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DESIGN, CONSTRUCTION, AND TESTING OF ULTRASONIC TRANSDUCERS WITH MODIFIED
RADIAL VELOCITY PROFILES
by
Paul Samuel Zerwekh
thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
APPROVED:
MASTER OF SCI ENCE
in
ELECTRICAL ENGINEERING
DR. R. 0. CLAUS
March, 1982 Blacksburg, Virginia
DR. P.H. Wllr
ACKNOWLEDGEMENTS
I would like to thank my thesis advisor, Dr. Richard 0. Claus,
for his unending help and support in this endeavor. I would also Ii ke
~~ to thank Dr. F. G. Gray and Dr. P. H. Wiley for serving on my advi-~ --::::' ~ sory committee.
Special than ks are due my fellow graduate students in the Acous-
tooptics Group, Dan Dockery, Avinash Garg, John Gray, Janet Wade,
and Tyson Turner, for their help and consultation.
I wish also to thank the faculty and staff of the Electrical Engi-
neering Department as a whole, and specifically Dr. I. M. Besieris for
his helpful suggestions, and Vicki Trump and the other secretaries for
their invaluable organization.
I express my gratitude to my family and numerous friends for
their support and encouragement. Finally, I thank my wife for her
love and understanding which made this possible.
ii
CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . ii
Chapter
I . . . . . . . . . • • INTRODUCTION
THEORETICAL DEVELOPEMENT 11.
111.
IV.
v. VI.
Propagation Theory . . . . Piezoelectric Effects . . . . . Tr-ansducer Equivalent Circuit Electric Field Analysis . . . .
4 18 21 23
TRANSDUCER DESIGN ANO CONSTRUCTION
Electrode Pattern Design Impedance Network . . Construction Details
Experimental Setup . . . . Data Acquisition Technique Results . . . . . . . . • Quantitative Fit of Data . .
. . . • • • • . • . . 24 . . . . . . . . . . 27 .......... 32
TEST AND RESULTS
36 . 39 • 42 . 48
SUMMARY
SUGGESTIONS FOR FURTHER RESEARCH
Appendix
A.
B.
TERMINAL SIMULATION CODE
DATA ACQUISITION CODE
page
page
, 4
24
36
51
52
54
55
c. PROPAGATION CODE 61
o. . . THEORETICAL NEAR FIELD PLOT CODE 64
E. . • . • . . . . . DAT A ?:_QT ROUTINE 66
iii
REFERENCES ................... 68
BIBLIOGRAPHY ................... 69
iv
Figure
1.
LIST OF FIGURES
page
Thoeretical Far Field for Uniform Aperture Excitation
2.
3 ..
4. .
5.
Theoretical Near Field Magnitude for Gaussian Distribution
Theoretical Near Field Magnitude for Uniform Distribution
6 ..
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
. . . . . . Material Models
Transducer Equivalent Circuit
Electrode Pattern Optimizer Output
Transducer Element
Transducer Housing
. . RF Gating Circuit
Ultrasonic Test Tank
Data Acquisition System
Far Field Radiation Pattern of Gaussian Transducer
Far Field Radiation Pattern of Uniform Transducer
. . . . . Axial Plot of Gaussian Radiation Pattern
Axial Plot of Radiation Pattern of Uniform Transducer
v
Fit of Near Field Gaussian Data
. . Fit of Far Field Gaussian Data
8
16
17
19
22
26
34
35
38
40
41
44
45
46
47
49
50
LIST OF TABLES
.Table page
1. Capacitance and Resulting Impedance of Electrode Rings 28
2. Normalized Gaussian Voltage Values for Electrodes 30
3. Divider Resistor Values 31
vi
Chapter I
INTRODUCTION
The theoretical consideration of the propagation of bulk acoustic
waves in various materials may be greatly simplified if the wave ampli-
tude distribution on a cross section oriented perpendicular to the direc-
tion of propagation does not change as a function of distance along the
axis of propagation. In addition, in many applications of bulk ultrason-
ic phenomena it is desirable to produce a radiation pattern which is
bounded radially and has a single amplitude maximum centered on the
axis of propagation. One such application is the study of the Schoch
critical angle phenomenon in which a Schlieren visualization technique is
commonly used to observe specular as well as secondary displaced re-
flections of an ultrasonic beam incident on a liquid-solid interface. The
theory which describes this effect has been developed for both optical
and acoustic fields but only solved in closed form for the special case
of incident waves with a Gaussian pressure profile. The corresponding
field measurements would also be greatly simplified by the elimination of
the diffraction sidelobe maxima found in the far field of conventional ul-
trasonic transducer radiation patterns. Similarly, experiments involving
multiple scattering may also be simplified by the absence of sidelobes.
The beam profile which meets these criteria is Gaussian as a func-
tion of beam radius. Gaussian beams are also unique in that they min-
imize the divergence of acoustic energy in the radial direction and have
2
recieved a great deal of analytical attention due to use as a model of
the intensity profile for laser and maser output beams [1].
In acoustics, Breazeale described a method for generating a pres-
sure distribution which is Gaussian in one dimension utilizing multiple
linear electrodes [2]. A radial profile which approximates a Gaussian
function may be produced by reducing the size of one electrode in a
conventional circular piston transducer due to the fringing of the elec-
tric field lines and lines of mechanical stress in the piezoelectric materi-
al [3]. Filipczynski and Etienne developed a method of minimizing side-
lobe maxima in the radiation pattern of a focusing 7 MHz transducer by
constructing a non-circular 'star shaped' electrode with a voltage which
when integrated around the radius of the transducer provides a Gaus-
sian distribution [4]. This distribution, however, is dependent on angu-
lar displacement and therefore is not a function of radius alone. A
summary of the effects of other electrode shapes on ultrasonic radiation
patterns is presented by Rose[5].
An ultrasonic transducer with a radial velocity profile which is
Gaussian to within 1.5 percent has been designed, constructed, and
tested and is described herein. Chapter 11 provides a general theoreti-
cal summary of the properties of Gaussian spherical waves. It is shown
that the far field distribution of an ultrasonic transducer may be repre-
sented by the two dimensional Fourier transform of its near field profile
and numeric techniques for the calculation of far field radiation patterns
are discussed. Also, a brief explanation of the piezoelectric properties
3
of crystals and an analysis of the electric and resulting stress fields
within the transducer are presented. The design and construction of
the Gaussian profile transducer are outlined in Chapter 111. A method
of optimizing the placement of electrodes, basic electrical network analy-
sis, and construction details of the transducer are discussed. Chapter
IV describes the experimental ultrasonic and optical techniques utilized
in the testing of the prototype transducer. The experimental hardware
and data acquisition methods are described and a comparison of emperi-
cal and theoretical data is presented. Following a summary in Chapter
V, Chapter VI outlines a number of possible future applications of
transducers with modified radiation patterns. Simplification of acoustic
imaging, modifications of transducer element phase distribution, and ap-
plications of reciprocity principles to receiving transducers are ex-
plored. Finally, the Appendices contain circuit diagrams and software
used in the design and testing of the device.
