LI P 'M R y
Michigan Stan
University

This is to certify that the
thesis entitled

Design and Performance of an
Ultrasonic Phased Array Tranducer

presented by

Usman Saeed

has been accepted towards fulfillment
of the requirements for

Masters degree in Electrical

 

Engineering and Systems Science

__llanie_K._Reinhard__

Major professor

0m 2/22/73

0-7 639

Awgtmg

 

DESIGN AND PERFORMANCE OF AN
ULTRASONIC PHASED ARRAY TRANSDUCER
By

Usman Saeed

A THESIS

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

MASTER OF SCIENCE
Department of Electrical Engineering and Systems Science

1978

ABSTRACT

DESIGN AND PERFORMANCE OF AN
ULTRASONIC PHASED ARRAY TRANSDUCER

By

Usman Saeed

An ultrasonic transducer phased array has been designed, fabricated
and tested for the purpose of investigating the potential advantages and
problems of incorporating such a transducer in a computerized ultrasonic
imaging system.

The fabrication was preceded by computer calculations based on a the-
oretical model. The performance testing of the ultrasonic phased array is
based on the study of field strength patterns of individual elements as'
well as the array itself. For testing the ultrasonic beam steering cap-
abilities of the fabricated transducer array, a programmable delay line
has been developed, which can provide the necessary delays for each of
the elements of the array to produce a constructive interference pattern
at the target point.

Several experiments were performed by observing the field strength
pattern of an individual central element and the seven element arrayunder
different operating conditions. Results lead to the conclusion that in-
deed an ultrasonic phased array can be fabricated by the method used dur-
ing this research. Both beam steering and focusing were performed suc-
cessfully, however, several important differences between observations and
predictions were noted; notably reduced lateral resolution and dynamic

range. Some suggestions for resolving these differences are discussed.

ACKNOWLEDGMENTS

I am deeply indebted to my research advisor, Dr. D. K. Reinhard,
for his invaluable advice, guidance and encouragement which he ren-
dered during the course of this research. His sound technical
advise and enthusiasm helped me in completing this research.

I would like to extend special thanks to Dr. E. James Potchen,
Chairman of the Radiology Department, for his keen interest towards
this research and to the Biomedical Research Support Group which
financed a part of this research along with the Radiology Department
at Michigan State University.

Special appreciation is extended to my major professors,

Dr. G. L. Park and Dr. R. 0. Barr, for their thorough review of the
thesis.

I would also like to extend my sincere thanks to Dr. G. Harris,
Mr. Philip Chimento, Mr. Bruce Johnston, Mr. David Gift, Mr. James
Siebert and my fellow research assistant, Mr. Mark Funk, for their

supportive efforts in this project.

ii

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION ...................................... 1
II. THEORY OF PHASED ARRAYS ........................... 6
2.1 Field Strength Expression for an Array ....... 8
2.2 Beam Steering 16
2.3 Beam Focusing 19
2.4 Expressions for a Circular Element
Transducer ................................... 23
2.5 Expressions for a Rectangular Element
Array ........................................ 25
III. ULTRASONIC PHASED ARRAY FABRICATION ............... 28
IV. EXPERIMENTAL RESULTS .............................. 33
4.1 Group 1 Experiments .......................... 33
4.2 Group 2 Experiments .......................... 39
V. CROSS COUPLING .................................... 56
5.1 Experimental Results ......................... 63
VI. CONCLUSIONS ....................................... 66
VII. RECOMMENDATIONS ................................... 70
REFERENCES ........................................ 71
APPENDICES
A. Programmable Delay Line .................. 73
B. Gates and Power Amplifiers ............... 77
C. Computer Programs ........................ 81

