SURFACE-WAVE WSDUCER
MODELING

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSH'Y
ALBERT 0. SIMEON
1974

 

     
    

  

L 18 R A R Y
Michigrm State
University

This is to certify that the

thesis entitled

S URFACE- WAVE TRANS DUC ER
MODELING

presented by
ALBERT O. SIMEON

has been accepted towards fulfillment
of the requirements for

Maya in 4:“;

Major professor

Date j/é/‘i’é?’

0-7639

 

 

 

 

 

 

 

  
 
   
     

  

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HUM} & SUNS
800K BINDERY INC.
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ABSTRACT
SURFACE-WAVE TRANSDUCER MODELING

by Albert 0. Simeon

It is the aim of this research to find useful equivalent circuit
models which are developed specifically for surface-wave transducers and
to show their relation to the already-proven crossed-field model of Smith
et a1 [1].

In order to accomplish this the surfacedwave problem.is somewhat
simplified to obtain a closed form solution. This is done by considering
only particle displacements in the sagittal plane. This solution is
applied in a new approach to defining dynamic variables and the
characteristic impedance for piezoelectric transmission lines: The
surface potential is selected as the cross variable. The power flux is
calculated from the surface—wave solution and is also related to the
cross variable directly. This defines the characteristic admittance of
the transmission line.

The response of an alternate phase array of Coquin and Tiersten
[2] is extended to frequencies other than the synchronous frequency.

This is made possible by considering only the short-circuit current
response at first which permits the conformal mapping of the semi-
infinite strip and the residual solution of the potential even if the
first zeros of its even part are not located halfway between two

electrodes.

f3

§?~J

5'0
6;, With the aid of reciprocity the admittance matrix of a basic

Albert 0. Simeon

section is obtained and the corresponding equivalent circuit consisting
of a transmission line section and voltage dependent current sources,
plus a parallel circuit for the electrical part consisting of a
capacitance in parallel with current sources, depending on the cross
variable which is the particular solution of the potential problem.

By duality the transmission line part of this circuit is changed
and the crossed-field circuit model of Smith et a1 [1] is obtained. The
difference lies in the frequency dependence of the characteristic
impedance. However, over any frequency range of practical interest the
performance of this circuit resembles theirs very closely so that all
the established analysis and design procedures based on the crossed-
field model apply here as well.

Applications included are the detection and excitation of surface
waves by means of the dependent generator model. The calculation of
radiation admittance and the scattering parameters is done with the dual
circuit.

The principal contribution of this research is that it relates
transducer models directly to surface waves and does not rely on the

equivalent bulk-wave behavior implied by other circuit models.

 

[1] Smith, Gerard, Collins, Reeder, and Shaw, "Analysis of Interdigital
Surface-Wave Transducers by Use of an Equivalent Circuit Model,"
1888, MTT-l7, No. 11, 1969.

[2] G. A. Coquin and H. F. Tiersten, "Analysis of the Excitation and
Detection of Piezoelectric Surface Waves in Quartz by Means of
Surface Electrodes," The Journal of the Acoustical Society of
America, Vol. 41, No. 4, Part 2, 1967.

SURFACE-WAVE TRANSDUCER
MODELING

By

Albert 0: Simeon

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1974

To my wife

Ilse

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation for the
active assistance given him by his major professor, Dr. Fisher.

He also acknowledges gratefully the valuable suggestions made by
the committee members, Drs. Ho, Hetherington and Nyquist.

Dr. Olin's help with the digital computer was invaluable and so

was Mrs. Green's competent typing and organization of the material.

Many thanks to both.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . .
LIST OF FIGURES AND TABLES . . . . . . . . . . . . . . .
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . .
ELASTIC SURFACE WAVES. . . . . . . . . . . . . . . . .

2.1 Motivation. . . . . . . . . . . . . . . . . . .
2.2 The Problem Statement . . . . . . . . . . . . . . .

PIEZOELECTRIC TRANSMISSION LINES . . . . . . . . . . . .

3.1 Equivalent Circuits for Transmission Line Sections
in General. . . . . . . . . . . . . . . . . . . . .
3 2 Selection of Dynamic Variables. . . . . . . . .

3 3 Complex Power Flux. . . . . . . . . . . . . . . . .
3 4 The Forced Electrostatic Problem. . . . . . . . . .
3.5 Transmission Line Models Based on Surface Potential
3 6 The Vertical Shear Wave Approximation . . . . . . .
3 7 Modification of 20 Through Consideration of the

Piezoelectric Interaction . . . . . . . . . . . . .
3.8 Summary . . . . . . . . . . . . . . . . . . . . . .

THE MODIFIED MASON MODEL . . . . . . . . . .

4.1 Description of a Surface~Wave Transducer and its
Simplified Physical Model . . . . . . . . . . . .
2 Development of the Model for One Electrode Section.
.3 The Determination of the Effective Coupling

Coefficient kc. . . . . . . . . . . . . . . . . . .
4.4 Concluding Remarks. . . . . . . . . . . . . .

THE DETECTION OF SURFACE WAVES BY AN ALTERNATE PHASE
ARRAY. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 The Residual Solution of the Potential Problem. . .
5.2 The Detection at Synchronous Frequency. . . . . . .
5 3 The Frequency Response of the Short Circuit Current
from Electrode to its Nearest Neighbors . . . . . .
5.4 The Norton Equivalent Circuit . . . . . . . . .

iv

Page
iii

vi

Mb

19

19
23
26
29
32
35

40
43

44
44
46
53
58
60

6O
63

71
81

6.

EQUIVALENT CIRCUIT MODELS FOR SURFACE-WAVE TRANSDUCERS .

6

6.

.1
2

.3

An Equivalent Circuit with Dependent Sources.
Analysis of the Detection and Excitation Problem
with the Equivalent Circuit . . .

An Equivalent Circuit Model with Ideal
Transformers. . . . . . . . . . . . . . .

The Admittance Matrix for an Array of an Even
Number of Interelectrode Spaces . . . . .
Calculation of the Radiation Admittance of an Array
Made Up of N Identical Interdigital Periods .
The Scattering Parameters of the Array at
Synchronous Frequency with a Tuned Load .

CONCLUSION . . . . . . . .

7.

1

Historical Context. . . . . . . . . . . . . . . . .

7.2 Summary of the Results. . . . . . . . . . . . . . .
7.3 Further Investigations. . . . . . . . . . .

APPENDIX A DERIVATION OF THE PROPAGATION VELOCITY . . .

APPENDIX B THE TENSOR TRANSFORMATION OF THE STRESS

COEFFICIENTS O O O O O O O O O O I O O 0 O 0

APPENDIX C DERIVATION OF THE TWO-PORT PARAMETERS

FOR A LOSSLESS TRANSMISSION LINE . . . . . .

APPENDIX D CALCULATION OF THE POWER FLUX. . . . . . . .

APPENDIX E SOLUTION OF THE POTENTIAL EQUATION . . . . .

APPENDIX F THE FREQUENCY RESPONSE OF A 20 ELEMENT

SURFACE-WAVE TRANSDUCER ARRAY. o o s o o 0

APPENDIX C CALCULATION OF THE Y-PARAMETERS FOR ONE

INTERDIGITAL PERIOD. . . . . . . . . . . . .

APPENDIX H DETERMINATION OF THE RADIATION ADMITTANCE. .

APPENDIX I PROGRAM FOR THE RADIATION ADMITTANCE OF A

REFERENCES 0 O O O O O O C O O O O O O O O O O

30 ELWT ARMY O O O O O O O O O O O O O O

Page
83
83
90

102
113
116
125

130
130
131
137

139

142

145
149

154

157

159

162

164

166

Figure

2.1

2.2a

2.2b

2.3

3.1a

3.1b

3.2

3.3

3.4

4.1

4.2

4.3

4.4

5.1

5.2a

LIST OF FIGURES AND TABLES

Layout of reference directions for surface waves
on semi-infinite piezoelectric slab. . . . . . . .

Relative particle displacement at the surface
decomposed into individual modes . . . . . . . .

Combined particle trajectory at the surface. . . .

Combined particle trajectory at four decay
constants of mode 1. . . . . . . . . . . . . . .

The equivalent circuit of a longitudinal
billk'wave transducer 0 o o s o o o o o o o o o a 0

Its actual layout. 0 O O O O O O O O O O O O 0 O O

A transmission line section with delay to.

Equivalent circuits for a passive linear
bilateral two-port. (a) The T~mode1. (b) The
N‘DIOdBI o o o a a s s o s o o o o a a o o o o o o 0

Equivalent circuits for a lossless transmission
line section which introduces a phase shift

6 - wtoo o o o a a o o o a a o o o o o o a o o o a

Layout of surface-wave transducers. (a) Overview.
(b) Detail showing field lines and equivalent
depth. 0 O O O O C O O I O O O O O O O I O O O O O

Free-body diagram of section under a half
electrode used for the development of the
Mason circuit. . . . . . . . . . . . . . . . . . .

Equivalent circuit for a half electrode of a
sheardwave transducer. . . . . . . . . . . . . . .

Determination of the open circuit voltage
between adjacent half electrodes for

U(x) - U e‘Jkx employing sections of the
equivalent sheardwave circuit. . . . . . . . . .

Geometry used for the Coquin and Tiersten .
anaIYS18 O O I O O O O O O O O O O O O O O O I O 0

Boundary conditions in the x-ay plane. . . . . . .

vi

Page

16

16

17

20

20

21

21

24

45

47

52

55

64

64

Figure

5.2b

5.3

5.4

5.5

5.6

5.7

6.1

6.2

6.3

6.4

6.5

6.6

6.7

Page

Map of the infinite strip in Fig. 5.2. The

corresponding points of Fig. 5.2 are indicated

by [ ]. The boundary conditions are also

indicated. . . . . . . . . . . . . . . . . . . . . . . 65

Illustration of the net induced emf in relation

to its nearest neighbors,
%[A-(B+C)/2]............. 68

The geometry for the field problem with no

external potential applied and w i mo. . . . . . . . . 76

Normalized frequency response of the short
circuit current from an electrode to its
nearest neighbors. . . . . . . . . . . . . . . . . . . 80

Norton equivalent circuit of a single electrode
and its nearest neighbors. . . . . . . . . . . . . . . 82

The normalized frequency response of the open
Circuit valtageo I O O O O O O O O O O O O O O O O O O 82

Connection of generators consistent with the
meaning of Eq. (6.2). The resulting short
circuit current is physically incorrect. . . . . . . . 85

Choice of circuit interconnection to obtain a
meaningful expression for the electric terminal
bahav1or O O O O O O O O O O O O O O O O O O O O O O 0 85

Equivalent circuit to determine ¢P(xn) when
terminals A-B are aborted O O O O O O O O O O O O O ' O O 87

Temporary equivalent circuit of the electrical
part of a basic section extending from the
mid-point of one electrode to its neighbor . . . . . . 37

Norton equivalent circuit of the electrical

part of a basic section between xn and xn+1.

The current is denoted by the symbol I/2

because it represents approximately one-half

the current into the electrode at xn . . . . . . . . . 89

Equivalent circuit of a section shown in Fig.

6.7. O O O O O O I O O O O O I I O I O O O O O O O O O 91

A basic section for which the equivalent circuit
in Fig. 6.6 is applicable. . . . . . . . . . . . . . . 92

vii

 

Figure Page
6.8a Equivalent circuit of twenty electrodes. . . . . . . . 94

6.8b Interspaces are broken down each into three
sections to account for difference in velocity
and Yo under metallized surfaces for materials
with stronger coupling . . . . . . . . . . . . . . . . 94

6.9a Frequency response of a 20 element array
terminated in a relatively large capacitor . . . . . . 98

6.9b Central peak of the frequency response of a
20 element array . . . . . . . . . . . . . . . . . . . 99

6.10 Method to obtain the dual circuit. . . . . . . . . . . 104

6.11 Dual of the upper part of the equivalent circuit
of Fig. 6.6. The negative of all dynamic
variables corresponds to the most convenient
reference directions . . . . . . . . . . . . . . . . . 105

6.12 The variation of the turns ratio with frequency.
n - 1/2. 0 O O O O O O O O O I O O O O O O O O O O O O 106

6.13 The modified dual of the upper part of Fig.
6.6. Both circuits yield the same circuit
equations. . . . . . . . . . . . . . . . . . . . . . . 108

6.14 Equivalent representations of an ideal
transformer. . . . . . . . . . . . . . . . . . . . . . 110
6.15 Alternate equivalent circuit for a basic

section of l interspace and 2 half electrodes.
The location of electrode A is point xn. . . . . . . . 110

6.16s Equivalent circuit used to obtain the y-
parameters for one interdigital period
according to procedure by Smith et a1. . . . . . . . . 114

6.16b Connection to find y13, y23 and y33. . . . . . . . . . 114

6.17 Normalized radiation admittance of a 30
element array. . . . . . . . . . . . . . . . . . . . . 119

6.18s Central portion of frequency response of
radiation admittance . . . . . . . . . . . . . . . . . 120

6.18b A comparison of the normalized theoretical
radiation conductance and the experimental results
for Yz lithium niobate. The array consists of
15 interdigital periods. n - 1/2. . . . . . . . . . . 121

viii

Table

5.1

Page

Normalized frequency response of a single section
for n 8 1/2, 2wo/3 < w < 2mo . . . . . . . . . . . . . 79

Conversion from engineering notation to tensor
nOtatiOn o m o o m o o o I o o o o a o o o o o o o o o 143

ix

CHAPTER 1

INTRODUCTION

Equivalent circuit models for surface-wave transducers were first
developed by Smith et a1 [10] in 1969, four years after White and
Voltmer [17] initially introduced the surface-wave transducer. Such a
delay would seem unusual. In the case of transistors, for example, the
first modeling was done by Shockley, right after the introduction of the
device. The delay in the case of the surface-wave transducer may be
explained because fundamental understanding of surface waves is not
possible in a one-dimensional representation of the particle motion as
it is in semiconductor devices. Particle motion occurs in three
dimensions in elliptical trajectories. In addition, there is the com-
plicated electromechanical interaction with the electrodes deposited on
the surface.

The particle motion of elastic bulk waves is one-dimensional.
Mason [5] had developed earlier equivalent circuit models for bulk-wave
transducers. In particular, a model for series excitation and one for
crossed—field excitation were developed. Smith et al found that the
crossed-field model gives consistently correct results if applied to
surface-wave transducers. The justification for this model rests
entirely on an analogy between surface waves and bulk waves and, of
course, its proven success in applications of design and analysis of

various transducer configurations.

It is the aim of this research to find equally useful equivalent
circuit models, but which are developed specifically for surface—wave
transducers, and to show their relation to the already-proven crossed-
field model of Smith et a1. It is hoped that this approach to modeling
will lead to a better understanding of surface-wave transducers. Con—
comitantly, predictive models might emerge which will lead researchers
to develop improved surface-wave transducers.

In order to accomplish this it will first be necessary to
simplify the surfacedwave problem somewhat because a general solution in
closed form is not possible. This is done in Chapter 2 without losing
the characteristic features of the surface—wave particle motion. In
Chapter 3 this solution of the surface-wave problem is applied in a new
approach to defining dynamic variables and the characteristic impedance
for piezoelectric transmission lines. In Chapter 4 it is shown how an
equivalent bulk-wave model may be developed from this approach. This,
however, is not pursued any further because it is possible to derive
surface—wave transducer models without referring back to equivalent
bulk-wave behavior. Towards this end, the analysis of surface-wave
detection by Coquin and Tiersten [4] is generalized in Chapter 5. With
the aid of the results of Chapter 3 and Chapter 5, a reciprocal
equivalent circuit model is obtained in Chapter 6. It employs dependent
generators, but is rather easy to apply as shown by several examples.

By means of various circuit theory techniques, the equivalent circuit is
manipulated to resemble the crossed-field model of Smith et al. The
similarities and differences are discussed. This form is also used in
various applications which tend to support the validity of the model.

By comparison with experimental results, it is finally possibleto adapt

the equivalent circuits developed here to crystal cuts other than the
one for which the simplified solution in Chapter 2 was performed.

Throughout this investigation, it is assumed that bulk-wave
generation is negligible; that there are no inherent losses in the
prOpagation of surface waves, i.e. internal losses or losses due to mass
loading by the air or the metallization on the surface. It is further-
more assumed that the generated waves do not spread, but rather stay in
a well confined beam of a width equal to that of the generating
electrodes. All these problems have been investigated in the literature
[9,18]. Their inclusion would detract from the principal understanding
sought here, and as comparison with the experimental results proves,
they are only secondary effects.

The results of this research are restated in Chapter 7, where
possible future investigations are also suggested from the vantage point

of the insights gained here.

2.1

CHAPTER 2

ELASTIC SURFACE WAVES

Motivation

In this Chapter a simplified treatment of the problem of
surface waves on piezoelectric plates will be given for easy
reference. A number of articles [1]—[3] have treated the problem
without such simplification, but the mathematics cannot be handled
then in general symbols. It rather involves the simultaneous
solution of two fourth order determinants. The problem is reduced
somewhat in complexity if the electromechanical interaction is
ignored [4], but this still results in the simultaneous solution of
two third order determinants.

In crystals with orthorhombic symmetry, there is no particle
motion transverse to the direction of propagation [6]. Here the
mathematics becomes tractable. It involves the simultaneous
solution of two second order determinants. These can be handled in
closed form.

The commonly used materials for piezoelectric surface wave
transducers do not have orthorhombic symmetry. However, the
coupling to transverse motion is rather small in certain frequently
used rotated Y-cuts of quartz [51-[6].

In the so-called ac cut, which is a rotated Y-cut of 31°

about the x-axis, the face shear is decoupled [5]. This results

2.

2

in negligible transverse motion. Coquin and Tiersten [4] bear this
out indirectly. For a rotated Y—cut of about 30° to 32° there are
only two decay constants. This is typical for a system with
motion in the sagittal plane only [6].

For the sake of clarity, generality will be sacrificed. By
using the elastic constants of the ac-cut the results correspond

very closely to those obtained in a broader treatment [4]-[6].

The Problem Statement

 

All transverse motion will be neglected and it is assumed
that the mechanical electrical interaction is small. Fig. 2.1
shows the reference directions. The wave propagates in the x-
direction. The elastic material is confined to the half space
y > 0. With the present assumptions the linear elastic equations

[6] become:

*‘3
II

1 C1181 + C1282 (2.1)

H
II

2 C123l + C2232 (2.2)

T6 = C6686 (2.3)

where T1 is the tension stress in the x-direction

T2 is the tension stress in the y-direction
T6 is the shear stress about the z-axis.
The S1 are the corresponding strains and the C11 are the elastic

stress coefficients. For a rotation of 31.62° (the angle when

C56 is exactly zero):

 

Figure 2.1 - Layout of reference directions for
surface waves on semi-infinite
piezoelectric slab.

C11

86.74, C12 = -7.65

C = 28.85 all in 109 N/mz.

22 127.84, c

66

These values were obtained from Ref. 4, by a method illustrated in
Appendix B. It should be realized at this point that without the
simplifying assumptions made before one would have instead to deal

with the following standard piezoelectric constitutive equations

[5]:

E .
Tij - Cijkl Skl - e , E - (2.4)

s
1 ' °1k1 Skl + 51k Ek ' (2'5)

D
Clearly Eqs. (2.1)-(2.3) are easier to handle. Let u be a dis-

placement in the x-direction, v a displacement in the y-direction.

Their relations to the strains become:

Bu
S1 — Rx (2.6)
By
S :- _. 2.
2 By ( 7)
av au
8 = -‘+-—— . 2
6 3x 8y ( 8)
The equations of motion are given by
8T 3T
.. 1 6
= -—-+-—-- 2.
Du ax 8y ( 9)
and
3T 3T2
" = —-— -—- . - . O
0v ax 3y (2 1 )

p is the density of quartz, p = 2.65 x 103 kg/m3. As is conven—

tional the particle displacements are next assumed to have the form:

u = Re[U chmkv e1<wt-kx)] (2.
and

v = Re[V e-aky ej(wt-kx)] (2.
k is the prOpagation constant of the wave, and

k =4%; . (2.

ll)

13)

The quantity ak is the assumed decay constant. Equations (2.11)-

(2.12) coupled with Eqs. (2.6)—(2.8) imply that the strains and

hence the stresses are also of the same general form given in

Eqs. (2.11)-(2.12). For the sake of convenience of notation let

the real part operator Re[ ] be implied throughout the following.

Then the partial derivatives may be replaced by:

3 a

§"‘-“9‘ho (2
at ‘ ‘
33—- ——) m1. . (2.
av

Equations (2.6)-(2.10) then become:

14)

15)

I6)

81 = 'jku, (2-17)
32 - -akv, ' (2.18)
S6 = -jkv - aku, (2.19)
2
'pw 0 = -jle - akTG, (2.20)
and
'2
-pw V = -jkT6 ' asz . (2.21)

When these are combined with Eqs. (2.l)-(2.3) the stresses become:

T1 = -jkC11u - akClzv, (2.22)

T2 = -jkC12u - akszv, (2.23)
and

T6 = -akC66u — jkC66v . (2.24)

With the aid of Eqs. (2.20)—(2.21) and

k2 = 002/sz (2.25)

the stress relations become the homogeneous equations:

2 2
(0Vp - C11 +~o C66)“ + jo(C12 + C66)v - O (2.26)

and
10(0 +c )u+(pV2-C +62C )v=0 (2 27)
‘ 12 66 p 66 22 ' °

If Vp were known these could be solved for the two possible o's
and the relative amplitudes u and v. This would result in two
modes: o1,u1,v1 and az,u2,v2.
To proceed the conditions for a free boundary are required:

No normal stress can be acting on a surface element:

at y = 0 T2 = O (2.28)

and T6 = O . (2.29)

When both modes are used Eqs. (2.23)-(2.24) become at y = 0:
-jkC12[U1 + U2] - kC22[olV1 + oZVZ] = O (2.39)
and
-kC66[o1U1 + oZUZ] - jkC66[Vl + V2] = O . (2.31)

Equation (2.26) gives for each mode the relative amplitudes if VD

were known:

10

 

EU}. {1. - 3““1‘Clz + C66)
8v V 2 _ 2 .
i I pr C11 + a1 C66
U1 and V1 could be expressed in terms of a mode amplitude
U1 ' Bur Bi
and

vi = 8V1 Bi ‘

For convenience the following constants are defined (see
8 .Eu
9
15 C66
K = 812 + 1.
and 2
0V
q-—__P——.
C66
Hence,
Sui - -301k9
and
8 . _ 2
V1 q 811 + “1

With this formulation the displacements are

—o ky
u = [BUIBI e

-0

2ky
+ BUZB2 e

—jk(x - V t)
l e p

and

-o

—o ky 2
+ BVZBZ e

V 3 [BVIBI e

ky -jk(x - th)

] e .

