AD—A0 73 425 ADVA#IC D ENOINCERINS LaB ADEL AIOC (AUSTRALIA ) F/S 20/1

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AN INTROCqXYION TO TIE DCSI6N OF SIaFACE ACOUSTIC WAVE DEVICES. (U)
APR 78 P 000SON

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DEPARTMENT OF DEFEN C E
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~~~~~~~~~~~~ LIA ~~~

DEFENCE SCIENCE AND TECHNOLOG Y ORGANISATION

’
ADVANCED ENGINEERING LABORATORY
~
DEFENCE RESEARCH CENTRE SALISBURY
SOUTH AUSTRALIA

TECH NICAL REPORT

AN INTRODUCTION TO THE DESIGN OF

SURFACE ACOUSTIC WAVE DEVICES

R. DOBSON

Approved for Public Release.

Commonwealth of Ausrr~~e
COPY No. APRIL 197
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DEPARTMENT OF DEFEN CE

DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

ADVANC GINEER ING LABORATORY

IECHN ICAL RE OTs

( 7//,. AEL-OOOl-TR~~ -~~ J
*

I
L
AN INTRODUCTION TO THE DESIGN OF SURFACE
!__— ACOUSTIC WAV ~~DEVICES,
,. ‘ (
~

~~~~~~~7t~bs]
~~~ ~~~~~~~~

S U M M A R Y

Surface acoustic wave devices such as bandpass filters,

matched filters and oscillators are in general one or two
orders of magnitude smaller than their conventional
electromagnetic counterparts and in many applications
provide superior performance. This note provides a brief
introduction to the fundamental properties of surface
acoustic waves and some of the more common surface acoustic
wave devices. The practical considerations for surface
acoustic wave device design are discussed and some results
are given.

1
Approved for Public Release

POSTAL ADDRESS : Chief Superintendent, Advanced Engineering Laboratory ,

Box 2151, G.P.O., Adelaide , South Australia, 5001.

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page UNCLASSIFIED _j
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rix r~~
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Number: AR 00 1477 ~~~~~~~t:

Report b. Title jfl

AEL-0001-TR UNCLASSIFIED
Number: Isolation:

Other c. Summary in
Numbers: Isolation: UNCLASSIFIED

3 [TITLE
AN INTRODU CTION TO THE DESIGN OF SURFACE
ACOUSTIC WAVE DEVICES

4 [PE RSONALAUT H OR ( S) : DOCUMENT DATE:

J
R. Dobson
I April 1978

6.1 TOTAL NUMBER

OF PAGES 67

6.2 NUMBER OF 69
R EFERENCES:

7 7.1 CORPORATE AUTHOR(S): 8 [ REFERENCE NUMBERS

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Defence Research Centre Salisbury
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AND NUMBER 9 COST CODE:
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0001-TR

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(Title(s) and language(s))

Defence Research Centre Salisbury

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Approved fo? Public Release

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13
J ANNOUNCEMENT liMITATIONS (of the information on these pages):
—
____________

No l i m i t a t i o n .

14 1 DESCRIFIORS: ] Acoustic signals Acoustic measurement I5ICOSATI CODES

a. EJC Thesaurus Elastic waves transducers
Acoustics 2001
Terms
Acoustic f i l t e r s 1402 H
Bandpass f i l t e r s H
Surface waves
b. Non-Thesaurus
icr
Surface acoustic waves

16 UBRARY LOCATION CODES (for libraries listed in t he distribution):

SW SR SD AACA

SUMMARY OR ABSTRACT :
(if this is security classified,the announcement of this report will be similarly classified)

Surface acoustic wave devices such as bandpass filters , matched

f i l t er s and oscillators are in general one or two orders of
magnitude smaller than their conventional electromagnetic
\
counterparts and in many application s provide superior performance.
This note provides a brief introduction to the fundamental properties
of surface acoustic waves and some of the more common surface acoustic
wave devices. The practical considerations for surface acoustic wave
device design are discussed and some results are given .

.
~~ ~~~
.-, ,‘ H
~

Security c ssification of th page:

~ ~ UNCLASSIFIED
I
 AEL-000 1 -TR

TAB LE OF CONTENTS

Page No.

1. INTRODUCT ION 1

2. SURFACE ACOUSTIC WAVES 1 - 5

• 2.1 Physical properties 1 - 3
2.2 The interdigital transducer 3 - 5
• 3. SURFACE ACOUSTIC WAV E DEVICES 5 - 10
3.1 Chirp matched filters 5 - 6
3.2 Phase coded sequence generator/correlator 6 - 7
3.3 SAW bandpass filters 7
3.4 SAW oscillators 8 - 9
3.5 SAW frequency synthesisers 9
3.6 SAW convolvers 10

4. PRACTICAL CONSIDERATION S 11 - 26
4.1 Substrate attenuation ii
4.2 Air loading attenuation 12
4.3 Energy coupling attenuation 12
• 4.4 Transducer coupling losses 12 - 13
4.5 Electro-acoustic regeneration 13 15
j
-

4.6 Reflections due to presence of metal electrodes is - 18

4.6.1 Electro-acoustic impedance mismatch is - 17
4.6.2 Mass loading of substrate 18
4.7 Wave velocity changes due to metallisation 18 20 -

4.8 Effects of crystal anisotropy 20 - 24

4.8.1 Velocity anisotropy 20 - 21
4.8.2 Diffraction and beam steering 22 - 24
4.9 Tanperature effects 24 - 25
4.10 Bulk wave generation 25 - 26
4.11 Direct transducer to transducer feedthrough 26

5. TRANSDUCER ANALYSIS 27 - 41
5.1 Equivalent circuit models 28 - 34
5.2 Electrical matching network 35 37
5.3 Transducer scattering parameters 37 - 40
5.4 Transducer electrode capacitance 41
 Page No.

6. TRANSDUCER DESIGN 41 - 46
6.1 The undirectional transducer 42
6.2 The laultistrip coupler 42 43 -

6.3 Transducer stripe to gap ratio 43 - 44

6.4 Transducer beamwidth 45
6.5 Transducer electrode metallisation 45 46 -

7. PRACTICAL RESULTS 46 - 49
7.1 Phase coded sequence generator/correlator 46 - 48
7.2 150 )‘*lz bandpass filter 49
L
8. CONCLUSIONS 50
9. ACKNOWLEDGEMENT 50
REFERENCES
p

LIST OF TAB LES

1. ACOUSTIC SURFAC E WAVE MATERIAL DATA 19

2. OPTIMUM TRANSDUCER DESIGN FOR VARIOUS SUBSTRATES 36

LIST OF FIGURES

1. Surface acoustic waves i

2. SAW particle displacement 2
3. SAW shear and compressional components 3
4. Interdigital transducer showing electric field components 4
5. Interdigital launch and receive transducers s
6. Chi rp matched filter

7. SAW phase coded sequence generation/correlator 6

8. Bandpass filters and their impulse responses 7

9. SAW delay line oscillator 8

10. SAW resonator oscillator 8

11. SAW frequency synthesiser 9

12. SAW convolver 10

13. SAW substrate attenuation 11

14. SAW air loading attenuation 12

 ~L-00O1 -i ’R
Al

Page No.

15. The unidirectional transducer 13

16. The triple transit effect 14
17. Dummy f i l l i n g electrodes 14
18. Impulse responses of a phase coded transducer without (top) 15
and with (bottom ) dummy f i l l i n g electrodes

19. Short circuit reflection parameter ‘S’ 16

20. A comparison of conventional and split finger transducer 16

geometry

21. A comparison of reflections from conventional and s p l i t 17

• fingered transducer geometries

22. Schematic of a tilted transducer pair 17

23. Wave velocity as a function of fraction of surface m e t al l i s e d 18 :‘~

24. A comparison of wavefront and distortion with and without 20

compensating dummy electrodes

25. Sur face and bulk wave velocities as a function of direction 21

of plate normal

26. fr,r/v~ as a function of direction of p late norma l 21 ‘

I
27. Schematic representation of beam steering and d i f f r a c t i o n of 22
a SAW launched on a crystalline substrate

28. Diffraction loss versus scaled transducer separation 23

29. Trade-off between diffraction loss and beam steering loss 24

30. Phase delay as a function of temperature for ST cut quartz 25

31. Transducer geometry for bulk wave cancellation 26

32. Distortion of ideal sin y/y response caused by reflection 27

at finger edges

33. Mason equivalent circuit for one periudic section 28

34. ‘Crossed field’ and ‘in line field’ approximations to actual 29

transducer electric field

k 35. A comparison of empirical results with theoretical va lues

from the ‘crossed field’ model
29

36. Experimentally determined scattering parameters compared with 30

results from ‘in line ’ and ‘crossed field ’ circuit models
37. Equivalent circuits for the ‘in line field’ and ‘crossed field’ 31
models
 AEL-0001-TR

Page No.

38. Radiation admittance as a function of frequency 32

(crossed field theory)

39. Tran sducer composed of N periodic section s , a c o u s t i c a l l y in 32

cascade and electrically in parallel

40. Apodized transducer cut into lateral imaginary strips to aid 33

analysis

41. Equivalent circuit for an apodized transducer 33

42. Input admittance for approximately rectangular band-pass 34

characteristics for an assumed beam width of 100 X

43. Transducer geometry, impulse response and fr equency response 35

44. Minimum achievable SAW delay l ine insertion loss with b i - 37

d i r e c t i ona l tra n sduce r

45. Scattering parameter modcl 37

46. Reflection loss Lil and transmission loss L2 1 versus 39 H

normalized electrical susceptance

47. Reflection loss Lll , transmission loss L2 1, and couplin g loss 40

L31 to elect rical port versus normalized load conductance
p

48. A multistrip coupler used as a track changer 42

49. Harmonic excitation for conventional X/4 transducer 43

electrodes as a function of metallization ratio

50. Effective coupling coefficient (k2 )for harmonic excitation of 44

split finger electrodes as a function of metallization ratio
51. Pulsed DC voltage breakdown in air between interdigital 44
elect rodes (atmospheric pressure)
• 52. Electrode efficiency in decibles as a function of aperture 45

53. Phase velocity and propagation loss for aluminium and gold 46
films on YX quartz

54. Phase coded array response to an electrical impulse 47

55. Expanded impulse response of phase coded arra y 47

56. Response of phase coded array to its matching waveform 48

57. Correlation peak of phase coded array 48

58. 150 P’*lz filter 49

59. 150 141z filter frequency response 49

 • _ _ _ _ _ _ _ _ _ _ _ _ _

- 1 - AEL-000 l-TR

1. INTRODUCTION

Although Surface Acoustic Waves (SAW) were f i r s t studied b y Lord Rayleigh in

1885, little interest had been shown in them u n t i l recently, except by those
studying flaw detection in materials and those studying the geophysical conse-
quences of earth tremors. SAW have been generated artificially for many years
now by conversion of energy from bulk elastic waves; this method is relatively
complicated, of low efficiency and is of little practical use. The great up-
• surge of interest in SAW came after 1965 when Whi t e and Voltmer ~ 6J * announced
the development of the interdigital transducer. This transducer is a metallised
interdigital structure which is fabricated directly onto the surface of a piezo-
electric substrate and interacts directly with the surface wave . As the inter-
digital transducer may be placed anywhere along the surface wave path , the device
may be used as a tapped delay line. This coupled with the fact that the wave
velocity is about 1/ 100 ,000 of the velocity of free electro-magnetic waves meant
tha t SAW technology offered great potential for the developmen t of novel devices
and the miniaturization of conventional devices . Early device development ,
however, was plagued by “second order” effects which considerably reduced device
performance and usefulness. Research over the past decade has enabled designers
to reduce most of these “second order” effects to a level where many SAW dev ices
now outperform conventional devices in many respects.
The maturing of SAW technology over this period has attracted an increasing
number of potential SAW users and device designers . This report is intended to
give a brief but broad overview of the subject; s t a r t i n g with the fundam ental
properties of surface waves, followed by a few of the more common SAW devices ,
with the bulk of the report being devoted to the practical design considerations
and the elimination of “second order” effects. A sectionalised reference is
included from which more detailed information may be obtained .

