not apodized. This is clear from (24) where it is seen bhat the short-circuit input admittance for an apodized t,ransducer 1 is determined by the sum of the subchannel admittances for transducer 1 alone. Of course if transducer 2 is not apodized then it will not be necessary to subdivide that transducer in order t o calculate y22.

-4fter the short-circuit admittances for the transducer pair on a substrat'e have been calculated it is a straight- forward matter to calculate the overall circuit properties for any desired electrical load terminations. Let Y , and Yz in Fig. 6 represent the source and load admittances including any tuning elements. Then the input admittance looking in electrical port 1 with electrical port 2 terminated in Yz is given by

An entirely analogous relation exists for calculating ( Y i n ) ?

for the admittance looking in port 2 with port 1 terminated in Y l . The overall transfer voltage &io is given by

including the effects of t,crnminating admittances Y1 and Yz . Often one is less interested in the transfer voltage ratio than the transducer power ratio Tvhich can be obtained from (28) by use of

of strength &-hrn. Therefore k F ( k ) , the transform of d f ( s ) / d x , is simply the sum of exponentials

M'( k ) = C h, { exp [ -jk ( x m + 1-urn/2) ] n

m-l

- exp [ - j k ( x , - W,&)]}. (30)

The evaluation of (17) from (30) is a straightforward matter, but the evaluation of the Hilbert transform in (18) is more involved. This can be accomplished by use of algorithms given in [l], and [a]. Another approach is t,o evaluate (17) a t discrete points over a range of fre- quencies (recall that k = w/v), and then evaluatc t,hc Hilbert transform (1.5) by use of fast-120urier transform operations. This can be done by use of thc. rclations [l13

n-here

and

sgn ( z ) = 1, z > 0.

= - l , z < o

= 0, Z = 0.

From the preceding i t can be seen that the Hilbert, trans- form can be accomplished by use of a forward and an inverse Fourier bransform operation.

For many cases of practical interest easy-to-use closed relations are possible for t'he Fourier transform and the Hilbert transform. This is part'icularly true if we takc a somewhat different view of the results in Fig. 2 . Let, US

set the L, = W, which will cause the pairs of impulses in Fig. 2(d) to merge with each other yielding altcmat,ing positive and negative impulses throughout. This would correspond exactly to the situation for the work discussed in [3]. Yote that for a uniform wt o f fingers the impulses in Fig. 2( d) a t the beginning and end of th r sequcncc. would be of half the strength of those i n thc interior n h r n L,,, = W,. The impulses at the ends in Fig. 2(d) corrc- spond to fringing at the outer m d s of thc first' and last fingers in Fig. 2(a). In the actual surfaw-wav(: dcvicc this fringing is probably much weaker than thc cross-field

P a v a i l model suggests, and the accurary may actually be im- ~- - (29) proved if these outer impulses arc' completely neglected. If I,, = U), and the end impulscs arc ncglcctcd, ~ v r sec

where P a v a i l is the available power of the generator and that the transducer in Figs. 1 and 2(a) will bc reprcwntcd P2 is the power delivered to the load Y2. by the d f ( ; r ) / d x function in Fig. S. Note that from this

V. CONCERNIIYG EVALUATION OF THE TLIAiYSFORJIS

point of view the df( x ) / d . ~ funct)ion uscd ronsist,s of a single impulse for each gap bet,n-een fingers. Each gap hctn-een fingers can be thought of us a sin& transvcrsd

We saw from Pig. 2(d) that @ ( x ) ldr, the derivat>ive of filter tap [12], [13], and the strrngths o f the impulscs in Y c> Y

the excitation function, is simply a sequence of impulsw Fig. S can be thought of as thr tap weights C1-11, [l:].

MATTHAEI et al.: SURFACE-WAVE DEVICES

t t 111

Fig. 8. Case of Fig. 2(d) with L,, = t o m and with irnpltlses a t beginning and end of sequence neglected. d f o t

dx , L J 1 1 ” X

For the remainder of this section \\-e will utilize the approach of using a single delta-function “tap” in @(S) /dz for each gap between fingers. Then the df(z) jdx funct,ion for an unapodizrd transducer wit,h uniform equally spaced y i ! 1 , $pL;j I I 1 fingers would be a sequence of positive and negative impulses of equal weights as shown in Fig. 9 (a) or 9(b) for the cascs of .lJ = 11 and S = 12 gaps respectively. (C1 1 1 Using this point of view i t is convenient to calculate the c P.5

magnitudes of t,he delta function weights (all equal) by URO of the formula have magnitude of 1 h,,,, 1.