Correspondingly, the boundary conditions given in Eqs. (2

(2.31) become

(2.3?)

B1 as

(2.33)

.34)

[6]):
(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

(2.41)

.30)-

11

' = 7
(01801 + JBvl)B1 + (a28u2 + JBv2)32 0 (2.4-)

and
(iglzsu1 + olg228v1)31 + (jglzsu2 + o2R228V2)BZ = o.(2.43)

These again are two homogeneous equations which must be solved
simultaneously with Eqs. (2.26)-(2.27). The determinant of their

coefficients is

q - 811 + G jak

= o . (2.44)

 

2
jak q — 1 + o s22

 

The unknowns which Eqs. (2.42)-(2.44) must yield first of all are
q and the a's. Express Eqs. (2.42)-(2.43) also as a determinant

set to zero:

olBul + JBvl aZBuz + ijz

= 0.(2.45)

 

 

jglz BUl + algzzavl jg128U2 + 022228v2
It is shown in Appendix A how the simultaneous solution of Eqs.

(2.44)-(2.45) leads to an implicit relation for q = szp/C66:

q3(1 - d) - q2(1 - e + 2f) + q(2f + £2) - r2 = o, (2.46)
g P 2

where d --—£- , e --—ll and r - gll - —33—-. (2.47)
g22 g22 g22

With the elastic stress coefficients for the rotated Y-cut
of 31.62°, Eq. (2.46) becomes:

0.7743q3 - 6.3029q2 + 14.9258q - 8.9444 = o, (2.48)

12

which has the following solution:

q = 0.9096 . (2.49)

qC__‘
v =\/—:°-°- . (2.50)
p o

The speed of propagation Vp is therefore

By definition

Vp = 3147 m/sec . (2.51)

From Eq. (2.44) the a's may be determined:

4 2 2
O :- 2220: + a [Q(g22 + l) " 1 - $52231]- + K ] + (q—gll)(q—l)9 (2052)

or
4 2
O = 4.4315a - 8.8432a + 0.18957 . (2.53)
a2 = 0.147 (2.54)
a1 = 1.40 . (2.55)

From Eqs. (2.38)-(2.39) the relative amplitude ratios 8 are

obtained:
Bui = -ia1(g12 + 1). (2.56)
BUZ = -j0.108, (2.57)
BUI = -j1.03, (2.58)
= - + 2 (2 59)
5V9 = -2.08, (2.60)
and

Bvl ' “0.123 0

13

The ratios of the mode amplitudes B as defined by Eqs. (2.33)-

i,
(2.34) [or Eqs. (2.40)—(2.41)], may be determined with the aid of

 

Eq. (2.42).
B o BU + ij
-§§ = - -1 1 + 1 (2.61)
1 “2302 ij2
B2
--= -o.75 (2.62)
Bl

Equations (2.33)-(2.34) allow the calculation of the surface
amplitudes in both directions for each mode. For mode 2 the peak

displacement in the vertical direction is

It is convenient to normalize all others with respect to V2:
V
02(2) =-—3 a 1 (2.64)
v2

(2) U2 8U2 B2 BU

2
c - —-= —~————-- ———-= 30.052 (2.65)
U BU B
c1(1) = Vl" E7l73l’g -j0.66 (2.66)
2 V2 2

V 8v B
c2(1) - 61" E—l—El = —o.079 . (2.67)
2 V2 2

The superscript of these normalized surface amplitudes refers to the

mode. The subscript l to the x-direction, 2 to the v-direction.

14

The values obtained here on the basis of a two—dimensional
formulation correspond very closely to the numerical analysis of
the AT cut (a 35.25° rotated Y-cut) in [6]. There, however, a
third mode is present because of shear about the x—axis which
represents particle motion in the transverse direction (z—axis)
with a rather small amplitude.

The j's in Eqs. (2.64)-(2.65) stand for phase shifts of 90°
with respect to the vertical particle displacements of mode 2.
This should be expressed in the time domain in Eqs. (2.40)-(2.41)

as follows:

-o kv
u(x,y,t) = |C1(1)| e 1 cos(wt - kx --%)
~o ky
+ [Cl<z)l e 2 cos(wt - kx +-% (2.68)
is the horizontal displacement, and
-o ky
v(x,y,t) = [C2(l)| e l cos(mt — kx - n)
+ [C2(2)| e 2 cos(wt - kx) (2.69)
is the vertical particle displacement.
k = w/Vp is the propagation constant.
w is the radian frequency.
0) .
Vp, a1 and [C1 I are.
Vp = 3147 m/sec (2.70)
= 1.40 (2.71)

15

a2 = 0.147 (2.72)
|c1(1)| a 0.66, [01(2)| a 0.052, [c2(1)| = 0.079,
[c2(2)| = 1 . (2.73)

In Fig. 2.2 v(0,0,t) is plotted against u(0,0,t), separately for
each mode and combined. This represents the trajectory of a
particle at the surface. Typical dimensions are of the order of a
few Angstroms. Fig. 2.3 shows v(O,4/ka1,t) vs. u(0,4/kal,t). In
this plot mode 1 has decayed to 1.8% of its surface value. Mode 2

is practically unaffected.

—4 n 7°xaz/“1
u(0,4/ka1,t) = 0.66 x e cos(mt - 5) + 0.052 e
x cos(wt + g) (2.74)
and
-4 '°X°2/“1
v(O,4/ka1,t) a 0.079 e cos(mt - n) + 1.0 e
x cos(wt + O) . (2.75)

With the a-values a1 = 1.40 and a2 = 0.147 this becomes:

u = 0.022 cos(wt + g) (2.76)
and

v - 0.66 cos(mt + 0) . (2-77)

The motion is almost entirely that of a shear wave, but it has
taken on a very slight clockwise rotation, as mode 2. This is

calculated at 0.4 times the decay distance of mode 2. On the basis

16

[,zedr'“"“““‘—-~l‘\ u .052

 

 

 

1.0 t a 0

 

 

v v

u(1)(0,0,t) - 0.66 cos(wt - n/Z) 6(2)(0,0,c) 0.052 cos(wc+,/2)

ll

v(1)(0,0,t) I 0.079 cos(wt - n) v(2)(0,0,t)
(a)

A

1.0 COS(wt + O)

0.61 u

t=0 u(0,0,t)=0.6lcosQutdn/2)

0.92 v(0,0,t)=0.92cos00t+0)

 

V

(b)

Figure 2.2 — (a) Relative particle displacement at the
surface decomposed into individual modes.
(b) Combined particle trajectory at the
surface.

17

 

.022

 

 

 

 

 

t - 0

Figure 2.3 - Combined particle trajectory at four
decay constants of mode 1.

18

of this it can be safely said that the dominant behavior of a

surface wave is that of a vertical shear wave.

3.

1

CHAPTER 3

PIEZOELECTRIC TRANSMISSION LINES

Equivalent Circuits for Transmission Line Sections in General

Equivalent circuits for surface wave transducers [7], as
well as "crossed-field" bulk-wave transducers [5] degenerate into
sections of piezoelectric transmission lines when the electric
terminals are shorted (see Fig. 3.1).

It is the intent of the following.treatment to show how such
a transmission line equivalent circuit is obtained. It is
necessary that in all cases a pair of complimentary variables must
be found. With their aid the characteristic impedance is defined.

Consider the section of lossless transmission line shown in
Fig. 3.2. Without loss in generality the cross variable may be
denoted by V and the through variable by I. It is well known that
in Laplace transform notation for a delay to the terminal quantities

are related by the following transmission matrix [8]

V cosh(sto) 2031nh(sto) V

1 2

E . (3.1)

 

 

 

 

 

 

 

 

 

 

 

 

I1 Yosinh(sto) cosh(sto) I

For convenience this is derived in Appendix C.

These "ABCD—parameters" are readily converted into "2-

parameters" (Appendix C):

19

20

.
ll

 

 

 

 

 

 

 

 

.FR.
[Zotun(wl/2v) i2 tan(nl/2v)
"iz H
“—5.0 -
Sin(mI/V)
I
F, 4' to l “T“- l“
A 7“ .L
3% g; + l
C V
r) ’1‘ -
(a)
l
4.
conducting
F surface v
l .2
m ./

 

—— '7
~
~-———-— ~-—~---——-————-— l ————————-—-———-.—
*‘-——- '

 

 

')
conducting surface

(b)

Figure 3.1 - (a) The equivalent circuit of a longitudinal
bulk-wave transducer.
(b) Its actual layout.

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- =1 -1
11 I1 11 1+ 2 2 2
-———p- ——-o- ——.
'i' "—;=' i—
+ - + -
V = V + V V - V + V
1 1 1 z a V+/I+ 2 2 2
-st -st
+ o + + +
I2 - 8 I1 V2 II e V1
_ sro _ _ st _
12 - e I1 V2 - e V1
Figure 3.2 - A transmission line section with delay to.
11 _ ~12
————a- I ~—————
2 - z z - z
11 12 22 12
+ I. +
v1 212 v2
1 (a) -r
1 2
-———O- ‘—
’Y12
+ +-
V V
1 y11 I y12 y22 + y12 2

 

 

 

 

 

 

 

(b)

Figure 3.3 - Equivalent circuits for a passive linear bilateral two-
port. (a) The T-model. (b) The n-model.

22

 

 

 

 

 

 

 

 

 

 

 

 

 

v ———i§1———— —~-A° I
l tanh(sto) sinh(sto) 1
a (3.2)
7. 7.
V .__.£L.... _____£L.__ -1
2 sinh(sto) tanh(sto) 2
or "y-parameters" (Appendix C):
Y -Y
I _.__£L__._ _____£L___ V
1 tanh(sto) sinh(sto) 1
a . (3.3)
'Y Y
-1 ____Jl.___ _._..£l___. V
2 sinh(sto) tanh(sto) 2

 

 

 

 

 

 

 

 

 

 

 

 

Laplace transforms and phasor transforms are related by the simple
expediency of setting 3 equal to jw. The delay to translates then
into a phase shift

8 = wto . (3.4)

Equations (3.2)-(3.3) then become:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v 2° 2° I
l jtane jsine l
. (3.5)
v 7° 2° -I
2 jsine jtane 2
Yo -Yo
Il jtane jsine v1
= . (3.6)
-Yo Yo
“‘2 3m; 3BR? V2

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.3 shows the T- and n-model for a bilateral linear network.

211-212 become here

3.2

23

 

 

 

6
. 2 —- - l
2 -2 =:9[°°°°-ll=-7—9—C08(2) ] (37)
, .
11 12 j sine sine j sin(2-%
or
20 -251n2-%) 6
z - z =-—— [ ] = j Z tan(—) = Z - Z - (3.8)
11 12 j 281n(§-)cos(-e—) o 2 22 12
2 2
Similarly y11 + y12 become
—‘ .2
y11 I y12 y22 + y12 ' 3 Yo tan(2) ° (3'9)

The resultant equivalent circuits are then shown in Fig. 3.4. It
is seen that Fig. 3.3a is indeed the Mason circuit with v = O and
a delay t - l/v, 6 . wl/v.

It should be pointed out here that although the Mason
circuit was developed specifically for bulk waves, it has been used

successfully for the equivalent circuit description of surface wave

' transducers. The appropriateness of this model is discussed in

Chapter 4. The present develoPment has shown that important aspects

of the Mason circuit are found in all lossless transmission lines.

Selection of Dynamic Variables

 

Surface waves, as described in the preceding Chapter,
exhibit time delay independent of frequency and propagate, as
formulated in Eqs. (2.68)-(2.69), unattenuated in the x-direction.
These are requirements for lossless transmission line behavior, but
in order to complete an equivalence satisfactory dynamic variables
must be determined. In the Mason model, Fig. 3.1, the dynamic

variables are force and particle velocity. In an electric trans-

24

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 I2
-———.- —-—-.
jZOtan(6/2) jZotan(6/2)

+ +
v1 Zo/jsine V2
(8)

I1 12
———-D- -——-—C--
Yo/jsine
+ +
Vl ontan(0/2) ontan(6/2) V2
(b)

Figure 3.4 - Equivalent circuits for a lossless
transmission line section which
introduces a phase shift 6 = wto.

25

mission line they are voltage and current. The common feature is
that the time average power flux of a wave travelling in the x-

direction is

1 + +*
V+ and I+ are the cross—variable and through-variable respectively

for a wave travelling in the positive x-direction. Since

V a I Z (3.11)

1 +2
P 'EIII

x Re[ZO] (3.12)

*
or conversely, since Re[Y0 ] - Re[Yo]:

Px =% |v+|2 Re[YO] . (3.13)
V and I should be taken generally as cross- and through-variables
rather than as voltage and current in particular.

In the case of surface waves, the power flux requirement is
satisfied as follows: Conveniently one of the two variables is
selected first. This could be done in various ways. One could
select, for example, the vertical velocity of mode 2 as the through-
variable. As another example, the piezoelectrically generated
surface potential could be chosen as the cross-variable. Both of
these will be considered later on. Whatever the choice, Eqs.
(3.12)-(3.13) will then determine uniquely Re[zo] or Re[YO], since

it is shown in Coquin and Tiersten [4] that Px follows from the

determination of the decay constants c and the amplitudes C1(j)

.‘l

which have been determined in the last Chapter.

3.3

26

The second dynamic variable is not uniquely determined. If

+
I were defined and Re[Zo] was found subsequently by

Re[ZO] a 2Px/|I+|2 (3.14)

4.
then V (a 201+) still requires ImIZO]. It turns out that this does
not really present a serious problem if Eq. (3.10) is generalized

to the complex power flux:

 

—- 1 + +*
PX ‘2‘ V I (3.15)
and similarly Eqs. (3.12)-(3.14) would become:
-— l + 2
Px - §-|I | 20 (3.16)
- l + 2 *
PX ' '2- 'V l YO (3.17)
from which follows
-- *
2Px 2Px
O II+|2 O 'v+|2

The complex power flux is obtained from Coquin and Tiersten [4]

very simply by leaving out the Re[ ] operator.

Complex Power Flux

 

For all rotated Y-cuts the power flux vector and the
propagation direction are co-linear [4]. This is not generally the
case [9], but such cuts are not used in practical applications.

Components of principle interest are [4]:

l . *
P1 . --E -/A Re(Tijuj )dx2 (3.19)
o

27

This is done with the understanding that the wave front is 1m wide.

For a width W and propagation in the x ~direction Eq. (3.19) is

1

modified to become the relation for the required complex power, i.e.
P w '1‘": 320
1 - 2 ljuj dx2 . ( . )

The tensor subscripts must be interpreted according to Table

8.8 in Appendix B. Then for Eq. (3.20):

‘ w d '* '* 0 2
Px --§ y[Tlu + T6v + T5 x ], (3. l)

0

where x1 = x, x2 = y, u1 = u, u2 - v and u3 f O in accordance with
the treatment of Chapter 2.

The quantities u and v must be taken here as the complex

amplitudes of Eqs. (2.68)-(2.69) multiplied by ej<wt-kx), from

t *
which 6 and v follow:

(1) ‘°1ky + C (2) e’°2ky] ej(wt-kx)

u(x,y,t) = [C1 e 1 (3.22)

and

’3 ky “a ky
v(x,y,t) ' [C2(1) e 1 + C2(2) e 2 ] ej(wt-kx)

, (3.23)

which imply

-a ky “a ky _ _
6* = -jw[Cl(1)* e l + C1(2)* e 2 ] e j<wt kx) (3.24)

and

‘1)* e_alky + c2(2)* e-azkyl e‘5(wt’k*)

9* - -36[cz . (3.25)

28

T1 and T6 are obtained from Eqs. (2.1) and (2.6), and 51’ $2 and
86 from Eqs. (2.6)-(2.8). Then,
= 32. 2!.
T1 = C11 S1 + C12 52 C11 8X + C12 3y , (3.26)
and
By au
T6 C66 S6 C66 3X + C66 3y (3.27)
Also,
5’31 - -jku (3 28)
ax ’ °
- ky a ky
3g , (1) “1 _ (2) 2 Rat-kx)
3y [ alkC1 oszl ] e , (3.29)
91 =- — kv (3 30)
ax j 9 0
and
- ky -a ky
.31, (1) Cl1 (2) 2 30,640.)
3y {-olkC2 e oszZ e ] e . (3.31)

These relations are worked out in Appendix D. It is found there

(2)
2

that in terms of C , the peak amplitude of the dominant mode,

P; is real and has a value of

_ l_ (2) 2 9
PK -72 wW[C2 ] x 90 x 10 . (3.32)

It is also seen that, had only the vertical motion of the dominant
mode been considered, the value for the power flux would have
differed from Eq. (3.32) by only 92. This suggests that a vertical

shear wave of suitable depth is a good approximation for the

3.4

29

surface wave. This is born out by the fact that Eq. (2.50)

gives a speed of propagation of

0.9096 x c
v -IV/ 66 (3.33)
p o

 

 

which differs from that of a vertical shear wave

V = -——- (3.34)

by only 4.6%.

The Forced Electrostatic Problem

 

Any piezoelectrical interaction has been neglected so far.
In quartz this is quite justified because the interaction is weak,
and the expression for the power flux developed in the preceding
section is quite accurate. To obtain the potential the solution
obtained in Chapter 2 is now used as a forcing function in Eq.

(2.5)

S a . (3.35)

Skl + 81k k

1 = e1k1

Since the divergence of D'is zero in the crystal (no free charges)
it will be convenient to perform this operation on Eq. (3.35) with

the result:

8
1,1 e11.1 sk1,i I e1k Bk 1 O ' (3'36)

D

The subscripts behind the commas donate a partial derivative. It
can be said, furthermore, that E'is -V¢ as in an electrostatic

problem since typical dimensions are much smaller here than one

30

wavelength in free space. It follows that

S

 

 

 

 

e11.1 sk1,i C1k ¢,k1 ' 0 ' (3°37)
In all rotated Y-cuts in quartz the dielectric constant matrix cijs
is given by [4]:
£11 0 0
s
61:) = 0 £22 823 (3.38)
0 e23 633
Since no variations in the z-direction are assumed Eq. (3.37)
becomes:
829 a°$ a a
611 ax2 I 622 ;;2 ‘ e11.1 32'3k1 I 821.1 3§'Sk1 ° (3°39)

In terms of engineering notation the tensor subscripts must be once

more interpreted according to Table A.8 in Appendix A:

2 2 BS

as as as
e a 9 + E a 9 _ e 1 2 3 4

11 3x + 612 3x + el3 3x + e14 3x

3x 8
as as as as as
5 __9. .__l .__E ._;1
I e15 8x I e16 ax I 821 3y I e22 8y I 823 ay ~
as as as
4 5 6
I e24 5§"'I e25 3y I e26 3;"' °

The matrix for the piezoelectric constants is also largely empty

for rotated Y-cuts:

"\1-

31

e11 e12 813 814 O 0
Give 0 O 0 0 e25 e26 , (3.40)
0 0 O 0 e35 e36

 

 

 

 

Furthermore, according to the treatment of Chapter 2, only 81’ 82

and S6 are not zero; which results in the inhomogeneous differen-

tial equation:

 

 

2 2 S S
E a 6(x,y,c) +y£ a Q(x,y,t) g e 3 1 + e 3 2
ll 2 22 2 11 3x 12 3x
3x By
336
+ 826 '3‘;- . (3.41)

The strains relate through Eqs. (2.6)—(2.8) to the solutions of the
simplified surface-wave problem in complex notation Eqs. (3.22)-
(3.23). The general solution is found in [4]; for this particular
case it will be simpler than there. For convenience it is worked

out in Appendix E. Of particular interest is the potential at the

surface.

+ eJ(wt-kx)

0 - es - ¢(x,0,t) (3.42)

It follows from the solution in Appendix E (Eq. E.8) that Is is:

(1) _ (1)
¢ . (’°11 I “1°26’C1 I J<e26 r“1312)“2

S
(1 I If"‘1) “511522

 

(2) (2)
(‘°11 I “2°26’C1 I 1“26 ' r"‘2°12)Cz

(1 I ‘92) “511522

-\/‘22
r - E——' . (3.43)

11

+

 

3.5

32

Transmission Line Models Based on Surface Potential

Two initial choices for dynamic variables are made in
connection with ¢+. First, it will be taken as the cross—variable.
This will lead to a n-circuit as shown in Fig. 3.4b. The next
choice is not quite as direct, but it leads ultimately to equiva—
lent circuits for surface wave transducers which are very similar

to the Mason circuit for bulk waves (see Fig. 3.1a):

I+ - ijellezz W¢I (3.44)
and
II - -16/e11622 W¢' , (3.45)

where I(x) is the through-variable as defined in Fig. 3.2

I(x) - I+(x) - I-(x) . (3.46)

and

' e3<wt+kx)

4 = 45 (3.47)

represents a generalization of the treatment where only waves in
the (+)x-direction are considered. For either approach, it will
first be necessary to evaluate Is in Eq. (3.43) in terms of C2(2)
so that the characteristic impedance or admittance may be

determined according to Eq. (3.18).