LIST OF TABLES

Table Page
1. Computer Simulation of Field Strength
Pattern of an Array ............................... 12
2. Computer Simulation of Field Strength
Pattern of an Array Element ....................... 14
3. Computer Simulation of Field Strength
Pattern on Steering ............................... 19
4. Computer Simulation of Field Strength
Pattern on Focusing ............................... 23
5. Output Voltage versus Axial Displacement
of an Array ....................................... 37
6. Output Voltage versus Axial Displacement
of an Array Element ............................... 39
7. Output Voltage versus Axial Displacement
of an Array ....................................... 43
8. Output Voltage versus Axial Displacement
of an Array Element ............................... 45
9. Vout versus Axial Displacement for Left
Side Beam Steering ................................ 47
10. Vout versus Axial Displacement for Right
Side Beam Steering ................................ 48
11. Vout versus Axial Displacement for Left
Side Beam Steering ................................ 52
12. Vout versus Axial Displacement for Left
Side Beam Steering ................................ 53
13. Induced Voltage for Different Elements
of the Array ...................................... 64

iv

Table Page

14. Induced Voltage for Different Elements

of the Array ....................................... 65
15. Induced Voltage for Different Elements

of the Array ....................................... 65

LIST OF FIGURES

Figure Page
1. System Block Diagram ............................... 3
2. Geometrical Details of an Array .................... 9
3. Simulation Plot of P(6) vs 6 for an Array .......... 13
4. Simulation Plot of P(O) vs a for an Array

Element ............................................ 15
5. Simulation Plot of P(e) vs a on Steering ........... 18
6. Focusing Effect .................................... 20
7. Simulation Plot of P(e) vs 6 on Focusing ........... 24
8. Rectangular Element Array .......................... 27
9. Fabrication Details for Transducer ................. 3O
10. Controlling Electronics of Group 1 Experiments ..... 34
11. Plot-of Vout vs Axial Displacement of an Array ..... 36
12. Plot of Vout vs Axial Displacement of an Array

Element ............................................ 38
13. Controlling Electronics of Group 2 Experiments ..... 4O
14. Plot of Vout vs Axial Displacement of an Array ..... 42
15. Plot of Vout vs Axial Displacement of an Array

Element ............................................ 44
16. Plot of Vout vs Axial Displacement for Left

Side Beam Steering ................................. 46
17. Plot of Vout vs Axial Displacement for Right

Side Beam Steering ................................. 49
18. Plot of Vout vs Axial Displacement for Left

Side Beam Steering ................................. 51

vi

Figure Page

19. Plot of Vout vs Axial Displacement for Left

Side Beam Steering ................................. 54
20. Equivalent Circuit Model of an Array Element ....... 58
21. Norton's Equivalent Model of an Array

Element ............................................ 58
22. Receiving Element Equivalent Circuit ............... 60
23. Plane Nave Model of a Transmitting/

Receiving Element of an Array ...................... 62
24. Programmable Delay Line ............................ 74
25. Block Diagram of Gates and Power Amplifier ......... 78
26. Circuit Diagram of Power Amplifier ................. 80

vii

CHAPTER I INTRODUCTION

Ultrasound has been used for the past three decades as an alter-
native or as a complementary technique to x-rays for medical diagnos-
tic examinations.(1) Inhomogeneities in a medium may be determined
and imaged by observation of reflected ultrasound waves, for example
flaw detection in metallic plates, sonar, etc. More recently, phased
arrays have been used to advantage as ultrasound transducers since
the ultrasound waves may be electronically steered and fbcused.(2)

This thesis deals with the fabrication, theory and performance
testing of phased array ultrasound transducers. The need for a study
of ultrasonic phased arrays arose in order to investigate the poten-
tial advantages of electronic beam steering and focusing for the
computerized ultrasonic system being developed in the Radiology De-
partment of Michigan State University.

Ultrasound transducers are used for generating mechanical stress
waves when excited by a pulsed electric field. If the transducer is
coupled to a medium, ultrasound energy is transmitted to the medium
in the form of waves. Commonly, the active element of the ultrasound
transducer is a solid state crystalline or polycrystalline piezoelec-
tric such as lead zirconate. In the polycrysalline form, these
materials are made in the form of a ceramic composed of a large number
of randomly oriented polarized domains. When such materials are heat-

ed and then cooled in the presence of a strong electric field, the

direction of polarization of these domains line up with the field and

1

2
remain lined up even after the field is removed. If a certain mechan-

ical stress is applied on such materials, it is their inherent char-
acteristic to develop a certain charge on the surface. This phenomona
is called the piezoelectric effect.(3) Conversely, if a voltage is
applied between the two faces of such a material it will cause a me-
chanical deformation and generate a mechanical stress wave. This
phenomona is called the inverse piezoelectric effect.