( b ) l i i N ;l2 M.0 c

- x x

N :l2 M . 3

r -2

Fig. 9. Impulse “taps” for unapodized transducers. ,411 taps

I I = 2/<, (7)”’ vc,s = 2 X - C (y) W ” C $ lis (32) where 11‘ is an integer. The Hilbert transform of this func-

tion squared with W as the variable of integration is

where, as before, v is t,he surface wave velocity, kc is the surface wave electromechanical coupling coefficient, L is defined in Fig. 9(a) and (b) , C, is the capacitance per unit length of the finger gaps, and S is the length of the gaps. For examplc, C, would correspond to, say, CZ3/s in Fig. 2(a), and for a uniform transducer Cos in (32) would have the same value as C, in Eq. ( 3 ) . Here w0 = V X / L is the radian synchronous frequency of the transduccr. The tap \\-eight magnitude in (32) is consistent with that defined by (3) and (4) with L, = W , = L so that the resulting merged impulses in Fig. 2 (d) would have double strength. For YZ lithium niobatc, and equal gap and space widths in the actual transducer, C, is roughly 2.5 pfd/cm, v = 3.49 X lo5 cm/s, and /cC is about 0.22

We will now present closed form expressions for kF ( k ) , (Gin)c, and electromechanical susceptance B1 for un- apodized uniform transducers having t,ap weights such as those in Fig. 9(a) and (b) , and also for unapodized transducers having reversals of the finger sequences so as to give tap weight. sequence reversals such as that in Fig. 9 (c). These results should also be useful for evaluating thc input and transfer admittances of apodized transducer subchannels as in (24) and ( 2 5 ) .

[lo].

x ( R a Z ( u , S ) ) = T b ( U , A 7 ( 3 5 )

where

Tb(UJAV) = sin ( 2 ~ 7 ~ ) - S sin ( 2 ~ )

2 sinZ ( U ) . (36)

Then for a transducer with AV taps alternating in sign (i.e., a uniform transducer with h’ gaps between A’ + 1 fingers), if X is odd, as in Fig. 9(a), then

kF(X-1 = 1 h,,, 1 Ra(u,:Y) (37)

and if M is even, as in Fig. 9(b), then

Equat,ion (37) is exact for an odd number of taps provided the sequencing is so chosen that the center tap is positive as indicated in Fig. 9 (a ) . If the opposite sequencing were used, a minus sign would have to be introduc.ed. I,ikcwise (38) is exact, for an even number ;V of taps if the sequenc- ing is chosen so that thr tap to the left of thc center line is positive as indicated in Fig. 9(b). Again if the opposite phasing is chosen, a minus sign should bc introduced.

For N either odd or even we find that

/ 2 \ WO / and by use of (18), ( 3 . 2 , and ( 3 6 ) that

ZCn (?(,.V) = sin ( S u ! sin ( U )

( 3 1 ) The results in (37) to (40) arc cxact for tap weights such as those in Fig. 9(a) and (h). For AiT of reasonably large

while for N even

k F ( k ) = j I hsap I [Ra(u,N

For either (41) or (42)

112

size and for frequencies near or in the passband of the transducer, they will be found to be in agreement with the approximations in [3, eq. ( 2 5 ) and ( 2 6 ) ] , if one takes into account the difference in notation in that paper. In [3], k, N , and C. correspond, respectively, to kc, N / 2 , and 2C,s herein.

In cases such as phase reversal transducers C171 and phase coded transducers Cl81 the unapodized transducer may have reversals in the sequencing of the taps such as suggested in Fig. 9(c). Such reversals can be accomplished by inserting additional tap weight sequences having opposite phase. Consider adding an additional tap weight sequence M to the basic sequence N , where the sequence M is displaced P units of L/2 to the right of the center line of sequence R; (case of P positive as in Fig. 9( c) ) or PL/2 to the left of the center line of sequence N (case of P negative). Having treated the case of one additional sequence, the generalization to adding more sequences to achieve more phase reversals is straightforward.