2. SURFACE ACOUSTIC WAVES

A detailed coverage of the theory and properties of electro-acoustic surface

waves is given in [5~ , and only a brief treatmen t of surface acoustic or
Rayleigh waves will be given here, from the device designers viewpoint .
2 .1 Physical properties
Surface acoustic waves are elastic waves which propagate along the
stress free surface of a body. Figure 1 shows a few cycles of a surface
wave travelling down the surface of a piezo-electric substrate, the wave
plot being obtained by mean s of an electrostatic probe.

- ~~‘-~—~ z. -
- _ -
~~~~~~~~~~~~~~~~~~~ -
-
- --
~ ---~~~~~~ ~~ 00
0 -~~
I_I
~~~~~~~~~

Figure 1. Surface acoustic waves

*11 Indicates a reference in the sectionalised reference. Note that not

all entries in the reference are referred to in this report.
 r ______________________________
• AEL~oool-rk - 2 -

Typical partic ’e displacement , illustrated in figure 2, results in two

component s t r a i n s , shea r and compre sslon al ; the largest of these , the
sh ear com ponent . i s normal to the propagating surface w h i l s t the compress-
ional componen t is parallel to the propagating surface and t o the propaga-
tion direction . The particle motion is retrograde ellipti cal and the
majority of the wave energy lies within one acoustic wavelength of the
surface, this is also illustrated in figure 2. Figure 3 is a plot of the

Mwdmu m Max CCW Maximum MaxCW

compmslon shea r expansio n shear

Figu re 2. SAW p a r t i c l e displacement

magnitude of each component as a function of depth in wavelengths ; note

that the shear component is shifted one quarter wavelength along the wave
norma l from the compressional component to illustrate the quadrature
relationship between these two. As the particles on the surface of a
substrate are less constrained than those in the body of the substrate
the velocity of a wave on a surface is less than the velocity of a bulk
wave ; thus the surface wave energy cannot propagate into the interior to
become a bulk wave and it remains on the surface as a surface wave.
Surface acoustic waves have a wave velocity that is typically 3000 mfs
or 1/100 000 of the velocity of electromagnetic waves, velocity being
independent of frequency throug h the low microwave range. This means
that devices implemented with surface wave delay lines will be corres-
ponding ly smaller than the same devices implemented with electromagnet ic
delay lines . Mother significant feature of surface acoustic waves is
that for a given time delay the propagation losses of a surface wave
device are significantly lower than the corresponding losses of a guided
electromagnetic wave device.
 • • .-•- -
~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ _ _ _ _ _ _

• - 3 -

‘° : ‘
\

,‘ \ x
or .
• ~
- Shear

‘I ~~~
06

4’. s
g’ a,
~~
O.
~~
I 10

0 •2
I -

/ /
1/ I, L. I, Lr -
‘fl 0-5 10 20
• Depth (wdvelenQths)

- 0-2 Coinpressional

Figure 3. SAW shear and compressional components

2.2 The interdigital transducer
If metal electrodes are placed on the surface of a piezo-electric
substrate as shown in figure 4 and a potential difference is applied between
adjacent electrodes then an electric field will be set up as indicated.
Note that this electric field may be broken into two orthogonal components
which will , due to the inverse piezo-electric effect, create a mechanical
stress with two orthogonal components. The stresses set up resemble those
of a surface acoustic Rayleigh wave, thus if an impulse of electrical energy
is applied between adjacent fingers then a standing wave will be set up on
• the piezo-electric surface which will then break up into its two components,
launching a surface wave in each direction. The periodicity of this
• surface wave will be equal to the periodicity of the applied electric field
or , the frequency of the wave will be the velocity of the surface wave
divided by the interdigital finger pair spacing.

- L
 A1iL-000I- (It - 4 -

_ _ __ _ _

/
//
A 7/ /
/
/
/ A’

7
/
/ (
_ / _ _ _

+ +
_ _ _ _ _ _ _ _

I (a .‘~ •

d4 , d
u ‘ I
~~~~~~~~~ ~~~~~~~~~~~ L
/ /PIEZOELECTRIC
/~~~

+E ~ 73

Figure 4. Interdigital transducer showing electric

field components

Consider now figure 5, where the transducer on the left is launching a

surface wave in the manner just described. As the wave propagating along
the surface causes mechanical stress in the piezo-electric substrate, then
there will be an electric field accompanying the mechanical stress due to
• the piezo-electric effect. This electric field may be “tapped off” any-
where along its propagation path by another interdigital transducer.
Thus the interdigital transducer may be used to both launch and receive
surface acoustic waves.

_ _ _ _ _
 - 5 - AEL-0001-TR

Figure 5. Interdigital launch and receive transducers

Each cycle generated by the launch transducer is as wide as the overlap

of the finger pair that generates it. Consider now a cycle of sine wave *

of half the width of the launch transducer shown in figure 5 propagating

under the receive transducer, the receiver transducer finger pairs integrate
over their whole overlap width and so the sine wave of half width will appear
across these electrodes as an electrical sine wave of half the amplitude of
that of a full width wave. Thus the amplitude contribution of individual
taps in a transducer may be controlled by varying the finger overlap this -

is known as apodization .
As distance along the surface corresponds to time, then the phase of a
signal may be varied by varying the position of the transducers. Thus
the interdigital transducer allows precise control of frequency, phase and
amplitude of signals on a surface wave delay line, or equivalently there
is a unique correspondence between the metallization pattern of the trans-
ducers and the device ’s response to an impulse of electrical energy.
• Interdigital transducers for the frequency range up to several hundred
megahertz may be produced using standard photolithographic processing
techniques used in the production of microelectronic circuits. Transducers
covering the range up to low microwave frequencies may be manufactured by a
more elaborate process using electron beam etching and laser interferometer
positioning techniques.

3. SURFAC E ACOUSTIC WAVE DEVICES

The principles of operation of a few of the more common surface wave devices
will be discussed.
3.1 Chirp matched f i l t er

d4.psrssvs troniduc r

~ IIl hIh IIIIU LIII IIII li i III 11111 11111

_________
-

— wwvwwwvwv’--
•
~~ flmt c,Uflg I*gflo I

• Figure 6. Chirp matched filter •

 -• _______

AEL-000l-TR - 6-

Figure 6 shows an interdkgital transducer which has its electrode

spacing linearly increasing from left to right. If an electrical Impulse
is applied to tht s transducer , then by the inverse piezo-electric effect
there w ill he mechanical distortion set up beneath the electrodes which
has the same periodicity as that of the fingers of the Interdig ital
electrodes. The standing acoustic wave so produced will then break up
into its two components and launch a wave in each direction . The wave
propagating to the right will he an up-chirped (low frequency to high
frequency) signal and the wave propagating to the left will be a down-
chirped (high frequency to low frequency) signal. This waveform is
• adequately received by the short length , wideband transducer on the left .
Consider now the case of an identical SAW device being used to receive
the chirp signa l generated above. If the up-chirped signa l is app lied
to the small transducer to the l e f t of the chirped transducer t hen the
• chi rp waveform w i l l he launched along the surface and pass under the
chirped transducer. Very l i t t l e response will be produced when the
surface wave has d i f f e r e n t periodicity to the p e r i o d ic i ty of the transducer
taps and only at one point will he periodicity of each cycle in the train
correspond to the periodicity of the tapping electrodes , at this point there
will be a great increase in the output response. Thus the chirped trans-
ducer may be used for generating a chirped waveform and also as a matched
filter for correlat ing this waveform .
3.2 Phase coded sequence generator/corrclator

•
L’ ’L l
~• I \ V V J\JWVVV\

*31~ ~
t ~~It ~~ I I O ~~*NSOININ

~~ W I ~ I$I* N~~O I
~~

Figure 7. SAW phase coded sequence generation/correlator

In spread spectrum communicat ions an information bearing carrier is

phase reversed by a pseudo-random binary bit stronm before transmission ,
this spreads the signal over a bandwidth many times greater than the infor-
mation bandwidth and produces a low ‘probability of intercept’ signal with
considerable anti-jam capability. At the receiver this phase-reversed
• carr ier mu st be rephaso-roversed in exactly the same points that the original
• phase reversals took place in order to regain the information from the
carrier. This corresponds to synchronizing the phase reversing pseudo—
random code generator in the receiver to that in the transmit~ er. The
conventional method for code synchronization is to let the receiver code
slowly drift past the transmitter code until correlation is achieved .
Synchronization by this method may take anything from a few minutes to tens
of minutes , depending on the code length , and has made spread spectrum
communication s unacceptable for a tactical press-to-talk applications.

-
 --— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- 7 AEL-000l-TR

It will now be shown how a surface acoustic wave correlator may be used to
overcome these shortcomings. Consider the case of code generation. In
figure 7 the transducers at either end of the substrate consist of N
finger pairs with a periodicity corresponding to the desired RF carrier.
If the transducer on the left has an electrical impulse applied to it then
a train of N cycles of sine wave will be set up which will then propagate
under the centre phase coded transducer. As these N cycles pass under
• each tap of the centre transducer then N cycles of sine wave will appear
at the output, the phase of which will be determined by the phase coding
of each tap. As the last cycle of sine wave finishes passing under one
tap, the first cycle will start passing under the next tap and so a
continuous stream of phase coded sine wave at the RF carrier will be
produced until the wave train passes under the last tap.
A device identical to that used for generating the phase coded sequence
may be used to correlate the sequence. Consider the phase coded sequence
generated above being applied to the transducer on the right of figure 7.
This sequence will propagate under the phase coded taps and at one point
only will the phase coding of the sequence correspond to the phase coding
of the taps and produce a correlation peak ; at all other times the
contributions of the phase coded taps will tend to cancel each other out.
The amplitude of the correlation peak is proportional to the number of
taps.
3.3 SAW bandpass filters

IMPULSE RESPONSE ENVELOPE ht Mu$ FREQUENCY RESPONSE

Sip (w-W 0)
—1 }4t
A
fl RECTANGULAR
~j

A h-Af
~1
j ~~~~ Sm(t— t)0 ) RECTANGULAR
It t0

~~~~~~~ \~~~AUSSIAN ___L \~~ **JSS AN

• Figure 8. Bandpass filters and their impulse responses

IL. ~~~~~~~~~~~~~~
•
r
AEL-000l-TR - 8 -
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

3.4 SAW oscillators

•
i~~
!
flfl 1 _ _ _ _ _ _ _ _ _ _ _ _

Condition for oscillation - d, - Integral number of wavelengths

Desegn to, d~ • nA.
However d, — (n+r)A, wh e reA , . A. ,r • O t l t 2 ~~3 etc

Thus oscillation will tend to occur at all values ol I, •

I.-2/1 1.-I/I I. t ,’2 / 1 I,.3/ 1

~~ 3/T ~~~~~
FREQUENCY RESPONSE OF IRANSDUCER OF LENGTH d~ •
Nulls occur at tos ,/T wtwre I• d I V • KA ,/V = k / f ,
Thus nulls occu’ at I~ +,I , /li #0
OR It k = n and di d ,
Nulls will occur at ( t . , / n ) l ~ . .ind will thus efl~ u.e oscillation only at I,

Figure 9. SAW delay line oscillator

A SAW delay line oscillator , figure 9, consists of a launch transducer
acoustically coupled to a receive transducer with the loop being closed by
an external sustaining amplifier. The conditions for stable oscillation

‘1
at one frequency only are also discussed on figure 9.

l l a Ull P
____________ _____________

~~
Figure 10. SAW resonator oscillator
a

Another type of SAW osc illator which has been developed more recently
is the ~‘esonator controlled oscillator shown in figure 10. This
oscilla~.or relies on reflections from periodic discontinuities placed at
half wavelength spacings to create a resonant structure.
SAW oscillators using ST cut quartz substrates have a stability which
is considerably better than the common inductor-capacitor (LC) oscillator,
but is not as good as the crystal controlled oscillator. SAW oscillators
do not rely on harmonic operation as do some crys tal osc illators and,

Mm..,. _________ ,.-

 -~~~~ --- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- 9 -

with current technology, may be fabricated for fundamental operation in

the low microwave range. The volume of a SAW oscillator may be an order
less than the equivalent LC oscillator.
3.5 SAW frequency synthesisers

S~~~ 4i O wls

.1 !J5
~~~~ IL
,1 1
-

ri
—
Yv-
•
~~~~~
1= 20 CYCLES 00 wis
(L
~
SPOaEE
~~ ~~ ~~

Figure 11= SAW frequency synthesiser

The details of a SAW frequency synthesiser are shown in figure 11.
An impulse generator is used to generate the harmonics of a stable
frequency source, and a SAW filter is used to filter out the desired
harmonic. If the SAW filter is constructed so that all the fingers are
of uniform overlap (unapodized) then the resulting frequency response will
be the familiar sin x/x pattern. By selecting the appropriate number of
fingers in the interdigital transducer , the nulls in the sin x/x pattern
can be made to have the same spacing as the harmonics of the frequency
source and so all of the undesired harmonics can be rejected . The
thought of having 20 filters to synthesise 20 different frequencies sounds
bulky and expensive by conventional standards, however , SAW filters are
very smallwith 10 fil ters/ca2 not be ing uncommon for the above
application, and in fact a SAW frequency synthesiser may be one or two
orders of magnitude smaller than a conventional frequency synthesiser .
SAW synthesisers are ideally suited to communication systems where a
• number of channels are separated by a fixed frequency difference.