The mechanical stress waves are longitudinal in nature. Their
velocity in a given medium depends upon the density of the medium and
its elasticity.(4) When an obstruction comes in the path of an ultra-
sound wave, part of it passes through the boundary but a part of it
gets reflected back. This happens with every boundary the wave meets.
An image of the obstructing object's boundaries can be created if one
is able to detect these reflections. This principle is used frequent-
ly in medical diagnosis equipment using ultrasound transducers. An
important feature of ultrasound waves is that it is a non-ionizing
radiation, and hence it is non-carcinogenic. This makes them presum-
ably safer than x-rays.

This research was a part of an investigation being carried out
at the Radiology Department at Michigan State University which may be
described as "computer control and processing of ultrasound signals
for clinical application."

The overall block diagram of the total system where this parti-
cular research fits in is shown in Figure 1 and is described in
brief as follows:

1) The transducer is made in the form of an array rather than a

single element. This gives it the feature of electronically

steering and focusing the beam. The system may also be used

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4
with a standard single element transducer.

The programmable delay line is used to provide necessary delays
for each element of the array to steer or focus the beam. It
has two modes of operation; the manual mode can be used to
manually control the delays by setting external switches. In
the automode the computer interface sends an initiating signal
called DECX. Once initiated in any mode the programmable delay
lines initiate monostables, which have been designed to gen-
erate a pulse of 0.3 microseconds at each rising edge of input
from the delay lines. These pulses are passed through the
gates. The purpose of gates is to avoid loading of the mono-
stable by the power amplifiers which are connected with the
gates. There are ten gates so we can get ten pulses delayed
by a time controlled by the programmable delay lines which can
also be automatically incremented by subsequent DECX pulses
coming from the computer. The ten pulses coming out of ten
gates are amplified by ten amplifiers and are applied to each
element respectively for exciting the transducer.

The transducer generates the ultrasound waves in pulses depend-

' ing upon the excitation pulses. For the time interval when

4)

the transducer is not transmitting, it can be used as a receiver
because the reflections can be detected by the transducer due

to its inverse piezoelectric effect. For example, the central
element of the transducer may be used as a receiver.

The reflected pulses are primarily of 1 MHz frequency and after
their detection are applied at the input of an amplifier. The
output of the amplifier (panametric 5050 Pulser Receiver) is

applied to a Biomation A/D converter which samples it at a rate

of 5 MHz.

5) The A/D convertor converts each sample to an 8-bit word and
stores these words for 400 microseconds (2048 words) in its
own memory. Once this memory is full it gives signal to the
block transfer controller (part of the computer) through an
interface to start transferring all these words to the comput-
er memory and simultaneously initiates the delay lines to
start a new cycle of excitations for the transducer.

6) From computer memory these words are transferred to the tape
and from the tape to the disc. Then, gray levels are assigned
for each word and then the image is displayed on a video
monitor.

This thesis is limited to design, fabrication and performance
testing of the ultrasonic transducer arrays fabricated for this research
and a description of the supporting electronics. A description of the
computer software, hardware and interfacing is to be found in Mark

Funk's master's degree thesis to be published.

CHAPTER II THEORY OF PHASED ARRAYS

The need for arrays of antennas arose when the microwave engineers
wanted to increase the directionality range of their antenna systems.