It can be shown that for N odd

(41)

) + rRa(u,W exp ( - P u ) l . (42)

if N i- M = odd, P = odd.

and if N + M = even, P = even. (43 )

If we let y = -1 and I P 1 5 AV - M , then the M-tap sequence centered at PL/2 from the center line of the se- quence N will cancel M of the taps of sequence N . If y = - 2 and 1 P I 5 N - M , there will be a net reversal in polarity of M t’aps out of sequence N . This is illustrated in Fig. 9(c) , for the case where N = 12, M = 3, y = - 2, and P = 5 . In this case N = even so that (42) would apply. If more phase reversals were to be included, this could be accomplished by adding additional terms like the terms on the right in (41) or (42).

By use of (17) , (37) , (38), (41), and (42) we derive for 11‘ either odd or even

(Gin)c = ___ [ R d ( u , X ) + 27Rn(u,L\-)R~(u,M) l h,,, P 2

cos (Pu) + - p R d ( U , M ) ] . (44)

By (18) we obtain

B1 = [Tb(u,N) + 2yTc(u ,N ,M,P) + y2Tb(u ,M)]

(4.5)

2

where ( 3 5 ) and (36) were used, and where using (J as the variable of integration

X ( R U ( ~ , N ) R ~ ( ~ , M ) COS ( P u ) ) = Tc(u,N,M,P) (46)

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS. MARCH 1975

and

- j3zTb (” - - I p 1)) 2

( (X a M))] - Ra2 U , ___ (47)

where = 1, if I P I S + M = 0, if 1 P I 2 -V + M = I , if \ P I S - M

= 0, if 1 P 1 2 _V - M

and where S 2 M. Note that for 1 P 1 = :V + M, can have any value while for I P I = S - M , pz can haw: any value, since the relevant Tb functions are zero in these cases.

Equations ( 3 5 ) and (36) , and (4fi) and (47) were derived by evaluating the Hilbert transforms using the approach in (31). They were 1atc:r checked against Hilbcrt transform results obtained using fast-l:ourier-transfornl methods.

The solid curves in Fig. 10 show the normalized (Gin)c and electromechanical susceptancr B1 charac:tt!ristics versus normalized frequency for the tap sequence in Fig. 9(b). The dashed curves in Fig. 10 show the corresponding results for the tap sequence in Fig. 9 (c ) which includes U

phase reversal. Note that thc cffect of the rrversal is con- siderable. These results were computed from the closed- form equations (44) and (45) and checked against fast- Fourier-transform results n.hich were computed for t.hc same cases.

In cases where a number of phase reversals involving, say, M , , MP, and M a taps centered at points P1L/2, P21,/2, and P&/2, respectively, within a t,ransducer having A: taps overall, bhe transform k F ( k ) can be obtained by simply adding the required addit.iona1 terms to (41) or (42) wit,h y = - 2 . When forming (Gin)c from (17 ) , additional t,crms similar to t,he second and third terms i n (44) will also result. These added terms arc of forms such as +r2Ra2(u,Mj) and + 2 y ? R ~ ( u , l l ~ ~ ) K n ( u , i l / l k ) cos ( ( P , - Pj)u) . In order to obtainB1, the Hilbert t,rctnsfornms of these terms must. be taken, which is easily arcomplishecl by use of ( 3 5 ) and (36), and (46) and (47) reinterpreted for this purpose.

VI. ANALYSIS OP I>INGER REFLFXTIOK EFFECTS

It is well known that the presenccb of mct,al fingers on piezoelectric material int,rodurcs rcflrctions. In the case

MATTHAEI d.: SURFACE-WAVE DEVICES 113

"" l I

V 0'5

1 I O 15

f /f,

Fig. 10. Solid curves show real and imaginary parts of transducer input admittance (neglecting input capacitance) for impulse "taps" in Fig. 9(b). Dashed curves are for case in Fig. 9(c) which has phase reversal.

of lithium niobate the presence of metal fingers shorting out the electric field of the piezoelectric material appears to lower the surface-wave impedance by about 1.7 per- cent [5]. If the fingers of the transducer are not split into double electrodes these small reflections uill tend to add in the vicinity of the synchronous frequency and can cause sizable disrupting effects [5]-[7]. In the cited references the method used for analyzing these effects was to use a Mason equivalent circuit for each finger and an equivalent T for a transmission-line section for each gap between fingers. This yields a very complex model and it is of interest to obtain a simpler model for purposes of analysis and design.