By means of the tensor transformation [4]

e (3.48)

ijk' - air ajs akl 8rat

the required piezoelectric constants for the AC cut are obtained:

33

0.171 c/m2

11

2

2
9

e11 = 39.21 x 10.-12 F/m is not affected by the rotation about the

x—axis, but 522 suffers a slight change, i.e.

'3
522 “Zr “23 crs (3°49)
or
' = cosze + 2 cose sine
e22 E22 823
+12 (350)
8116533 0
or

12 F

' - 39.71 x 10' /m . (3.51)

822

The potential at the surface becomes

_ (0.17+1.4x0.106)(:19.66) - j(0.lO6+l.4x0.163)(0.079) c (2)

‘D
(1 + 1.0063 x 1.4) x 39.46 x 10'12 2

8

+ (O.17+0.147x0.106)(19.052)+j(0.106+0.l47x0.l63) c (2)

 

_12 2 . (3.52)
(1 + 1.0063 x 0.147) x 39.46 x 10 .
This reduces to
Is . -)2.225 x 10I9 02(2) - 30.277 x 10I9 02(2)
+ 30.215 x 10I9 02(2) + j2.867 x 10I9 02(2) . - (3.53)

34

Hence,

(2) (58 mV for c (2) = 10'10m). (3.54)

8
I3 = j5.8 x 10 C 2

2

The numbers were left in the same order in which they appear
in Eq. (3.43). It is seen that the slowly decaying "shear" mode
still furnishes the main contribution to this particular property
of the surface wave, but it is strongly opposed by the electric
interaction of the "longitudinal" mode which exists closer to the
surface.

o is now selected as the cross variable

600 = ¢I(x) + {(20 . (3.55)

The characteristic admittance according to Eq. (3.18) is then
Y 29 / 2 I
o x |¢S| . (3.56)

From Eqs. (3.32) and (3.54) follows the value for Yo.

x 90 x 109/[5.8 x 108 0 (2’12 (3.57)

(2) 2
I 2

YO 3 wW[C2

Y0 = aw x 2.68 x 10'70 (3.58)

Typical values for W are 3mm and for m 109 rad/sec. Then

Y0 = 0.80 and the corresponding through variable would be I+¥Yo¢+
where II - 0.8 x 58 x 10"3 m 50 ma for a wave in the x-direction.
The line potential was based in Eq. (3.54) on a peak vertical
displacement of 12 at the surface. It should be noted in Eq.
(3.58) that the line admittance increases with frequency. This

result defines now for any section of line the equivalent circuit

35

shown in Fig. 3.4b, or if Zo - l/Yo is used, the circuit in Fig.
3.4a.
Alternately a through—variable may be defined first as in

Eq. (3.44). Zo follows from Eq. (3.18):

2 2 2
20 = 2Px/m £11522 w |¢S| (3.59)

with the values obtained in Eqs. (3.54) and (3.32) this becomes:

 

7 = 6w x 90 x 109 x 1024 .9 (3 60)
'o 2 2 2 16 °
w W x 39.21 x 39.71 x 5.8 x 10
14
z ‘ 1.72 x 10 9 . (3.61)

 

o (0W

20 decreases with frequency. For a wave to the right, of peak

(2) ‘ + ————- +
amplitude C2 of 1A, I - jw/elcz W¢ has a peak value of 6.87uA
for w - log/sec and W a 3 x 10.-3 m. The corresponding cross-

variable is then

V a Z I . (3.62)

20 at w - log/sec, W = 3 x 10.3 m is 57.3 MD. V+ has then a peak
amplitude of 394 volts. With the aid of Eq. (3.61) the equivalent
circuit shown in Fig. 3.4a is now defined for any given length of

transmission line.

The Vertical Shear Wave Approximation

 

It was pointed out earlier that the speed of propagation of
a vertical shear wave differs from that of the surface wave by only
4.62. Furthermore the major contribution to the power flux (Eq.

3.32) comes from the last term in Eq. (D.3), Appendix D:

36

C66 w...
3 “——

(2) 2
Px 4oz ICZ I

. (3.63)

The error here was only 8.8%. This contribution is the same as the
power flux of a vertical shear wave of effective thickness h, where

h can be found from an expression corresponding to Eq. (3.20).

— W .*
PX = - ‘2’ [T6 V ] h (3.64)

,*
For v the value of the surface wave at y = O is taken, but for

mode 2 only (Eq. 3.25):

-* (2)* e-J(wt-kx)
2

v - ~jw C (3-65)

T6 follows from Eq. (3.27), consistent with the present approxima-

tion.
T=C s-c fl--kc (366)
6 66 6 66 ex 3 66 ”1 °

which becomes through Eq. (3.23):

v(x,0,t) = C2(2) ej<wt-kx) (3.67)
a (2) j(wt-kx)
T6 —jk 066 02 e (3.68)
The power flux, as defined in Eq. (3.64), is then
- . 11 _ <2) <2)* _
Px - 2 [ jk C66 C2 C2 ( jm)] h, (3.69)
where
2p 32_ 1 A
T'k-w/Vp , and 03.,‘E-E. (3.70)
Hence,
’1? “Ilia |c (2)|2hb (3 71)
x 2Vp 66 2 ° °

37

This is to equal Eq. (3.63):

2
C66 Wm C66 Wm

TE—‘Th- (3’72)
2 D

It follows that the effective thickness h of the equivalent shear

wave should be:

v
h _ p - 10700 m

2wa2 w

 

. (3.73)

It decreases with frequency. In terms of a wave length the

effective thickness is

l
4no

 

= 0.541 . (3.74)
2

In the Mason model (see Fig. 3.1a) particle velocity is
taken to be the through-variable. There longitudinal waves were
considered. In that case the natural choice for reference
directions would be such that the forward and reverse waves would
subtract as shown in Fig. 3.2. For a shear wave the particle
velocity is in the transverse direction. The displacement at the

surface given by Eq. (3.23) is, using the "mode 2" approximation:

v+ - 02(2) ej(wt-kx) (3.75)
The reverse wave would have displacement:
v' - 02(2) e3(“°Ikx) . (3.76)

By Eq. (3.66) the shear stress is

8v

T6 " C66 ‘53; . (3.77)

38

This results in

T + . -jk C (2) ej(mt-kx)

6 C (3.78)

2 66

and

T r (2) ej(wt+kx)

6 I 3k C2

C66 . (3.79)

The velocities add and the stresses subtract. One could take
various choices now. Here, arbitrarily, the cross-variable will be

defined as

v = -T6Wh = -T6+Wh - T6-Wh a VI + v‘ (3.80)

and the through—variable has to be taken then as

I - II - I , (3.81)
where I+ and I_ must be in accordance with Eq. (3.64)

(2) ej(mt-kx) a 0+

II = 1602 (3.82)

and

I" = -ij2(2) ej<thkx> - ~47 . (3.83)

The quantity (WhT6) is the counter force acting externally on the
section. For this reason, the power flux expression contains a
minus sign. Accordingly V+ equals -T6Wh. The characteristic

impedance is now

(2)
z _ v _ 6 66 C2 "h _ E366wh (3 84)
o If; v' (2) m . °
jw C2

+ -T Wh ch

 

By Eq. (3.74): h = l/4ua2, k - ZnIA and hence Zo becomes

39

C W
Z a 66

o 2a2 w

 

. (3.85)

This last expression (Eq. (3.85)) resembles Eq. (3.61) where the
surface potential was used. There, also, the numerator is pr0por-

tional to C66 if the dominant term for the power flux is considered

(Eq. 0.3, Appendix D):

C Wm IC2(2)|2

Px " ““2““ ' “2‘5;— - (3°86)

By Eq. (3.59) 20 is

2 2 2
2o - 2Px/m 611522 W [0 I . (3.87)

S

According to Eqs. (3.53) and (3.43) the largest contribution to Is

 

 

is:
(2)
3 e26 C2 (3.88)
VE11522
Because of cancellations Is is less:
2
3 826 C2( )
48 = 0.215 x . (3.89)
V811522

The characteristic impedance in Eq. (3.61) is then in general

symbols approximately:

 

C66 ”“ 1 611°22

z = ——————— - ~————-———- - (3.90)

° 202 wzc c w2 0 2152 e 2

1 2 ° 26

c w
66 22
Z. '" 2378" (“-2 2) - (3°91)
W 26

.7

40

Comparison of Eqs. (3.91) and (3.85) shows the relation
between the purely mechanical representation of a piezoelectric
transmission line section and the representation through the surface
potential. It should be noted that both expressions are prOpor-
tional to C66 and, therefore, to sz. This differs from the treat-
ment by Smith et al [7], [10], [11]. There Z0 is proportional to

Vp. That result follows directly from Eq. (3.84):

kC Wh C Wh

 

7. = --—————-—66 I 66 , (3.92)
O a) V
p
c = v 2 p - (3 93)
66 p °
is next replaced to yield

This is Mason's expression for Zo which would apply here if h were
to be taken as a constant. But h‘a A/4naz results in the correct

expression Eq. (3.85) used here.

Modification ofuzo Through Consideration of the Piezoelectric

~“m .-

 

 

_—.—'—..—

In this section it will be established that for the vertical
shear wave C66 must be modified when the electric field is not
disregarded. This treatment is taken from Berlincourt et a1 [12].
The piezoelectric constitutive Eqs. (2.4)-(2.5) are for this

special case:

26 2 (3.95)

41

But the region is charge-free so that the divergence of D is

zero. In this case then

an BS BE
2 6 2
3x - O - 226 3x + E2 ax (3°97)
may be combined with Eq. (3.95) to eliminate E2. Then
3T6 . CE 356 _ e 3E2 (3 93)
3x 66 3x 26 3x ' °
8T
But-3;— is the net force per unit volume acting in the y—direction:
3T 2
3x6 . 342' , (3.99)
at
. av . ,
56 is here S6 3x so that Eq. (3.98) becomes.
2 2 EE
3 v E 3 v 2
p—-—-C —--—-e -——-. (3.100)
at2 66 ax2 26 3x
Let the speed of propagation for an independent E—field be
. C
VpL - -§é-. Then Eq. (3.100) may be regarded as an inhomogeneous
3E
E 2
wave equation with (e26/C66) Siw-as the source term.
2 2 e 3E
3 V - 3 V . 26 2 (3.101)

 

 

1
2 E 2 2 E 8x
p) at C66

If E is forced externally then V E is indeed the speed of propa-

2 9
3E2 32v
gation. Through Eq. (3.97) S;f-and-——§ are interdependent:
8x
3E2 e26 32v 3 ,
57"7"? <40”
2 3x

Equation (3.100) then becomes

42

 

 

2
2 e
E 26
p a z . C66 [1 + E l 3 g , (3.103)
at E2C66 8x
D
which defines C as
66
e 2
D E 26
(:66 =- c66 [1 + ———-C—E-—] . (3.104)
82 66
The corresponding speed of propagation
D
C
vD a (~99 (3.105)
p o
is then higher, i.e.
D E e26
V - Vp l + E . (3.106)
€2C66

For the values used before (e262/C66e) z 0.01. This means from
Eq. (3.104) that 20 should be taken 1% higher in Eq. (3.61) and in
Eq. (3.85). The velocity Eq. (3.106) would increase by only l/ZZ.

Smith et a1 [11] state that the changes in impedance and
velocity are the same:

Z
0 1 2
T - 1 + '2- kc . (3.107)

'<l'<
o

B
S

The quantity kc is the electromechanical coupling coefficient. In

the present case this would be

e
k --—J¥i—— . (3.103)

C
Vezcea

Equation (3.106) agrees with Eq. (3.107). However, the change in

impedance should be pr0portional to 1 + kcz. This follows directly

from Eq. (3.104) and Eq. (3.85). Smith et a1 [11], on the other

3.8

43

hand, base their conclusion (Eq. (3.107)) on the expression for 20
given in Eq. (3.94) where Zo seems to be proportional to Vp. The

correct statement is

2
Zn - Zm(l + kc ) . (3.109)

The quantities Zm and Z0 are the uncorrected and corrected impedance

values.

Summary

A method has been described by which an equivalent circuit
may be obtained for a section of piezoelectric surface wave trans-
mission line. The expression for the characteristic impedance
followed from the value of the power flux and a suitably chosen
dynamic variable. By means of the vertical shear wave approxima—
tion, it was finally shown how, in retrospect, the initially
omitted electric interaction may be accounted for to produce
slightly higher values for the characteristic impedance and the

speed of propagation.

CHAPTER 4

THE MODIFIED MASON MODEL

4.1 Description of a Surface Wave Transducer and its Simplified

fhysical Model

 

The typical layout of a surface wave transducer is shown in
Fig. 4.1. Alternate electrodes are interconnected. When the trans-
ducer is used for excitation of surface waves, the array is connec-
ted to an electric signal source. Conversely, the surface poten-
tial difference can be detected by the array when a surface wave
travels through it.

The electric field distribution between adjacent electrodes
is rather complicated. It will be dealt with in subsequent
chapters. In the current literature it is assumed that only the
field component in the y—direction contributes significantly to the
surface potential [7], [l3]. Mason's bulk wave transducer model
[5] with crossed-field excitation (Fig. 3.1) is there employed for
an interspace and half of its two nearest neighbors. According to
the results of Chapter 2, the surface wave behavior is in several
respects similar to that of a vertical shear wave. The equivalent

thickness was developed in Chapter 3, Eq. (3.74):

h - A/4wa2 - 10700/m . (4.1)

The surface potential Eq. (3.43) also obtained its major contribu-

tion from the vertical displacement producing an E-field in the

44

45

12W 2 flag/J; “‘

 

 

 

 

 

 

 

 

 

(a)

+ + —

JL/ v 11 /

/ /:
4’

/ l /‘ \M:

  

_.__A

h . depth of t - typical pathlength

equivalent shear wave

 

 

 

(b)

Figure 4.1 - Layout of surface-wave transducers.
(a) Overview.
(b) Detail showing field lines and
equivalent depth.

4.2

46

y—direction:

 

¢s m . (4.2)
“511522
Expression (4.2) is too large, but this is taken into account by
suitable correction factors.
The main justification for this equivalent circuit is,

however, that it predicts observed behavior [7], [13].

Development of the Model for One Electrode Section

The reference directions are shown in Fig. 4.2. It
represents a "free-body" diagram. The thickness is h = A/4naz, the
width W and the length L. It is assumed that a vertical shear wave
can exist throughout this volume and that the E-field consists of
E2 only, where E2 is assumed to be constant within the volume. For

this case the constitutive Eqs. (2.4)-(2.5) are:

T6 - C6686 - e26E2 (4.3)

D2 ' e26s6 + 5252 (“-4)
The standard assumption [12] is that the particle displacement 5(x)

may be described by a standing wave. The same may then be said

about the particle velocity é(x) and the strain Eq. (3.66):

.33.;
S(x) 8x . (4.5)
Let all of the variables be represented by complex exponents.

By Euler's relation a standing wave may be expressed as the super-

position of two waves:

47

 

 

Metallized

 

“it-LII l llJ'd //
, /’//
— +/7 f- _— .— F‘ —— u— I
l
I
l
. “,1 1...,
-Th(U) l l T6(0)
I
.J_ _. _ _. ._ __ a. ‘w.

 

 

V

a — h ._ .— -.._

‘ x
y U21 1 U2
hz

.1 14 /

 

 

Figure 4.2 - Free—body diagram of section under a half
electrode used for the deve10pment of the

Mason circuit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

48

+ K

 

 

28

 

 

 

K2

 

 

 

8(X) - K1 e"jkx
It follows that U1 and U2 are
U1 1 1
U e-jkL ejkL
2
These relations yield K1
U1 and U
jkL _
K1 e l
..__;L.__
ZJsinkL
_ —jkL
K2 e 1
By Eqs. (4.5)-(4.6) the strain is:
86(x) 3X jw 3x j

With the aid of k/w = V

or

36(0)

 

 

36(L)

where the column matrix has been replaced

equations are multiplied by C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P
l 1
56(0) " V“ T
P P
-jkL jkL
e e
56‘” " v T“
P P
_ 1.. .1.
V V
1 p p
ZJsin(kL) -jkL jkL
_ e e
V V
P P

66

to obtain

 

 

 

 

 

 

 

 

- k -jkx k jkx
K1( m ) e + K2933) e .

K1

 

 

 

_e-jkL

by Eq.

(4.8).

 

 

 

 

U

2

 

(4.6)

(4.7)

and K2 in terms of the terminal velocities

(4.8)

(4.9)

, 86(0) and 86(L) may now be expressed as:

(4.10)

(4.11)

 

These

the stresses acting on a

49
surface to the left, providing E2 is held equal to zero:

T6(O) —cos(kL) 1 U
C66/v

76331n(kL) '
T6(L) -1 +cos(kL) U

(4.12)

 

 

 

 

 

 

 

 

 

 

 

 

2
If E2 is not zero, then the quantity -e26E2 is added to both
equations. The internally acting forces at the left and right

surface relate to those stresses by Wh for the left surface and -Wh

for the right surface:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F -cos(kL) l U e WhE
l l 26 2
. th66/v -
jsin(kL)
F2 1 -cos(kL) 02 -e26WhE2
(4.13)

The electric current I/2 is related to the displacement D2(x) by

L L
I/2 - M], D2(x) dx - jwMjp D2 dx . (4.14)
0 0

The current is denoted by I/2 because the section in Fig. 4.2 shows
only one-half of one electrode.

By Eqs. (4.8)-(4.9) 86(x) is

 

 

 

 

K1
86(x) - %— {-e‘Jk" e31“) , (4.15)
K2
0r
ejkL -1 U1
86(X) ' EEV;§IEEE {-e-jkx ejkx] , (4.16)
-e-.jkL 1 U2

 

 

 

 

 

 

 

 

50

-cosk(L-x) coskx

86(x) ” ijsinkL U1 + ijsinkL U2 ° (4°17)

Equation (4.4) combined with (4.14) results in:

L 'Uleze 02826
1/2 III ijfo [W cosk(L-x) + W coskx + €282]dx (4.18)

or

I/Z - ~We26 01 + We26 02 + jwwLezEz . (4.19)

The actual field is not simply 22. Let t be %-the effective path

length to the adjacent electrode of opposite polarity such that

tEZ - V/2 , (4.20)

where V is the actual potential difference. Then Eqs. (4.13) and

(4.19) may be reexpressed as follows:

 

 

 

 

 

 

 

 

 

 

 

z z
0 0
“F1 jtan(kL) jsin(kL) e26w U1
2 2
F - -———il——- -———11-—— e w -U (4 21)
2 jain(kL) jtan(kL) 26 2 ° '
jwe WL
I 2 Vh
‘5) ’e26" '926" h '5?

This choice of algebraic signs is necessary to obtain for an elec-
tric short circuit (V - 0) transmission-line behavior identical to

that derived in Section 3.6, Eq. (3.5). Here as there (Eq. (3.84)),

20 . Wh C66/Vp - C 6 W/2a m . (4.22)

6 2
Since —F1 must now be used as the cross-variable at x - 0, it is

apparent (see Fig. 4.2) that the cross-variable is always the force

51

applied to the face of a "free body" to the right. This is
consistent with the reference direction taken for the power flux
in Chapter 3. Particle motion in the y-direction is represented by
through-variable arrows pointing to the right for consistency with
Eq. (3.5) and Fig. 3.4.

The depth h of the shear wave is not a constant, it rather

decreases with frequency:

h 8 0.52) .

On the other hand, the effective path-length t, chosen to make

V/2 - th,

is in no way related to frequency. But, the device will be used at
a frequency where the separation of the electrodes is roughly half
a wavelength. The quantities t and h will then be of the same

order of magnitude. It is required to make the identification
h = t

in order to form the equivalent circuit in Fig. 4.3, on the basis
of satisfying Kirchhoff's laws. The turns ratio N on the basis of

Eq. (4.21) must be of the form
N - e26 W . (4.23)

For convenience, the turns ratio is actually described as follows:
N = Vp ke Wlezp , (4.24)

if kc is defined as kc before in Chapter 3 as

52

RL kL
jzotan(§v—) jZotanCEV—) U
1 p p 2

——a—
F21

 

 

 

 

 

 

 

 

 

 

 

A (electrode)

f
2c
T V/ 2

. v —: B
(common reference node)

 

Figure 4.3 - Equivalent circuit for a half-
electrode of a sheardwave
transducer.

4.3

53

k ' ‘—-—*——— . (4.25)

and “€66 is expressed as

Jazz'vf; 9

P

then Eqs. (4.23)-(4.24) are identical.

However, it was seen in Chapter 3, that the surface
potential of a surface wave is smaller than the value produced by
the vertical shear wave, which has been used for the development of
the equivalent circuit. To correct this error, an appropriate
value will be chosen for the piezoelectric coupling coefficient ke
in the next section.

The expression waZWL/h relates half the current into one
electrode to half the interelectrode potential difference. It is,
therefore, equal to jm times the capacitance between one electrode
and its two neighbors. The capacitance to one neighbor, however,
is of more practical interest and is defined as C,

eZWL

2t ’

 

C - (4.26)

The values for the elements in the T-circuit are obtained

from Eq. (4.21) as in Section 3.1, Eq. (3.8).