Chapter II
THEORETICAL DEVELOPEMENT
2 .1 PROPAGATION THEORY
The manner in which an ultrasonic beam propagates through a
homogeneous material is determined to a large degree by the magnitude
and phase of the velocity distribution at the source. This section is a
survey of the propagation of beams from two types of sources, those
with uniform harmonic velocity over a circular aperture, and others
having a distribution which is Gaussian as a function of radius. Both
sources are considered to be monotonic and of constant phase over the
aperture.
Acoustic beam propagation may be described mathematically by· the
wave equation shown here in cylindrical coordinates due to the circular
symmetry of the beams to be considered;
2- _i_a2uce.z,t) 7-u(p,z,t) 2 2 • O, c 3t
(1)
where U • Displacement,
p • radial distance from axis of propagation,
z a distaace along axis of p't"Opagation,
t a time,
4
5
c
Ev • bulk modulus, and
d =- density.
For monotonic harmonic fields, the wave equation may be transformed
into the complex Helmholtz equation
v2u(p,z) + k2u(o,z) =- 0,
with the boundary constraints
and
Here,
U(p,O) .. U (p) 0
I U(p,z)I = 0.
;l-+<D
k -2Ilf
c
where f • operating frequency.
(2)
(3)
(4)
(5)
6
Fields sufficiently far from the transducer (source) so that the
condition
2 ~ « 1, z
(6)
holds may be calculated from an expression involving the two dimension-
al Fourier Transform of the initial distribution. For cylindrical coordi-
nates, it is convenient to use the Fourier-Bessel Transform (Hankel
transform of zero order) denoted by
B{f(p)} = ~r f(r)J0 (2Ilrp)dr, 0
J 0 (p) • Zero Order Bessel Function.
More specifically, the displacement is given by
U(~,z) = exp(jkz) j>.z
= 2IT T·
(jko2] I exp I 2 B{U (p)}I ' z 0 p '
L h
(7)
(8)
7
In the case of the uniformly excited aperture, the initial field may be
represented by the circ function for which
J 1 (Jiap)
ap/2
The far field distribution is then given by
( ~) ka2 [2J1 (ka /2z)l U(p,z) = exp(jkz)exp j 2z J j8z ka /2z _I·
(9)
(10)
Figure 1 shows this theoretical distribution for a distance corresponding
to a distance of 800 wavelengths from the transducer. If the aperture
distribution at the transducer is given by a Gaussian function of radi-
us,
U0 (p) • ~exp [-p~], ; rra.l a
( 11)
then the far field distribution is
.,, <D c ., ID .... -i
'::T 0 ID ., ID ... -· (") OJ .,, OJ ., .,, ID
Q.. -0 ., c :i -0 ., 3 )>
"O ~ ., ~ c:: ., ID m
x (") .....
OJ ..... 0 :i
~
):.
()
~ -
I --;
-
c:: 0
t:)
Ill
,.,., ,,, ):
. •
~
~
):.
t:)
c...
--
c:: ~\
~ Cl
) - 0
B
9
U(P,z) = ~ exp(-jkz - $(z)!xp [ -o\] / IIa(z) ~ a(z)
(12)
where,
a(z) • (13)
and
-1 [ 2z l +(z) a tan ka2 ' (14)
which has an amplitude distribution which remains Gaussian.
Solving the Helmholtz equation for fields near the transducer for
arbitrary transducer velocity distributions in closed form is a difficult
task. The problem is therefore approached here using a numerical
technique outlined below. First, The Helmholtz equation in cylindrical
coordinates may be written as
10
( 15)
If we let
U(p,z) a $(p,z)exp(-jkz), ( 16)
and assume that changes in the magnitude of the field with respect to
distance along the axis of propagation are small compared those with
respect to radial displacement, then
- 0 • ( 17)
11
It is convenient at this point to separate variables and introduce to fol-
lowing notation;
~(p,z) a R(p)Z(z) =
aR(p) ao
az(z) az
I • R m
I ~ z .
n
R Z m n ( 18)
(19)
(20)
(21)
12
Equation ( 15) then is given as
( l I II J -jkz - R Z + R Z - 2jkR Z e m 0 p m n m n m n '
(22)
which implies
l a'z + R"z - 2jkR z - o . p m n m n m n (23)
The separated variables may be approximated by truncated Taylor Ser-
ies expansions. For R,
? ' + tio- R" R(p +lip) = R-...Ll :: R + tio R
~~ m m 2 ~
(24)
and
R(p - Ap) a R m-1
13
A 2 II
- R - AO R1 + ~ R m m _ m
Adding (24) to (25) gives
Also,
II
R m _ Rm+l + Rm-l - 2Rm
tip
' - Rm+l - Rm R :n tip
For Z,
I - 2t.zZ
n
(25)
(26)
(27)
(28)
14
which gives
(29)
Substitution of (26), (27), and (29) into (23) yields the difference equa-
ti on
- z rR -~[l(Rm+l - Rm-1) + n l m 2k p 2tio
(30)
Recombining variables such that
U =- Z R (31) n,:n n m
gives the field difference equation
u :u n+l,m n,m _ill
2k
15
A Fortran program was developed using this difference equation to cal-
culate the magnitude of the field in the region from the surface of the
transducer to a distance corresponding to approximately 50 wavelengths
from the transducer. This distance is 2.5 cm for a 1.3 cm dia. trans-
ducer operating at 2.25 MHz. The program is listed in the appendices.
Figures 2 and 3 show the near field distributions predicted theoretically
for the Gaussian and uniform excitations. The program is listed in the
appendices.
"'M -· <D
c ., (I)
I-.)
.. z (I)
OJ ., -c 0.
(I) -0 ..,
I
!I( 1
0 l0
0.0
0
75.0
:l
""'0
'"'N
~ TU
JE:
5,0.
DO
.I
. ,..
;JI
25.x
r.
:x;
' c:
J,•
m8;~
'"f;
llfi
!!-
~.~Ji!!!
.,., (0 c ., Cl> w ..
50.0
0 i~.oc
~ ~
(1) 0 ., (1) ,_
("\ Ill z (1) Ill ., "Tl
:ii a.
~
I»
(0
:J
..+ c 0.
Cl> -0 ., c
2 :>
....