We have investigated the approximate analysis of these reflection effects by use of the circuit in Fig. 4 with steps in the transmission-line impedance added to account for the steps in surface-wave impedance created by the metal transducer fingers. It should be recognized that this is an approximation as compared to the model used in [5]-[7]. That model has separate electrical connections a t each finger, and therefore views these reflections electrically from a number of different points. Though the proposed modification of the circuit in Fig. 4 introduces the proper reflection effects, these effects are only viewed electrically at the center point of the line so the results cannot be exact.ly the same. It should be recalled, however, that if we assume there are no steps in impedance between the finger and gap regions of the transducer both models will give identical results.

I n order to evaluate the amount of error in the proposed approximation an example having an odd symmetry f(z) function was analyzed using the proposed method, and also using the more involved procedures of [5]-[7]. For this case of an odd symmetric f ( z) function the equivalent circuit in Fig. 4 takes the form in Fig. 5 where 2, is the wave impedance for the region under the finger electrodes and 2, is the wave impedance for the region under the gaps between fingers. The real part of the input admittance was calculated for Z,/Z, = 1 (no finger reflections) and

f/f,

( C )

Fig. 11. Real art of input admittance of transducers computed with and witxout inclusion of finger reflection effects. Dotted curves were obtained using approximate methods discussed in Section VI.

for Z,/Z, = 0.983 (the value used in [5]) for cases of unapodized uniform transducers with 20, 40, and 60 fingers. As in [5], the difference in ve1ocit.y between gap and finger regions was neglected. The results are pre- sented in Fig. 11 (a)-( c) where the solid-line curves were computed using the methods of [5]-[7] n-hile the circles were obtained using the model in Fig. 3 . Note that the amount of error increases as the number of fingers in-

114

creases, but t’hat the approximation is still good, for prac- tical purposes, for the case of 60 fingers. In these test examples t’he finger and gap widths were equal. For t8he cases where Z,/Z, = 1, both programs gave identical results, as expected.

The analysis of pairs of transducers such as those in Fig. G using a model such as that in Fig. Ti in order to account for finger reflection effects can be accomplished most easily by multiplying together the ABCD transmis- sion paramet’er matrices for the sections of transmission line and other parts of the circuit [19J. The ABCD param- eters for the center portions of the circuits in Figs. 4 and Fig. 5 can be obtained from [Z, Fig. 8).

VII. CONCLUSIONS The minimum-eomplexit,y equivalent circuit in Fig. 4

provides a simplified means for analyzing crossed-field models of transducers such as that in Fig. 2 (b) and for deriving or computing the circuit properties of interdigital devices. The approximation illustrated in Fig. 5 for in- cluding the effects of finger reflections should in many cases permit much sinlplification in the analysis of devices where finger reflection effects are important. It is interest- ing to note that Smith [7] has shown that if finger reflec- tion effects are included in the analysis that the crossed- field model is remarkably accurate even for quite small- scale effects.

The closed-form formulas in Section V for k F ( k ) , (Gin)c , and B1 are potentially quite helpful for the rapid analysis of various kinds of t,ransducers. For example, they can be used for minimizing the amount of computa- tion needed for obtaining the input admittance and transfer properties of the many subchannels used in analyzing apodized transducer pairs.

The use of short-circuit admittance parameters to char- acterize transducer pairs, as discussed in Section IV, is seen to he very desirable. This is basically because accord- ing to the crossed-field model when one transducer is shorted, the other transducer will see only matched acoustic loads. Short-circuit parameters are also found to be convenient for treating the subchannels used in analyz- ing apodized transducers, bec.ause the short-circuit ad- mittances of parallel-connected channels are additive.

ACKNOWLEDG3IENT The authors wish to thank Brian Hanson of the Uni-

versity of California, Santa Barbara, who assisted greatly

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, MARCH 1975

in programming and carrying out most of t’he fast-Fourier- transform check examples used to check the closed-form expressions in Section V .

REI~ERENCES [l] 11. A. Leedom, R . Krimholtz, and G. L. hfatt,haei, “Equivalent

circuits for transducers having arbitrary even- or odd-symmetry piezoelectric excitation,” IEEE Trans. Sonics and liltrasonics, vol. SU-18, pp. 128-141, JUIY 1!171.

[2] K. Krimholtz, “Equivalent circuits for transducers having

on Solzics and TTltrasonics, vol. SU-19, DD. 427-436. 0ct.ober arbitrary asymmetrical piezoelectric excitation,” I E E E Trans.