The Determination of the Effective Coupling Coefficient ke

 

The turns ratio in the Mason circuit, Fig. 4.3, is propor-
tional to the coupling coefficient ke' In this section the surface
potential, Eqs. (3.43) and (3.54), will be related to ke by means

of the Mason equivalent circuit. Since the surface potential is

.14

 

54

known a value for ke may be obtained which yields results consis-
tent with surface waves. This is true despite the fact that the
Mason circuit was based on the shear-wave approximation.

In Fig. 4.4 two half-electrode sections are separated by an
inactive transmission line section. This is an approximation since
the area between the electrodes also has an applied electric field.
This field is largely in the x-direction, as shown in Fig. 4.1b.
The piezoelectric constant, 216’ which relates E1 to T6 is zero
(Eq. (3.4)). This approximation is, therefore, consistent with the
vertical shear-wave approximation.

The section is one half of a period D in the transducer
structure Fig. 4.1a. The origin is chosen at the mid-point between
any two electrodes. It was established at the end of Chapter 3
that the presence of an electric field has little effect on VP, the
speed of propagation, and on 20’ the characteristic impedance. It
is, therefore, reasonable to assume that a wave can travel through
this section with a constant velocity and negligible reflections.

Let the complex amplitude be:

U(x) - u e’jkx . (4.27)
The open circuit voltage is
. __i_ - -.__i_
voc ijC ["1 ”2] ijC [U3 U4] ’ (4'28)
kD kD(1-n)
5 2" j 4
where U1 I U 8 U2 I U 8 (4.29)
kD RD 1-
_j T j (4 n)

and U4 I U e U3 I U e . (4.30)

on-..

U
2
_ r_-.———- -‘ ---o *-—-“-“‘ x :3 0
F‘“ _‘ :_——_’{» T )—-—-.

55

 

 

 

.—.-' .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x=~D(I-n)/4 L ] x=l)(l—n)/4 xJU/l.

 

 

 

Figure 4.4 - Determination of the open circuit voltage
between adjacent half electrodes for
U(x) - U e'ka employing sections of the
equivalent shear-wave circuit.

56

The quantity U4 - U3 is multiplied by the complex conjugate of the

quantity U1 - U2. It follows that Eq. (4.28) becomes

kD kD(l-n)
J'_’ j 4

VOC "335 Re[e _ e ] (4.31)
i.e.
. .99.. a; - _kv_____n.<1- >_ ’
0C ij [cos 4 cos 4 ] . (4.32)

This expression gives the open circuit response for any frequency

w I kVp . (4.33)

At present the attention will be focused on the special case
when w is the so-called synchronous frequency mo. At this
frequency the wavelength A equals the period in the comb structure
D. It is clear that, when w I ”o and the electrodes become very

narrow, the open circuit voltage, V is twice the surface

OC’
potential. This is used to specify the turns ratio N.

The Open circuit voltage for infinitely narrow electrodes is:

_ gig lim _1_ 52 _ k0g1-g)
V0C jm n+0 C [cos 4 cos 4 ]}u (4.34)

By trigonometric expansion, and the fact that for small a cosezl

and sinese the open circuit voltage becomes

V I - NC

00 2(2) 81“ 2kg [11m L130] - (4.35)

n+0 4C

(2)

U is also expressed here as ij2 , consistent with the vertical
shear-wave approximation. For m I ”o k becomes 2n/D. The ratio n
of the electroded part to the total length of the section is

n I 4L/D. For very narrow electrodes the capacitance is

57

proportional to L, since a change in L no longer affects t.

- EWL
C -§E- (4.36)

The exact nature of C will be studied in the next Chapter. At this

time it will be sufficient to estimate the path length 2t to be
somewhat longer than D/2. Under these circumstances Eq. (4.35)

becomes

N c(2)

IVOCI z fi-E-fi 2 . (4.37)

To find the correction factor for the turns ratio N (Eqs. (4.23)

and (4.24)), let

N I A e26 W , (4.38)

where A is the required correction factor. The open circuit

voltage should be twice the surface potential, so that

we
_ 26 (2)
|0S| A 2e 02 . (4.39)

 

Equation (3.89) gives the actual surface potential

e
_ 26 (2)
|6S| 0.215 ‘;"C2 (4.40)
It then follows that the correction factor A is
A I 0.14 . (4.41)

The effective coupling coefficient kc in the turns ratio N should

then be (Eq. (3.108)):

ke I 0.14 826/J5C66 I 0.014 . (4.42)

The equivalent circuit is now specified except for an analytical

 

.-'-I . 7"
‘; L 3 '

4.4

58

expression for the capacitance C. This will be obtained in a
different context in the next Chapter. It could certainly be

determined experimentally without difficulty.

ConcludingyRemarks

 

The development of the last three sections pointed out both
the attractiveness of the Mason model and its flaws. The strength
of the model lies in the orderly, analytical relation of the final
matrix, Eq. (4.21), to the initial assumptions. The ease with
which circuit sections were cascaded (see Fig. 4.4) to calculate
the open-circuit voltage (Eq. (4.31)) illustrates the versatility
and usefulness of the equivalent circuit. Its weakness lies in the
fact that there are many farfetched assumptions associated with the
deve10pment of the model; i.e. in both setting up the problem and
in taking the matrix equations to the equivalent circuit form.
First, in order to obtain the desired result, a surface wave is
approximated by a vertical shear wave of finite thickness, even
though the mechanical boundary conditions (T6 I 0 at the surface)
would not permit such a shear wave to exist in a finite medium.
Secondly, the thickness h is frequency dependent, but it has to be
canceled against the frequency independent representative path
length t in Eq. (4.21), in order to facilitate synthesis with
passive elements. Finally, the values of the capacitance and the
turns ratio obtained in the mathematical development have to be
modified in order to give correct results.

In the following chapters, an equivalent circuit for a

complete section of an alternate phase array, as shown in Fig.

 

59

4.4, will be derived. In this derivation many of the assumptions
made previously are relaxed. Even though the subsequent develop-
ment lacks some of the elegance of that which led to the Mason
circuit, it is conceptually and theoretically sound and results in
an equivalent circuit which lends itself readily to analysis and

design.

CHAPTER 5

THE DETECTION OF SURFACE WAVES BY AN ALTERNATE PHASE ARRAY

In this Chapter the frequency response will be formulated by

extending the techniques used by Coquin and Tiersten [4].

5.1 The Residual Solution of the Potential Problem

 

It was established in Chapter 3 that the speed of prOpaga-
tion of an elastic wave in quartz changes very little with an
applied electric field. It will be assumed here that the velocity
under the electrodes, where Ex is zero at the surface, is the same
as between the electrodes or far away from them. In Chapter 3 a

surface potential was deve10ped and is of the form

ej(wt-RX) ’

¢P(x9°9t) ‘ ¢+ ‘ ¢S (5.1)

for a wave traveling in the positive x—direction and far away from
any electrode structure. As in Coquin and Tiersten [4], a residual
solution 6c is introduced so that right under an electrode at peak

potential 60

¢C(x.0.t) + °s ej(wt-kx) . o0 e . (5.2)

However, it should also apply at any point within the material and

be of the following form:

¢<x9Y9t) ' ¢C(x)y’t) + ¢P(x9Y9t) - (503)

60

61

According to Coquin and Tiersten the electric current
injected into an electrode of width W, extending from x = a to

x I b, would be in the present notation

dx . (5.4)

b C
I I -jw622Wf fl- +
a yIO

3y

 

The proof will be developed here for completeness. It is
assumed here, as there, that the DIfield outside the piezoelectric
slab is zero. The error in this assumption can readily be
corrected by appropriately increasing the final interelectrode
capacitance. The conduction current into the electrode is then

b

I I ijJ‘ D2
a

By Eq. (2.5) this becomes

y-0+ dx . (5.5)

 

b
I I ijj; (e2k1 Skl + £22E2) + dx . (5.6)

3'90
But E2 may be expressed as
P C C
E I-fl-_ii.-_a_g._.EP-§_¢—_ (5.7)
2 3y 3y 3y 2 3y

where E2P is the field in the absence of any metallization nearby.
Therefore
I I -jmczijrb %%2- + dx + ij b(e2k13ki+322E2P) + dx.(5.8)

a yIO a yIO
It is assumed that the strain wave is not affected by the electrode.
The integrand of the last term is sz|y_o+, the D2 component at the

surface in the absence of any metallization, which by continuity of

62

the normal D—field is zero.
It seems that the integrand of Eq. (5.4) would be rather

difficult to obtain. The potential ¢ anywhere is a solution of

2 BS askl

2
.2.1 .§_; 1 __JSL ____.
E11 ax2 + E22 3y elkl ax + eZkl ay ° (5'9)

By Eq. (5.3) this may be rewritten as

2 2
c 9493+. fine 1233-5 1.13., 3.51.1.
ll 2 22 2 11 2 22 2 lkl 8x
3x 3y ax By
33
kl
+ e2k1-—§;— . (5.10)

Since the right hand side is zero it follows that the residual

potential is a solution of Laplace's equation

2 C 2 C e I
2_%—.+.§—$——§ - 0 , a - -;ll , (s 11)
8x 3(ay) 22

in an x - ay coordinate system. The boundary condition is given by

Eq. (5.2),

ej (wt-kx)

¢C(x.o.c) - ¢o - 9S , (5.12)

under the electrodes, and in the interspace by

. (5.13)
Y'0

This relation follows from Eq. (5.4) since, for any interval a to b
on the interspace, there is no externally injected conduction
current, yet the mathematical proof for Eq. (5.4) applies there

equally well.

5.2

63

In Coquin and Tiersten [4], Eqs. (5.11)-(5.13) are solved for
the geometry shown in Fig. 5.1 at the synchronous frequency “0' In
this thesis that method will be extended to other frequencies. Such
an extension is a significant contribution to the theory of the
detection of surface waves because it will make possible the con-
struction of useful equivalent circuit models. In order to
appreciate the details of the technique, it is essential that the
reader become familiar with the method used at mo in [4]. This

development is presented in the next section.

The Detection at Synchronous Frequency

Consider a surface wave traveling to the right at synchro-
nous frequency through the infinite array. It is assumed that the
velocity of propagation Vp I walk is constant, and that the wave is
unreflected and unattenuated. From the discussion at the end of
Chapter 3 these assumptions are reasonable for the AC cut in
quartz. Let the mid-point of the electrode, at potential ¢M with
respect to the piezoelectric bulk, be the origin as shown in Fig.
5.1. At the synchronous frequency ”0 a wavelength A is exactly
equal to D, the periodic distance of the array. The portion of D
which is metallized is denoted by nD. The usual value of n is one-
half. The electrodes to the left and right are at a potential ¢L
and ¢R respectively. Because they are directly interconnected
¢L I ¢R and this in turn equals -¢M. All electrodes at potential
¢M are also directly connected. The potential difference V between
the electrodes at potential ¢M and those at ¢L is 2¢M I ¢M - ¢L°

The residual potential right under the central electrode

 

"1}” [11/4 ~n0/4 1“”..0/4 05. 0):.
/ / ,/ ,7 ,7
4 ' / / / ¢R/

 

 

 

 

I
1

 

 

Figure 5.1 - Geometry used for the Coquin
and Tiersten analysis.

 

 

 

 

 

C M
‘_D/4 -nDC4 ¢ + I ¢ 1- °S cos kx qD/4 D/4
x

C+

=0——-.-

 

 

fl ay

 

 

Figure 5.2a - Boundary conditions in the
x—ay plane.

65

is by Eq. (5.2)

t ej(wt*kx)

C Jm
9 (x,o,t) ¢M e — 95 (5.14)
Equation (5.4) becomes
nD/4 C
. - §$L_.
I jug“ 511522 N] 3(8),) _ dx . (5.15)
-nD/4 ay-O

 

Only the even part of the integrand can contribute to the integral.
The pertinent part of Eq. (5.14) is therefore in complex amplitude

form (phasors):

C+
¢ (x,o) I ¢M - 03 cos kx . (5.16)

C+

The superscript (+) denotes that o is the even part of ¢C.

c c
¢C+ _ 9 (X) 3 Q (-X) . (5.17)

 

In general ¢C+ is not known, but since the even part of the surface

potential 0 cos kx is zero at synchronous frequency at x I :D/4

S
and ¢R I ¢L I -¢M there is a virtual ground at x I tD/4 for all

values of y. It follows that ¢C+ must be zero there

Wen/m) - o . (5.18)

A further boundary condition on the field ¢C+(x,y) is that on

nD/4 < x <D/4 and -D/4 < x <-nD/4

C+
9.1... .0

3(8)) ’ (5’19)

ay=0

because there is no current injected into the unmetallized surface

area. These conditions are illustrated in Fig. 5.2.

66

 

 

 

 

 

 

 

 

 

 

 

 

Ci-
0 3 d)” - 05 1111(1))
I-qD/4,0| Inn/4.0!
(~K,'J) /,/I [010' (K,O)
A I I -...,._.._._.... n
-6C+ - «J MO
*‘ - O —" l I‘--- ' r r9
u bu
[21)/ml
I'D/4.0] I l [DI/1,0]
(RAF) / \ (+K.K')
. \
@C 1 o ¢L+ :3 0

Figure 5.2b - Map of the infinite strip in Fig. 5.2. The
corresponding points of Fig. 5.2 are indicated
by [ ]. The boundary conditions are also
indicated.

67

The mapping function

V; sn(wlm) I sin(2nz/D) (5.20)

is used to map the geometry of Fig. 5.2 in the z = x + jay plane
into the w I u + jv plane, where the infinite strip in Fig. 5.2
becomes a rectangle shown in Fig. 5.3. The quantity m, the modulus

of the sn-function, must be

m - sin2(n-3—) . (5.21)

The reader familiar with [4] will find it difficult to identify
each step used here in comparison with that article. It was found
more convenient to use the notation of Abramowitz [14] for the
Jakobian Elliptic Functions. Figure 5.3 shows the result of the
mapping. By Eq. (5.20) the value of x in the boundary condition

Eq. (5.16) must be replaced by

x I %; sin—1[/m sn(u|m)] , (5.22)

so that Eq. (5.16) becomes:

C+

- ¢ cos sin-1[/E sn(u)] . (5.23)

¢M s
sn(u) is used from now on instead of sn(u|m) where the context

leaves no doubt about the modulus. Equation (5.23) may be

expressed in shorter form by recognizing that [14]:

 

cos sin_1[\/u-) sn(u)] I \/l - m sn2(u) , (5.24)

which by definition is-dn(u), another elliptic function. The
quantity K is the complete elliptic integral of modulus m. It is a

quarter period of the sn- and the dn-functions. The quantity K' is

68

 

 

, (7+
[Mlellc 22 "1%;—
yIO
X“. , \SF
4 R: 0 1,41;ng -_-4_._-.,.
\i Eg‘d/ Q/

high0 net emf

h x§
A C > n

(u-I 2(1)
0

no net emf

 

“-f

 

 

 

 

w <w<2m
O O

 

intermediate case

Figure 5.3 - Illustration of the net induced emf in
relation to its nearest neighbors,

712- [A - (n + C)/2].

69

the complete elliptic integral of m', which is related to m by

m' I 1 - m . (5.25)

The value of the dn-function at u I 0 is one, and at a quarter

 

period K is
dn(K) I Vm' I V1 - m . (5.26)
Laplace's equation becomes in the u-v plane
32¢C+ 32¢C+
2 + 2 I 0 . (5.27)
an av

Following the method of separation of variables the solution is of

the form
C+ _ _‘!_ Vf‘ ‘21 , _ nnu
¢ Ao(l K') + [:3 An sinh[K (K v)] cos(7E-) . (5.28)
nIl
Equation (5.15) becomes in the u-v plane
K 3 C+
1 - flew/FT)! —L— du (5.29)
11 22 -K av V‘O

19:31
From Eq. (5.28) it is seen that 3V is also a Fourier series in u

for any value of v:

C+ A I—- I
29.... - .2. ' .212
3V K' + 2:. sn(v) cos( K . (5.30)

nIl

C+
0f principal interest is the average value over u of 2%3— [see
A
Eq. (5.29)], the coefficient ao I --E$ [see Eq. (5.30)]. It turns

out to be independent of v.

A0 1 B C+

70

Equation (5.29) is therefore:

+
By Eq. (5.28) A0 is, in turn, the average value of ¢C (u,o), which

is the upper boundary condition:

¢C+(u’°) ‘ ¢M ‘ 05 dn(u) 9 (5.33)
hence,
K
A0 - i- t,“ - .3 66(6)] du . (5.34)
-K

The equation relating the electrical quantities I and V I 2¢M to

the surface wave is then:

K
I - 30)“ 811522 WK/K'lv - JwV £11522 w °S[-11zrf dn(U)dU] 0 (5035)
-K

By the integral relations listed in Abramowitz [14]

“/dn(u)du I sin-1[sn(u)] , (5.36)

but sn(K) I l and sn(-K) I -1, so that the square bracket in Eq.

(5.35) becomes

K
'%T J[ dn(u)du I-%T . (5.37)
--x

With this simplification Eq. (5.35) becomes

1 I ijV I jchllezz Wu Os/K' . (5.38)

The quantityvenc22 WK/K' has been identified as C, the capacitance
of one electrode to its neighbors. The relation shows how the

current into the electrode depends on both the applied potential

5.3

71

difference V and the amplitude of the surface wave 0 . The short-

8

coming of Eq. (5.38) is that it holds only at the synchronous
frequency. At other frequencies the current I will still depend on
V by ij but the second term will have to change drastically, for
it is reported by other investigators [15,16] that, for example,
the response is zero at twice the synchronous frequency, or it is
also clear that the response should peak somewhere near synchronous
frequency. None of this can be deduced from Eq. (5.38). However,

with care, the preceding procedure can be generalized to other

frequencies. This is done in the next section.

The Frequency Response of the Short Circuit Current from Electrode

to its Nearest Neighbors

 

Recall that the ultimate objective of this research is to
obtain a useful equivalent circuit for an individual section of the
array. The correct overall effect can then be accounted for by
cascading individual sections. In the next Chapter such a section
will be properly defined and its equivalent circuit will be
developed. To lead up to that development the frequency response
of the short circuit current from an individual electrode to its
next two neighbors will be studied here. In the following Chapter
this idea will be extended to account for the current to all
electrodes of opposite polarity.

In order to study the response at frequencies other than the
synchronous frequency ”0 Eq. (5.38) must be generalized. Referring
back to Fig. 5.2 it is seen that ¢C+ I 0 at x I iD/4. This is so

because of two reasons: First of all, at x I iD/4

72

Zn

kx I +
' A

 

:4U

1.
1 2 . (5.39)

so that cos(kx) is zero there, and secondly D/4 is the mid—point
between the applied potentials ¢M and -¢M I ¢R’ so that it is a
virtual ground from the point of view of the externally applied
potentials with respect to the piezoelectric bulk.

If the frequency is somewhat higher than mo, A/4 will fall
short of D/4, but the mid-point between the electrodes at potentials
¢M and -¢M respectively would remain at D/4. The latter problem
can be resolved easily by considering the short circuit response,
i.e. the current when ¢M I ¢R I 0. Then ¢C+ will be of the form
-Acoskx and the lines given by ¢C+ I 0 in Fig. 5.2 would occur at

A/4. The next difficulty is now that the current I determined

SC’
in this fashion, would be a current between the electrode and the
bulk of the piezoelectric material since it has been assumed to be
the reference node all along. At the synchronous frequency this
causes no problem; the current injected into it by one electrode "88
equal to that taken out by one of its neighbors. A connection to
the bulk was implied but it made no difference.

An extension of that reasoning to other frequencies is to

assume at first that a fictitious short circuit current still flows

from each electrode to the bulk based on Eq. (5.4).

nD/4 C
. ‘1" "" 2.9..—
ISCM +jm £11222 W 3(ay) dx , (5.40)
-nD/4 ayIO

 

but really there is no such connection, so the actual current from
the electrode to its neighbors [see Fig. 5.1] depends on the

difference between the induced emf on the electrode under

73

consideration and the average emf induced in the two neighbors.

if again only the even part with respect to the origin is
considered the waves shown in Fig. 5.3 illustrate that point very
well. Areas above the x-axis are positive and correspond to the
fictitious current from electrode to the bulk. Areas below the
axis represent current from the bulk to the electrode. The actual
current from the center electrode is then

I _ ISCM " (ISCL + Isca)’2 (5 41)
so 2 ‘ ‘

 

To be more general consider a superposition of a wave to the right
and the left. At the surface under the electrode, 6C is simply the

negative of op:

¢C(x.o) - 46$" 63“" + .S" elk“) . (5.42)

Equation (5.40) becomes now for the three electrodes:

 

 

nD/4 C
.. ‘/ ' 39;—
I CM + jw £11522 W‘]r 3(ay) dx , (5.43)
-nD/4 ayIO
nD/4-D/2 C
.. m/ 132.—
ISCL 115 22 W Jf 3(ay) dx , (5.44)
’UD/4-D/2 ayIO
nD/4+D/2 C
- Mr;— §_L_
ISCR 115 22 W.]” 3(ay) dx . (5.45)
-nD/4+D/2 ay=0

In order to comply with Eq. (5.41) a change in variables in the
last two integrals will result in the same limits of integration

as that of the first integral (Eq. (5.43)). Equation (5.44) will

74

get the integral

 

 

nD/4
__.§__ C -
/ 3(ay) ¢ (x D/Z’aY) 9* (5.46)
‘ ‘nD/4 ayIO
and Eq. (5.45) will get the integral
nD/4
’ 3 ¢C<x + n/z.ay) dx . (5.47)
3(ay)
' ‘nD/4 ayIO

 

The actual short circuit current by Eq. (5.41) is now:

nD/4
3 C 1 C
ISC I +juMc11522 W[§3?;;3'Jr [0 (x.ay) - 5'0 (x-D/2,ay)
-nD/4

--% ¢C(x+D/2,ay)]dx] . (5.48)
ayIO

The value of ¢C(x,o) is given in Eq. (5.42). In general, it is

reasonable to assume that ¢C is of the form

jkx

¢C(x.ay) . -l¢+(x.ay) e" + ¢'(x.ay) ejkx] (5.49)

+ -
where the wave amplitudes ¢ and ¢ may be functions of both x and
ay, but, because all electrodes are at zero potential they must be

periodic in x with period D/2, and they are symmetric about XIO.