-0 ., 3 0 Cll ..... ., Ci c ..+
0 :J
LL
18
2 .2 PIEZOELECTRIC EFFECTS
The preceeding discussion of wave propagation assumed the boun-
dry condition
(33)
the generation of which is the sole purpose behind the present Gaussian
transducer design. The material most commonly utilized to produce
bulk ultrasonic waves in nondestructive testing and similar applications
is crystalline quartz. While other materials have greater piezoelectric
coefficients, quartz is relatively rugged and inexpensive. The present
application, furthermore, involves generation of ultrasound in fluids
which do not support shear modes, thereby eliminating the need for
generation of strain in more than one dimension. X-cut crystalline
quartz produces such one dimensional strain and may be modeled as
shown in Figure 4 a . Because the distribution of charge in the lattice
of a non-piezoelectric material is symmetric, when an electric field is
applied some bonds are lengthened while others are shortened resulting
in zero net strain. Crystals which lack symmetry, however, behave as
shown in Figure 4 b
When an electric field is applied to such a material, both bonds shorten
resulting in a net strain determined by the piezoelectric coefficient, the
mechanical complience of the crystal, and the external loading applied.
More specifically, if the x-di rection is normal to the face of a transduc-
19
E
a) NON-PIEZOELECTRIC SOLID
E .. b) PIEZOELECTRIC SOLID
Figure 4: Material Models
20
er element, then the stress induced by an applied voltage is given
by[6]
where
.. d11V0 exp (jwt)
511dt
stress in X-direction
d11 c piezoelectric strain coefficient
s11 = compliance
v 0
w
=
=
thickness of transducer
amplitude of applied voltage
operating frequency
(34)
The resulting strain is not only a function of the compliance of the
crystal, but also of the loading. For an unloaded crystal, the strain is
given by
(35)
21
and the displacement is the strain times the thickness of the crystal.
The other limiting case is a crystal which is loaded with a medium of in-
finite density or rigidity in which case the induced stress produces no
deformation. To calculate the actual particle displacement or velocity
under a given loading, the longitudinal wave equation must be solved in
both the crystal and the loading media and the two solutions must then
be equated at the boundry.
2.3 TRANSDUCER EQUIVALENT CIRCUIT
The theoretical overview of piezoelectric transducer's would be in-
complete without a brief discussion of the electro-mechanical transfer
function which represents the operation of the transducer. An equivi-
lant circuit model due to Mason is shown in Figure 5 It may be seen
from this model that the transducer undergoes an electrical resonance at
the operating frequency corresponding to the damped mechanical natural
frequency of the crystal. At this point, the electrical input impedance
of the circuit is determined primarily by the ratio of the mechanical
impedance in the crystal to that in the loading media and to the static
capacitance introduced by the transducers electrode structure. This is
important in the electrode network design introduced in the following
chapter.
hC :1 0
22
jZ (t~ - csc(kd)) 0 2
t u
Figure 5: Transducer Equivalent Circuit
Z "'~c 0
k • w/c A • Area p "' Density c = Speed of sound h • Piezoelectric/dielectric
constant ratio
23
2 .'I ELECTRIC FIELD ANALYSIS
The final consideration which is pertinent to the design of velocity
profile modified transducers is the determination of the electric field
distribution in the transducer element. For a thin crystal, we may ne-
glect fringe effects and assume that the electric field within the crystal
is a function of displacement in the plane of the transducer surface al-
one. For a circular transducer with concentric annular electrodes the
solution to Laplace's equation is a logrithmic function of radius. If a
sufficient number of electrodes are included so that the interelectrode
spacing is small with respect to the circumference of the rings, the so-
lution may be closely approximated by a linear function of radius. The
complete field for a circular transducer with multiple concentric elec-
trode rings may then be represented by a piecewise linear function of
radius on the surface of the transducer.
Chapter Ill
TRANSDUCER DESIGN AND CONSTRUCTION
3 .1 ELECTRODE PATTERN DESIGN
The first practical step in the construction the prototype Gaussian
transducer was the layout of the proper concentric annular electrode
pattern needed to generate the desired piecewise linear approximation to
a Gaussian voltage distribution on the active surface of the piezoelectric
transducer element. The degree to which this distribution fits a Gaus-
sian function is determined by the width of each electrode ring, the to-
tal number of electrodes, and the location of the electrodes on the radi-
us of the transducer crystal. Since the desired Gaussian voltage
function attains a particular value at only one mathmatical point, the
electrode width should in principle tend to zero. The photo-etching
techniques used in our transducer construction, however, do not have
infinite resolution but require a minimum electrode width of approximate-
ly 0.5 mm. The degree of fit to the desired Gaussian shape may also
be improved by using a large number of electrodes, thereby increasing
the total number of segments in the approximation. This unfortunately
requires that the interelectrode spacing be small, increasing the possi-
bility of electrical breakdown between adjacent rings when high voltages
are applied. It was found, however, that with as few as 5 electrodes
the mean absolute error in the fit between the desired continous func-
24
25
tion and its piecewise approximation may be reduced to less than 1.5
percent of the peak.
The radii of the rings are the variables over which the greatest
control may be exercised during desig_!l. In order to fit a piecewise li-
near function of a given number of seqments most closely to a smooth
function, the number of electrodes should be proportionately related to
the curvature of the function. The initial placement of the eletrodes
was made on this basis, then an interactive computer graphical techni-
que was used to reduce the error in the fit. First, the desired Gaus-
sian, the fit produced by the initial ring placement, and the error bet-
ween the two are plotted on the same graph for comparison. The
electrode positions are then moved in the direction of the locations of
maximum error and this procedure is repeated until the error has no
pronounced maxima. The electrode spacings calculated using this optim-
ization technique for the prototype Gaussian transducer are shown in
Figure 6 .
c NO
R!-1A
LIZE
D VO
L7AG
E
..,, (0
~ ., ti> en
I,.;
rn
(I)
O"
::;:
..,.. n
~
,.... r-:
..,
:'%'!
> x
0 r
z 0.
~
- ·-· (I
) ,....
>
:;;::
....;
co
s -0
2
u:
Qi
N
0 :::::
:: ,....
-'-
' r
,.... v.
t"
l c::
II
(I)
t/)
-:i
.., ['!
'::
c :l
~
> ..., rr:
0
0 c:
:;;::
;,.;
......
,....
"O
._
6 c::
.
,.... v:
-:::::
:: >
3 V
1 ....
;
-N
3
~
(I)
3 ....
0 ..,
!,...)
!,.
..)
I,.)
0
N
0 N
-
c: 0
: -.
.)
3 ,....
3 "O
N
c: -..
.J ,..