As a solution they connected a number of half or quarter wavelength
stubs in parallel. Later on, the need for variable directional antennas
lead into the development of phased array antennas, where a certain
phase delay was introduced for each element depending upon its location
in the array system. As a result of this development, it was possible
to electronically steer and focus the beams of radiation from these
arrays. The need for ultrasonic phased arrays arose in the analogous
fashion and such transducers have been developed in order to steer and
focus the beams of ultrasonic radiation.(1)

This chapter deals with the theoretical aspects of the ultrasonic
phased arrays which includes the derivation of expressions for the
field strength of an array and steering and focusing effects by intro-
ducing phase delays in the field strength expression. The second half
of this Chapter includes the computer simulations from a simple theoret-
ical model for the transducer which was fabricated and tested in this
study.

In order to understand the theory of ultrasonic phased arrays, it
is necessary to start the discussion with a brief introduction to the
acoustic fields. When the particles of a medium are displaced from their
equilibrium position, then, as a result of this action, internal restor-
ing forces arise as a reaction. It is these elastic restoring forces

6

7

between the particles, combined with inertia of the particles, which

leads to the oscillatory motion of the medium. As a result of this vi-
bration in the medium, traveling waves are generated by the medium which
travel out of the generating medium. It is this basic phenomenon which
is responsible for the production of acoustic fields.(2)

The materials used for generating acoustic fields are normally class-
ified as piezoelectric materials such as were discussed in Chapter 1. A
subset of materials exhibiting strong piezoelectric properties are the
ferroelectric materials. Lead zirconate is a good example of such a
material and it was used in the research done to fabricate the ultrasonic
phased array.(3)

Since the piezoelectric materials are anisotropic, the equations
relating the electrical to mechanical properties will necessarily involve
a large number of coefficients. The six basic coefficients involved in
these equations are elastic stiffness coefficient 'c', absolute permit-
tivity 'e', two piezoelectric strain coefficients '9' and 'd' and two
piezoelectric stress coefficients 'e' and 'h'.

Strictly speaking, the equations for a piezoelectric material should
be written in terms of tensor notation.(4) However, in order to simplify
matters, we assume that when a field is applied parallel to one of the
axis the stress or strain will also be in the same direction. Under this
assumption the four basic equations relating two electrical quantities,
displacement 'D' and field strength 'E' to two mechanical quantities,
strain '5' and tension 'T',describing the direct piezoelectric effect are
as follows: (i) D eTE + dT

ESE + eS

(ii) D

(l/eT)D - gT
(I/eS)D - hS , where

(iii) E

(iv) E

8

d = Strain developed/Applied force

e = Stress developed/Applied force

9 = Strain developed/Applied charge density
h = Stress developed/Applied charge density
8 = Dielectric constant

These types of basic equations can be used to derive important re-
sults like the field strength expression for one radiating element of

(4)

an array of ultrasonic antenna.
2.1 Field Strength Expression for an Array

This section describes a simplified expression for the acoustic
field strength pattern of an array of radiating elements. The first
analysis is based on an approximation which holds considerably well for
far zones of the field(5)and point source elements. Later sections of
the Chapter discuss the field strength expressions for elements of fin-
ite dimensions. Theoretical consideration of cross coupling is treated
later in this thesis.

The radiation pattern produced by an array of radiating elements
will first be derived by considering each element as a radiating point
source separated by the same distance as in the actual array fabricated.
Then, the overall radiation pattern is the superposition of field pat-
terns produced by all these elements and the patterns produced by all
their point sources.

Referring to Figure 2, note that each element is shown as a point
source separated by distance 'd' and 'e' is the angle of observation of
the resultant field strength P(e).

In the far zone approximation, we approximate the path difference

between any two elements or point sources as d SIN(0).

 

 

 

 

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Figure 2 Geometrical Details of an Array

 

 

10

2(n)(path difference)
wave length

2(3)(d SIN(O))
A

$0, the phase difference (u)

 

C
II

We define P(o) as the acoustic field strength amplitude of each radiat-
ing element.