¢i(x t D/2.ay) - ¢t(x.ay) (5.50)

The integrand of Eq. (5.48) is then, if the argument of the ampli-

tudes is left out:

+e-jkx(ejkD/2 + e-jkD/Z 1 - jkx

) - 5'6 e

_[¢+e-jkx + ¢-ejkx _‘l

2 ¢

e-jkD/Z + ejkD/z

( )1 . (5.51)

75

or

+

-[¢ e-jkx[1 — coskD/Z] + ¢-ejkx[l - coskD/2]] . (5-52)

The identities

 

l - c0529 I 2 sinze (5.53)
and
kD/4 = 2“ .-9 =-1&- (5.54)
A 4 Zwo

simplify Eq. (5.52), the integrand of Eq. (5.48), to

2¢C(x,ay) sin2(%§— . (5.55)
o

The short circuit current from the middle electrode is then

nD/4
2 am 3 C
va 11:22 W sin (55;) Jf -3?;;7 p (x,ay) dx . (5.56)
-nD/4 ayIO

The integral is solved by the procedure outlined in Section 5.2.
As mentioned earlier the infinite strip shown in Fig. 5.2 does not
extend from -D/4 to +D/4 but rather from —A/4 to +)/4 (see Fig.

5.4) and ¢C+ on the electrode is now by Eq. (5.42)

¢C+ = -(d>S+ + ¢S-) cos kx . (5.57)

By Eqs. (5.35), (5.37) the result will then be

-36¢elle 22 W(¢S + 68 )[./x (6)1316 2(“0) . (5.58)

The elliptic integral K'(m) is now a function of frequency since
the infinite strip in Fig. 5.2 changes width with frequency (see

Fig. 5.4). The validity of Eq. (5.58) extends only to a frequency

76

 

 

A/4

 

 

 

1/4 Ci +
_,-.¢ :»(Pp+¢v)cnskx
4V4 ” °
+l)//.
J . 41 i
T L I l
\ [tn 3:: ¢L :2 tr,“ 1.. 0 .....__..
\ ,. (2+
9*“
Bay
Vt C+
M. t. O 3’ O M.-
V

 

 

+“[(x.av) ' I[¢+(x,av) + 67(x,ay)]eos kx

 

Figure 5.4 — The geometry for the field problem with no
external potential applied and w I mo.

77

less than that which would produce a quarter wavelength equal to

nD/4, the edge of the electrode, i.e.

A > [)0
211V 21tV
01’ -——-B- > n --—R
(L) (A)
O
or -9— < l/n .
mo

Furthermore, since the boundary condition requires on the range
nD/4 < x < A/4

that-3%z—T— I 0 there is also an upper limit on )/4:

3 C+
Y
A/4 < D/Z - nD/4 ,

which defines the edge of the next electrode. In terms of a

frequency restriction this means

 

The combined statement of the frequency range over which Eq. (5.58)

is valid is then

l
2-n

 

<

Eli-z
A
JIF‘

. (5.59)

o
The frequency dependence of K'(w) is caused by the modulus m'Il-m
of the elliptic functions (Eq. (5.21)),
, 2 n
m I 1 - sin (n? . (5.60)

Here, this relation must be modified, because n represents no

78

longer the ratio of electrode width to interelectrode distance but

rather electrode width to A/Z, half a wave length. Hence

m'(w) - 1 - 313(91— (5.61)
mo

The complete elliptic integrals K' are listed in Abramowitz [14] as
a function of

“1111)

a.
2w
0

In Table 5.1 the normalized form of Eq. (5.58), F(w/wo), is computed

for n I 1/2, where

I’m/mo) a ISC/jmox/enezz we); + 05'). (5.62)
i.e
Fan/mo) - 53— sin2(12'—3-)[n/K'(wlwo)] . (5.63)
O O

The frequency response is plotted in Fig. 5.5 for n I 1/2 for
frequencies between Zoo/3 and Zoo. Most applications fall in this
range. It is seen that the value of F at (o/wo I l is equal to

that obtained in Section 5.2. In addition it should be noted that
the functional form of Eq. (5.58) is identical to Eq. (5.35) if

m I mo and V I 0. It is furthermore seen that the response is zero
at w I 2wo as it must be. The peak for the short circuit current
response occurs near w I 1.2 mo, i.e. higher than the synchronous
frequency. This will not be the case in the open circuit voltage

frequency response, which is derived next.

79

TABLE 5.1 - Normalized Frequency Response of a Single Section

for n I 1/2, 2wo/3 < w < 2wo

~-.~--

 

 

who a = 1119 K'(w/w ) Foo/6 ) [Fm/m )/(6/.. )1-‘9—
0 2w 0 o o 0 K
7-1;};.M..v~0—36° 2.01327 1.12915 1.41144
1.0 45° 1.85407 1.69443 1.69443
1.2 54° 1.741499 1.95804 1.63170
1.4 63° 1.66272 1.73131 1.23665
1.6 72° 1.61045 1.07835 0.67397
1.8 81° 1.58054 0.34165 0.18980
2.0 90° 1.57080 0.0 0

 

80

f. 2..

1"”

 

 

Figure 5.5 - Normalized frequency response of the short
circuit current from an electrode to its
nearest neighbors.

F(w/wo)
n I l/2
7[ 1.0
/
/
/
/
/
/’
./

//

_L/ j:
1.0 2.0
w/wo

5.4

81

The Norton Equivalent Circuit

 

By Thevenin's theorem the product of the short circuit
current times the output impedance is the Open circuit voltage.
The output impedance is obtained from Eq. (5.38). All independent

sources are set to zero, in this case 0 then the ratio of the

S’
externally applied voltage to the current is the output impedance.
This is clearly 1/ij. A Norton equivalent circuit for a section
consisting of a single electrode at x I 0 with the potential

referred to its neighbors is then shown in Fig. 5.6. The voltage

current relation for other than short circuit terminations follows

directly:

1 - ijV - 1667611622 u(cps+ + ¢s-) F(m/mo) . (5.64)

It is seen from this that the open circuit voltage response equals
. + _
. I
v0C (K /K)(0S + 98 ) F(w/wo)/(w/mo) . (5.65)

The value of C used here is given by VEZIEEE W K/K'.

The normalized open circuit voltage response is plotted in
Fig. 5.7. The peak has indeed shifted to the left in relation to
the peak of the short circuit current response.

The technique used here in this Chapter is an important
extension of the work by Coquin and Tiersten, since it will lead
to equivalent circuits which apply specifically to surface waves

and are not based on analogies to bulk waves.

82

I

afi———-I
_.. electrode

 

 

 

T 1:... .. -
ISC ‘ on/ 11622 WI¢S+¢S]F(w/wo)
—L—- ' ———-—-—-—— '
V
i :. common node of
neighbors

Figure 5.6 - Norton equivalent circuit of a single
electrode and its nearest neighbors.

.-.. -. —~-—-

: I w/
1.0 2.0 “’o

 

 

Figure 5.7 - The normalized frequency response of the
open circuit voltage.

CHAPTER 6

EQUIVALENT CIRCUIT MODELS FOR SURFACE-WAVE TRANSDUCERS

The transducer models presented in this Chapter have been
developed specifically for surface waves and do not rely on an equivalent
bulk-wave behavior. The two circuit models derived here are one with
dependent generators and the other, derived from the first, with an ideal
transformer similar in form to the Mason model. The starting point is

Eq. (5.43) in addition to various surface-wave transmission line models

of Chapter 3.

6.1 An quiyalent Circuit with Dependent Sources

With Eq. (5.58) an expression was obtained for the short-
circuit current from an electrode centered at x I O to its next
neighbors. This led to a Norton equivalent circuit from which Eq.
(5.65) was derived, which in turn gave rise to the frequency
response plot [Fig. 5.7] of the open-circuit voltage of the
electrode with respect to its neighbors.

In order to arrive at a valid equivalent circuit for a
suitably small section, that can serve as a building block for a
larger array, the constraint that the current exchange only occurs
between an electrode and its next neighbors [see Eq. (5.41)] will
be lifted. The nature of the final equivalent circuit will be such
that short-circuit calculations based on it result automatically in

the current between electrodes rather than current from an

83

84

electrode to the grounded bulk implied by Eq. (5.40).

Nevertheless, the current from an electrode to the grounded
hulk is obtained first. By comparing Eqs. (5.40), (5.56) and (5.58)
the conclusion can be drawn that the current to the bulk for an

electrode at x I 0 will be:

SCH - 3w 611:22 Win/K'(w/wo)](¢s+ + $5-) . (6.1)

If the electrode is located at a point xn I nD/2, i.e. n half
periodic lengths to the right of the origin, the equation becomes

-jkx +jkx

In - jw/cllazz W[n/K'(w/wo)][¢s+ e “ + 08' e “1. (6.2)

An array of N electrodes shorted to the bulk is shown in Fig. 6.1.
All odd numbered electrodes are interconnected first before this is
done. The same is done with the even numbered electrodes. Thus
the total current into the bulk would be 11 + 12 + I3 + . . . . IN.
This is suggested by the mathematical formulation of Eq. (6.2), but
it is physically not the case. By analogy to Eq. (5.41) the actual
short-circuit current from the upper half of the array [see Fig.

6.1] to the lower half is determined by one-half the difference in

emf's, which leads to

l
I ”-2-[1

- + - + a a a a a '-
SC I 1 I I

1 2 3 4 (6'3)

N] °

By Kirchhoff's current law this result is simply achieved by
connecting the upper electrodes directly to the lower electrodes,
bypassing the bulk node. The strength of each generator must also
be cut in half. Figure 6.2 shows this connection as a hatched

line. To guarantee the correct output impedance between terminals

85

 

 

 

 

 

* 12 14 $11“ /
12+14+°°°

Figure 6.1 - Connection of generators consistent with the meaning of
Eq. (6.2). The resulting short circuit current is
physically incorrect.

5:34

 

 

 

 

A
\.
I g -
' : + \\ ISC Ils I23
I -1 /2 I =1 /2 ¢
:91 1 s3 3 A \ +138
I ' \
| I bulk no __ v
j { longer connected _‘__ AB '
- I i ”-5 i
l - E9. ,
I l 2 /
"32312/2 154°14/2 ¢B
/
I ' /
I -— ./
+ .1
B

Figure 6.2 - Choice of circuit interconnection to obtain a meaningful
expression for the electric terminal behavior.

86

A-B a capacitance of value C must be added as shown for each pair

of current sources. The terminal voltage V is now the difference

AB
between the potentials referred to the bulk, ¢A and ¢B° The effect
of the field above the slab must finally be included so that the

value of C is given by [16]:

= 1
C (EO+V611622)WK/K , (6.4)
and the strength of each current source is

Isn - jw £11522 W[n/2K'(w/wo)][¢s+ e.jne + ¢S- ejne] , (6.5)

where

e - kD/2 = ww/wo . (6.6)

The current sources depend on the surface potential at points xn.

In Chapter 5, it was assumed that under short-circuit con-
ditions the transducer behaves as a piezoelectric transmission line,
without losses or reflections and a constant velocity of propaga-
tion. Consistent with this assumption will be the equivalent
circuit in Fig. 6.3 from which ¢P(xn) may be determined, where

+ -jkx _ +jkxn

I’ 4..
¢ (xn) - ¢ e n + ¢ e . ¢ e jne +

s s s ¢s e

The circuit, however, applies only if the electric terminals
A—B are shorted. Since the array is a linear, passive system the
reciprocity theorem must apply to any three-port network formed
from it. For this purpose cut out a section between the center of

two adjacent electrodes at xn and x respectively. The current

n+1

sources Isn and Is<n+1) should then be halved. The appropriate

lb

 

 

0

Y llszln‘é
o

 

'D
vi (xn)

87

I!
0 (xm‘)

 

 

 

 

 

 

 

[iY Van(9/21
l o

 

 

 

1Yotun(0/2)

 

 

 

 

 

q_ _- Y /isinu
CT | o
liYoiun(0/2)

 

 

 

 

 

ontun(9/2)

 

 

 

 

 

 

 

 

Figure 6.3 - Equivalent circuit to determine ¢P(xn)
when terminals A-B are shorted.

 

  

 

—r~-~
‘ A
.F————1 +.
[sn P
-2-- = M (xn)
Y a ’MVZIICQZ NIH/4K'(m)]
~—_ o~- ~—*O V
All
//l‘\\ -_ to bulk
9 +1 P
I -_q%“) . Y° (xn+l)
I 1P.
+ “—"‘ '-
l_ 4- - n

 

 

 

Figure 6.4 - Temporary equivalent circuit of the electrical
part of a basic section extending from the mid-
point of one electrode to its neighbor.

88

capacitance is then also C/Z. The electrical part of the section
is shown in Fig. 6.4.

The symbol 7 has been introduced for the sake of brevity.

Y ' Jw £11622 Win/4K'(m/wo)] (6.7)

It may be interpreted as a transconductance. The difficulty with
the circuit model shown in Fig. 6.4 is that it is not a one-port
network but rather a two—port network because of the presence of the
connection to the bulk. However, it was seen in the development
that led to Fig. 6.2 that the contribution to the short-circuit

current out of terminal A and into terminal B is

ISO - Ian/2 - 13(n+1)’2 - y[¢P(xn) - ¢P(xn+1)] . (6.8)

Since the output impedance is determined by C/2 the Norton equiva-
lent circuit for this section will be that of Fig. 6.5. The

terminal equation for this is

1/2 - ij vAB/z - y¢P(xn) + y¢P(xn+1) . (6.9)

The equations for the transmission line part under shorted condi-

tions are by Eq. (3.6)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1P ) Yo -Yo ‘ P< )
(xn jtane jsine ¢ xn
I X o (6.10)
-Y Y
P o o P
-I (xn+l) jsine jtane ¢ (xn+l)

 

 

Equations (6.9) and (6.10) are next combined. By the reciprocity

theorem Y13 - Y31 and Y23 - Y32:

89

 

 

 

1/2
_.___. x
n
O 0: 40A
+.
P P
___J___ on YM (xn)-¢ “n+1”
/"'"\ V
AB
. d ’B
xn+1

Figure 6.5 - Norton equivalent circuit of the electrical
part of a basic section between xn and xn+1.
The current is denoted by the symbol 1/2
because it represents approximately one-half
the current into the electrode at xn.

90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P Yo -Yo P
l (xn) jEEhB' jsins -Y ¢ (xn)
-Y Y
P o o P
-1 (xn+1) a EEIEE jtane Y x ¢ (xn+l)
1/2 ‘Y Y ij/Z VAB
(6.11)

Because the effect of the electrical part on the transmission line
part is now known, the complete equivalent circuit, valid under all
conditions, is now obtained and shown in Fig. 6.6.

When such sections are interconnected, proper polarities
must be observed. All current sources associated with terminal A,
i.e. (xn), point up; all current sources associated with terminal
B, i.e. (xn+1), point down. In the next section an example will be
presented to demonstrate the utility of this equivalent circuit
model. It will be used to study the detection of a surface wave by
an array and the excitation of surface waves from the electric

terminals.

529}X§1S of the Detection and Excitation Problem with the Equivalent
.91.:ng

Consider an array made up of twenty identical electrodes.
The field distribution at the end sections is quite different from
that for the other sections. It is therefore uncertain whether to
account for 19, 20 or 21 sections. For a large array it won't

matter. In this case the mathematically most convenient value will

be chosen. The array has an equivalent circuit as shown in Fig.

91

 

 

 

 

JYntan(0/2)

 

 

 

 

 

Y /)sln6
o

ip(x )

n+1
a.”

A A

 

 

 

 

 

 

ontan(0/2\

 

 

 

+-
YVAB
P
Q (‘nvl)

 

v

V

Yoptxn+1)

P
70 (xn)

V'v

X
n+1

Figure 6.6 - Equivalent circuit of a section shown in
Fig. 6.7.

91

 

 

 

 

 

)Yotan(0/2)

 

 

 

 

Y /}san
o

 

 

 

Y¢P(xn+1)

P
70(xn)

 

 

 

ontan(G/2)

 

 

n+1

Figure 6.6 - Equivalent circuit of a section shown in

Fig. 6.7.

92

 

 

 

 

 

Figure 6.7 — A basic section for which the equivalent
circuit in Fig. 6.6 is applicable.

93

6.8. In the electrical part of the equivalent circuit, the value
of the current sources y¢P(xl) and y¢P(x20) has been doubled. This
is mathematically convenient and corresponds to an assumption that
the 19 sections of the array are terminated on both ends by half-
sections. Apart from this convenience, it is also consistent with
the results obtained from Fig. 6.2, which originally gave rise to
the present equivalent circuit. The procedure to be extracted from
this is stated below:
Each electrode has a dependent current source of
strength 27¢P(xn) in the lower part of the equivalent

circuit, and a source of 2yV in the upper part. If

AB
the particular electrode is connected to node A, the
terminal labeled with a (+) sign, the reference arrows
must point up. For node B, they point down. Now,
without resorting further to matrix equations, the
model is put together by giving each space between
the electrode centers a n-section of characteristic
admittance Y0 and transit angle 6 = nm/mo in the
upper part, and a capacitor of value C/2 in the lower
part. Then each electrode is accounted for as
described above.

This procedure extends easily to unequal electrode spacings and

apodized arrays [ 7], i.e. arrays where the "acoustic aperture" is

changed by letting adjacent electrodes overlap only partially. An
apodizing function w(x) is defined [13]. It determines the

fraction of overlap. The generators in the upper part of Fig. 6.8

each become

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘v’ /j5|ne Y /lr-lne Y WISH“? V Using
1 O X 0 X H i 0
n-l an n+l nil
A A . . H
' "t " “TB [‘1
1 T.
. a an n [You-nu“) _) gYntnuw/Z) )Yotanw/Z)
J 3A
4
P P P P
2yl¢ (x )-¢ (x )+¢ (x ) ..~¢ (x )1
_____ l i 3 10
Inc N
VAB

Figure 6.8a - Equivalent circuit of twenty electrodes.

 

ijIJll )2 NIH/4K'(uhoo)l
t = (to + V: ll c2?) wK/K
8 a flm/u ’ kD/2
0
'7 (AC cut)

Yo - MN x 2.68 X IO U

 

 

 

b

 

 

 

 

 

iY Mtanw /" )

 | _—;:}isin8m
l

a \ /0

 

 

 

 

 

Y /isin0 Y Iislne
n o m m

13%] i; :_

. /4
1Y (sn(Q /Z) \iY inn(8 /2)
o o m m

 

0

 

——-— ----~ -

 

Figure 6.8b -

Interspaces are broken down each into three
sections to account for difference in velocity
and Y0 under metallized surfaces for materials
with stronger coupling.

95

+1
Ign a 2Yw(xn) vAB (-1)n , (6.12)

with the reference arrows pointing up and the total strength of the
current source in the lower part becomes
N

. n P
ISC = "ZY £31 (-1) ¢ (xn) w(xn) . (6.13)

Each capacitance value is also multiplied by w(xn), which has some
value between zero and one.

In the present discussion the sections are identical so that
w(x) - 1. But here too a more sophisticated model could be employed
to account for the difference in Vp and Y0 under the metallized
parts as compared with the unmetallized interspaces. By the dis-
cussion of Section 3.7 this is not worth the effort in the case of
quartz, but in materials with stronger coupling, such as lithium
niobate, the section between the electrode centers could be broken
down into three parts [see Fig. 6.8b]. The change in velocity, and
therefore of Y0, under the surface metallization has been investi-
gated for lithium niobate by Campbell and Jones [ 3]. Their
tabulated values could be used in connection with this equivalent
circuit. Smith et al have produced a similarly composite equivalent
circuit based on the Mason model [12]. Usually these complications
are ignored. Back to the analysis problem, for n - 1/2 (half the
surface area is metal plated) the elliptic integrals K and K' are
equal. If the width is 2mm then the total capacitance 10C equals
only 1 pF. A high frequency probe with an input capacitance of 9
pF is connected to the array. This will lower the measured voltage

to one-tenth its theoretical open circuit value, but because of

96

this it will lower the strength of the current generators Ign so
that it is quite reasonable to assume that the wave is unaffected
by the array. The form of ¢P(x) throughout is taken to be

-jkx _ _
¢P(Xn) = ¢ + e n - 68+ e 3(n 1>e .

S (6.14)

The value of x1 was taken to be zero and e a nw/wo. The terminal

voltage is then

+
0.021¢
_ _fi____§ _ '19 j28 _ _ 'jl98

VAB ij [1 e + e .. e ] , (6.15)
or

o 02 0+ -jzoe
V - ' Y S 1 - e . .021 e—j19e/2 sin(lOe) ¢ +
AB ij 1 + 8-39 wC cos(e/Z) s '

With the form of y and C inserted, this becomes

x' + n x 0.01 Si“(1°"“/wo)

AB K . ¢S . 2K'(w/wo) . cos(ww/Zwo)

 

 

. (6.16)

At w - ”0 the value of sin(lOnw/wo)/cos(nw/2mo) is 20. At n - 1/2
K' - K - 1.85407, so that VABMS+ has a peak value of 0.16944.