~
\...>
~ ~
V1
V'
-0
- O' t::'
N 1.11
i3 3
::" :--
9Z
27
3 .2 IMPEDANCE NETWORK
Once the desired electrode pattern design was completed a method
of introducing the proper voltages to' the individual electrodes was de-
veloped. The first step in this process is the determination of the
electrical impedance of each of the electrodes. It is assumed here that
when the transducer is used in immersion testing it is loaded relatively
little and that the majority of the electrode impedance is due to the ca-
pacitance between the electrodes and the transducer ground plane.
Although an exact calculation of the electrode capacitance is difficult
due to fringing of the electric field within the piezoelectric crystal, an
upper limit may be found from the following approximate relation.
(36)
where R = mean electrode radius, and
w • electrode width.
The impedance at the 2.25 MHz operating frequency may then be calcu-
lated for the electrodes. Table 1 shows the capacitance and impedance
values for the electrode pattern shown in Figure 6
It was decided that the reactive impedance of the electrodes was suffi-
ciently high that a purely resistive divider network could be used to
28
TABLE 1
Capacitance and Resulting Impedance of Electrode Rings
Electrode Radius (mm) Capacitance (pF)
(Mean)
0.15 1.32 2.79 3.94 5.16 6.25
0.003 0.048 0. 101 0.143 0. 187 0.226
Impedance (MegOhms)
53.1 3.32 1.58 1. 11 0.85 0.70
29
produce the desired Gaussian voltage distribution. Table 2 lists the
desired nonnalized voltage values at the electrode positions. Two
points were considered in the choice of resistor values. First, a suffi-
ciently low impedance is required to override the capacitive coupling ef-
fects. Second, the total series resistance must be high enough to avoid
excess power- dissipation and loading of the transducer driver amplifier-.
To satisfy these constraints, the value of the total series resistance was
chosen somewhat arbitrarily to be 60 kOhms. The values of the indivi-
dual divider resistors were then calculated to produce to proper voltage
values and are listed in Table 3
30
TABLE 2
Normalized Gaussian Voltage Values for Electrodes
Electrode Radius (mm) (Mean)
0. 15 1.32 2.79 3.94 5.16 6.25
Normalized Voltage Value
1.00 0.83 0.46 0.21 0.08 0.0
31
TABLE 3
Divider Resistor Values
Resistor Value ( kOhms) Percent of Total
Center 1 9.6 16 2 22 37 3 15 25 4 7.8 14 5 4.8 8
Ground
32
3.3 CONSTRUCTION DETAILS
The electrodes were placed on the transducer element using vapor
deposition and photo-etching techniques. Three electrode metals, alu-
minum, chromium, and gold, were applied in various combinations to
form the desired pattern. The metals were deposited directly on the
quartz in a vacuum chamber by vapor deposition at a distance of appox-
imately 10 cm. The artwork for the electrode pattern was done in ink
at five times actual size and reduced photograpically to produce a nega-
tive mask. The transducer was then coated with Kodak KPR 3 photore-
sist, dried for 15 minutes at 90 °C, and exposed to ultraviolet light
with the mask in place.
Initial attempts were to deposit aluminum directly on the crystal. It
was found, however, that although the resulting coatings were quite
uniform, considerable undercutting occurred during etching with either
sodium hydroxide or ferric chloride. Two succesful combinations re-
sulted using a thin base layer of chromium before deposition of contact
layers of aluminum or gold. The aluminum was deposited in our labora-
tory; the gold was commercially plated on the crystal before shipment.
Ferric chloride was used to etch the aluminum, however, the best re-
sults were obtained when gold was etched with a mixture of potassium
iodide and iodine followed with alkaline ferricyanide to etch the underly-
ing layer of chromium.
Once the electrode etching was complete, the resist was cleaned off
with resist solvent and electrical leads were attached with conducting
33
epoxy and supported with a layer of non-conducting epoxy which was
dome shaped to avoid resonant modes external to the crystal. A thin
sheet of aluminum foil was used for the ground plane/front wear plate
and was bonded to the side of the transducer with conducting epoxy.
The epoxy was blended with excess resin so that it would remain semi-
viscous to damp surface modes on the crystal and, in addition, to at-
tenuate shear modes in the event that the transducer be used in a sur-
face contact application for the generation of bulk longitudinal
ultrasound in solids. Figure 7 shows the details of the transducer ele-
ment construction and Figure 8 illustrates the housing details.
The 30 gauge electrode leads were soldered into the voltage divi-
der and the network was attached to a coaxial line. The transducer
assembly was then placed in a 1.3 cm inner diameter PVC cylinder and
cast in place with an epoxy mixture which was loaded with filler materi-
al to disperse ultrasound emision from the electrode side of the crystal
and to increase the output efficiency of the transducer. The completed
prototype was then tested as outlined in the following chapter.
SUPPORTING EPOXY
CONDUCTIVE ADHESIVE
/ ELECTRODE LEADS
WEAR Pl.ATE
Figure 7: Transducer Element
ANNULAR ELECTRODES
PIEZOELECTRIC CRYSTAL
-
(,.) ~
PVC CASE
RESISTOR ~ETWORK
35
TRANSDUCER ELEME~
COAXIAL CABLE
Figure 8: Transducer Housing
FILLER-LOADED EPOXY
Chapter IV
TEST AND RESULTS
11.1 EXPERIMENTAL SETUP
The transducer was initially tested with both gated rf signals and
pulses. Although negligible difference in the results for these two cas-
es were obtained, the design assumptions specify cw operation so the
final experimental apparatus was designed for gated rf excitation. Fig-
ure 9 shows the circuitry for the device used to generate the gated
2.25 MHz signals which are amplified and used to excite the transducer.
The comparator produces a TTL compatible pulse train of adjustable
phase at the frequency of the input rf. This pulse train is fed into a
counter, the outputs of which are connected to two sets of binary mag-
nitude comparators. One comparator resets the output flipflop when the
desired number o.f cycles of rf have been passed to the output buffer.