When all point sources (elements) are radiating with no relative
time delays, the total field strength for an odd number of elements
in an array is given as
5) P(e) = P(o) (e'iNu + e'HN'l)u +...+ 1 +...+ eHN'l)u + eiN”)

where 'n' is the actual number of radiating elements; n = 2N + 1
Equation 5 is a geometrical progression and can be simplified as

P(e) P(o) (e'iNu(1 + eiu +... + eiZNu))

P(O) (e-iNU(eI(2N + 1)U - i)/(eIU _ 1))
P(o) (e-i(2N + 1)U/2(el(2N + 1)U/2 _ e-i(2N + 1)U/2))

eiU/2(eiU/2 _ e-iU/Z)

 

which may be written as
6) P(e) = P(o) SIN((2N + 1) U/2)/SIN (U/2) ..... (1)
for an even number of elements in the array, if the field strength is
calculated at ad from the center.
P(e) = P(o) (e-iNu + e-i(N-1)u +... + 1 +... + ei(N-2)u + ei(N-1)u)
where n the actual number of radiating elements; n = 2N.
The preceding expression is a geometric progression and can be
simplified as P(O) = P(o) (e'i"”(el2N“ - I/eiu - 1))
simplifying
= P(o) e-iNueiNueiu/2((eiNu_e-iNu)/2i/(eiu/2)/21)
= P(o) e'iu/2(SIN Nu/SIN u/2).
So, the magnitude of the expression P(e) is given as

7) |P(8)| = P(o) (SIN Nu/SIN u/2)

11
The expression for a single radiating element of an array antenna is
derived from the fundamental equations of the acoustic fields as men-

(4)

tioned in the introduction of the Chapter and the result is stated
below.

The pattern from a single element from finte width 'a' is
8) P(o) = SIN(u')/u' .....

where u' = Phi * a * SIN(e)/A

where 'a' is the width of radiating element, refer to Figure 1.
So, the overall expression for the field strength become for 'n' even

elements; n = 2N

9A) |P(9)|

A0(SIN(u')/u') * SIN(Nu)/SIN(u/2) (Refer to Equations 7, 8)
and for 'n' odd elements
98) P(e) = (SIN(u')/u') * SIN((2N + 1)u/2)/SIN(u/2)
(Refer to Equations 6, 8)

These expressions are an approximation for far field strength pat-
terns in which cross coupling is neglected.

Using the above expressions, two computer calculations were perform-
ed for plotting the field strength pattern of an array with seven elements
and of a single element of the same seven element array. Some of the

results are indicated in the following few pages.
Computer Calculation 1

This calculation shows the evaluation of field strength pattern of
a seven element array. The data used is as follows:

Spacing between the elements (d) = 1 m.mts

Velocity of ultrasound (V) = 1.5 x 106 m.mts/sec.

Frequency of ultrasound (f) = 106 Hz

Width of each array element (a) = 0.4 m.mts

12
Delay time of firing between adjacent elements (t) = 0 seconds
The following table shows the variation of field strength (in st)
verSus Theta (6) in the range :45 degrees obtained by using the Comput-

er Program 1, Appendix C

Table 1 Computer Simulation of Field Strength Pattern of an Array

 

Theta (a) Field Stren th P (6)

(degrees)_ ~(st)
1) -45 -19.22
2) -40 -50.22
3) -35 -18.04
4) -30 -17.16
5) -25 -36.05
6) -20 -13 77
7) -15 -15.55
8) -10 -12.97
9) - 5 - 2.46
10) o o
11) 5 - 2.46
12) 10 -12.97
13) 15 -15.55
14) 20 -13.77
15) 25 -36.05
16) 30 -17.16
17) 35 -18.04
18) 40 -50.22

19) 45 -19.22

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Figure 3 Simulation Plot of P(e) vs 6 for an Array

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14

Computer Calculation 2

This calculation shows the evaluation of field strength pattern of
a single element of a seven element array. The data used is as follows:
Spacing between the elements (d) = 1 m.mts

l

Velocity of ultrasound (V) = 1.5 x 106 m.mts/sec.

Frequency of ultrasound (f) = 106 Hz

Width of each array element (a) = 0.4 m.mts

The following table shows the variation of field strength (in st)
versus Theta (6) in the range :45 degrees obtained by using the Computer

Program 1, Appendix C.