-1

I
U
 -~~~~~~~ - -
~~~~~ ~~~~~~~~~~~~~~~~ ~~~~~ U

AEL-000l-TR - 10 -

3.6 SAW convolvers

P0 7

P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
j POR T

Figure 12. SAW convolver

A schematic diagram of a surface wave convolver is shown in figure 12. *

It has a transducer at each end and detector electrode plates on the top
and bottom in the centre of the substrate . The SAW convolver is unusua l
in that it relies on elastic nonlinearities , which become evident when the
device is operated at abnormally hi gh power levels , to generate produc t
terms. The convolution of two signals

f (t) * g Ct) =
J _oo
f (t - r) g (r) d r • -

may be thought of as the integration of the product of g ( r) and a time

reversed f Ct) as they are slid past each other . The same process may
be seen to occur on the surface wave convolver of figure 12. Consider an
RF carrier ( w 1). modulated by f (t) being launched from port 1 and the same
carrier modulated by g (t) being launched from port 2. As these
contradirectional waves pass each other the nonlinearities of the device
gives rise to a product term which is then integrated by the large
detector plates of port 3. This output (at w 1) is the convolution of
f (t) and g (t) time compressed by a factor of two. If the duration of
f (t) and g Ct) is less than the transit time under the pick up plate,
then the integration limits may be taken to be infinite. The convolver
of figure 12 is a special case in that f (t) and g (t) modulate the same
carr ier frequency , this is known as a degenerate convolver. Convolvers
in which f ( t) and g (t) modulate different frequency carriers have the
detector plate replaced by an interdigital array which operates at the
difference of the carrier frequencies. SAW convolvers using simple
piezoelectric substrates to generate non-linear effects are relatively
inefficient and most practical convolvers are semiconductor coupled and
rely on the nonlinearities in the semiconductor.
SAW convolvers may be used in a wide variety of applications including
waveform correlation and the generation of real time Fourier Transforms
of signals.
 - 11 - AEL-000l-TR
I
4. PRACTICAL CONSIDERATIONS

The ideal surface acoustic wave may be thought of as a parallel wave the
width of the launching transducer travelling unattenuated down the length of
the substrate, with the conversion of e lectr ica l energy to acous tic energy and
back again being one hundred percent efficient. The deficiencies of practical
devices will now be discussed .
4.1 Substrate attenuation
The attenuation of a surface acoustic wave due to the substrate
material has been found to be approximately proportional to the square of
the operating frequency 1 501 . Both magnitude and frequency dependence of
the at t enuation are important parameters to be considered when high
percentage bandwidth filters are being designed . Curves of attenuation
against frequency for Quartz , Lithium Niobate and Bismuth Germanium Oxide ,
the most common surface wave substrates, are illustrated in figure 13.
Note that Quartz has more than twice the attenuation per given delay than (
has Lithium Niobate. (Both have delay times of approximately 3 Ps/cm.)
For devices designed to operate in the lower VHF region, the attenuation
will be negligible except where very long time delays are being considered.

ATTENUATION IN VACUUM

Z ILL
// /
~
W
0.4 - 0 LiNbO 1
~
.

I/ P
ty. x BiGeO
fi
/7
it *0

0.2 - a Q UA R T Z

I • I . . I__
~~
. I • I
200 400 700 2000 4000
FREQUENCY (MH i )

Figure 13. SAW substrate attenuation L 501

~~pirical expressions for the substrate attenuation curves of
figure 13 have been derived 1 501 :
Y cut, Z propagation LiNbO3 loss (db/Iis) — 0 88F~~’ ... .(1)
001, 110, Bi,2Ge02. - loss (db/ttLs) — 1.4SF~~9
Y cut, X propagation Quartz loss (db/Ps) • 2.1SF3 ...
.(3)
where F is the operating frequency in (
~-I~

_ —~~~~~~~~~~~~ -
• •
.
~~~~~~~~~~~~~~~~~~~
-- ~~~~~~~~~~~~~ •
 AEL-0001-TR - 12 -

4.2 Air loading attenuation

Curves of surface acoustic wave attenuation as a function of operating
frequency are illustrated in figure 14 ( from 1 501 )where it can be seen that
attenuation is proportional to the operating frequency. Again the effects
are neg lig ible in the low VHF range , but should be considered in the UHF and
m icrowave frequencies. Fortunately, air loading can be eliminated by
vacuum encapsulation or by using a light gas (e.g. Helium)environment.
Empirical expressions for the air loading attenuation curves of figure 14
have been derived 1 501 :
Y cut , Z propagating LiNbO3 loss (db/ps) 0.19F ..
....(5)
•

001 , 110 , BiGeO2 0 loss (db/ps) 0.19F

Y cut , X propaga ting Quart z l oss (db/ p s) = 0.45F ....(6)

where F is the operating frequency in GH

~

_ _ T I
~~~~~
0.5 - C

/
£
£
7
/
-=04 - / -
U /
SLOP C . -o
1
.
-. •~~
I l u lO _ _ _ _ _
0
t —03-
MH I _ ~~’$eC
-

~~O 2
• LlNbO
~ I
.4 ‘ 8 1it0 * 0
*0
~~~ lb~~

500 000
I
1500 2000
I
]
F EQU( NCY lMH.)
~

Figure 14. SAW air loading attenuation [501

4.3 Energy coupling attenuation
If a surface acoustic wave must pass under a large number of taps, as
occurs in a correlator for phase coded signals, then it should ideally
remain unattenuated down the entire path length. To achieve this , the
coupl ing coefficient of each of the taps must be kept very low , otherwise
energy will be taken off at each tap along the path length, resulting in a
severely attenuated signal for the final taps. The degree of signal
attenuation due to this cause may be calculated from the power scattering
parameters given in the section on transducer equivalent circuit models.
4.4 Transducer coupling losses
• These losses are those associated with the launching and receiving of
• the surface wave. The most common type of interdigital transducer is
bidirectional and launches waves in both forward and reverse directions,
the reverse wave usually being absorbed by an acoustic absorbing material
• such as silastic or black vacuum wax. Bidi-rectionality results in a 3db
loss in launching and another 3db loss in receiving due to reciprocity.
 ~~~~~~~ -~~~~~ — -~~~~~~~~~~ -~~~~~~~~~~~~~

- 13 - AEL-000l-TR

msuLArING
UYU

~~~~~~~~~~~~~~~
iIt[[1i
Figure 15. The unidirectional transducer

The unidirectional transducer (figure 15) overcomes these losses to a

large extent and will be discussed in the section on transducer design.
Further losses occur in the electrical matching network used to match the -
-
electrical source to the transducer. A typical value for this loss being
about 4db overall. If a broadband response is required , then the
electrical circuit Q must be damped appropriately, which will cause even
further losses.
4.5 Electro-acoustic regeneration
As a surface acoustic wave propagates under a bidirectional transducer,
an electric field appears between adjacent fingers due to the
piezo-electric effect. This field is applied to all of the electrically
connected fingers in the array which , in turn, causes a mechanical stress
on the surface of the substrate due to the inverse piezo-electric effect -

•
whi ch then becomes a new surface acoustic wave propagating hidirectionally -
from the transducer. This effect occurs in both the launch and receive

•
transducers. The wave which is relaunched in the opposite direction to
the incident wave is called the reflected wave.
Consider the case of a wave which is incident on an ideally terminated •
bidirectional receive transducer. Only one half of the incident wave
energy will be absorbed, the other half will be relaunched bidirectionally
(see figure 16).
~~~~ AEt-0001-TR • - 14 -
- - - •—-—

0dB — 6dB

=
-
~~~

Figure 16. The triple transit effect

The r efle c ted wa v e , which will now be 6db down on the incident wave ,
will propagate back toward the launch transducer where it will be
reflected back to the receive transducer. This re-reflected wave will
be 12db down on the incident wave and has done three transits down the
substrate when it reaches the receive transducer. It is known as the
triple transit wave and will produce a delayed spur ious response 12db down
on the main response.
Electro-acoustic regeneration can cause severe problems in transducers
which have groups of taps spaced periodically as occurs in comb filters and
phase coded transducers. In these cases , reflections occur between groups
of taps, which causes stop bands to be generated at harmonics of the group
• spacing frequency (this corresponds to harmonics of the chip rate in a r
phase coded transducer). Judd et. al. (421 , have shown that a successful
method of removing these stop bands is to place dummy electrodes between
the tap groups as in figure 17. The improvement in response by using
dummy electrodes is shown in figure 18.

Figure 17. Dummy f i l l i n g electrodes

_
--
U-
 -- — -• - - — •~- - -w-- - ~
~~~~~~~~ ~~ ’
- 15 - AEL-0001-TR

I :

H
F igure 18. Impulse responses of a phas e coded transduce r
w ithout (top) and with (bottom) dummy filling
electrodes 142 1

E l e c t r o - a c o u s t i c regeneration and i t s assQciated problems may be

v i r tually elim ina ted by carefu l des ign. Firs tly , i f a unid ir ec ti onal
transducer ( figure 15) is used , most of the energy launched at one end
w i ll be received a t the o ther end , prov id ing the el ec tr ical ne twork is
correctly matched • To match the electrical network correctly, the
capac it ance of the in terdigi tal transducer array mus t be made to resona te
wi th an external inductor at the frequency of acoustic resonance of the
surface wave transducer . To reduce reflection s in the case of a
bid irec tional transducer , a damp ing resis tor may be placed across th e
electrical matching network which virtually shorts out the p iezo-electric -

•
field. This has the disadvantage of severely attenuating the received
signal; however , in many cases th is may be en t irely accep table.
•

Transducer scattering parameters may be used to determine the level of

reflections due to this cause for various load conditions.
4.6 Reflections due to presence of metal electrodes
4.6.1 Electro-acoustic impedance mismatch
• Each interdigital electrode edge encountered by a surface wave
represents a discontinuity in the propagating medium , this causes
a fraction of the surface wave energy to be reflected which
results in spurious responses . The presence of a conducting
electrode on the surface of a piezo-electric propagating medium
causes changes in both the electro-acoustic impedance and wave
velocity. These parameters change because the electrodes short
circuit the electric field , the effect being more severe in the
• substrates with high coupling coefficients. Note that reflections
due to this cause are independent of load and are not related to
- electro-acoustic regeneration . Eintage [561 , has carried out an
•

• analysis of interdigital transducers which includes reflections due

to the electrode short circuiting effect , and concludes that
reflections due to this cause become significant when (4N S ~ v/vao) --
becomes of the order of unity . N is the number of finger pairs
and the parameter S is a reflection coefficient , illustrated in
figure 19 , as a function of the fraction of substrate surface
 -~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AEL -000l -TR - 16 -

Figure 19.
metallised.
FWiTIO4 OF ~~~~~~ NETALL IZED . .

Short c i r c u i t r e f l e c t i o n para m e t e r ‘S’ 1 561

The amplitude reflection and transmission
coefficie nts for a short circuited single finger pair are given
by:
I1 ~
-:
Reflectio n coefficient i sin 211 S (Av / voo~ = (7) ....
Transmission coefficient = cos 21r (i~v/voo~ .. .. (8) S

Reflectio ns due to acoustic impedance mismatch may be reduced by

increasing the fraction of surface meta llised (reducing S) or may
v i r t u a l l y be eliminated by use of split fingers 1 38] or a tilted
transducer geometry 1 52). The split finger geometry is shown in
figure 20 where it can be seen that the finger spacing and width are
both X/8 instead of the usual X/4, this causes the most efficien t
reflection frequency to be at twice the oper ating frequency which
can be considered as being out of band for most applications. -

•
snvsnhsono) •urtocs wo v, tronsd ucpr qSOIIiItr-y
~

Split tsn gsi- troniducs, pse.,i.try

Figure 20. A comparison of conventional and split

finger transducer geometry

-
- •_ ~~~~~~~~~~~~~~~~~~~ —
~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 — 17 — AI I.—( PO ll — l I ~
~

Figure 21 (from I 551 ) presents a compari son of r e f l e c ti o n

• coefficients at the fundamental for both conventional and split finger
geometries. Note that t he reflec ti on c o e f f i c i e n t of the split finger
geometry is independent of the number of fingers. A disadvantage of
the split finger geometry is that the upper frequency limit is halved
for a given fabricat ion process , unless harmonic operation is utilised.

CONVENTIONAL SPLIT

•I -
~~
—
~~~~~~
- ________ —