This value is essentially independent of the number of electrodes
used, provided the load capacitance seen by the array is nine times
its own output capacitance. The value is identical to that
obtained in Chapter 5. The advantage of many electrodes is the
decrease in output impedance and the very selective frequency
response. For general values of w/wo the function has been
programmed in Appendix F. The quantity K'(w/mo) is by Eqs. (5.59)-

(5.62) the complete elliptic integral of modulus

m' 8 l - sin2(nnw/2wo) . (6.17)

97

According to Abramowitz [14] K' may be calculated with the

"geometric—arithmetic" mean aN:

n/K' = 23 (6.18)

N O
The geometric-arithmetic mean is found to great accuracy within
very few steps of the following algorithm. As an example, w/wo is

taken to be 1.2:

D
u
H
V
II

sin(nnw/2wo) = .809017 (6.19)

 

a + b
o 0
a1 = -——§-——-= .904508 b1 - Vaobo = .899454 (6.20)
a1 + b1
a2 = —~—E——-- .901981 b2 8 Valb1 = .901977 (6.21)
a2 + b2
a3 = - 2 = .901979 b3 = \laZb2 = .901979 (6.22)

The quantity 2a = 2a3, so that n/K' = 1.803958, for m/mo = 1.2.

N
The frequency response could only be computed for the range

%-< w/mo < 2, because the validity of Eq. (6.1) has not been
established outside of that frequency range. Figure 6.9 is a
computer plot of Eq. (6.16). The general shape of the frequency
response is quite similar to that obtained by Tseng [16],who used a
flat—field approximation between the electrodes for the excitation
of surface waves. It will be shown next that, within a constant of
proportionality, the frequency response of the array used for

excitation of surface waves is the same as that developed here in

Eq. (6.16) for the detection of surface waves.

98

~o-m-mom~.~
~01mo-m¢m.~
ooom~o50¢m.m
Noammmdmmm.a
~oomcmw~om.~
Noumeasqnm.~
nonw0¢o¢~a.m
moawoommmm.~
Nocmoaooao.m
woumomsoo~.m
oo-aomm~no.m
mo-wommo-.m
donmmwsmoo.a
soummmeomm.~
donmnm¢¢oo.~
doomsqomam.a
acumsmuomo.~
Nouma0¢¢mo.w
~01mo~o-o.m
No-mmflm¢~a.~
~oumn~om~¢.m
Noummmoomo.~
soauoodno~.a
mo-mmmom~n.a
Noum:umomo.a
mouumaaoa~.~
noums¢~o-.~
~01mm¢oomo.~

+

S
N A
B
A
v

w
/
w

 

Figure 6.9a - Frequency response of a 20 element

array terminated in a relatively

large capacitor.

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99

i
1 .
i
1

‘\ 0‘
19,,
e
-.-A----- .

rail “Hunt ”.4...“
v V

Figure 6.9b - Central peak of the frequency response of a 20

element array.

100

Let it be assumed that the transducer is terminated on both
sides by an infinite piezoelectric line with a characteristic
admittance Yo equal to that of the transducer array.

The line generators are each of strength

n+1
Ign (_1) 2Y VAB 0

(6.23)
[see Fig. 6.8], all reference directions pointing up.

By the principle of superposition the response to the
generator connected to xn is considered first with all others set
to zero. The total input impedance seen at point xn is, because of

the matched conditions, 1/2 Yo. The line potential at this point

is therefore:

n+1

¢P(xn) ‘ (-1) 2y VAB/ZYo . (6.24)

From this point a wave spreads in both directions. At a point x to

the right of the array the line potential will be:

 

YV -Jk(x-x )
¢np(x) - (-1)‘“+1 _ng e “ . (6.25)
o
The complete response at x is then by superposition:
20
P P
was? 6“ (x).
{#1
or
yV _ jkx jkx jkx jkx
¢P(x) - —YAB e JKX [e 1 - e 2 + e 3 . . . . - e 20]. (6.26)
0

Once more, let x1 = 0 and kxn+1 - kxn - 6, the result is the

identical progression, but with positive exponents, as in Eq. (6.15).

101

It sums therefore to

 

yV _, _ jZOG
¢P(x) a: TA}: 8 ka }‘—‘e"5’6’— o (6.27)
o 1 + e
This simplifies to
¢P(x) - -3 63199/2 YVAB e‘jkx sin(109)/Yo cos(e/z) , (6.28)
or
jl9flw/2m _
¢P(x) . -j e ° vAB A(w/wo) e ka/Yo . (6.29)
The function A(w/wo) is
1n(10rm/w )
a l§£2££9g1.3 ' S O
A(w/6o) cos(B/Z) 167211622 th/AK (w/wo)] cos(flw,2wo) . (6.30)

From Eq. (3.58) the value of Y0 is for the AC cut:

Y0 = w” x 2.68 x 10'78 . (6.31)

This results in a cancellation of the term wW:

j19nw/2w
P - - 0
¢ (x) ' JF11¢22 e V

 

-jkx 5
AB e x 4.7 x 10

Zn sin(10nw/wo)
k'(m/wo)cos(nw/Zwo) .

 

(6.32)

Comparing the last fraction with Eq. (6.16) shows that the shape of
the frequency response is identical here. At w - ”o the absolute

value of Eq. (6.32) is

|¢P(X)/VAB| - 1.2 x 10‘3 (6.33)

6.3

102

which means it takes an excitation of 83 volts to produce a surface
potential of only 0.1 volt, but the associated line current is
rather high. At w = 1.5 x 109/sec and W - 2.5 mm, typical values,
the line admittance Yo equals 1 u, so that [IP(x)I is 100 milli—
amps. However, the purpose here was merely to demonstrate the
versatility of the equivalent circuit, not to judge the efficiency

of the transducer.

An Equivalent Circuit Model with Ideal Transformers

In the last section it was shown how readily the equivalent
circuit developed in this Chapter lends itself to mathematical
analysis. The dependent generators are no drawback. In the
excitation problem they offered, in fact, an advantage over the
ideal transformers found in the Mason model because of the principle
of superposition, by which all but one of the sources may be set to
zero at any time. This facilitates the calculation of the response.
No direct equivalent procedure exists for transformers which are
passive circuit elements. Nevertheless, for the sake of interest
an equivalent circuit will be developed in this section that
resembles the Mason model with its ideal transformer. The
principal difference lies in the fact that, in Chapter 4, the
equivalent circuit is developed for half-electrodes; here, the
equivalent circuit will be centered on the interspace with one-half
of the adjacent electrodes on either side as illustrated in Fig.

6.7. The Mason model of Smith et a1 [10] is centered around the

lnterspace as in the present treatment.

103

In order to accomplish this the dual of the upper part of
the circuit of Fig. 6.6 is found first. The technique is
illustrated in Fig. 6.10. The admittances become impedances with
the same numerical value; meshes become nodes. When a current
source pointing right is encountered by a path linking the mesh
centers, it is replaced by a voltage source with a "- +"
orientation. The voltage across the current source becomes the
current through the voltage source. The reference directions must
be such that the power relations are maintained for that source.
The short linking two sections becomes an open circuit. The current
through the short will be the voltage across the open circuit. The
equations resulting from the dual circuit shown in Fig. 6.11 are
identical to those obtained from the original circuit in Fig. 6.10.
However, the physical units are all wrong. This can be easily
corrected by multiplying ¢P(xn) by the factor -jw/;;IE;; W,
similar to Eq. (3.44). The quantity jw/EIIEEE'W¢P(xn) has units of
electric current and it points to the right [see Fig. 6.11]. In
order to maintain the equality in any conceivable circuit equation,
all possible ratios of "voltages" to "impedances" 20' = YO must be
multiplied by the same factor -jw/EIIE;; W. In particular,

consider yVAB/Zo', where y is defined in Eq. (6.7),

2 2
J”£11522 "YVAB _ “’ E11522 w

z‘ -
O mWXZ.68810

 

 

17
7 mevm . (6.34)

By Eq. (3.61) the value of 20 is determined uniquely for this new

definition of electric current. It must be

104

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.10 - Method to obtain the dual circuit.

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

jZ 'tan(6/2) jZ 'tan(e/2) _ P
-¢P(x ) o 0 ¢ (xn+1)
+- +-
YVAB YVAB
z I
P o P
-1 (xn) jsin8 l -1 (xn+1)
Z ' - Y
o o
;; i ;

 

 

Figure 6.11 - Dual of the upper part of the
equivalent circuit of Fig. 6.6.
The negative of all dynamic
variables corresponds to the
most convenient reference
directions.

106

r n/4K'(m/wo)

a" a-U-

% L l 1 1
I I t

% 4~ :m+U>
.6 .8 1.0 1.2 1.4 1.6 1.8 2.0

(00)
IO

 

Figure 6.12 - The variation of the turns ratio
with frequency. n = 1/2.

107

20 = 1.72 x 1014/ww {I (6.35)

[or the AC cut in quartz. An examination of Eq. (6.34) shows that

the first part of the tripple product has exactly the right value:

“511822" = 6w
2.68 x 10'7 1.72 x 1014

 

 

1

=2.— 0 (6036)
o

The voltage dependent voltage source now has a strength of

n
4K'(w/wo) . VAB . (6.37)

It is only slightly frequency dependent over the frequency range of
interest. Fig. 6.12 shows a plot of n/4K'(m/wo). A reasonable

approximation for the function would be u/4K' which for n - 1/2

equals
n/4K' - 0.4236 . (6.38)

Such an approximation, however, is not essential if an ideal
transformer with a "slightly frequency dependent turns-ratio" is
acceptable.

The ideal transformer comes about as follows: First, by the
Blakesley shift, the dependent voltage sources are moved into the
center leg. This situation together with the new dynamic variables
and line impedance is depicted in Fig. 6.13. Since the rigorous
procedure of Chapter 3 was strictly adhered to, the new line
voltages, VP(xn), will have units of voltage.

Next, the lower part of Fig. 6.6 [or 6.5] must be examined.

It contains a dependent current source of strength

108

 

 

 

 

 

 

 

 

 

 

 

 

 

“fill—‘3; WP (x11) W 511822 W¢P("n+1)
-——«»- ‘ -———¢—
.____._._+ 20.9 20.6 ____.

+ T +

z ,e
O
p P
V (xn) V (xn+1)

___.1;___.v
4K'(w/wo) AB'

 

F a

Figure 6.13 - The modified dual of the upper part
of Fig. 6.6. Both circuits yield
the same circuit equations.

109

P P P
yi¢ (xn) - 6 (xn+1)] "ZET?E7J;T [ijellczz W6 (xn)

— 1 v w¢P(x )1 . (6.39)

M 811622 n+1

Any ideal transformer with turns ratio a:l may be represented by
the dependent voltage—current—source configuration shown in Fig.
6.14. The converse is also true. This follows directly from the
fact that the transmission matrix for an ideal transformer may be

implemented by the circuit shown in Fig. 6.14.

V1 a 0 V2

. x (6.40)

 

 

 

11 0 1/a I2

The final equivalent circuit follows immediately. It is shown in

 

 

 

 

 

 

 

 

 

Fig. 6.15. The "dots" must be reversed for every alternate section,
because the dependent generators in Fig. 6.6 would be reversed also.
Since the frequency variation of the turns ratio is not strong,
replacement by the constant n/4K' will lead to good results. This
then is the justification for the usage of the Mason model [10].

It should also be noted that the new "line current" jw £11522W¢P(xn)
is much smaller than in calculations performed on the basis of the
model in Fig. 6.6. In the excitation problem at the end of Section
6.2 the line voltage ¢P was 0.1 volt and the line current 100 ma.

In this new model the line current would be for the values given

there

12

ImJEIIEEE w¢p| . 1.5 x 109 x 40 x 10’ x 2.5 x 10‘3 x 0.1

= lSuA . (6.41)

110

 

.£_4. a l
+- a
V1
(a) (b)

Figure 6.14 - Equivalent representations of an ideal transformer.

_..__ p P
“£11522 ”4’ (xn) Mcnezz ”‘1’ (“n+9

C Z , 8 Z , 8 ‘F3

+ ° T ° I +

X X
n

 

 

 

 

 

 

 

 

 

 

 

 

n+1

Z , 6

o
P P
V (x ) V (x )

n :3: n+1
c/2 I
LBJ?
4K (w/wo )

_' ~0. 42: l0 (for n-1/2)

 

9

Figure 6.15 - Alternate equivalent circuit for a basic section
of l interspace and 2 half electrodes. The
location of electrode A is point xn.

111

The line—impedance 20 would be

14
z = 1'72 x 10 = 45.9 Mg. (6.42)
0 mW

 

The line—voltage VP must then be

P P
V = Z0 x lw/gllezz W¢ | = 688 volts. (6.43)

In spite of the vastly different values, the effect on any
measurable quantity is the same. Neither model is therefore
superior to the other from a physical point of view; unless the
interspace section shown in Fig. 6.8b is used in conjunction with
the first circuit, in which case that circuit will account for the
effect of the metallization, whereas the present circuit cannot do
that. Otherwise, any preference should be based on mathematical
advantages offered by a particular model in the context of the
external terminations under study.

As mentioned in Chapter 4, for very narrow electrodes and a

single wave of the form
P + -j V—ix
4 (x) = 63 e P (6.44)

a potential difference V B of 2¢S+ should be observed across the

A
basic section [see Fig. 6.7]. The equivalent circuit of Fig. 6.15
will account for that effect correctly, as shown presently. The

current into the dot of the primary is

_jwoxn _jwoxn+1
---——— + V + V
1p, = 167611622 wtos e P - 63 e P 1 . (6.45)

At the synchronous frequency x is greater than xn by half a

n+1

wave length, so that

112

 

f““““ + P
Ipr jw 511522 N 2¢S e . (6.46)

At the synchronous frequency K'(m/wo) = K', the secondary current

is then n/4K' times the primary current, and VAB becomes

a V I

vAB (n/4K ) Ipr/jwvellezz WK/ZK . (6.47)
The effect of the field above the piezoelectric slab on the value
of C/Z has been ignored. The quantity 2K' will cancel, and VAB

becomes then:

-jwoxn/V

. ,f“"" + p
VAB (n/2) jm 811522 N 2¢S e /jwV611522 KW . (6.48)

For very narrow electrodes K is n/2, so that VAB reduces to the

desired result:

~jwoxn/Vp

V - 2¢ e . (6.49)
This certainly does not represent a proof of the validity of the
equivalent circuit, but it is gratifying to see that the postulates
which led to these equivalent circuits do not create a conflict
with this expected physical behavior of the device. More
supporting evidence for the correctness of the theory presented
here will be established in later sections of this Chapter where
the radiation conductance and scattering parameters of the array
are calculated on the basis of the last model [see Fig. 6.15].

The outcome fits the experimental results of Smith et a1 [10]. In
order to perform these calculations, it is first necessary to

obtain the three—port admittance matrix for the whole array.

6.4

113

The Admittance Matrix for an Array of an Even Number of Inter-

elegtrode Spaces

 

The equivalent circuit of Fig. 6.15 is similar in form to
that used by Smith et a1 [10]. The turns ratio and the character—
istic impedance are frequency dependent in this investigation; in
Smith et a1, they are not. Since, however, the configuration is
the same, the identical steps may be followed for the derivation.
It will be particularly convenient to change the impedance level by

a factor of the square of the turns ratio a

a2 . [n/4K'(w/wo)]2, (6.50)

so that the resulting characteristic impedance is

4K'(w/w°) 2

l/Go ' R0 5 Zo[-—T—] , (6.51)

and the new line-current will then be
1 - 3.1 c c wu/4x'(6/w )]¢P(x ) = ¢P<x > (6 52)
n 11 22 o n Y n ' °

First, two adjacent sections are cascaded as shown in Fig. 6.16 and
the y-parameters for this combination are obtained. The symbol 6
stands here for the transit angle for an interdigital period. It
has therefore twice the value of the 6 used previously. The
calculations are based directly on the procedure suggested by Smith
at al and are repeated for completeness in Appendix G. The results

are listed below in Eq. (6.53):

114

jRotano/o 11311...

l..—
L__4 +

R0 v
W —————*
jainB/Z n+1

In. jRotan(6/4)

 

 

 

 

 

 

 

 

 

1:1

 

 

 

 

a”...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ ..
VAB
(a)
In In+Ix In-In-i-l
-——-—- -———>5 -—————u~
jRotan(0/4) —+—+1Rotan(6/4) fr JRotan(6/4)
I 7 I
x X
4 R0 4
Vnno m vn+1
=0
9 V18 VAB
(b)

Figure 6.16 - (a) Equivalent circuit used to obtain the y-
parameters for one interdigital period
according to procedure by Smith et al.

(b) Connection to find y13, y23 and y33.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G -G
o o

In jtane jsine -jGotan(6/4) vn

-G G
-1 - ° ° jG tan(6/4) v

n+1 jsine jtane o n+1 '
I -jGotan(6/4) jGotan(6/4) ij+j4Gotan(6/4) VAB
(6.53)

When N such interdigital periods are cascaded, the resulting

current IT will be by Eq. (6.53):

IT - -jGotan(B/4) V1 + jGotan(6/4) VN

+ (ijT + j4NGotan(6/4)) VAB (6.54)

The total capacitance NC has been denoted C The reason for this

T.
simplicity is that in Eq. (6.53)

y31 - -y32 . (6.55)

This causes cancellations of the contribution from all Vn except
V1 and VN.

Equation (6.54) is sufficient for the determination of the
overall admittance matrix, since for VAB - 0 all sections behave
like a transmission line, and because of the required symmetry about

the main diagonal y13 and y23 are known through Eq. (6.54). The

final result is then:

6.5

116

 

 

 

 

 

 

 

 

 

 

 

 

c -0
O 0
I1 jtan(N0) jsin(N6) 'jcota“(°/4) v1
_GO GO
"IN ' j81n(N8) jtan(N6) jcota“(9/4) “ VN ’
1T -jGotan(6/4) jGotan(6/4) jch+j4Ncotan(0/4) vAB
(6.56)
where
CT = NC , (6.57)
0 . 2n(w/wo) (6.58)
and
1 a 2
Go " '2: [m] . (6.59)

This admittance matrix for the whole transducer may now be
used to determine various properties of the array. Since further
verification of the theory developed in the last two Chapters is
desirable, the radiation conductance will be calculated in the next
section since it can be compared with experimentally measured

frequency response curves for that quantity.

Calculation of the Radiation Admittance of an Arrangade Up of N

Identical Interdigital Periods

For the purpose of calculating the radiation admittance the

array is terminated in Go on both ends, so that

I - - G V
o

1 (6.60)

1

117

and IN I Go VN . (6.61)

With this modification, Eq. (6.56) is rewritten as:

 

 

 

 

 

 

 

 

 

 

 

 

0 Y11 + Go Y12 Y13 v1
0 - 912 Yll + co -Y13 x vN . (6.62)
IT Y13 ’Y13 Y33 vAB

The input admittance under these conditions is the radiation
admittance:

._A_

y - 1 /v
A33

a T AB . (6.63)

The quantities A and A33 are the determinant of Eq. (6.62) and the
co-factor of Y33 respectively. These are worked out in Appendix H.

The result of that calculation is summarized below:

8 NB 2
Ga ZG°[tan 4 sin-E— (6.64)
and
B - 0 tan 9- [4N + tan 3 sin N9] + 6,0 (6 65)
a o 4 4 T ° °

These results are identical to those found in Smith et a1 [10]

except for the frequency dependency of Go given by Eq. (6.59):

1 n 2
“o " E; [4'K"'("““6/6°)]' z (6 ) "I.— ‘j’ [’3‘7—8 <6 6 12‘6“)

The last substitution has been made on the basis of Eq. (3.61):

20(6) - 1.72 x 1014/mw 0 . (6.67)

118

These relations are programmed for the digital computer in

Appendix I. The resulting frequency response plots [see Figs. 6.17,
6.18s] for N I15 show negligible difference between this theory and
the assumption of a constant Go made by Smith et al, who used the
Mason model for the corresponding development. The reason is that
for larger arrays the central peak, where G8 has a significant
value, is so narrow that variations with frequency of Co are not
noticeable, although Go increases monotonically with frequency
because of (m/wo) as well as the factor [n/4K'(m/wo)]2, [see Fig.
6.12].

The direct comparison of the shape of the frequency response
of G8 with experimental data gathered [10] for a transducer laid
out on YZ lithium niobate is made in Fig. 6.18b. An important
conclusion to be drawn from this comparison is that it is now
possible to deduce the value of the characteristic impedance Zo for
YZ lithium niobate without resorting to the complicated calculations

performed in Chapter 3. It is seen from Eqs. (6.64) and (6.66) that
. 9.. __.._.____1' 2 EL 2
GaZo(wo) 2(wo)[4K'(W/wo)] [tan(2mo) sin(ISum/wo)] . (6.68)

At w I ”0 this becomes numerically:

w 2
Gazo(mo) ZLZET] x 900 I 323 . (6.69)

(I)
O

The experimental result [10] for a transducer with N I 15,

w I 1.25 mm, C I 8.5 pF and fo I 105 MHz yielded for G8 I 4.2 m8.

T
It follows that for YZ lithium niobate the characteristic impedance

is

119

"x Bazo (.0)

b /.\ / " 1004

V

O
A‘A.
‘9

 

2“ {F
i
0 0'0"“ I
o 0 0.0..” .. ..‘ Ed
qn«o.‘os o—o one-o} *‘W b
it! 0.8 0.9 10 1.1 1.2 __ “"9,

 

 

 

I100 <5

Figure 6.17 - Normalized radiation admittance of a
30 element array.