The other comparator sets the flipflop back to the on state to allow the
rf to pass when the total count sequence has been completed. The
output flipflop controls a pair of bi-mos analog switches which feed the
output current buffer. In addition to the gated rf output, the output of
the comparator is made available for synchronization purposes. Quanti-
tative measurements were made by scanning the field distribution gener-
ated in a water tank using an aperture-stopped Harisonics G 0204 pie-
zoelectric detecting transducer. The 6.35 mm diameter of the
36
37
transducer was effectively reduced by placing an aperture mask in
front of it. Several methods were used to produce the aperture. The
initial measurements were made using an aperture in a piece of aluminum
foil coated with soft wax. It was found, however, that although this
technique attenuates the signal outside the aperture by 8-10 dB, it is
not nearly sufficient for near field measurements where the beam profile
is relatively narrow, due to the integrating effect of the effectively
large aperture. The second device used for the aperatu re consists of a
one cm thick plate of aluminum with a hole drilled at an angle greater
than the acoustic angle of complete internal reflection for longitidinal ul-
trasound incident on aluminum in water. This angled aperture techni-
que works well in the near field, but is quite directional due to the
fact the projected area of the aperature depends on the alignment of
the l. 5 cm long hole, making it unacceptable for measurements off axis
in the far field. A third material, closed cell polyurethane foam, was
found to block transmission to an acceptable degree, however the re-
flection from the surface of the foam became a problem for near field
measurements. The best solution to the aperature problem was found to
be a metallic cone which presents an angle greater than the total reflec-
tion angle. In the center of the cone is a conical hole of slightly more
acute angle. This hole forms the desired aperture and provides a rela-
tivly large angle of acoustic signal acceptance. This aperture and en-
tire receiver were used to test the transducer using the system de-
scribed in the next section.
~ ,
... ,
RF IN
TRIGGER LEVEL IN
I 'YCOMPARATOR
PULSE COUNTER
-
ANALOG SWITCHING
PULSE TRAIN LENGTII l NPU'f
DIGITAL CUMPARATOit
UIGITAI. COMPARATOR
SEQUENCE LENG"l11 I NPU'f
I
• I -I I _rr--
RESET
FL Ip I 1''LOP
SET
Figure 9: RF Gating Circuit
CURRENT -.. DRIVER
.
w ()C)
39
'1.2 DATA ACQUISITION TECHNIQUE
Figure 10 shows the water tank with two axis carriage used to
carry out the measurements on the transducer. The recieving transduc-
er is mounted in a holder designed to attach onto a magnetic mount
which fastens on a steel plate on the bottom of the carriage.
The transducer under test is mounted either in the bottom of the
tank or on the side of the tank depending on the type of scan to be
taken. This was found to be more satisfactory than fixing the receiver
and scanning the prototype due to the fact that in the latter case when
the carriage rotates slightly, the transducer mounted on it may be misa-
ligned. For profile cross sectional scans, the transducer is mounted in
the bottom of the tank pointing upward, while for axial scans it is
mounted in the middle of one side of the tank pointing inward horizon-
tally. Figure 11 illustrates the system used to control the scans and
acquire the data form the receiving transducer.
The motion of the receiving transducer is controlled by the two
Cybernetic Micro Systems CY-500 stepper motor controllers which re-
ceive commands from the TRS-80 Microcomputer via a parallel interface
card on the S-100 buss. Schematic diagrams for the stepper control
circuitry and S-100/TRS-80 interface are included in Appendix I. The
rf signals from the receiving transducer are narrowband filtered and
demodulated using a Realistic DX-160 receiver in the AM mode. The
output from the DX-160 is fed into an lthaco Dynatrac 361-A lock in
amplifier, the output of which is sampled by the DI A at 225 kHz, prov id-
40
STEPP/NC MOTORS LEAD SCREWS
MOUNT/NC. PLATE
Figure 10: Ultrasonic Test Tank
DATA LINK
'
s-100 SCANNER -TRS-60 , . -BUSS DRIVE
' ~· ~ r-.
A/D LOCK IN ENVELOPE 1-1 L IVER -
~ in \) [; CONV. AMP. ref. DETECTOR RTURE ~ ~ r---
~ " ~ ,,,.,---. sync. -
RF RF ENI - -, GENERATOR MODULATOR AMPLIFIER SDUCER
.... _
I '
Figure 11: Data Acquisition System
42
ing a quasi-real time digital value to be sampled by the microcomputer.
The linearity of this system is on the order of 1 percent over the 25
dB dynamic range dictated by the 8 bit resolution of the AID. The
support circuitry for the TRW TDC1001J AID is shown in Appendix I.
The microcomputer samples the data at each point in the scan after
waiting for approximately one second for vibration in the tank due to
carriage motion to dissipate. The data acquisition program is included
in Appendix 11. The current software samples a square matrix of 2500
(50x50) points at user selectable intervals. The data are then convert-
ed to 3 byte ASCII-Decimal strings and uploaded via an RS-232-C serial
converter and modem to the VPI I BM 370 Computer for processing. The
results of scans taken with this system are discussed in the following
section.
11.3 RESULTS
As predicted in Chapter 2, the radiation pattern of the Gaussian
transducer is void of sidelobe maxima. The far field radiation pattern
of the uniform circular transducer, while not showing the degree of si-
delobes predicted, does contian substantial off axis maxima. The fact
that the lobes are not as pronounced as calculated is due to fringing of
the lines of stress in the transducer which smooth the edges of the
field making it not truly uniform over the transducer face. A compari-
son of the far field radiation patterns for the two types of transducers
is shown in Figures 12 and 13 . The near field scans are taken at 40
/
43
wavelengths from the face of the transducer; the far field data is taken
at 800 wavelengths. The wavelength of the 2.25 MHz ultrasonic waves
in water is 0. 5 mm making the ratio of transducer radius to wavelength
12. 7. Both the Gaussian and uniform transducers have the same radius
(6.35 mm) and were excited in the same manner.
The most striking result, however, may be seen in Figures 14 and
15 . Again, as predicted, the Gaussian radiation pattern remains quite
uniform even very near the transducer surface. The radiation pattern
from the uniform tranducer contains points where destructive interfer-
ence completely cancels the signal, sidelobes begin to form, and the
beam is much more divergent than the Gaussian beam.
..... N .,, GI ., .,, (!)
:::::: ~
('")
t:"!
1--1 < :':l
:;:
r 0 c;
> ..; ,... ,..., z
RECE
IVER
OU
TPUT
a:.:;:
:: i 2~ .
c~.
j sc . ·
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---
-I
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! I I I ~-!
• I
8; I
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,... ... -,
:..... '
~: I I I
~~ 8i I
§-l
. I
,.... ' 01
I I =..
.: -' ::l . =i
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~-: I
,... :
ei I ' ! ~1 I I gr - -
'Tl -· '° c ., CD .... (...) .. 'Tl
I» ., 'Tl -· ~ ~ ::0
Al ~
Al ,. -· 0 :i
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Al .. .. CD ., :i
0 - c: :i -· -0 ., 3 -i ., Al
:i
Il
l ~
c: n CD .,
200.
::JO
x '"O
0 ..... 0 - C>
QI c Cll
Cll
QI :i
;o
QI a. -· 0 :i
'"O
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9tr
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8 8.
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(_")~ er: ld > ~ 'c> ldo L\) lJ.11(>" CY
(] 0
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n n
z=l30>.