-K

NUM&R OF ELEC7*OD S N
~

Fi gure 21. A comparison of r e f l ection s from convential and spli t

fingered transducer geometries 1 55]
An alternative to the split finger geometry, the tilted transducer
geome try, is shown in figure 22. A wave launched from one of the
transducers passes under only a few of the fingers of the second
transducer , thus reducing the reflection s considerably. The effect
of the wave travelling under the busbar appears to be negligible.
The disadvantage of the tilted transducer geometry is that a wider
substrate is needed . In the cases where neither the tilted nor the
split finger geometry is acceptable , substrates with low electo-
mechanical coupling coefficients, such as quartz, may be used to
reduce reflections.

. TRANSMITTER I R~~EIVER
ARRAY
I ARRAY

Figure 22. Schematic of a tilted transducer pair

~ —-- — -— -- ~~~~~
- - •-
-“ —

AEL-0001-TR - 18 -

4.6.2 Mass loading of substrate

A second, less severe source of reflections is the mass loading
of the metal electrodes on the surface of the substrate. In
determining the electrode thickness, the designer must trade off
the resistance of the electrode with the mass loading of the
electrode. Aluminium has been a very common electrode material due
to its low mass density and high conductivity. Typical thicknesses
for alumin ium are of the order of a few thousand Angstrom units.
An alternative electrode metal which has been extensively used is
gold which is usually deposited in thinner layers than is aluminium .
The split fingered and tilted geometries reduce the effect of
reflections generated due to this cause also. Mechanical
reflections in interdigital transducers have been analysed by Skele
1 49] .

4.7 Wave velocity changes due to surface metallisation

The presence of metal electrodes on the surface of a piezo-electric (
substrate causes a slowing of the surface wave velocity due to the short
circuiting of piezo-electric field .

V00 —
0

Veo UNM ETALLISED

~ -

VM t .O
FRACTION OF SURFACE METAWSED

Figure 23. Wave velocity as a function of fraction

of surfave metallised [56]

Figure 23 illustrates the mean phase velocity and resonance velocity of

waves travelling under a transducer as a function of the fraction of
surface that is covered with metal. These curves are taken from Emtage
[561 who has carried out an analysis of interdigital transducers which
includes this often neglected second order effect. The change in surface
wave velocity, A , of a substrate caused by depositing a high conductive
metal film on the~ surface has been determined for many of the comaon
substrates. This parameter is usually listed as An y00, v0o being the
velocity of the wave on the unmetallised substrate. Values of txv/vt’o for
the comon substrate materials are listed in Table 1. Note that Av/y.o H
is often used as a measure of the electro-mechanical coupling coefficient.

. - L- - - - - - -- -
 - 19 - AEL-000l -TR

TABLE 1. ACOUSTIC SURFACE WAVE MATERIAL DATA

N411i14 1K0 UNOI l~IS0~ 1~N0 1 111101 I

I G.0 I 1l!17~ 1II~~, 11117~ 1014 G~ i Gii.
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1511101117
IUCIIONLNTIC IA ACOIJSIIC I ISI IW A 177 1
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Cou rt.iy of Air Fore. Csmbrtdi. RsINSre h Liboralorl..

-ì
_ _ _
 -- - - ‘-— - — ---— •— ---•-—---.--— •—~
—-—‘I.—------- -~~----~‘.~~~~~~~~~ ••----.-•= .--~----- ---- —-- - —,---‘-----—--- •---.- •~—-—-~ —.---- -
• v•—---

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~~

AEL-000l-TR - 20 -

The effects of velocity changes due to surface metallisation are most

severe in apodised structures, where different sections of the wave front
• pass under different amounts of netal , thus causing some of the wave front
to be retarded more than the rest . Tancrell and Williamson ( 531 , have
overcome this problem by the use of dummy fingers which do not interact
w i t h the wav e b u t p rov ide t he wh o le wa v e f r on t w i t h a n eq ua l amount of
metal to pass under (see figure 24). This effect is more prominent in
substrates with high electro—mechanical coupl ing coefficients such as
Lithium Niobate.

—H ~~
N •’• t~~~~ ? PHA5E

_ _ _ _ _ _ _

[thj vcj1:th ___

L_______
f FINGER

Fi gu re 24. A compari son of wavefront and distortion with

and without compensating dummy electrodes

4.8 Effects of crystal anisotropy

The materials commonly used for surface acoustic wave substrates are
anisotropic crystals. This means that fundamental physical quantities,
such as velocity, power flow angle , electro-mechanical coupling
coefficien t and other electrical and mechanical parameters, are functions
of the direction of propagation. The effects of crystal anisotropy are
well covered in 1 501 and 1 511 , however, in the interest of completeness ,
they will be covered briefly here.
4.8.1 Velocity anisotropy
The obvious effect of velocity anisotropy is the change in
fundamental frequency for taps placed at a given spacing. A less
obv ious effect is the change in electro—mechanical coupling
coefficient k with direction. The electro-mechanical coupling
coefficient is defined as the ratio of mutual elastic and
dielectric self energies. The coupling coefficient for a given
configuration is relatively difficult to determine directly; but
it may be obtained fro. a knowledge of the change in wave velocity
due to a change in the electric field boundary conditions . A
simplified expression for the coupl ing coefficient:

k2 ~‘ 2 I Av/voo I
Curves of surface wave velocity and An/voo versus plate norma l
direction for Lithiwi Niobate are illustrated in figures 25 and 26.
These curves will give the designer some idea as to the tolerance to
spec ify for the substrate crystal orientation for a given device.

— IL~~
 —
- , - —-- ---——-—. --
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- 21 -
-•—-—--•--
• - —----—. ----
~ ~ —-—-- ,---------—-— -- -~~---- •-,-“,-- ••

AEL-000l-TR
• —---- —..--- ~--- -
~‘11

SURFACE AND SULK WAVE VELOCITIES

I I ; I I
~

3410

OF
~~~TC NORMAL .y( SIISSS)

Fig ure 25. Surface and bulk wave velocities as a function

• o f di r ection of plate normal 1 511

3S.IO” - -

25 .10’

20.10 ’

12.10 ’ -

04.10 ’ -
•

I I I I I I I I I
IS ~4 90 $26 ‘ 62
PLAT E NORMA L DIRICT ION p I deq ..esl

Figure 26. An/voc as a function of direction of

p late norma l 1 501

~~~
TTT
~~. -~~~~~ Z~~- ~ -
•
~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ _ _ _
_ _ _ _ _ _ _ _ _
~ — - - — --
_— - -—- i- - -- .- - •— •-- - — --
-‘
—
.‘

AEL-000 1-TR - 22 -
~ ~ ~~~~

-1.8.2 I)it’t’raction and beam steering

Diffraction occurs in both isotropic and anisotropic materials.
However, diffraction in an anisotropic medium is a function of the
direction of pr opa gation an d losses due to t h i s ca u se m ay b e
conside rably greater or less than in the analogous isotropic case.
Beam steering is confined to anisotropic materials. Both
diffraction and beam steering are illustrated in figure 27.

Wl
li / CIflTNL IN
7 I
I
I)

_i
//

z
/ ~~~ *C( ~~~~ ~
—. —

/
: ~~~~~

~
_ _

- -
- : ~~~~ ~~~-~~~~ *u~~ I
~~~~~~ W

5* 510* 11

Figure 27. Schematic representation of beam steering and

diff raction of a SAW launched on a crystalline
substrate 1 501
In this diagram , a launch and receive transducer are each placed -
-

al on g an ax is which is at an ang le 0 to a reference crystalline axis.

A wave is launched along the 0 axis , however, instead of the power
propagating along this axis , it propagates at an angle of 0 to this

•
-~

axis. This phenomenon is known as beam steering. Practical

devices are designed so that propagation occurs along a pure mode
axis , that is , an axis for which • = 0. Un fortunately, th er e w i l l
always be some error in the alignment of the angle 0 and some beam
steering will result. In order to specify a tolerance on the
alignment of angle 0, the slope of the power flow angle d~ /d0 must
be known . This will give a direct measure of the seriousness of
beam steering. Values of dO/dO are given for most popular
substrates in Table 1. Also given in Table 1 is the parameter
dO/dp . This parameter determines the effect of a misoriontation of
the perpendicular to the crystalline surface in direct analogy with
dO/dO for a misor ientation in the plane of the plate. Note that
only in Lithium Tantalate does this perpendicular misalignment have
any serious effect.
The loss due to beam steering may be readily derived by simple
geometric methods.
Beam steering loss (db) -20 logt o (1- ~ A l dO/dA l a/t) . . (10) ..
where alignment error of crystalline axis (radians)
t = distance between launch and receive transducers
(wave lengths)
£ transducer width (wavelengths)

—-— •
- -- ~~~~~~~~~~~~~~~ -
 - - -~~--—- --

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- 23 - AI3L-0001-TR

Diffraction , illustrated in figure 27 by the changing beam

profile and eventual beam spreading , is also a function of crystal
orientation and must be considered simultaneously with beam steering .
In fact , the designer must trade off the losses caused by each of
these in order to optimise a particular design. A thorough
treatment of diffraction is given by Szabo and Slobodnik 1 sil .

4 -

2 = Dlstsnc b.twONn laUnCh and

~
r.c.lv.r tnanld u c11 (w.v.Mnqth.)
S .
~~
L - Tran.ducs, ap.rtuv.
(w.vSI.ngtPls)

--
.0 .D4 •1 .4 I 4 10

( 44 ,(1.aØIaoJ
TRANSDUCER SEPARATION
~~
Figure 28. Diffraction loss versus scaled transducer
separation 1 511
I
Diffraction loss is illustrated in figure 28 which is a universal
diffraction loss curve for all parabolic materials (materials for
which velocities near the pure mode axes can be approximated by
parabolas). The use of these curves for the known non-parabolic
surface wave substrates leads to only slight errors. The near
field diffracti~n loss never exceeds 1.6db (the near field being
defined as a < L 2/ 11 + a#/aol). The far f ield diffrac tion loss
may be expressed as:
Far Field Diffraction Loss (db) = -10 log1o- X -l.6/x2
-
....(11)
where x = a / L I 1 + a~,ae i
Ag

The above expression is valid for: ~ ~‘I ~ / I 1 +

~~Iao
I
Figure 29 illustrates the trade off obtained by siwiltaneously
plotting beam steering and diffraction losses. Several values of
transducer width and frequency are included . The parameter d /dO
may be obtained from Table 1 and superimposed upon figure 29 to
obtain a comparison of trade-offs for the co on substrates. It
may be seen that by increasing transducer lengths and decreasing
the operating frequency, beam steering may be made negligible for
the lengths of substrate currently available.

1-.i-J
 -
-- -
~~~