Table 2 Computer Simulation of Field Strength Pattern of an Array Element

 

Theta (a) Field Strength P (6)
(degrees) (st)
1) -45 -6.48
2) -40 -5.22
3) -35 -4.10
4) -30 -3.22
5) -25 -2.62
6) -20 -2.13
7) -15 -1.50
8) -10 -0.95
9) - 5 -O.52
10) 0 O
11) 5 -O.52
12) 10 -0.95
13) 15 -1.51
14) 20 -2.13
15) 25 -2.62
16) 30 -3.22
17) 35 -4.11
18) 40 -5.22

19) 45 -6.48

15

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2.2 Beam Steering

In order to steer the ultrasonic radiation beam being generated
by the ultrasonic transducer, one needs to provide appropriate delays

(5)

to the excitation pulse for each element. This delay depends upon
two factors, first,the amount of steering one wants to provide, i.e.
the amount by which one wants to shift the central lobe of radiation,
and, second, the physical location of the element within the array.
For example, suppose one wants to shift the central lobe to the right,
then the last element starting the count of elements from left would
get the maximum phase shift or time delay. Primarily, the delays are
provided so that all radiations reach a particular preselected point
at the same time to provide a constructive interference, so that one
can get the maximum intensity of the ultrasonic beam. In order to
steer the beam for checking the properties of the transducer through
small angles like $5 degrees, one may use a fixed delay line, which
are simply lumped models of a transmission line. (See Appendix A

for the details of the fixed delay lines used in this study.) For
controlling the beam steering electronically through a range up to
245 degrees, one needs programmable delay lines or phase shifters (see
Appendix A for a description of the programmable delay lines used in
this study). In order to obtain computational results for beam steer-
ing, we can simply subtract an element phase shift due to a time de-
lay from 'u' in the expression (9A) and (9B) of Section 1, resulting
in the following expressions for 'n' even numbers.

10) |P(e)| = (SIN (u')/u') * SIN N(u - 2nft)/SIN (u - 2nft)/2

where f frequency of radiations

t time delay required between adjacent elements

17
2nft = phase delay of firing

and, similarly, for 'n' odd elements

11) P(B) = SIN(u')/u' * SIN ((2N + 1)(u - 2nft)/SIN (u - 2nft)/2)

Using Expressions (10) and (11) a computer calculation was performed;

some of the results are indicated in the following few pages.

Computer Calculation 3

This calculation shows the field strength pattern of a seven ele-

ment array with a time delay of firing of 0.3 microseconds between

consecutive elements to steer the central lobe of the ultrasonic beam

by 30 degrees.
The data used is as follows:
Spacing of elements (d) = 1 m.mts
Velocity of ultrasound (v) = 1.5 x 106 m.mts/sec.

6

Frequency of ultrasound (f) = 10 Hz

Width of one array element (a) = 0.4 m.mts

Delay time of firing between adjacent elements (t) = 0.3 u seconds

The following Table shows the variation of field strength (in st)

versus angle Theta (6) in the range :45 degrees, obtained by using the

Computer Program 1, Appendix C.

18

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Table 3 Computer Simulation of Field Strength Pattern on Steering

 

Theta (6) Field Strength P (e)

4_(degrees)_ (st)

1) ~45 ~13.27

2) ~40 ~15.73

3) ~35 ~42.71

4) ~30 ~16.87

5) ~25 -18.75

6) ~20 ~27.60

7) ~15 ~16.98

8) ~10 ~25.46

9) ~ 5 ~18.55

10) O ~17.25
11) 5 ~24.94
12) 10 ~12.83
13) 15 ~20.52
14) 20 ~ 9.12
15) 25 ~ 1.85
16) 30 ~ 0.26
17) 35 ~ 2.31
18) 40 - 8.64
19) 45 ~39.99

2.3 Beam Focusing

In order to focus the ultrasonic radiation beam being generated
by an ultrasonic transducer one needs to provide appropriate delays to
the excitation pulse for each element as was necessary for beam steering,
the only difference being that the delays required for focusing primar-
ily depend upon the location of the element within an array(7) and depth
of desired focusing. Simple focus requirements reveal the details. Con-
sider a 5-element array focused at depth r on the central axis, as shown

in Figure 6. In order to focus the beam what we physically mean is to

 