120

~oownno~o~.~u
~U.mou_oon.du
~o.mmOn~oo.uc
wo.mmnnonm._o
Nuowooouus._u
~u.w.0n~0n.ao
muowo-soo..c
~oowoo~moo.~s
mu.wnonnno.~c
~o.wno~o-.~o
~o.mu~ocn~.~o
~5.wonon~..~u
mo.momooo..~u
mo.mno~on..~a
nu.woomnoo.~u
mu.unoo_oo.~o
woowdwogoo.~n
muownomom~.du
~u.~o~ooon.av
~o.mo--c..o
~uowanooo~.du
duowsuanno.on
do.m.-.~n.~o
mu.w.-noo.oo
.o.muso~cn.~u
0.0
auomocoaoo.~
do.wanooao.o
doowuadunn.~
“oowm-oon.o
~o.mo~nona.a
~o.wooo~on..
~u.w«o~noo.~
~0.w~uOOco..
~0.wom~.n~.—
~o.wmnauno.~
no.wn.ouoo.~
mu.m-no~o.u
woownaoano.a
~o.wo~.-o._
mo.wu.~owm..
~u.wnnonoa..
ao.mon.~aa..
No.m.onooa.~
wo.mnna._o..
mo.w~o.a_a..
~o.wo.wo....
~u.w~nmo~n..
wo.wnmmmu~.a
No.uomso~_.a
wo.wm-~qo..

c
U
0

9L
a]

Po.w_~oono._
oo.un~r~a~.o
oo.wu“nnmc.o
.O.m~nOona.d
~O.w.m~oo~.~
«u.w.uo~ou.n
”o.w—_m~o~.:
no.wo~—O~o.a
do.m~onn~u.~
_o.w_von-.o
Noowssssnu.u
~u.w-ov:~..
~u.w0on0ac..
~o.w~uoOoo.~
~u.wuumoou.~
~o.woono>o.~
No.wnoou-.~
~oown‘oooo.~
~0.won-no.~
Neowoouoo~.~
No.wuooOno.~
~uowonOOou.n
~u.woo-nd.n
~oouon0hou.n
No.un~on-.n
~o.wouuum~.n
wo.w¢~n~u~.n
~o.moonno..n
~u.wo-~no.n
~0.mp~uooc.~
nu.w~no~4m.~
~o.wa~0ono.~
~u.uou.»m:.~
~o.wo_.~o~.~
~o.mu~u.ofl.~
~0.mo~sood.~
uo.wonanow.d
~0.wnvpno¢.~
~O.wnoooQ~.u
~o.u~muuo_._
.o.w.~omo~.o
.o.wuoamoo.~
«o.wo-na..o
.o.wr_a~oo.e
«o.w_~o.ao.n
.o.w~nn.eo.~
.o.u.o..am.a
_o.wnoo.o_.
Oo.munom~o.
0c.u~qowcw.
oo.w..mvuu.
in.

o

—m.g..-

u

f.
a

A

s
0
la

 

(L95

 

 

Figure 6.183- Central portion of frequency response

of radiation admittance.

121

.lml'la

.NxH I c .nvowuoo Heuawfivuousa ma mo ououosoo
menus ugh .ouopoae sawnuwa N» sou monsoon Housoaquoeuo «do one

oueouoavooo sowuuwvsu Huowuouoonu vowuuslhoo use no eonwuueaoo < 1 awH.@ enough

 

 

 

 

6:: a oHa mod ooa
o \3 mo.a so.~ o.a so. as.
Jr W V p I] [u .6
D
. .qIJqlloavnnuxnuui.
nv nu .
Au
.d.~
MICH .v o o
as
0v
0.
.r..
nu
mica: °
um
.
o .6.»
O
MICHAv
an
. .o
o x633 as o
O 3N N . 3
. N: 3\3rn3c«maflvawu~N—llsllluAva I
secs
0 0.

nioH_v
us

 

.uosuu Scum ac

on\aoaoo~ue

122

20(60) - 323/4.2 x 10"3 - 77000 n.

It is reasonable to assume the same functional relation to w and W
here as in the AC cut of quartz. Hence,

77000 x 26 x 105 x 106 x 0.00125

wW ’

 

20(6) - 20(60) wow/6w -

. 6.35 x 1010

wW

 

20(m) (6.70)

To complete the equivalent circuit of Fig. 6.15 only C/2 is
required in addition to Zo(w). From the experimental data in this

case that would be

C/2 I CT/30 I 0.28 pF . (6.71)

Otherwise it could be calculated from:

(a +'Ve s ) WK
c/2 - ° 2;} 22 . (6.72)

 

The dielectric constants are listed in Warner et a1 [19]. They are

also required to obtain the surface potential from the line current:
P P
I (x) . jun/811822 W6 (1:) . (6.73)

In lithium niobate the propagation velocity shows a greater
change under the metallized parts of the surface [13] than in
quartz. Such a change in velocity implies by the investigation in

Section 3.7 a change in 20:

v = v 1 + k 2 (6.74)
po pm c

123

2
and Zoo I Zm(l + k6 ) , (6.75)

where the subscripts m and 0 mean metal and interspace respectively.
It follows from these equations that the relative change in 20 may

be determined if the change in Vp is obtained

2 v . .
m pm

In Campbell and Jones [:3], both Vp0 and me are given for various
cuts of lithium nibbate. Since even here that change in Vp is only
a few percent it does not really matter whether Eq. (6.70) is used
for 200 or low since the relative change in 20 would cause
reflections and that can be determined from Eq. (6.76) with the
data obtained from [13]. However, the T-model in which 20 is used
will not accommodate a variation in 20. On the other hand, the
modified n-model [see Fig. 6.8b] will account for both a change in

20 as well as a change in Vp through the quantities 6° and 9m

respectively:
0 - —"-’-— nD/4 (6 77)
m V °
pm
and
w
80 " V'— (1 - n) D/2 . (6.78)
pa

The quantity Dn is the length of an interdigital period times the
fraction n which gives the relative length of metallization. To
complete this n-equivalent circuit Yo must be determined from Eqs.

(6.70), (3.56) and (3.59):

124

*4
I

. 2
ZPx/I¢Sl ,

and
2 2 2

2o ' 2px,“ c11522|°s| w ’

hence
2 2

or

Y - w x 6 35 x 1010 (6 80)

0 w 811622 . C O

The dielectric constants [19] are also required for the transcon—

ductance 7 shown in Fig. 6.8:
Y ' JmVellczz Win/4K'Cm/w071 . (6.81)

This completes the n-model for YZ lithium niobate.

It has been shown in this section that the equivalent
circuits deve10ped in this investigation are quite adaptable to
other piezoelectric materials. Originally they were developed for
the AC cut in quartz only as a matter of mathematical convenience
because the solution of the surface wave problem could be performed
for that particular crystal cut in closed form. Later on in
Chapter 3 the result was used for the determination of Yo and 20.

In this Chapter, by comparison of the computed radiation conductance
with experimental results, it became possible to determine Yo and

20 independently of the complicated procedure of solving numerically
for the power flux of a piezoelectric surface wave. The circuit
model may thus be determined by direct experimental procedure,

which is analogous to finding transistor equivalent circuits from

 

6.6

125

measurement rather than from a detailed knowledge of the geometry

and physical properties of the material.

The Scatterinngarameters of the Array at Synchronous Frequency

With a Tuned Load

 

In the detection problem of Section 6.2 it was assumed that
the array does not reflect nor attenuate a surface wave. The
assumption was justified there because the load constitutes, for
all practical purposes, a short circuit in which case the array
behaves by assumption as a transmission line section.

In this section a resistive load in parallel with an
inductance in resonance with CT at the synchronous frequency will
be considered. It will be of interest to find out how much energy
can be extracted from the wave at synchronous frequency. The
scattering parameters are found from the admittance matrix of the

two-port resulting from the termination described above by

calculating [8]:

A

. ,. 9 . y -
[s1 - [1 - G—i-‘EJ [I +345] 1 . (6.82)
O O

I is the identity matrix.
The matrix §1k is derived from the admittance matrix of the
three-port (Eqs. (6.56)) by eliminating V from the first two

AB
lines by means of the third line:

JG tan(e/4)[V - V l
o l N
- . (6.83)
AB 0L + Museums/4)

 

V

The elements of §1k are then

‘.

126

GOG + jGoztan(e/4)[4N+tan(6/4)tan(N9)]

 

 

L
yn - yzz . GLJMIKNO) - 4NGotan(Ne) tan(6/4) ’ (6'84)
and
c c + jG 2tan(8/4)[4N + tan(6/4)sin(N8)]
. . _ ° L ° (6 85)
y12 y21 GLjsin(N9) - 4NGosin(N9) tan(6/4) '

Unlike the elements of the three-port matrix these y-parameters

remain finite at synchronous frequency, where
sin(Ne) tan(6/4) I tan(N6) tan(6/4) I - 4N . (6.86)
Let

2
u I GL/16N Go . (6.87)

With this definition [yik] is at synchronous frequency

91k - co . (6.88)

-a a
The matrix [I + 91k/co] is in terms of a:
1+0 --a

[i +§1klcol - . (6.89)

-o 1 + o

Its inverse is required in Eq. (6.82) which becomes:

1+3 3

 

 

 

 

 

 

1 ' a a 1 + 2c 1 + 20

[S] I x , (6.90)
- a !;JLJ;_
“ 1 a 1 + 20 1 + 2c

 

 

 

 

 

127

 

 

 

or
v-
1 1 l
8 (IS )=__..____. , (6.91)
11 22 V + l + 2c 1 + G IBNZG
l L o
and
+ 2
v G /8N c
.. 2 2a L O
S (I S ) I -—— I I . (6.92)
21 12 V1+ l + 23 1 + GL/BNZGO

If CL is zero, there will be complete reflection off the array. It
must be remembered, of course, that there is still the parallel
inductance in resonance with CT' This corresponds exactly to both
the experimental and theoretical results of Smdth et a1 [10]. By

Eq. (6.64) the radiation conductance is at synchronous frequency

2
G8 I 8N Go . (6.93)

With this identification S11 and 812 are rewritten respectively as
power scattering coefficients

2 2 - 4-
p11 ' S11 ' 1/(1 + GL/Ga) ' P1 (P1 ’ (6'9“)

and
2 '2 2
p21 ' S21 (Gt/Ga) ’(1 + GL/Ga)
+ +
- P2 ’P1 . (6.95)

The load power will be found from
P - P - P — P , (6.96)

01' +

1
2 2
(l + GL/Ga) (l + GL/Ga)

2
P - (CL/Ga) P

 

 

(6.97)

128

(ch/ca)

 

PL = 2 P1+ . (6.98)

(1 + GL/Ga)

The result of maximizing this is
G I G . (6.99)

under these optimum conditions, Eq. (6.98) implies that only one-
half of the incoming power can be extracted from the electrical
terminals. By Eqs. (6.94)-(6.95) one-quarter will be reflected and
another quarter transmitted. Numerically the optimum termination

for the lithium niobate transducer discussed earlier would be
RL I 1/G I 238 Q .
a

If the identical transducer were laid out on AC quartz it would be
by Eq. (6.68) 2700 times larger, because that is the ratio of the
20's of quartz to lithium niobate. At frequencies above 100 MHz
practical impedance levels tend to be lower. Any realistic
termination to the quartz transducer would therefore appear as a
short circuit to the line, supporting the initial assertion made by
Coquin and Tiersten that the wave is unaffected by the quartz
transducer.

It should be pointed out here that these are not original
conclusions. They are the same as those found in [10], as they
should be by necessity, since the definition of the quantity Go in
Eq. (6.51) led to the identical equivalent circuits in terms of

this Go [see Fig. 6.16 and [10]].

129

The development was made here for two reasons. The detailed
calculations shown here are not given in that reference and they
are an excellent example of the usefulness of the equivalent
circuit model on which they were originally based. Furthermore,
the equivalence of the results emphasizes that it was never the
intention of this investigation to disprove the well-established
analysis and design procedures of Smith et a1 [7,10], but rather to
put their equivalent circuit model on a firm theoretical basis which
admittedly it was not. It relied entirely on analogy to bulk waves,
but not directly either, for the equivalent circuit developed in
Chapter 4 by that analogy has a somewhat different frequency
response from that used by Smith et a1 and the one developed here
in Chapter 6. Smith et a1 anticipated the correct form without
deriving it. They did not, however, realize the complexity of R0
given in Eq. (6.51), which combined with Eq. (6.70) is for Yz

lithium niobate

'
R . 6.35 x 1010 "K Wu.) 2

o wW [ n l . (6.100)

They use a constant for R0, which gives good results for large

arrays as seen by the comparison made in the last section.

 

CHAPTER 7

CONCLUSION

7.1 Historical Context

 

The objective of this investigation was to develop useful
equivalent circuit models for piezoelectric surface-wave transducers
from surfacedwave theory. Because of the complex nature of the
subject, many simplifications had to be made along the way. Never-
theless, they clarified rather than obscured the logical development
of the subject, and they were never so gross as to make direct
numerical calculations meaningless.

Before this research was undertaken, a successful circuit
model indeed existed, and it has been widely used. It is the
Mason model as adapted by Smith et al. However, it derived its
justification really only from the fact that it correctly accounted
for observed physical behavior. Its theoretical justification was
based on the analogy between surface waves and bulk waves, for
which transducer equivalent circuits have been in existence for
some time.

The circuit models deve10ped here account for the observed
physical behavior, as they must, but their detailed logical
development also specifically links them back to piezoelectric
surface waves.

It has been shown, furthermore, how the elements of the

130

 

 

7.

2

131

equivalent circuits may be obtained from either the geometrical
layout of the transducer, plus a knowledge of the fundamental
physical properties of the material; or from direct measurement of
the input admittance of an array when terminated by an infinite
piezoelectric transmission line on both sides. This dual approach
to modeling is analogous to the modeling of transistors, where
either a knowledge of the geometry and the nature of the semicon-
ductor material, or a few key measurements, will lead to the Ebers-
Moll model. There, also, many simplifications are made in the
theoretical development of the model, but it is very satisfying to
have one mathematical model whose origins can be traced back to a
fundamental starting point, well founded in physical theory; even
if some physical phenomena are not completely accounted for by the

model. The value of this research must be viewed in that light.

Summary of the Results
In Chapter 2 a simplified solution of the surface-wave
problem has been performed in closed form for the AC cut in quartz.

The results are numerical values for the propagation velocity

Vp = 3147 m/sec , (2.70)

and the decay constants of the two dominant modes, together with
their complex amplitudes. This information is expressed in terms

of the normalized particle displacements at any depth y below the

surface:

 

132

u(x,y,t) I 0.66 e—1°4ky cos(mt - kx - n/Z)

-0.147ky

+ 0.052 e cos(wt - kx + u/Z) , (2.68)

for the horizontal displacement and for the vertical displacement:

-1.4ky

v(x,y,t) I 0.079 e cos(mt - kx - a)

as

e-0.l47ky cos(mt - kx) ; (2.69) g

+ 1.0

the quantity k here is the propagation constant w/Vp. The equations

show that the dominant behavior of this surface wave is similar to

 

that of a vertical shear wave.

'12::?“

In Chapter 3, these expressions were used to calculate the

time average power flux for the AC cut

2
Px I %-mW x 90 x 109 x [C2(2)] , (3.32)
and the surface potential
48 - 35.8 x 108 02(2) (3.54)

It was shown that these two quantities lead to a physically proper
expression for the characteristic admittance of the piezoelectric

transmission line

2 -7
Yo I 2Px/|¢S| I wW x 2.68 x 10 , (3.58)

if the surface potential is selected as the cross-variable:

96:) - $00 + 4'60 . (3.55)

where

¢i(x) 3 ¢ 1 ej(mt ; kx) .

S (3.47)

133

An alternate possibility was developed with the value for

the characteristic impedance as

2 2 14
2o - 2Px/m 511522 w2|4s| - 1.72 x 10 /ww . (3.61)

Here the through-variable is related to the surface potential by

It - 116/611822 W¢i , (3.45)
and I
I(x) - I+(x) - I-(x) . (3.46)

By analogy to a vertical shear wave, transverse shear stress
and vertical particle velocity were also considered as dynamic
variables in order to justify the bulkewave model used in the
literature. Such a model was developed in Chapter 4. Since better
models are developed in Chapter 6, it is of no further interest
here except, perhaps, that it is shown in Section 3.7 that for this
choice of dynamic variables the small relative change in character-
istic impedance and velocity is not the same for both quantities if
the vertical component of the electric field is forced upon the
wave externally. This statement differs from views commonly held.

In Chapter 5 the theory of Coquin and Tiersten was extended
to obtain an expression for the short-circuit current from an
electrode to its nearest neighbors in an infinite array with a

fraction n of the surface metallized:

Isc - Juarez, was“ + os'm/K'mn ainzg-ww-J) . (5.58)

The quantity K'(m) is the complete elliptic integral of modulus

 

134

m'(w) I 1 - sin2(nnw/2wo) . (5.61)

The validity has been established for the frequency range
1/(2 _ n) < w/mo < l/n . (5.62)

The quantity “0 is the so-called synchronous frequency, where one
wave length equals a periodic distance of the array D.

An essential assumption which led to the expression of the
short-circuit current at frequencies other than ”0 is that this
current is proportional to one-half the difference in the emf's
between the electrode and its nearest neighbors. In Chapter 6 this
concept was extended further. Here the emf's of all electrodes
connected to each common terminal are summed, and the current from
one common terminal to the other depends on one-half the difference

in the combined emf's. This is stated in the form

18C - é— Z («1)k+1 1k , (6.3)

where 2'Ik is the contribution from one electrode:
-1- I - We e ' war/210(6)] 4pc: ) - 2 Pa) (6 5)
2 k 11 22 n W n ° °

Together with the transmission line sections of Chapter 3 this led
to an equivalent circuit representation of the transducer array as
shown in Fig. 6.8.

It was shown how weighting by an apodizing function of each
dependent current generator makes this circuit model applicable to
arrays with non—uniform electrode overlap. Different values of
propagation velocity and characteristic admittance for the parts

with surface metallization can be accounted for by breaking up the

135

transmission line representation of the interspaces [see Fig.
6.8b].

By finding the dual of one section of the equivalent circuit
array, an alternate equivalent circuit was produced for a section
consisting of one interspace and its adjacent two-half electrodes
[see Fig. 6.15]. The turns ratio of the ideal transformer is
frequency dependent, but the line impedance may be reflected through

this transformer so that it becomes

. 4x'( ) 2
80 20(6) t—-—-———fl“’1 . (6.51)

where Z0 is the characteristic impedance of Chapter 3
Zo(m) ' 20(w°)(mo/m) . (3.61)

Under these circumstances the through variable is given by

P
In I Y9 (xn) . (6.52)

This equivalent circuit was used to obtain the y-parameters for one
interdigital period [see Fig. 6.16] which led to the calculation of
the radiation admittance Y8 and the scattering parameters of an
array of N equal interdigital periods in cascade. The purpose of
finding the radiation admittance was to adapt the equivalent
circuits obtained in this investigation to other crystal cuts by
comparing the normalized calculated frequency response with
experimentally determined frequency response plots. This was done
in particular with YZ lithium niobate. It was found that the
characteristic impedance corresponding to that deve10ped in Chapter
3 for the AC cut in quartz [Eq. (3.61)] is several orders of

magnitude lower here

136

10
z (w) _ 6.35 x 10

o ow

 

. (6.70)

This should be viewed as a measure of the stronger electromechanical
coupling in lithium niobate.
The evaluation of the scattering parameters showed that under

conditions of optimum termination,

RL I 1/Ga , (6.99)

one-half of the power contained in an incoming wave is extracted by
the electrical terminals, one-quarter is reflected and one-quarter
is transmitted. It was seen that while the optimum value for RL is
only about 2009 for a typical lithium niobate array, it becomes
unreasonably large for a quartz transducer. A practical lower
value of RL will leave the wave largely unaffected in its travel
through the transducer.

It has been shown in various applications in Chapter 6 that
the physical behavior predicted by the new models corresponds very-
well to either measured performance or the theoretical predictions
by other authors based on the Mason model. If, in fact, the
equivalent circuit of Fig. 6.16 is used with the value of R0 for YZ

lithium niobate,

I
6.35 x 1010 4x (w/mo) 2

R0 I w W [ fl ] , (6.100)

 

then it is clear in retrospect that the model deve10ped by Smith et
a1 is but a special case of the development presented here. This is
true because, while their characteristic impedance was determined

for the single frequency mo, the characteristic impedance R0

7.3

137

developed here is frequency dependent and reduces to their value at

it) .
0

Further Investigations

 

It would be of interest to show experimentally that under
certain wide-band applications the frequency dependence of Ro shown
here would predict the correct behavior, where a constant Ro would

not. The radiation conductance,

2 8 N8 2
Ga "§;'[tan z-sin 5—- , (6.64)

should be measured since the results of the experiment, for small
N, will clearly demonstrate which theory is correct. The small N
is necessary to achieve the required bandwidth. For large N, for
example N I 15, there is no significant difference between this
theory, that of Smith et a1 and measured performance.

It would be desirable, furthermore, to extend the frequency
range for which this theory is valid. At the lower end of the
frequency scale the expressions obtained here for the short-circuit
current might be valid down to DC. At higher frequencies, when
half a wave length is shorter than the width of an electrode, there
will be partial cancellation of the emf developed under an electrode
and the current should decrease for that reason. In either case,
the present theory cannot be justified whenever a quarter wave
length measured from the center of the electrode falls under a
metallized part because of the boundary conditions in the conformal
mapping problem of Chapter 5. The higher frequency end would

particularly be of interest because third harmonic generation is,

 

138

in fact, used to generate waves in the gigahertz region. However,
the scope of the present investigation must be kept within
reasonable bounds. Such research is therefore deferred to some

future date.