AEL-000l-TR - 24 -

I I . I I I u l fi l Il

~~~~~~~~~~~ f l

‘
~:

I
TRA NSDUCER WIDTH (WAVELENGTHS )
I 1111111
10
I I

B(ps)
~~~~1
1 1 1 1 1 1 1

100
I
t..o
I
M14Z

1 1 . 1 1 . 1
1000
I
p
.
DELAYTIME FOR 3dB BEAMSTEERING LOSS (ALIGNMENT ERROR Q•1e)

Fi gure 29. Trade-off between diffraction loss and beam

steering loss 1 501
4.9 Temperature effects
When consideration is bei n g g iv en to the design of a sur f ace wa v e
correlator for long pha se coded sequences , etc. , the temperature
coefficient of surface wave delay can be the limiting parameter in
determining the peak to sidelobe ratio. The temperature coefficient of
delay is a function of both t emperature coefficient of l inear expansion
and t emperature coefficient of sur face wave velocity. Un fortunately,
most common substrates have a positive temperature coefficient of linear
expansion and a negative temperature coefficient of wave velocity,
resulting in a nett positive temperature coefficient of delay. A
notable exception to this is quartz which has a negative temperature
coeff ic ien t of de lay for the Y cut , X propagating substrate and a zero -

f irs t order temperature coeff ic ien t of d elay for the ST cu t, X

•

propagation substrate. See figure 30. Temperature coefficients of

delay for the common surface acoustic wave substrates are given in Table 1.
Investigations are being carr ied out to develop a temperature stable,
high coupling coefficient surface acoustic wave device. A number of
approaches have been used , one involves the depositing of a resistive
ma terial on the substra te , through wh ich a curr en t is passed to keep the
substrate at a constant temperature. Another approach has been to use a
thin layer of one material on a second substrate material, the two
materials having counteracting thermal properties.

-- -.
_ — • --—_ - — - - —
- - -- --•--
~ _- -- —~
 ,- - -
- 25 - AEL .~~~ 1-I ~~

r t r r - - — — -T r- ~1

10 0 - S u ~~~~A o I ’

So ,, 0 Ø .
-

0
60
’
0’ -
-
~~~~~~~~~~~~~~~~~~ ~
0

-Z0 0 20 40 60 60

~~~ C PC.
T(MPERA T

-‘-

Figure 30. Phase delay as a function of temperature for

ST cut quartz 1 481
4.10 Bulk wave generation
In the generation of surface acoustic Rayleigh waves , both shear and
longitudinal strains are set up by the launch transducer . Instead of all
of this strain energy propagating as surface waves , some of it may
propagate away from the excitation region as bulk waves . Direct bulk wave
generation may also occur if an earth plane is located on the opposite side
of the substrate to the launch transducer. The most serious of the bulk
wave responses are the ones occurring within the surface wave passband.
These cause spurious responses which are typically 20db to 50db below the
desired responses and place serious limits on high performance devices.
Bulk wave excitation in anisotropic crystals, like surface wave
excitation, is a function of crystal axis orientation. Fortunately, the
axis of maximum surface wave coupling and the axis of maximum bulk wave
coupling are not necessarily collinear (see figure 25). Selection of the
correct crystalline axis is a major factor in reducing bul’ wave responses
1 471 .
Interdigital transducers may excite both shear and lon g itudina l bulk
waves. At the higher frequencies, the shear bulk waves will be directed
from the top surface of the substrate to the bottom curface where they will
be reflected . Roughening the bottom surface will help to scatter this
wave. A more effective method is to machine the reverse side of the
device so that it forms an angle with the top surface. This causes bulk
waves which are reflected from the bottom surface to be skewed on arrival
at the output transducer 1 54). Another method of reducing bulk wave
spurious responses is by the use of the split electrode transducer
geometry - this prevents reflected energy from being scattered into bulk
waves at the frequencies of interest. —

In the past, bulk wave generation has been a major problem in the
implemen tation of h ighly dispersive filters . La Rose and Vas ile , L 441 ,
have demonstrated a method of bulk wave cancellation using a launch
transducer which is split in two (figure 31); the two halves being phased

..__L~
~~~~~~~~~
- —— - - - _ _-_ - - -a.
-~ .- -
___
__ - - -- --
 1I
~~

A l l -UUUI — FR — —

so that bulk waves generated by each half cancel , whilst the two surface
waves generated by these transducers are phased up by placing a slowing
device in the path of one half of the aperture . A thin metal coat ing is - -

used as the slowing device.

LI
_
I -1
______

_ 1 1
~~
Figure 31. Transducer geometry for bulk wave cancellation 1 441

sulk wave responses may be considerably reduced by placing launch and

receive transducers on adjacent tracks with a multistri p coupler 1 631
coupling between these tracks. The multistri p coupler will cause the
surface wave to change tracks but will leave the bulk wave on its
original track, thus missing the receive transducer . Unfortunately —

multistrip coupler lengths become impracticable on low coupling

coefficient substrates such as quartz.
4.11 Direct transducer to transducer feedthrough
In practical surface acoustic wave devices , insertion losses of 40db
or more may be typical , particularly in the case of phase cod ed transducers
which use ST cut quartz for its zero first order temperature coefficient of
delay. Quartz has a low electro-mechanical coupl ing coefficient , and for
devices which have more than five percent fractional bandwidth must be
damped to realise the desired bandwidth , which implies high insertion loss.
If no precaution s are taken in these high insertion loss devices, then
-
-
direct feedthrough between the input and output transducers may produce a
lar ger signal than the desired acoustically coupled signal. Direct -:
feed throug h may be kept to a level of about -70db with careful layout and
shielding, or with a great deal of attention to earthing and special
shielding 100db of isolation may be achieved . Isolation has been found to
• improve as transducer separation is increased as would be expected from
normal capacitive coupling theory. By placing an earthed cap over a
dev ice, isolation is found to improve with closeness of the cap to the
dev ice , with good isolation (60 - 70db at 150 ~4Jz) occurring with a
spacing of about 0.25 to 0.5 imu.

_ _ _ _ _
----
_
--
 -— — —--— - - -- - — ~~~~~~
~~~~~~~~~ ~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~

-
- 27 - AL~L-000l -TR

5. TRANSDUCER ANALYSIS

The electric field at any point in an interdigital transducer is the sum of

the field due to the exciting voltage applied to that transducer and the field
associated with the wave travelling under that transducer. The field
associated with the wave travelling under the transducer is complex being
modified by reflections from each finger pair it passes under.
Smith et. al. 1 681 , have carried out a simplified analysis of interdigital
transducers which assumes the surface wave substrate is non-piezo-electric in
that there are no reflections between finger pairs and the electric forcing
field is not modified by the field of the existing sur face waves. The
distortion in transducer response caused by neglecting these ‘piezo-electric
feedback’ effects is shown in figure 32. - -
-

•
..
-
• -

to — ——
_

~~~

0.N 0.~~ 1.0 1.02 1.04

Frs~ jsncy.

. Figure 32. Distortion of ideal sin y/y response caused

by reflection at finger edges 1 56)
Ingebrigtsen, 1621 and Emtage , 1 561 , have both carried out analyses of
interdigital transducers which include the ‘piezo-electric feedback’ effects.
Both analyses agree well with experimental results. (Note that this
distortion does not occur with split fingers as the reflection from finger
edges tend to cancel.) Although it is desirable to be able to accurately
• predict the effects of ‘piezo-electric feedback’ or for that matter any second
order effect, it is even more de sirabl e to elim ina te these second order effects
— (by the methods discussed in the previous sections) . If these second order
effects can be reduced to a level where they do not have any appreciable effect
on the tran sducer response , then the simplified analysis of Smith et. al. may be
used with good results.

U
A EL-0001-TR - 28 -

5.1 Equivalent circuit models

lR 0 tan -
—
~

:1
PORT I
(ACOUSTIC )
R0
•

L
—
~~~~~~~~~~~

PORT 2
( ACOUSTIC )

1
3n 3n
_ (ELEC TRIC)
8 - 2ir w/w 0 ~ PERIOD SECTION TRANSIT
ANGIE
R 0 ELECT R ICAL EQUIVALENT OF Z 0
Cs .EIECTRODE CAPAC I TANCE PER SECTION

Figure 33. Mason equivalent circuit for one

periodic section ( 671

Surface acoustic wave transducer network models are based on the Mason
three port bulk wave circuit model shown in figure 33. Although this
circuit model was originally derived for bulk wave transducers having one
electrical and two acoustic ports, a cascade of Mason equivalent circuits
may be used to represent a surface wave transducer which has one
electrical and many acoustic ports. Smith et. al. 1 681 , have proposed
two diff eren t t ran sducer circuit models based on two different electric
field approximations; the ‘crossed field ’ model which assumes the
dominant electric field is norma l to the surface of the substrate and the
‘i n l ine field’ model which assumes the dominant electric field is along
-
-

the sur face of the substrate - see figure 34.

 - 29 - ~~L 0OO1-~~~~~ ~~~~~~~~~~

L- L—

t4
~
-1-l
~
-
~ ~~
+

(a)

CROSSLO ~*LO
(b)

(c) _ _ _ _ _ _ _ _ _ _ _ _ _ _

approximation s
Figure 34. ‘C rossed field ’ and ‘in line field ’
to actua l transducer electric field
researchers as to which is the
There is some disagreement amongst various Holland and
circumstances.
correct circuit model to use under which model should be used for
Claiborne 1 21, conclude that the ‘crossed field ’
coupling coefficient) with a large
Lithium Niobate (hig h electro-ulechaflical
should be used for Quartz (low
number of f inger s and the ‘in line ’ model small number of fingers.
electro-mechaflical coupling coefficient) with a , shown in figure 35
irical results
Smith et. al. (681 , have published emp model for Lithium Niobate.
which support the use of the ‘crossed field’

(b)

OV PITt NOISt *L P~~ X5

h theoretical
Figure 35. A comparison of emp ir ical resul ts w it (671
model
values from the ‘crossed field’

~~~~~~~~~~~~~~~~~~ ~
 -- _ _

AEL-000I-TR - 30 -

Milsom and Redwood (651 , have proposed a model intermediate between the
‘crossed field’ and the ‘in line field’ models and claim that this model
best suits the particular polarised ceramic PZT4. Bristol 1 551 , is quite
adamant that the ‘crossed field’ model is the only model to be used and
claims tha t the use of any other model will result in significant er ors .
In support o f h i s cl a im , Bristol gives results of a comparison of the
‘in l ine field ’ model , the ‘crossed field ’ model and a model consisting of
50% of each of these models , with experimentally determined results , the
scattering parameter P11 being the critical parameter see figure 36. -

Further support for the ‘crossed field ’ model is given by Hartmann et. al .

(ST .X)QUAR TZ
~~~ 14 ,4

30

20 0
10.51’
-

~~
=

10 - a-O— CRO5S ED FIELD MODEL

a —I = IN LINE FIELD MODEL -
-
e 1
I
10 100
0.01 0.1 —

Fi gure 36. Exp erimentally determined scattering parameters compared

with results from ‘in line ’ and ‘ crossed field’ ci rcuit
models 1 55)

12 01 who derives substantially the same results as the ‘crossed field’

model by an entirely different approach using conservation of energy
principles (Parsavall’ s Theorem) . Emtage [ 561 , has derived results
which are consistent with the ‘crossed field ’ model , providing that ‘piezo-
electric feedback ’ effects are neglected . From the above, it appears that
the ‘crossed field’ model has the strongest case, both empirically and
theoretically, and will be the one used in this paper. Equivalent
circuits for both the ‘crossed field ’ model and the ‘in line field’ model
are given in figure 37.