. I--· - ... ----.... c.11 .CXl U.00
Figure 15:
SOURCE
·---, ----·- ,-··--·-·--,-·· -··-···· --r · -- r·-· · ··-- --1 ·-·· -- -, ts.on 24.oo :12.nn 411.IJ(J •m.oo bt-i.OO 64.00
RlCFIVER LOCAlION
Axial Plot of Radiation Pattern of Uniform Transducer
~ --.!
48
II.II QUANTITATIVE FIT OF DATA
While the data presented in the last section indicates that the radi-
ation pattern from the prototype transducer behaves as a Gaussian, a
more quantitative measure of how closely it actually fits a true Gaussian
function is desired. Figures 16 and 17 show the results of this fit pro-
cess.
The symmetry of the data was checked by first finding the cen-
troid of the data set then collapsing the three dimension data into two
dimensions by plotting the field magnitude against the distance from the
centroid. The data files for the prototype transducer, however, were
erased before the fit routine was debugged, so the fit shown was done
on a manual scan which was taken before the automatic drive system
was completed. The near field standard deviation of the near field data
from the fitted Gaussian is less than two percent of the maximum. The
standard deviation for the far field scans is less than four percent.
(.) ( )
(JJ_ •I
() (.)
l>.
" >(.) l:(~
N (l)
I· :1 og I- . :J-i (JN
rr: laJ >o ..... n hJu) U-·· hJ 0::
0 0 (I)
g
NEAR FIELD (40 WAVELENGTHS)
(-"
-~
·-,-·--··-,- -----·· r - ----------,-- ···-- ·r-- ··-·--·--,--· · -·-.----·--r--·-- · (ll.OO 4.CXJ o.m 12.00 IG.OU 20.0U 24.00 20.00
fWfT I VF I~ I f"lf'Rl I ON 1111
Figure 16: Fit of Near Field Gaussian Data
~ c.o
.,, (,0 c
-.
.., ID
~~
..... •
-...I .,, .... 0 -
~
. . ,_
, _,
,.-~Ci I
•
I <
.......
:-i ~ !
.,, QI
-;
""-'
I ..,
- -i
.,, ID ~
::::JA~
'rJ
--~I
> =>
~I
;=
-c .....
: "2
j
C'>
QI c Ill .
Ill
QI :>
I ~
I co:
-
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.....
c::;
3 N
: -
:;: 8 i
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0 I
0
0 QI .... QI
I ~
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< c."
li to:
0
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~ -J
tr. -
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81
o~
Chapter V
SUMMARY
It was shown that there are several advantages to generating an
ultrasonic beam with a radial velocity profile which has been modified
from the piston radiator. In particular, it was shown that if the veloci-
ty profile is made to be a Gaussian function of radius, the beam is well
behaved in both the near and far fields. The Gaussian beam remains
Gaussian in the near zone, while the uniform piston radiator produces a
near field beam which is quite irregular, and, in fact is zero at some
points. In the far field, the Gaussian radiator produces a radiation
pattern which is void of the sidelobes found in the far field pattern of
the uniform piston.
An ultrasonic transducer which produces such a Gaussian beam
was constructed using a circular x-cut quartz disk with concentric an-
nular electrodes on one surface and a ground plane electrode on the
other. The transducer element was 1.27 cm in diameter and had a re-
sonant frequency of 2.25 MHz. An ultrasonic test tank and micropro-
cessor data acquisition system were designed and built to test the
transducer, and plots of the resulting data were generated. For com-
parison, a uniform piston transducer was constructed and tested in a
similar fas ion. Finally, data from scans along diameters of the Gaussian
transducer were fitted to Gaussian functions to show that the standard
deviation of the near and far field radiation patterns from true Gaus-
sians are less than two and four percent of the maximum, respectively.
51
Chapter VI
SUGGESTIONS FOR FURTHER RESEARCH
This project, like many, produced more interesting ideas and ques-
tions than it answered. Therefore, in the hope that some of them may
eventually be addressed, a brief list is included here.
1. The data acquisition system could be improved by design-
ing a more precise envelope detector and replacing the lock in
amplifier with a synchronous sampler circuit to further isolate
the signal from rf noise.
2. The Gaussian beam may possibly be used to simplify
acoustic imaging. Since the far field radiation pattern is given
by the Fourier transform of the initial distribution, the pattern
produced when an object is imaged in an ultrasonic field may
be calculated from the convolution of the transforms of the
beam distribution and the transmission or reflection distribution
of the object. Detailed knowledge of the beam profile at the
object (known for the Gaussian) should make this process more
direct.
3. In addition to the amplitude, the phase of the velocity dis-
tribution may be varied to provide a transducer which may be
made to focus at a point selectable by the driving electronics.
4. Finally, the transducer may be used as a receiver with a
weight function built in. When an ultrasonic transducer is
52
53
used as a receiver, it integrates the field over the area it sub-
tends. For some applications it may be desirable to weight this
integration, and the concentric annular electrode design should
make this possible.
5. The propagation code could be modified to use simultane-
ous difference equations in place of the foward equations which
eventually introduce instability in the solutions. The more sta-
ble difference technique could be used to study reflection and
scattering, and also predict the effect of placing acoustic lens-
es in the beam.
Appendix A
TERMINAL SIMULATION CODE
This routine lets the TRS-80 act as a terminal for initial placement
of the receiver.
10 CLS
20 B=&HEE
30 AS=INKEYS
32 l=I •1
34 IF 1=25 GOTO 100
40 IF As="" GOTO 30
45 PRINT AS;
50 IF AS="U" B = 238 :GOTO 30
60 IF AS = ''V" B=239 : GOTO 30
70 OUT B,ASC(AS)
80 GOTO 30
100 1=0
110 PRINT at 104,INP(&HED)/2.55
120 PRINT at 897, ""
130 GOTO 40
54
Appendix B
DATA ACQUISITION CODE
This routine in BASIC drives the stepper motors and collects data
from the AID converter. It then uploads the data to the I BM 370 CMS
operating system.