1972. W. It. Smith, H. M. Gera~,d, J. IT. Collins, T. M. Iteeder, and

by use of an eqnivalent, cirrrlit model,” I E E E Trans. Microwave H. J. Shaw, “Analysis o f interdigital sltrfnce wave transdllcers

Theory anrl Trchniptcs, vol. 31TT-17, pp. 856864, November 1969. T. Krairojananan and 11. Itedwood, “Piezoelectric generation and detection of ultrasonic: surface waves by interdigital elec- trodes: An electrical equivalent circuit,” Electronic Letters, vol. 5 , No. 7, 3 April 1069, pp. 1:34-18.i.

order effects i n surf:tc.e-wave device.,” I E E E Trans. on Sonics W. S. Jones, C. 8. ITartmann, and T. I). Sturdivant, “Second-

awl Vlh-asonzcs, WJI. SLT-I?, pp. 368-377, July 1!)72. W. l!. Smith, H. 11. (:ward, and W. l(. Jones, “Analysis and design of dispersive interdigital sllrfac3e-wave transducers,” ZEEE Trans. on Alicrowacc T h m r y und Tcchniqzm, vol.

W. R. Smith, Jr., “hlinimizing n~rdtiple transit erhoes in sllrfare

No. 73 C H 0 870-8SU. Instit,ute of I‘:lec+rical and E1ec.tronic.s wave devices,” 1.W.j U1lra.sonic.s Symposilcnl Procrwlings, Cat.

. I

MTT-20, pp. 458-471, J I I I ~ 1972.

Engineers, New York, pp. 41C-413. [S] A. J. Bahr and I : . E. Lee, “ I~ :~~~ iva le r~ t - c . i r c~~ i t model f o r int,er-

digital transdtlcws with varying electrode widths,” Elec.

[!l] Weinberg, ,\rPtwork Annlj/sis nn(l S!/nlhcxis, Mc(;raw-JTill Rook I,ettrrs, vol. 9 , No. 13, 28 June 1973, pp. 281-282.

Co., Inc., p. 282 (1962). [lo] 13. Tancrell and 31. c + . I-Toll:Lnd. “AcotI-;tic. surface-wave

filters,” Proc. IEEE, vol. .iO, pp. 3!1;$-409, AZarch 1!)71.

McGraw-IIill Book Co., New York, 1963, pp. 267-268. It. Bracewell, 7’hc Fo7rric.r 7’ransforn1 anrl I ls Applicnlions,

11. R. Kallman. “Transversal filters;.” Proc. I.IZ.E., vol. 28. pp. 302-310, Jttly 1940. H. l’ludin, Jr., “,4utomatic eqrlalimtion using tranversal filters,” IEEE Sptrtrum. vol. 4, pp. TiB-T,!), Jan. 1967. It. €T.,,Tancrell, “Analytic design of surface wave bandpass filters, I E E E Tram. on Sonics and lTltrasonics, vol. SU-12, pp. 12-22, January 1974.

digital acorlstic-surfacew~~ve filter svnthesis,” 1973 lilfrasonics G . L. hfatthaei and 1). Y. Wong, ‘‘Some iechniqnes for inter-

Symposium Prowcdings, Cat. No. 7 3 C H 0 807-SSU, Institute of 1’:lectrical and E1ectronic.s Engineers, New York, pp. 427-432.

H , J. Shaw, “lksign of srlrface wavedelay lines with interdigital W. It. Smith, H . XI. (:ward, J. H. Collins. T. M. Iiecder, and

transducers,” I E E E Trans. h lTT. vol. 3lTT-17, pp. 865-873, November 1069.

I ransdllcers,” 1.972 I:llra.sonics & / ~ n p o s i ~ ~ w ~ Proc~cdings, Cat. T. J1. Bristol, “Synthesis of periodic unapodized srlrfacc wave

No. 72 C H 0 708-8SU, InstitlIte o f llec4rical and Electronics Engineers, New York, pp. 377-380.

of digitally coded acollstic sruface-wave matrhed filkrs,” W. S. Jones, C. S. ITartmann. and L. T. Clait)orne, “13valuation

ZEEE T r o m . on Sonics and fTlh-a,sonirs, vol. SU-18, pp. 21-27, January 19iI. G. L. )latt,haei, L. Yollng, and R. M. T. Jones, Microwave Filtcrs. I,,lpPtlanrr-.Iintching Nctulorks, an0 Coupling Struc- turps, ~I(Graw-HiIl I h o k Co., New Yc~rk, 1964, pp. 26-40.