-
••
~~~~~~~~
_
—
~~~~~~~~~~
_
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - --
 - 31 - AEL-000 1 -TR

1F- Ra (
a.’) Ra (
w o) ( sin x ) 2

rh
~
=

Ra ( i 2 ) l I
(sin 2x 2x)

Ixa (‘)T
(cJ) = Ra (w o)
;

•
I -

‘In Line Field’ Model R a (‘-~‘o) = (~~_ ) k

2
w oCs) ’
(

)) (sin x )2
Ga (W~ = Ga
~
C

TT
I~”~1 ’
Ga (W~1 1
Ba (4 = Ga (c o) (sin 2 x — 2x)
~

‘Crossed Field’ r k ” l Ga (w o) = ( -) k2 (JoCs)N 2

~
where C = NCs
T
x = _ _ _ _

Figure 37. Equivalent circuits for the ‘i n line field ’ and

‘crossed field’ models

Their im iiittances may be expressed as:

For the ‘in line field’ series equivalent circuit :

Ra ( 3 ) = w o) (sin x/x)2
Ra ( .. ..(12)
Xa (‘~3) = Ra (~3 o) (sin 2x - 2x)/2x2 ....(13)
Ra ((
J o)= (4/w).k2ft.~ oCs ....(14)

For the ‘crossed field’ shunt equivalent circuit:

Ga (4 = ‘‘o) (sin x/x)2

Ga ( ....(15)
~ ~
Ba (‘4 = Ga ( 2
~~o) (sin 2x - 2x)/2x.
• 4/i).k~ ~‘oC sN2
Ga (4 ~o)= ( ....(17)

where x = N~ (‘ - w o)/c~ o
~
• and Cs = capacitance per periodic sec~ ion
C1. = N.Cs total transducer capacitance

N = number of periodic sections

k = electro-aechanical coupling coefficient

- •— . .
- -
•
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~ = ~~~~~~~
-

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~~~~~~ - £.J ~~~

 AI L-0001-TR - 32 -

EXPERIMENT
10
THEORY
- B~ (w)+wC y

95 100 lOS 110 ItS

FREQUENC Y IN MH z

Figure 38. Radiation admittance as a function of frequency

(crossed field theory) 1 671

Figure 38 illustrates the measured radiation admittance and susceptance 1

~~

as a function of frequency compared with that calculated from the

‘crossed field’ model (from [68)).

Ii ‘0 2 N—i ‘N 12

(AC~~~~~~ ) U~ T I C)

Figure 39. Transducer composed of N periodic sections,

acoustically in cascade and electrically in
parallel

A transducer may be treated as a network of Mason equivalent circuits,

• the most simple of which is the uniform unapodized transducer which is
depicted in figure 39 as a cascade of Mason circuits. The analysis of
apodized transducers has been treated b y Tancrell and Holland 1 251 , in V
this case the transducer comb is cut into lateral imaginary strips as in
fi gure 40 , these strips are used to determine which of the electrodes - -

interact acoustically and the equivalent circuit of figure 41 is then

der ived .
_ _ - - — -
• ~~~~ — ~~~
-

- 33 - AJ
~L-0 0I-Ii~

‘5 II0fflhl Illll ._ IIIHIIOh

11011111 ~~~
___________
- -
‘.4 11111111
0110 oil
1—2 III” Ill
+ ZI I-I

$
I
—
lIii

___________________________ — Hil
111111
_______________________________

I I hl hi f TT ~~

_________

I.

Figure 40. Apodized transducer cut into latera l imaginary

strips to aid analysis 1 121

z f l h i l i t I l I I l D z
~r~~~~~~~~~

-
•
~~~

zt L { J Ef E~~ I I i r ri 1 113iz
1r1
•
••
~~~~~~~~~~~~~~~~ -~~~~ •~~~~~~~ • ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

EL iI I I. I 1 1 1 1 1 I I I J11J z
~ ~
.
~~~~~ ~~~
-

~~~~~~~~~~~~~~~~~ ~~~~
-~~
•~~~~~ .
~~~~

Z11 r
~~~~ .:—w~-i
L~~~i~~ &- 1 1 1 I
~ ~
1
1 1

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

z 4 ~~ -t L~~ hJi- ~~~~~~~~~~~~~~~~~~~~~~~

I l l I l l I l l I l l I l l
N(T~~ NII n.I 2 3 4 5
~ 7 8 9
~ II 2 II
~ Is

Figure 41. Equivalent circuit for an apodized transducer 0 121

-
~~
j
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- ~~~
— --- —
‘—•- 4
~~~~~~~~~~~

AEL-000l-TR - 34 -

The analysis of dispersive surface wave devices , such as phase cod ed

correlators and linear FM pulse compression filters , has been carried out
by Hartmarnt et. al . (201 . It has been shown that the input admittance
of a dispersive transducer is equa l to the t ime-bandwidt h product of the
dispersive transducer multipl ied by the input admittance of a non
dispersive transducer having the same bandwidth , centre frequency and
beainwidth as the dispersive transducer.
Thus for a dispersive transducer:
C (~J o) — (4/R)k2 (“ o C (fo/~ f) 2 ? i~ f
s
. ..- (18)
where 1 duration of impulse response
s~ f Bandw id th
For the case of phase coded transducers which have widely spaced taps
consisting of a few fingers each , the impulse response will be a train of
RF bursts and the bandw idth will be the bandwidth of a single tap. The
fractional bandwidth ( f/fo)of a tap consisting of N finger pairs
should be taken as I/N. ~ If each tap in the transducer is a single finger
pair then it w i l l have a time-bandwidth product of uni ty and the time-
bandwidth product of the entire transducer will be eqi~al to the number of
taps. Hartmann has also derived a set of universal admittance curves for
tran sducers on both Quartz and Lithium Niobate. These curves shown in
figure 42 are for non-dispersive transducers with an acoustic beamw idth of
100 wavelengths and should be scaled to actual device bandwidth.
Dispersive transducers may be treated by multiplying admittance values by
the t ime-bandwidth product . -:

¶
1
\
~~~~ -
‘
~ ~ ~~~~~~~~~~~

4-.
.
44 ~~~ ~~~~~~~
4
4

-4 4
4-.

_
z— 1 0
________
YZ U Nb O 3 ‘-
———ST QUARTZ
-6
10

\
u1
~0’l 1 10 100
FRACT10NAL BANDW IDTH (“Se)

Figure 42. Input admittance for approximately rectanguar

band-pass characteristics for an assumed beam
width of 100 X 1 201

-
 -~~~~ --
_
~~~~~~
‘
~~~~~~
uITU
~~~~~~
- 35 - M~L-000l-TR

5.2 Electrical matching network

For minimum dev ice insert ion loss , the source immittance should be the
— complex conjugate of the inpu t transducer imm ittance and the load
immittance should be the complex conjugate of the output transducer
immittance . To achieve max imum bandwidth , for a fixed number of fingers
N the Q of th e r adiati on imaittance should be minimised . The radiation
Q of the ‘crossed field’ model and the ‘in l ine field ’ model may be
expressed as:
l/v.~lo C Ra (co o) ‘in l ine field’ model
1
=

= c~ o C /Ga (‘c o) ‘crossed f i e l d ’ model

T
By substitution , both of the above expressions simplify to:
= R/ (4Nk 2 )

This indicates that the material with the highest coupling coefficien t
w i l l have the maximum fractional bandwidth under minimum insertion loss
conditions.
To a first order approximation , the overall bandshape of a surface
wave device may be considered as the product of the electrical bandshape
and the transducer acoustic bandshape, the transducer acoustic bandshape
being determined by taking the inverse Fourier transform of the electrode
pattern. This implies that the overall bandshape is determined by the
smaller bandshape of the two and also that , for maximum bandwidth under
• minimum insertion loss conditions, the transducer electrical Q should
equal the transducer acoustic radiation Q.
Consider the case of an unapodized transducer consisting of N periodic
• sectio n s , of X = V/f,, each, as in figure 43.

Transducer N finger pairs

k NX — .4

1\J\
X=V ff,,

p__ Impulse Response

N periods of f,, = N x t -

_
Inverse Fourier transform
of impulse response —
frequency response of
• transducer
~0
f,,-f0 f0+f0
N
Note: fractional bandwidth between units = 2/N
Q~~ N

Figure 43. Transducer geometry , impulse response and

frequency respons e

_______________________________________________________________________________________________
_ I
~1
~~ ~1 ~~~~~~~~~~~
- --- ~~~~
~~~~~~~~~~~~~ —
 _ _
-— - - -- - -~~~ - T- - - - -

M~L-0O0I-TR - 36 - ~~~

As there is a one to one correspondence between the transducer finger

locat ion and its impulse response , th e l at t er w i l l be a pulse train
consisting of N cycles of frequency f,,. The inverse Fourier -transform of
this impulse response will be the transducer frequency response which , for
this case, is the familiar sin x/x response centred about f,,. The zero
crossings of the sin x/x pattern occur at multiples of l/T where I is the
duration of the pu lse train .

No w T =
N.t = N/f,, = N X/V
where t is the period corresponding to a frequency fo
V is surface wave velocity

Thus the first nulls in the transducer frequency response occur at

fo ± fo /N or the fract ional bandwidth between the n u l l s
= 2/N. Note also
that the 3db response points occur at approximately f,, f,,
/2N or the ‘Q’ ±

of the transducer acoustic response is approximately equal to N.

Substituting N for in expression (19), thus equating the radiation
Q with the transducer acoustic Q (the condition for maximum bandwidth
under minimum insert ion loss conditions) , gives:

N2 = ir/4k2 .. ..(20)
Smith ( 691 has shown this condition also to be the condition for minimum
phase dispersion and has determined the optimum transducer design , based
on (20), for various substrates. These appear in Table 2.

TABLE 2. OPT DIJM TRANSDUCER DESIGN FOR

VARIOUS SUBSTRATES (69 1 _ _ _ _ _ _ _

Piezo- Cut and V ~~ N -~ ‘---~~ A

I

(
electric Voo T.~ --
Aperture ~ ‘° ~

L.iNbO3 YZ 2.46 4 108 0.24 2.88 0.69

Bi13 GeO2,, ( i i oj j i i oj 1.15 6 183 0.17 633 1.08
ZnO XZ 0.56 8 99 0 .12 3.74 0 .45
Quartz YX 0.11 19 53 0.053 3.06 0.16
PZT t 2.15 4 - 0.23 4.55 1.04
CdS XY 0.31 12 54 0.09 5.82 0.52
LiTaO3 ZY 0.82 7 31 0.14 2.86 0.40

Note that transducers with wider bandwidths than those given in Table 2
may be readil y obtained , although the electrical circuit Q must be damped ,
sometimes severel y, to obtain the desired bandwidth. Output tran sducers
may be loaded to reduce electrode reflections and also to reduce energy
coupl ing and hence signal droop in long transducers. Figure 44 illustrates
the minimum achievable insertion loss versus fractional bandwidth for
various substrates.

L -
~~~~~~~~~~~~~~ — — ---- -
-
—.- ---- - --
- - -
~~~
- -— - - ______

- ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~

• ~~~~~~~~~~~~ -- .~~ ~~~~~~~~~~~~~~~
~

- 37 -
AEL-000l-TR

1 1 T lT 1 r i
~ ~~~~

: SIQ ~~dl

• 1:: 1.- -- 2
~
• - -

,
/ --

~:
i i ititul 1 1 tu tu
F,actvjna
~
kndra h ‘P rcenft

Figure 44. Minimum achievable SAW delay line insertion loss

with bidirectional transducer (20)

• An alternative to the matching network described above is the

transmission line approach as described by Ueighway et. al. (61). In
this case , the transducer is considered as a distributed network , its
characteristic impedance is determined and impedance matching transformers
are used to match the load and source respectively, the other end of the
transducer is terminated in its characteristic impedance. The transmission
line approach results in lower insertion loss than the lossless matching
network for dispersive structures with high compression ratios. It also
considerably simplifies the matching problem .
5.3 Transducer scattering parameters

R0 I 2

‘ ‘‘
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~
2E j nc E G
~~~~~
TRANSDUCER ~~ R 0

~~~~~ j 2

Z~
_ F—
~~
Figure 45. Scattering parameter model (671

In order to determine the fraction of power absorbed , reflected and

transmitted by a transducer from an in ciden t surface w ave , the transducer
scattering parameters must be known. Using the model of figure 45, a set
of power scattering coefficients will be defined for a simple unapodized
transducer:
P11 = fraction of incident power reflected at port 1
P21 fraction of incident power transmitted to acoustic port 2
P31 - fraction of incident power coupled to electrical port 3

-- —- - —- - - — ~~~~ - - - --

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -.-
—— -.
— ~~~~~ —
--•--- .~~‘-—.—-- .-—----- ——- - - - -.-- - --- - - . ---
-~~~~~~~~~~

—- --
4
-
 Pr.— —-
~
.- —, ._-—- —-— ———-----— —,--- ••—• •——————-•-- -~~~~~
-- —

M1.-000 I-T R - 3$ -

From t ransducer symmetry and reciporcity I’ll = I~22 , I~l3 I’23, I’ij = Pij. =

Scattering loss Lij = -10 log,,, (Pij) dB.