""'.t!.ll
!

' "f— I
3";

1‘
I

 

APPENDIX A

DERIVATION OF THE PROPAGATION VELOCITY

 

 

140

It is shown here that Eqs. (2.44) and (2.45) lead to Eq. (2.46).
When Eq. (2.45) is multiplied out with the definitions of the 3's

inserted we obtain:

(01 - o2){822[(q - 311) + «121m - gm) + (1221 + 812(o102K2)}
+ (0102822 + 812) {01K[(q - 811) + 0.22]

" 02K[ (‘1 - 811) + 312]} - 0 0 (A01)

This leads to an equation as described in [6] on page 670:

2 2
0102K[822(q '- 311) + 812(K ' 1)] + 0.1 02 (1 ' K>322
2 2 2
+ (a1 ’+ a2 )(q - 311) 822 + (q - 311) -322

+ 312 K(q - 311) I 0 . (A.2)
Equation (2.44) is of a form shown in Eq. (2.52):
a4+qza+b-O. (A.3)

In factored form this must equal:

2 2 2 2 4 2 2 2 2 2
(o - 01 )(a - o2 ) ' a + 6 (-ol - a2 ) + n1 o2 . (A-4)

It follows from Eq. (2.56) that we can make the following identifications:

2
+ I ..
01 “2 822 (A05)

 

2 2 (q - 311)(q - 1)
“1 “2 '

 

(A.6)
822

 

161

These expressions are inserted into Eq. (A.2) to obtain, after some

manipulation:

2 3;; 2 2
(q - 311M - 822 [322(q - 311) + 312 ] . (A.7)

This is a cubic in q which simplifies eventually to Eq. (2.46).

 

 

 

APPENDIX B

THE TENSOR TRANSFORMATION
OF THE STRESS COEFFICIENTS

143

The stress coefficients shown in Eqs. (2.1), (2.2) and (2.3) are
really fourth rank tensors. The subscripts l to 6 used in engineering

are a short form which convert as follows to tensor notation:

TABLE B.l - Conversion from Engineering Notation

to Tensor Notation

 

Engineering;Notation Tensor Notation
l 11
2 22
3 33
4 23 or 32 because of
symmetry
5 13 or 31
6 12 or 21

 

In order to obtain the stress coefficients for a rotated Y-cut by an

angle a one must use the transformation according to [5]:

 

 

'1 o o T
air - 0 case sine . (3.1)
L0 -sine cose_
The relation of the stress coefficients in the rotated system C'ijkl
to those for the standard crystal axes is then given by:
‘ I
c ijkl airgjsaktaeu crstu ' (3'2)

As an example it will be shown here that for e - 31.62“ 0'56 is zero.
This is the coefficient that relates vertical shear strain 56"312) to

face shear stress Ts (-T13).

 

 

144

' - ' - -
C 56 C 1312 “as“zu Clslu “32“22 c1212
(3.3)

+ “33“22 C1312 + “32“23 C1213 + “33“23 C1313
C' - -sin6cos6 C + C0829 C - sinze C + sinecose C (B 4)
56 66 56 65 55 '

There is further symmetry in the stress coefficients in engineering

notation
C11 I C11 . (3.5)
With values inserted which were taken from [4] C'56 is:
c' - cosBsin9(57.94 - 39.88) - 17.91(cos26 - sinze) (8.6)

56

in units of 109 N/mz.
This becomes zero at a rotation of 31.62“. The other constants

were obtained in a similar fashion.

 

APPENDIX C

DERIVATION OF THE TWO-PORT PARAMETERS
FOR A LOSSLESS TRANSMISSION LINE

 

146

All pertinent quantities are defined in Fig. 3.2. Express first

the terminal quantities at port 2 in terms of the cross—variable "waves"

at port 1:

 

 

 

 

 

 

 

 

 

 

 

I- q I- “St at '1 r-“Bt 8t —1 P "
+ - o +
V2 V1 e + V1 e e e V1
- - . (0.1)
+ -sto _ sto -sto sto _
_12_ _Yovl e - Yovl e _ _Yoe -Y°e .. _vl _
Next this relation is inverted to yield:
‘ P 8t 8t r- -
f’ + o 67
V1 -Yoe e V2
1
I -——-—-2Y o (C.2)
_ o -sto -sto
V -Y e e I
L 1-J L— O J b.2—

 

 

 

The "through-variable waves" at port 1 are related to the "cross-variable
waves" by Yo:

o . (C.3)

L‘fi .

 

 

 

 

The first equation is added to the second in relation (C.2) to result in
V1:

+ _ 1 3:0 -st

sto -sto
+ (20/2) [e - e ] I2 (0.4)

V1 - cosh(sto) V2 + Z0 sinh(st°) 12 . (C.5)

Also from relation (C.3) I1 is obtained:

147

st -st st -st

+ - o o l o o
11 - Il - I1 - (Yo/2)[e - e ]V2+§-[e +e ]I2
(C.6)
ll 8 Yo sinh(sto) V2 + cosh(sto) 12 (C.7)

Equations (0.5) and (C.7) form the desired transmission equations as
stated in Eqs. (3.1).

Next regroup these equations as follows:

1 x V1 - cosh(sto)V2 - 0 x I1 - 2° sinh(sto) {-12} (C.8)

0 x V1 + Yo sinh(st°)V2 - 1 x 11 + cosh(sto) {-12} . (C.9)

If the equations stated above in matrix form are next premultiplied by
the inverse of the coefficient matrix of V1 and V2 one obtains the 2-

parameters. The y-parameters are obtained below that.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F’ 1 - 1 r- _ 1 '- -1
V1 Yosinhsto coshsto 0 Zosinh(sto) 11
1
Y sinh(st ) (6'10)
0 o
L_Vz‘ .0 l _ _l cosh(sto) _ :12.
rv-l - 2o 20 1 1 1
l tanh(sto) sinh(sto) l
. (0.11)
2 Z
V -—-—£L———- o -1
L. 2‘ _sinh(sto) tanh(sto)‘ _ 24
_ l 1 _ _. _ _ r W
11 . cosh(sto) Zosinh(sto) 1 cosh(st°) V1
1
. (0.12)
Zosinh(st0)
.712. :1 0 _J L_O Yosinh(sto)J va

 

 

 

 

 

 

 

 

 

 

 

Y

-Y

O

tanh(sto)

O

L81nh (B to)

148

-Y 1
O

sinh(sto)

Y
o

 

tanh(sto)"

 

 

(0.13)

APPENDIX D

CALCULATION OF THE POWER FLUX

 

150

The integrand of Eq. (3.21) becomes:

- ky - ky - ky
Tluir + T662: - -jw[cl(1)* e 01 + 010”. e 02 ]{-jkc [C (1) e :11

(2) “2 y (1) “1 (2) 2
+ 01 e ] + C12I'“1kcz e - aszZ e I}

"a RY ‘ k?
+ c2(2)* e 2 ] C66 {-jklc (1) e “1

-e “a kY “ RY
+ 02(2) e 2 1 + {-a1k01(1) e 1 — a RC (2) e “2 1} (v.1)

-2 ky
<1)c (1)* e “1 + (01(1)*C <2)

' “”kcl1lc1 1

+C

'(e +0 )kY
(1) (2)* 1 2
1 1 C1 )3

~2a ky
+ 01(2)01(2)* e 2 ]

-2a ky
(1)C (1)* e 1 (2)C <1)* + C <1)c (2)*

+ j""‘C12["‘1c2 1 + (“zcz 1 “1 2 1 )

'(o +u )ky * 'Za RY
x e 1 2 + a C (2)0 (2) e 2

2 2 1 1

<1)C <1)* e'zfiky c (2)C (1)* +

(1) (2)*

*(o +u )ky * '20 Ry
x e 1 2 + 3201(2)02(2) e 2 ]

(1)c (1)* e‘z‘fiky (1)* <2)

- wkC66[C2 2 + (C2 C2 + C e

(1) (2)*
2 C2 )

-2o ky
+ c2(2)c2(2)* e 2 1 (v.2)

151

We integrate this as prescribed by Eq. (3.21).

<1)C (1)* c (1)*c (2) + c (1)C1(2)* C (2)C <2)*

1 1 1 1 1 1 1
——-[ + + 1
X 11 2 231 (:1 + (:2 252

 

 

(1) (1)* (2) (1)* (1) (2)* (2) (2)*
._-[C C1 + azcz C1 + 01C2 C1 + C2 C1 ]

 

 

 

(1) (1)* (2) (1)* (1) (2)* (2) (2)*
c1 02 + 8201 c2 + alcl c2 1 + c1 c2

+ C66 '2— ‘ 23 1(81 + 02) 21 l

(1) (1)* (1)* (2) (1) (2)* (2) (2)*
Y_‘_"_ [C2 C2 + C2 C2 + C2 C2 + C2 C2 1 (D 3)
66 2 231 a1 + a2 _ 262 °

 

 

+C

Expression (D.3) is a special case of Eq. 28 in [4]. For the values

derived in Chapter 1:

c (1) —jO.66

1
c1(2) - 30.052
c2(1) - -o.o79
02(2) I 1 (the vertical amplitude of mode 2 is

taken as reference) '

C I 86.74 x 109 N/m2

11

c - -7.65 x 109 run2

12

c66 - 28.85 x 109 mm2

01 I 1.40

oz I 0.147

 

152

 

 

 

 

 

 

 

2 2
—- _ 1_ (2) 2 0.66 12 x 0.66 x 0.052 0.052
Px 2 “"lcz I [2x1.4 + 1.55 + 2x0.1a7](86’74)
+ {—0.66 x 0.079 + 0.147 x 0.66 + 1.4 x 0.052 x 0.079 + -0.052](_7.65)
2 1.55 2
+ [0.66 x 0.079!+ 0.147 x (-0.079)(0.052) + 1.4(-0.66) +‘9;92£4(28.85)
2 1.55 2
0 0792 -0 079 x 2 1 9
+ [2:17.- + 1.55 + W] (28-85)} " 1° (”"0
- 1 (2 2
Px - 71.1402 )| 13.5 - 3.8 + 0.8
+ 0.2 - 005 + 002
+ 008 - 1702 + 008 1

These numbers are arranged in the same order in which they appear in
Eq. (D.3). It is seen that by far the largest contribution to the power

comes from the vertical component of mode 2:

p (2) - l-a)W|C (2)|2 x 98 x 109 watts . (D.6)
2 2 2
Equation (D.5) yields

P; - %~,wW|02(2)|2 x 90 x 109 watts . (D.7)

P2(2) taken alone actually results in a value which is slightly too high

(8.81). This suggests that a surface wave is in a lower energy state
than a shear wave.

It should be noted furthermore that P; is real. which will pro-
duce a real 20 (or Yo) so that the through-variable and cross-variable

are in phase, whatever their choice.

 

153

10

(2) might be of the order of only 10- mPx will be

2
rather small. For W I 3 mm and w I 109/sec we obtain

Since C

P I 1.3 mW.
x

APPENDIX E

SOLUTION OF THE POTENTIAL EQUATION

 

 

155

Equations (2.6), (2.7) and (2.8) are inserted into Eq. (3.41)

2 2 2 2 2
2.1.+ .1_1. é.2.+.§_!;. 2.2.
e t 'e (e +e )+e 2 . (E.l)
ll ax2 22 ayz ll ax2 axay 26 12 26 3y

The right-hand side is assumed to be known, f(x - th,y) say. The
solution will consist of two parts: the complementary and the particular
solution. The complementary solution is found from the homogeneous

differential equation:
2 2
”n “a
811 2 + 622 2 ' ° ° (3'2)
ax 37

 

 

Because of the assumption that the x-direction is unbounded ¢ in complex

‘J(kX-mt)

form will vary as me ‘with x and t so that the homogeneous

differential equation becomes:

2

a 0 e
2“ - _11-12 ¢ - 0 . (E.3)

8? €22 “

 

The solution is of the form

e (e '
_[ell ky ell ky
A e 22 + B e 22 .

c), - (8.4)

B must be 0 since ¢ is zero for large values of y. The complementary

solution is then

- 1‘.
r Y .1 (wt-kx) .

¢n I K e (E.S)

For the particular solution let either gar-3 - all: or '3'; a-azk and use

the principle of superposition, solving for each mode separately:

2

2 2 2 2 2 2
‘k ‘11‘1 + “1 e221‘ ’1 ' ‘°11k u1 ’ j“‘1“ (‘28 + e12)"1 + e26k “1

111 (E.6)

156

2
(“11 ‘ e26‘“1 )“1 + j“1‘“26 +“12)”1

 

 

¢1 I 2 . (E.7)
e11"“1 622
Similarly
2
(“11 ‘ e26“2 )“2 + j“2‘“26 + e12)"2
42 - 2 . (8.8)
‘11 ' “2 “22
The total solution is then
¢'¢n+¢1+¢2° (E-9)

In order to determine the coefficient K in ¢n (Eq. (E.5)) the simplifying
assumption is made in [4] that D2 I 0 for y < 0. Since it must be

continuous, the following boundary condition follows from Eq. (3.35) for

 

 

 

 

 

y I 0:
an av 32.
n - 0 - e -—- e -—- e . (E.10)
2 26 8y yIO 26 3x _0 22 3y 0

Applying this the result is

2 (1) (1) _ 2 (i) (1)
K _ 2E: e28“1C1 + 3 e26C2 _ ta (“11 e26“1 )°1 +-"“1(“26‘*“12)c2

r"“" 1 2
1-1 611°22 611 ‘ “1 e22

(E.11)

 

APPENDIX F

THE FREQUENCY RESPONSE OF A 20 ELEMENT
SURFACE-WAVE TRANSDUCER ARRAY

158

By means of the geometric-arithmetic mean Eq. (6.16) is
programmed below for the special case of n I 1/2, K' I K I 1.85407.

The result is shown in Fig. 6.9.

C‘.".IHIS POUSRA‘Q CUHPUYFS THE FREQUENCY REJPUNSE CF A £0 iLEHENI
C.‘0..SURFACE HAVE TRANSDUCER ARRAY: WITH HALF THE SURFACE AREA
CeeeeeHEIAlllLO. [HE LOAD IS A OPF CAPACIIURnQX THE VALUE OF C-UUT
UlNENSlUN V‘BODoONE‘801
KIT
I-l
ONI.675
lO ALF'UH‘Oo7853961h
BISIN‘ALF’
A'Ie
zc AI-0.50(A¢8)
BI'SORIIA'BI
A'Al
b-Bl
C'5"8”Ze
lFlC‘.OOOCOl)ZlolloZO
2| VIIl‘.01.A'SINlIC..ALF)ICOS‘Zo‘ALF’
VII)IABSIV(|)D
22 OflflllI'OH
[01.1
lF‘K'l’25025n75
25 ONIUHOO.025

 

DJ 70 86
75 JNI0H¢0.005
8b [FlABStOH-loi'1.E-5D 40:40:30
40 VII)I3.36885/ZO-
GO TO 2?

30 IFiK'1135035090
J9 1FlUH°ZoUlOObODbO
00 CALL PLUIOCVDOHEOI’
KIZ
l'l
05.0.9
”0 lFCOU’lolllQolLDQS
95 CALL OLDT“VOU"ED‘O,
STOP
END

APPENDIX C

CALCULATION OF THE Y-PARAMETERS
FOR ONE INTERDIGITAL PERIOD

 

160

Consider Fig. 6.16s. If vAB I 0 the two sections obviously
degenerate into a transmission line. Hence yll’ y12’ y21 and y22 are

determined. If Vn and V are set to zero the resulting I for an

n+1

applied VAB will give y33. In the same circuit In will yield y13 and

--In+1 will determine y23. By reciprocity y31 and y32 are then also
known. In Fig. 6.16b the voltage vAB has been reflected across the
ideal transformers. Because of symmetry it is possible to identify Ix
as one-half the input current, except for the portion through the
capacitor 0. Furthermore, it is seen that In and In+1 are equal.

Kirchhoff's voltage law in the left mesh is

vAB - -jRotan(O/4) 1N + Rbe/jsin(6/2) , (0.1)

for the center loop it is

 

szB - 2jR°tan(6/4) 1N + 2Rbe/jsin(6/2) + 2jRotan(6/4) 1x . ' (0.2)
From this IN is eliminated:
2vAB - IxRol2/jsin(6/2) + jtan(e/4)] . (6.3)
or
w _ 1 R 1 - 81820)“) G 4)
AB x o jsin(0/4) cos(6/4) ’ ( '
or

2vAB - Ix/jGOtan(6/4) . (0.5)

The required ratio for y33 is

ZIx/VAB I j4G°tan(6/4) , (0.6)

161

hence

y33 I ij + j4Cotan(6/4) . (0.7)

The value obtained here for Ix is now used to eliminate it from Eq.

 

 

 

 

(0.1).
2tan(6/4)
vAB - -jRotan(6/4) IN 4" m VAB , (C.8)
or
jRoIN _ 2tan(6/4) _ 1 (c 9)
VAB Zsin(6/4) cos(0/4) tan(6/4) tan(674) ’ °
or
2
I -jG sin (6/4) I
N o N+l
VAB - sin(0/4)cos(6/4)- -jGotan(6/4) - VAB ’ (6°10)
hence

y13 - -y23 - -jGotan(6/4) . (0.11)

APPENDIX H

DETERMINATION OF THE RADIATION ADMITTANCE

163

Solution of Eq. (6.62) for the purpose of determining the
radiation admittance.

The cofactor A33 is

2 2

The determinant is expanded in terms of the last column:
A " Y33 A33 + 2"13["Y13("11 + co) " Y13Y12] ' (“'2’

The radiation admittance is then

, 2
A 2"13 (’11 + Y12 * Go)
Y - _.__ - Y _ (He3)
a A33 33 (Y + G )2 _ Y 2
11 o 12

01'

ZGotan20/4Il/jtanNe-lljsinNe+l]
Ya - jch + j4NGotan6/4 +

 

2 2 . (H.4)
[lljtanNe+l] - [~1/381nN8]

The terms in [ ] are considered separately. By a2 - b2 I (a+b)(a-b)

they simplify to:

1/[ l 1 1] _ 1sin(Ne)

jtanNe + jsinNO + cosN0+jsinNe+l (H.5)

01'

-jN6/2
jsinNe _ jsinNee - 1__ jsingNe/Z)
1+ejNe 200s(N6]2) [2 2cos(Ne 2)] jsinNe ' (“'6)

After expanding sinNe as 2sin(Ne/2)cos(Ne/2) Eq. (H.6) becomes
13%2!§-+ sin2(Ne/2) . (H-7)

This is re-inserted into Eq. (H.4) with the desired result:

Ya I 200[sin(No/2)tan(e/4)]2 + ijT + j4NGotan(6/4) + jGosin(Ne)tan2(6/4).

(H.8)

APPENDIX I

PROGRAM FOR THE RADIATION ADMITTANCE
OF A 30 ELEMENT ARRAY

165

Computer program for the normalized radiation admittance of a

30 element array.

w I w/wo . (1.1)

0820(80) I gi-[n/K'(w')]2[tan(fiw'/2)sin(30flw'/2)]2 . (I.2)

Bazo(mo) I %%-[u/K'(w')]2 tan(nw'/2)[6O + tan(nm'/2)sin(600w'/2)]. (1.3)

COOIOIQAulAIICN ADMITTANCE
(000.00F A 30 ELrfiENI ARRAY.

6.;

37

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REFERENCES

10.

11.

12.

13.

REFERENCES

H. F. Tiersten, "Wave PrOpagation in an Infinite Piezoelectric
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C. C. Tseng and R. M. White, "Propagation of Piezoelectric and
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James J. Campbell and William R. Jones, "A Method for Estimating
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G. A. Coquin and H. F. Tiersten, "Analysis of the Excitation and
Detection of Piezoelectric Surface Waves in Quartz by Means of
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Warren P. Mason, Crystal Physics of Interaction Processes, Academic
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u. Deresiewicz and R. D. Mindlin, "Waves on the Surface of a
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W. Richard Smith, H. M. Gerard, and W. R. Jones, "Analysis and Design
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Rabindra N. Chose, Microwave Circuit Theory and Analysis, McGraw-
Hill Book Company, 1963. r

T. L. Szabo and A. J. Slobodnik, "The Effect of Diffraction on the
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Smith, Gerard, Collins, Reeder, and Shaw, "Analysis of Interdigital
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IEEE,MTT—17,No. 11,1969.

Gerard, Smith, Jones, and Harrington, "The Design and Applications
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No. 2, April 1973.

D. A. Berlincourt, D. R. Curran, and H. Jaffe (W. P. Mason, Ed.),

Physical Acoustics, Vol. 1A, Academic Press: New York, 1964.

 

R. H. Tancrell and M. G. Holland, "Acoustic Surface-Nave Filters,"
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167

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M. Abramowitz and 1. A. Stegun, "Handbook of Mathematical Functions,"
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R. F. Milsom and M. Redwood, "Interdigital Piezoelectric Rayleigh-
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C. C. Tseng, "Frequency Response of an Interdigital Transducer for
Excitation of Surface Elastic Waves," IEEE, ED-lS, No. 8, August
1968.

R. M. White and F. M. Voltmer, "Direct Piezoelectric Coupling to

Surface Elastic waves," Appl. Phys. Lett., Vol. 7, No. 12, December
1965.

Larry A. Coldren, "Optimizing Loss-Compensated Long Delay Devices,"
IEEE, Vol. SU—ZO, No. 1, January 1973.

A. W. Warner, M. Once and G. A. Coquin, "Determination of Elastic
and Piezoelectric Constants for Crystals in Class (3m)," J. Acoust.
Soc. Am., Vol. 42, October 1966.

 

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