100 CLS
120 PRINT "TANK SIGNAL MAXIMIZER"
125 INPUT "RESOLUTION";N
130 P=238
135 0$=''+"
140 SL=S
145 QS=INKEYS
150 IF QS="C" GOTO 750
155 IF QS="R" GOTO 125
160 C=C•l
170 IF C=10 P=239
180 IF C= 20 P=238: C=O
190 AS=DS
200 GOSUB 500
210 AS="O"
220 GOSUB 600
230 S=INP(237)/2.55
235 PRINT S
55
240 IF SL>S GOSUB 700
250 GOTO 140
500 OUT P,ASC(AS)
510 OUT P, 13
520 RETURN
600 F 0 R I= 1 T 0 N
610 GOSUB 500
615 NEXT I
620 RETt..; RN
700 IF Os=''+" Os="-" :RETURN
710 OS ="•":RETURN
750 As=" R 239"
760 GOSU B 1210
770 GOSUB 1290
56
775 INPUT "STEPS/SAMPLE I SPS I 25 x SPS , 50 x SPS";NP,N2S,N1$,N3$
780 As="N "•Nl$
790 GOSUB 1210
800 GOSUB 1290
810 As="-"
820 GOSUB 1210
830 GOSU B 1290
840 As="F 1"
850 GOSUB 1210
860 GOSUB 1290
870 As="G"
880 GOSUB 1210
890 GOSUB 1290
900 D IM G ( 50 I 50)
910 FOR I= 1 TO 15*NP:NEXT
920 AS="N "•N2S
930 GOSUB 1210
940 GOSUB 1290
950 A$="•"
960 GOSUB 1210
970 GOSUB 1290
980 FOR 1=0 TO 49
990 FOR J=O TO 49
57
1000 FOR K=l TO NP/2•50: NEXT K
1010 C=INP(237)
1020 PR I NT I ; J; C
1030 G ( I I j) = c
1040 As="G"
1050 GOSUB 1290
1060 NEXT J
1070 GOSUB 1210
1075 FOR K=l TO . 9*NP: NEXT K
1080 As="-"
1090 GOSUB 1290
1100 As="N "•N3S
1110 GOSUB 1290
1120 As="G"
1130 GOSUB 1290
1140 FOR K=l TO 20*NP: NEXT K
1150 As="N "•N2S
1160 GOSU B 1290
1170 As="•"
1180 GOSUB 1290
1190 NEXT I
1200 GOTO 1420
1210 FOR N=l TO LEN(AS)
1220 FOR K= 1 TO 10: NEXT K
1230 OUT 239,ASC(MIDS(AS,N, 1))
1240 PRINT MIDS(AS,N, 1);
1250 NEXT N
1260 PRINT
1270 OUT 239, 13
1280 RETURN
1290 FOR N=l TO LEN(AS)
1300 FOR K=l TO 10: NEXT K
1310 OUT 238,ASC(MIDS(AS,N, 1))
1320 PRINT Ml DS(AS, NI 1);
1330 NEXT N
1340 PRINT
1350 OUT 238, 13
1360 RETURN
58
59
1420 AS=INKEYS
1430 IF AS="&" GOTO 1560
1440 IF (INP(&HC1) AND 32)= 0 GOTO 1440
1450 IF A$<>'"' OUT &HCO,ASC(AS): PRINT AS;
1460 IF (INP(&HC1) AND 1)=0 GOTO 1420
1470 8=1 NP ( &HCO)
1480 IF B<>O PRINT CHRS(8);
1490 GOTO 1440
1560 FOR J=l TO 2500
1570 K=INT((J-1)/50)
1580 L=J-50*K-1
1581 Q=G(K,L)
1582 A=INT(Q/100)
1 583 8 =I NT ( ( Q - 100* A) I 10)
1584 C = I NT ( ( Q- 100* A - 1O*8 ) • 48)
1585 A=A •48
1586 8=8+48
1587 T$=CHRS(A)•CHRS(8)•CHRS(C)
1588 FOR I =1 TO LEN (TS)
1590 PRINT MIDS(TS,1,1);
1600 IF(INP(&HCl) AND 32 ) = 0 GOTO 1600
1610 OUT &HCO, ASC(MI DS(TS, I, 1))
1620 NEXT I
1630 IF (INP(&HC1) AND 32)=0 GOTO 1630
1640 IF J/10 <> INT(J/10) OUT &HCO,ASC(","):PRINT ",";:NEXT J
60
1670 OUT &HCO, 13
1680 IF(INP(&HC1) AND 1 )=O GOTO 1680
1690 BT=INP(&HCO)
1700 IF BT<>17 GOTO 1680
1710 PRINT
1720 NEXT J
1730 FOR 1=1 TO LEN(N2S)
1740 IF(INP(&HC1) AND 32 )=O GOTO 1740
1750 OUT &HCO,ASC(MIDS(N2S,l,1))
1760 NEXT I
1770 PRINT N2$
1780 GOTO 1420
Appendix C
PROPAGATION CODE
This routine uses a complex foward difference equation to calculate
the magnitude of an ultrasonic beam in the near field.
IMPLICIT COMPLEX*8(U, T)
DIMENSION AMAG (500,50), U (500,51)
DO 10 K=l, 500
10 U(K,51)=CMPLX(0.0,0.0)
c READ DATA
c
READ(S,*) (U(1,l),1=1,50)
READ(5, *) RD I DZ, AK
DR=RD/49.0
SUM=O.O
DO 15 1=1,50
T4=U(1,I)
C WRITE(7, *)T4
c for diagnostic use
AMAG (1, I )=SQRT( REAL(T4*CONJG (T4)))
15 SUM=SUM•AMAG(1,l)/50.0
WRITE(7 ,200) (AMAG (1, I) I 1=1, 50)
C WRITE(7,*) SUM
DO 100 K=1, 249
61
RO=O.O
SUM=O.O
DO 50 1=1,50
J=I
62
T2=CMPLX(1 .O/(DR*DR) ,0.0)*( U ( K,J•l) •U ( K, I B( 1-1) )-(2.0,0. 0)
l*U(K,J))
T1=T2
c check for center of field,
IF (RO.EQ.0.0) GOTO 30
T1 =CMPLX ( 1. 01 ( R0*2. O*DR), 0. 0)*( U ( K,J •1) -U ( K, I B ( 1-1)))
30 CONTINUE
U ( K •1,J) =U ( K,J) -CMPLX(O. 0, DZ/(2. O*AK) )*(T1 •T2)
RO=RO•DR
T3=U ( K •1, I )*CEXP( CMPLX(O .0, -AK*Z•DZ))
AMAG ( K •1, I) =SQRT( REAL(T3*CONJG (T3)))
50 SUM=SUM•AMAG(K•l,I)
C U(K•l, 1)=U(K•1,2)
Z=Z•DZ
WRITE(7 ,200) (AMAG ( K, I), l=l ,50)
C WRITE(7, *) SUM
100 CONTINUE
200FORMAT(1X,E10.5,1X,E10.5, 1X,E10.5, lX,El0.5, 1X,E10.5)
STOP
END
INTEGER FUNCTION IB(I)
IB=I
IF(l.EQ.0) 18=2
RETURN
END
63
Appendix 0
THEORETICAL NEAR FIELD PLOT CODE
This routine plots the theoretical data generated by the near field
propagation code.