Smith et. al. (681 presents simple expression s for the above scattering
electrical coefficients at synchronism as a function of load on the
electrical port. These, however , lead to errors , particularly in Lii
for frequencies near but slightly away from resonance . Gerard has
extended the analysis of Smith et. al. to include frequencies close to the
resonant frequency and are given as:

1 + (N5/2) 2 ,2 N6 Yi ....(2l)
P11 -
9 —
~~

D (N& , ? ) -:
—

121 2
P21
N& , ‘ )
—
D ( ....(22)
~
A
2Y
P31 =
D (N& , V) .. ..(23)
f-fo
• where 6 ~~2ir
~~~fo ~
= real part of
A
Yi - imaginary part of Y
2 normalised electric port load

= + J ‘ CT
~ ....(24)
Ga (4/ir).k2 ~
- ‘o Cs N
2

D (N6 , 2) = 1 + 2 c?r - 2~ N& /2) + ~~ I~ (1 (N& /2)2 )

For the case, f = fo (5 = 0), the above simplifies to the expressions

given by Smith et. a l . , i.e.
For a purely react ive load at acoustic synchronism :

P1 1 = - •...(25)
1~~ ~z
P21 — ~~~ 3 ....(26)
1 -
-

P31=0 ....(27)

where a — (X
L
- ) /Ra ‘in line field model’
~~~~ T
a - oC ) /Ga ‘crossed field model’
+
~~ t
X load reactance
L
—

- load suspec tance

For a load tuned to resonate the transducer capacitance at acoustic

synchronism :

1 P11
(1 ~ b) ’ ....(28)
-

....(29)

~~~~~~~~~~~~~~~~~~~~~~~~~ _ _ _ _ _ _ _ _ _ _ _ _ _ _
 -

- 39 - AE L - 000I - TR

P31 =
2b
.- ~~~~~ (30
where b = R / a ‘in l ine field model’ H
L~
b = G / a ‘crossed field model’
L~
= load resistance
= load conductance
Note that the simplified expressions of Smith ét. al. are in good agreement
with the more complex expressions of Gerard except for Lii and even this is
in good agreement over the range b = 0.5 to 2.0 and a = -2 to +2.
Figure 46 illustrates reflection loss and transmission loss versus
normalised electrical susceptance compared with Gerard ’s model and the
model of Smith et. al.
It -r
~~. ~

,-‘
.._. ...4 ~
-‘\ . ‘j i £XPtRiMENT
l.~~~~~
\
i,
,
~~~ / ~~~~~~~

j — THEORY r.
~~~
4. ~
S

~~~ d\.
2
-.
L~.
~-L 25 ~ /~~(~~
~~~
I ~~~~~~~~~~~~~~~~~~ -
~~~~
~~~~~

A
-10 -I -6 —. -2 0 2 4 6 S 10
NORMALIZED SUSCEPTANCE . (B , ’

(
a)

55 68

•
-.

4
~~~~~
~~ — THEORY
•,* EXPERIMENT

2-
I I I I

-10 -e —a -4 0 -2
2 4 6 S ~~
NORMALIZED SUSCEPTANCE , 16L. C-r)I’4.
~~~~ %
(b)

Figure 46. Reflection loss Lil and transmission loss L2l

versus normalized electrical susceptance
(a) Smith et. al .168) (b) Gerard 1 59)

I--
 - --- —
-- -

AEL-0001-TR - 40 -

Figure 47 illustrates reflection loss, transmission loss and coupling

loss versus norinalised conductance for both Gerard’s model and the model -
-

of Smith et. al.

Note also that the above expressions do not include reflection s due to
the short circuiting of the electric field by the metal electrodes.
These are given in paragraph 4.6.1.

IS -
:
:
:: EXPER IMENT
IS - — THEORY
14
12- H

z •

4 •
•
S

~~~

S 3~
I £ I I I _L I_ I I __i
~
t . I -

Di LO 10
a
NORMALIZED CONDUCTANCE,GLi’G.

(a)

20
-
\ — THEORY
.,& ,e EXPER IMENT
6 -
N8’0.9
14 -
12 -

• 10- •
z
1,
. 6~~ a ~
U, U
9 6-
•
a- £

1 iiitnl I I
0.1 02 0.3 1.0 2 3 10
NØRMAUZED CONDUCTANCE , G.j G

(b)

Figure 47. Reflection loss Lii , transmission loss L21 , and

coupling loss L3l to electrical port versus
normalized load conductance (a) Smith et. al.
1 68) (b) Gerard IS91
 --
=- --———— ~~~~ - --— ~~— -
——- -~~~~~~ --

- 41 - AEL-0001-TR

5.4 Transducer electrode capacitance

Farnell et. al. ( 581 , have derived expressions for the capacitance of
interdigital comb structures for a variety of configuration s including
layered structures, single interface structures and plated structures. I -

Only the single interface structure , the most common struc ture, will be
considered here. This is defined as a ‘transducer deposited on the free
surface of an infinitely deep substrate with alternate fingers driven in
• opposite phase’.
If it can be assumed that the transducer finger structure extends
i n f i n i t e l y in both direction s along the substrate surface and that the
finger width is much greater than its thickness , then the capacitance of
a single electrode pair may be expressed as:

C = (6.5 R2 + l.08R + 2.37) (Er + 1) pF/n • . . .(31)

where R is the ratio of electrode width to distance between electrode
centres.
er is the relative dielectric constant.
For the common case where finger and gap width are equal , i.e. R 0.5,
the above expression simplifies to:

C = 4.53 (Er + 1) pF/m .. ..(32)

For t he case of phase coded sequence generators or correlators which have
large gaps in between sets of taps, the capacitance may be taken as
equivalent to an array with the sane number of fingers as if the large
gaps were not there (2 1 .
I

6. TRANSDUCER DESIGN

As surface acoustic wave devices are essentially tapped delay lines, all
device specifications must ultimately be transformed into the time domain. - -

Simple delay lines and phase coded correlators are already time domain problems
and are readily handled in the time domain. Filters having a certain desired
band shape in the frequency domain , however, must have their response
transformed into the time domain. - -

The impulse response of a device is the convolution of its two transducers

impulse responses; or alternatively the frequency response of a device is the
product of the frequency responses of its two transducers. Now as the impulse
respon se h(t) of a dev ice and its frequency response H(~~ are a Four ier
transform pair, then its frequency response may be determined from a knowledge
of its transducers electrode patterns and hence impulse responses.

= h(t) exp (-jw t) dt

To obtain a desired frequency response, the inverse Fourier transform of the

response is taken , giving the impulse response which may then be used to
determine electrod e position and overlap.

h (t) =
~~ 1:H(9 ju’t) d*~d
exp (
AEL-0001-TR - 42 -

A l i m i t a t ion in practice is that the impulse response must be of finite

duration due to restriction on available lengths of substrate material . A
limit on the length of impulse response will cause a corresponding l imit on
the filter skirt response. In cases where a truncation of a given impulse
response is necessary, a weighting function is usually used to ensure a
gradual reduction of the response rather than an abrupt truncation which may
cause serious ripple in the filter passband and high sidelobes in the stopband .
The design of surface wave filters is discussed to some length by Hartmann et.
al. 1 201 and Tancrell and Holland (251 .
6.1 The unidirectional transducer
The common biphase interdigital transducer , having alternate fingers
driven by electrical signals 180° out of phase, has the disadvantage of
being bidirectional . Signals applied to one of these transducers will be
launched equally in both the forward and reverse directions this will -

result in a 3db transmission loss. By reciprocity there will also be

3db loss in receiving these signals. The resulting reflected signal - -

gives rise to the triple transit response. Several unidirectional

interdigital transducers have been developed. However , most have some
undesirable feature, such as bandwidth limitation or fabrication
complexity. The unidirectional transducer developed by Hartmann et. al.
(601 , appears to have none of the drawbacks of other unidirectional
transducers, except for fabrication complexity. It is a three phase
transducer having three electrode groups , each being driven by a signal
which is 120° out of phase with the two other electrode signals (see
figure 15). This transducer can provide a 20db or better front to back
ratio over a 20% bandwidth and has been used in appl ication s for low
inser t ion loss delay l ines provid i ng a loss of t he order of 1 2db . -

6.2 The multistrip coupler

0
IIS~~~~
.dvcs,

M.,U •tflpe

TricS A

—
—
Su~s~rs~s

Figure 48. A multistrip coupler used as a track changer

The multistrip coupler shown in figure 48 is a broadband directional
coupler which may be used to transfer surface acoustic wave energy from
one acoustic path to another acoustic path with low loss. Multistrip
couplers may be used for ‘magic Is ’ , unidirectional transducers , surface
wave mirrors with very low reflection loss , reduction of bulk wave
responses and many other applications. One appl ication of great
significance is in the field of surface wave filters. In general , if
one tr ansducer , either the launch or receiving transducer , is apod i ze d ,

_ _ _ _ _ _ —~~~ - - ~~~~~~ - — —— - --- — - - - - — -— -

~~~~
~ -— - — -.———-- --—-—-— - ----—------,-— , -—- -
—- - - - - ------- -- ----J

I
— ----.----‘
-— - ~ -— ~ ~ —- -

- 43 AI~L-0O0i-TR

then the other transducer must be unapodized . This is because the wave
front of an apodized transducer is not uniform in amplitude. A
multistrip coupler placed between the launch and receive transducer, now
on different tracks, will average the wave front from the launch
transducer causing a uniform wave front on the second transducer and allow —

both input and output transducers to be apodized. This means that for the
case where the input and output transducers are identical , the frequency
response will be the square of the Fourier transform of the impulse
response of a single transducer. This will have the effect of doubling
(in db) the stopband rejection and doubling the skirt steepness but not
significantly changing the insertion loss. Also as the multistrip -

• coupler causes the surface wave to change tracks but not so the bulk wave,
-

then the bulk wave from the launch transducer will miss the receive
transducer and thereby eliminate its purious response. See Tancrell,
( 24] , and Deacon et. al. 1 191 . The optimum length of a multistrip
coupler is proportional to the inverse of the electro-mechanical
coupling coefficient . Multistrip couplers on Lithium Niobate are quite
small and efficient due to that material ’s high electro-mechanical
coupling coefficient . Unfortunately, quartz has such a low electro-
mechanical coupling coefficient that multistrip coupler lengths are
impractical. The design and application s of multistrip couplers are
discussed to some length by Marshall et. al. (631 and (641 .
6.3 Transducer stripe to gap ratio
Transducer stripe to gap ratio determines the electric field
distribution in the surface wave substrate and hence the degree of
excitation of the fundamental and harmonic surface waves. It also
determines the degree of reflection caused by the metal electrodes short
• cirtuiting the electric field , see figure 17. An analysis of the
excitation of surface waves at harmonics of the transducer fundamental
frequency for the conventional and split finger transducer configurations
has been carried out by Smith (661 . Results for the conventional single
finger geometry are illustrated in figure 49 where surface wave excitation
as a function of metallization ratio is shown for harmonics up to the I -

eleventh. Similar results are given for the split finger geometry in
figure 50.

1.20
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ ,‘
~, M= 1 (FUNDAMENTAL )
g~~~~O.8O /
M
o .4o< )
•
~~~ ~ ~~:
\;
1 i\ . ..
i
c’
~~~~~

S( $
1/ ’

0.00
0
i
~ 0.20
~~ 0.40
~~ 0.60 0.80
~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ’

1.0
STR I PE TO SPACING RATIO (fl)

Figure 49. Harmonic excitation for conventional X/4 transducer

electrodes as a function of inetallization ratio (66)

____________________________________________ — — -— - — - — —
— — -~~~~ — -~~ — --~~— ~ ~~~~~~~ -
I ~I~ I.~~~ —~~ --- -- --- - . — - — _ _ _ _ Ig•—----_-_- - - ~~~~~~~~~~~~ .
~~~ IL
I ~~ -
~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~ - ~~~~~~~
 AEL-000l-TR - 44

1 .0
Z irQ 5 ELECTRODES
~
—
2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~

~ \ M 5 AND 7 ‘-, - AND 11

~
\ ~~ ~,-~~
/

0. 4 \
’
/

,/
f ~~
a / /
1...
0.2
/
\ _ ~~

\ (.