C THIS PROGRAM DOES A 3 D PLOT OF A 50 BY 500 DATA ARRAY
REAL C(lO) ,WK(lOOO) ,XF(l02), YF(102), IP, Y(500, 100)
INTEGER N,NN,IER,l,J
DO 5 1=1,250
READ(5,*) (Y(l,J),J=l,50)
5 CONTINUE
FC=4.0
DO 10 1=1,50
YF(l)=(Y(SO, I))
XF(l)=(l)*FC
10 CONTINUE
c N=SO
CALL PLOTS(0,0,50)
CALL SCALE(XF,8.0,N, 1)
CALL SCALE(YF, 6. 0, N, 1)
C YF(51)=0.
Y F(52) =YF (52) *5.
XF (52) =XF (52)*5.
64
65
CALL AXIS(. 5,. 5, 11 HBEAM RADIUS, -17 ,4.0,0. ,XF(N•l ),XF(N•2))
CALL AXIS(. 5,. 5, 11 HMAGN ITUDE , li ,4. 0, 90.0, YF(N•l), YF(N+2))
CALL PLOT (.5,.5,-3)
DO 20 1=1,250,2
DO 15 J=l ,50
YF(J )=(Y (I ,J) )/2. 55
15 CONTINUE
IP=.04
CALL PLOT(IP,IP,-3)
CALL LINE(XF,YF,N,1,0,1)
20 CONTINUE
CALL PLOT(O. ,0, 999)
STOP
END
Appendix E
DATA PLOT ROUTINE
This routine plots the data aquired by the microprocessor system.
C THIS PROGRAM DOES A 3 D PLOT OF A 50 BY 50 DATA ARRAY (INTEGER)
REAL C ( 10) , WK ( 1000) , X F ( 102) , Y F ( 102) , I P
I NT E G ER N I N N I I ER ' I , j , y ( 50, 50)
DO 5 I =1, 50
RE AD ( 5 , *) ( Y ( J , I ), J = 1 , 50)
5 CONTINUE
READ(5,*) NPS
FC=25. 4*( FLOAT ( N PS)/ 4000.)
DO 10 1=1,50
Y F ( I ) =FL 0 AT ( Y ( I , 25) )
XF( I )=FLOAT( I )*FC
10 CONTINUE
c N=50
CALL PLOTS (0, 0, 50)
CALL SCALE (XF, 8. 0, N, 1)
CALL SCALE(YF,6.0,N, 1)
YF (51)=0.
YF(52)=25.0
CALL AXIS(.5,.5,17HRECEIVER LOCATION,-17, 8. 0 I 0. , x F ( N + 1) Ix F ( N +2))
66
67
CALL AXIS( .5, .5, 17HRECEIVER OUTPUT I 17 I 6. 0 I 90. 0 I y F ( N + 1 ), y F ( N + 2) )
CALL PLOT (. 5,. 5, -3)
DO 20 1=1,50
DO 15 J=l, 50
Y F (J) =FLOAT ( Y (I, J)) /2. 55
15 CONTINUE
IP=.04
CALL PLOT(IP,IP,-3)
CALL LINE(XF,YF,N,1,0,1)
20 CONTINUE
CALL PLOT(O. ,0,999)
STOP
END
REFERENCES
[1] A. E. Siegman," An introduction to lasers and masers ", McGraw-Hill, 1971
[2] F. D. Martin and M. A. Brazeale, "A simple way to eliminate diffraction lobes emmitted by ultrasonic transducers " J. Acoust. Soc. Am. 49 1668 (1971)
[3] M. A. Breazeale, F. D. Martin, B. Blackburn, Reply to "'Radiation patterns of partially electroded piezoelectric transducers"', J. Acoust. Soc. Amer. (to be published).
[ 4] L. Filipczynski and J. Etienne, " Theoretical study and experiments on sperical focusing transducers with Gaussian surface velocity distribution ", Acustica 28 , 121 (1973)
[5] J. L. Rose," Ultrasonic Field field- analysis and approximation parameters ",Br. Journal of Nondestructive Testing 17, 109(1975)
68
BIBLIOGRAPHY
B. A. Auld, Acoustic Fields and Waves in Solids, Wiley, 1973
W. M. Ewing, Elastic Waves in Layered Media, McGraw-Hill, 1957
R. J. Hallermeier and W. G. Mayer, "Light diffracted by ultrasonic surface waves of arbitrary standing wave ratio", J. Acoust. Soc. Amer., Vol. 47, 1970
H. Kogelnik, "Modes in optical resonators", in A. K. Levine (ed.),Lasers: A Series of Advances, Vol. 1 p 295, Marcel Dekker, New York, 1966
H. Kogelnik, "On prpagation of gaussian beams of light through lenslike media", App. Optics, 4: 1562 (Dec., 1965)
W. T. Maloney, G. Meltz, and R. L. Gravel, "Optical probing of the Fresnel and Fraunhofer reqions of a rectangular acoustic transducer", IEEE Transactions on Sonics and Ultrasonics, July, 1968
C. H. Palmer, R. 0. Claus, and S. E. Fick, " Ultrasonic wave measurements differential interferometry ", Applied Optics 16, 1849(1977)
M. A. Plonus, Applied Electromagnetics, McGraw-Hill, 1978
H. F. Pollard, Sound Waves in Solids, Pion Limited, 1977
W. Sachse and N. N. Hsu,'' Ultrasonic transducers for materials testing and their characterization", in Phsical Acoustics, Vol. 14, W. P. Mason and R. N. Thurston, eds., Acedemic Press (New York), 1979
D. W. Sweeney and C. M. Vest," Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data ", Applied Optics 12 , 2649(1973)
K. von Haselberg and J. Krautkramer," Ein Ultraschall-Strahler fur die werkstoffprufung mit verbesserten nahfeld ", Acustica 9,359(1959)
69
The vita has been removed from the scanned document
DESIGN, CONSTRUCTION, AND TESTING OF ULTRASONIC TRANSDUCERS \YITH MODIFIED
RADIAL VELOCITY PROF! LES
by
Paul Samuel Zerwekh
ABSTRACT
In materials evaluation applications requiring the interrogation of
modified far field patterns of an ultrasonic transducer, it is desirable to
use a transducer which produces a beam with a Gaussian profile. A
transducer with a velocity profile which is Gaussian as a function of
radius and independent of angle is described. The transducer has I
been constructed by depositing a circularly symmetric metallic multiple
electrode array on a 12. 7 mm diameter x-cut quartz disk. Each elec-
trode is independently connected to an impedance network optimized to
produce the Gaussian distribution with less than two percent maximum
error. A computer aided electrode design and normalized three di men-
sional ultrasonic measurements of the far field distribution are present-
ed.