I ~~~~~~ _. £ • £
0 ~~~~ ~~~ ~~~~~

0 0 . 2 0 . 4 0 . 6 0 .8 1 .0
STRIPE TO SPAC ING RATIO (fl)

Figure 50. Effective coupling coefficient (k2 )for harmonic

excitation of split finger electrodes as a
function of metallisation ratio (661
1-lickernell et. al. (40) have empirically determined the electrical
discharge breakdown voltage between adjacent interdigital transducer •
electrodes as a function of electrode gap. These results shown in
figure 51 indicate that for the majority of appplications electrical
breakdown will not be a problem.

2 00C

I~J
0.
-

-—
i— 1 000 _ _ _ _ _ _ _ _ _ _ _ _
— _ _ _ _ _ _ _
—
~~~~~~~-
.
-I—
~~~~~~~~
- —

800 — — — ~~~~~~~~~~~~~~~~~~~
— —
—
LI TH IUM __

A
_ _ _ _ _ _ _

NIOBATE

400 ---- ~~~~~~~~

_- - —
~~~~~~~~~~
i2- :-

— QUARTZ ~~~~~~~~~~~ — —

20C
1 2 4 6 8 10 20 40 60 100
ElECTRODE SEPARATION (MICRONS)

Figure 51. Pulsed d.c. voltage breakdown in air between

interdigital electrodes (atmospheric pressure) ( 401

L ~~~~~~~~~~~~~~~~
_
~~~~~~~~~~~~~~~~~~~~~~~~~~~
— - — — _ _
 - 45 - AEL-000l-TR

6.4 Transducer beamwidth

The main factors determining the required transducer beamwidth are the
loss due to beam steering and diffraction and the desired transducer
radiation immittance. If transducers are made too wide , then each
finger w ill have a significant resistive loss. Typical transducer
widt h s are o f th e or d er o f 50 to a few 100 wavelengths . Note that for a
given substrate material, the radiation admittance is proportional to the
transducer width and the number of periodic section s squared .
Larkin 1431 , has derived expressions for transducer efficiency as a
function of transducer aperture and sheet resistivity for both ST Quartz
and Lithium Niobate. These results are shown plotted in figure 52.
Maerfeld et. al. 1 45) , have carried out a similar analysis of the
resistive losses in multistrip couplers.

-2O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

•
10 100 1000
(a)
-

—2S I
\ I I I ~~~
I • I I I ~
10 100 w 1000
(b)
Figure 52. Electrode efficiency in decibels as a function
of aperture [ 431

6.5 Transducer electrode metallisation

To determine the thickness of transducer metal electrodes, the
designer must trade off electrode resistance with the mass loading of the
electrode on the substrate surface. Figure 53 illustrates the effects of
metal film thickness on surface wave velocity and attenuation for both
Gold and Aluminium on a Quartz substrate. This figure combines the
results of Pouliquen and Vaesken (461 , and Coabon an d Rouzeyre [ 391 .

‘ - - rn r r. Ir H I ft T 1 ~~~~~ f i j [ f l i - -
. -

--
-

-
-

-:
.
.
 -

Al~L-uoo1-rR

-
- 4 , -
~

-
Values of attenuation (~lb/cm) may be linearly interpolated for
frequencies other than those shown .

F
- -

- VOQUANt1 1
- -- ~~~~~~~~~~~~~
- ~
-

- 1
~~
r~~~~~

~~~~~~~
oot u
I

I
-

F / ALUMI ~dU0
I
r—-—PN0,auAtmw
I 104 M•4.I
—4— ------ —- - — - - I

I /-k !
004

v~~~~ o~~
_ _ _ _

SSI 001 001 004 000 00 0

,aa *Uc*M*MAVS,tMtt,. ,
~ ~

I:i gurc 53. Phase velocity and propagation loss for alum inium
and gold films on YX quartz (391 (4~ 1
Pouliquen and Vaesken also give results of metal fil m resistance as a
function of film thickness. They conclude that metal films below about
500 to 600 are not mechan~ ca1ly continuous , consisting of small islands
of atoms. Below about SO A , these islands do not touch and the
electrical resistance of the ~ ilm is infinite. ~\s the metal film
thickness increases about 50 A , the small islands of metal atoms begin to
contact one another and the resistance falls rapidly until at about 500
to 600 the film becomes mechanically continuous . Aluminium has been
R

used extensively as the transducer electrode metal because of its low mass
density and high conductivity. Unfortunately, Aluminium has a great
chemical affinity with ox y g en and cannot be deposited I n thicknesses of
less than a few thousand Angstrom units X ithout quickly oxidis ing . If
ve ry t h i n metal electrodes (500 to 1000 A) are d e s i r e d , then (~old may be
used as t~o electrode materi al. c,old usually requires a keying layer of
about 50 A of Chromium to be deposited on the substrate to aid bonding .

7. PRACTICAL RESULTS

Several SAW devices have been des igned and fabrtcated at the Defence
Research Centre Salisbury, using the princ i ples discussed in this paper. At
this stage there is Insu fficient time to inc lude a complete analysis of any
part icular device ; however brief results from two of the devices will ho
given here.
7.1 Phase coded sequence generator/corrolator
Th,0 first devices fabricated wore a pair of phase coded sequence
generator/correlators similar to the one discussed In paragraph 3.2.
They have a centre frequency of 70 Phi and a chip or phase reversal
rate of 10 1h z with 127 phase coded taps . The dovicos w r e fabricated

-- .- --
_
 - 47 - AEL-000l-TR

on ST quartz because of its stable temperature coefficient. X/4 fingers

were used and the f i n g e r and gap width were both 11. 28 x lO~ ~~, with
f i n g e r overlap being about 150 X . (ol ~I me t alli zation of 750 A thickness
was used , however , from experience gained sin c e t h at t ime better r e s u l t s
~
would have been achieved by using 1000 to 3000 A of A l u m i n i u m .
F i g u r e s 54 and 55 i l l u s t r a t e the r e s u l t o b t a i n e d when using the devi ce
as a code generator. F i g u r e 54 shows the output of the phase coded a r r a y
when an electrical impulse is a p p l i e d to one of the end t ransducers , and
figure 55 is an expanded version of figure 54 showing the details of the
phase reversals.

A.1 s
~L
L1& - A
~~~~ ~~~~

~~ _~~~~ II ~~~~~~~~~~~~~~~~~

2 u s/div

Figure 54. P h a s e coded a r r ay response to an e l e c t r i c a l impulse

50 n s/ d i v —

Figure 55. Fxpande d impulse response of phase coded array

k . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-
--
- .-~
_
_ _____
___
- _ _ _ _
 - -

AEL-000l-TR - 48 -

Figures 56 and 57 show the results when the second device is used to
correlate the waveform of the first device. Figure 56 shows the
correlation peaks standing out clearly from the cross correlation noise.
Figure 57 is an expansion of t h e cross c o r r e l a t i o n peak showing the
almost ideal triangular shape . The cross correlation noise is
considerably worse than predicted from theory ; this is caused largely
by the two devices being made at separate t imes with separate
photographic masters , resulting in slig htly different centre frequencies.

Figure 56. Response of ph ase coded array to i t s matching waveform

~i.

1-’
=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
50 ns/ div

Figure 57. Correlation peak of phase coded array

 7. 2 150 MHz bandpass f i l t e r
A bandpass filter having 150 mHz centre frequency with a bandwidth of
10 MHz was fabricated. This device , shown in figure 58 has two
transducers which have been weighted in a sin x/x pattern and coupled
together by a multistrip coupler. The X/8 split finger configuration has
been u sed ‘iving the end transducer finger and gap widths of
2.85 x 10 m. The aperture or finger overlap is about 60X . The
• mult istr ip coupler has~ ll0 st r i ps each of 4.28 x 10 6m wide . Aluminium
metallization of 2000 A thickness has been used and Lithium Niobate was
chosen as the substrate material because of its high coupling coefficient .
Insertion loss for this device is about 13db. Figure 59 shows the filter
response.

- I
‘- S
-

Act u t 0i04

Figure 58. 150 MHz f i l t e r

10

20

.030

40

- 50

140 145 150 155 160

Figure 59. 150 Ph-li f i l t e r frequency response

AEL-0001-TR - 50 -
8. CONCLUSIONS
The fundamental properties of SAW devices have been reviewed and the
prac tical de si gn con sidera tions h ave been discussed , with several devices -:
having been successfully designed and fabricated using the principles outlined
in this paper.
Although SAW technology is still a developing technology, it has reached a
level of maturity where many useful devices can be readily designed and
fabr icated by a photo lithographic process which lends itself to mass
production. Matched filters, band pass filters and oscillators all fit into
this category and all exhibit superior performance at greatly reduced sizes
when compared with their conventional couterparts.
Over the next decade it is expected that many new devices will be developed
and that many improvements will be made to existing devices by the development - -
of better substrates , new transducer geometries and a better understanding of

-
-

the mechanisms which now degrade device performance.

9. ACKNOWLEDGEMENT

The author gratefully acknowledges the enthusiastic support of

Mr D.W. Neal and staff of Micro-Engineering Group who fabricated the SAW
devices, and of Computer Aided Processes Group and Drawing Office staff who
generated the ar twork.

I
F- --- -- -.- ----—.-----— - -—-- - - - ---- — — -- ----.- ‘-—------ --- ---- -- —

- 51 - AEL-0001-TR
~~~~~~~

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V
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 - - - -
~~~~~~~~~
— - - - - -
-— _ _ _ _ _ _ _ _ _ _

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~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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~
 ’
- ss -
AFL-OOOi Th
~~~~~~ W ~~~

No. Author Title

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45 Maerfe ld , C., Gordon, K. and “Resistive Losses in Acoustic

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46 Pouliquen, J. and Vaesken, G. “Effect of a Metallic Thin Film

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47 R istic , V.M. “Bulk Mode Generation in Surface-

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48 Schulz, M.B. and Holland , M.G. “Surface Acoustic Wave I)elay

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49 Skeie, H. “Mechanical and Electrical

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50 Slobodnik, A.J. “A Review of Material Trade-Offs H
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51 Szabo, T.L. and Slobodnik , A.J. ~‘Acoustic Surface Wave Diffraction

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52 Tancrell , R.H. and Meyer, P.C. “Operation of Long Surface Wave

Interdigital Transducers”. 1971
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53 Fancrell, R.H. and Williamson , R.C. “Wavefront Distortion of Acoustic

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57 Engan, H. “Excitation of Elastic Surface

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58 Farnell , G.W ., Cerma k, L.A. , “Capacitance and Field Distributions

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60 Hartmann, C.S., Jones, W.S., “Wideband Unidirectional Surface

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61 Hei ghway, J., Taran t , D.W . , “Simple Approach to the Desi gn of

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63 Marshall , P .G., Newton , C.O. , “Theory and Design of the Surface

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66 Smith, W .R . “Circuit Model Analysis for

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67 Smith, W.R., Gerard , H.M. “Analysis and Design of

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