 

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Figure 6 Focusing Effect

21
produce a constructive interference at the focus point. We can accomp-
lish this by adjusting the phase delays in such a way that the crests
of radiations generated by all the elements reach the focus point at
the same time. This would mean that as the distance of No. 1 is max-
imum from the focal point it needs the minimum delay and two needs more
delay than 1 and so on; the last element 5 would need the same phase
delay as No. 1 and so on. The method for finding the phase delays
necessary for each element are discussed in later parts of this section.
A mathematical analysis of such a situation is necessary for computer
simulation and is described below.

The pressure of spherical ultrasound waves at any observation is

given as = eIkr/r
where k = 2n/A ; A being the wave length
r = distance of point of observation from the source

If we introduce a phase delay,as would be necessary for focusing, then
the pressure is = e“D = 2"i * r/A)
where 0 represents the phase delay introduced.
The distances r1, r2, r3, rt are given as

f (x2 + fl

1‘1: / (TX ' ZdIYl‘YTI

 

Y‘

 

 

 

 

r2= / ((x - d)2 + yz)
ra= / ((x + dT‘ WTF
ru= / ((x + 2d)2+y2)

So, the pressure is given as
12) P (mm) = “END: I 2 ' 7‘ (r1 " r”AI/"la. 31((02 + 2'17 (r2 - r)/A/r2)

+ ei(Da)/r3+ ei((D.. + 2 - n (r3 - r)/A))/r,
+ ei((Ds + 2 - n (rt - r)/).)/rs))I

where the ri are taken to be equal in the denominators. In order to find

22

the values of phase delays; D's refer to Figure 6.

ds

d1 (2 * d) SIN(Arc TAN(2 * d/z))

d2 (1 * d) SIN(Arc TAN(1 * d/Z)) d6

where Z is the third coordinate value of the location of the focus point

along the z-axis so,

t1 = ts d1/Velocity of ultrasound waves in the medium

t2 = tr dz/Velocity of ultrasound waves in the medium
where t1, t2, t1, t5 describes the time delay of firing for elements

1, 2, 4, 5, respectively, so

13A) 01 05 2 * n * f * t1

13B) 02 DC 2 * n * f * t2
where 01, Dz, Dr, 05 represents phase delays for respective elements,
where f is the frequency of ultrasound waves and the velocity of ultra-
sound waves in water is about 1,500 meters per second.

Using Expressions 12 and 13, a computer simulation for focusing

was done and some of the results are listed below.
Computer Calculation 4

This calculation shows the effect of focusing on the field strength
pattern. The data used is as follows:

Spacing between elements (d) a 1 m.mts

Depth of measurement (y) = 100 m.mts

Focal point or beam focusing point (z) = 100 m.mts

6

Velocity of ultrasound (v) = 1.5 x 10 m.mts/sec.

6 Hz

Frequency of ultrasound (f) = 10
The following Table shows the variation of field strength (st)
versus the x - axis variation keeping the y and z axis dimensions constant;

obtained by using Computer Program 2, Appendix C.

23
Table 4 Computer Simulation of Field Strength Pattern on Focusing

 

Axial Distance (x) Field Stren th P (6)
(m.mts) (st§
1) ~40 ~0.65
2) ~35 ~0.34
3) ~30 ~0.81
4) ~25 ~0.93
5) ~20 ~0.80
6) ~15 ~O.56
7) -10 ~0.28
8) - 5 ~0.08
9) O ~0.00002
10) 5 ~0.08
11) 10 ~0.28
12) 15 -0.55
13) 20 ~0.80
14) 25 ~0.92
15) 3O ~0.80
16) 35 -0.33
17) 40 ~0.65

2.4 Expressions for a Circular Element Transducer

When the individual elements of an array cannot be modeled as point
sources or as infinite lines of finite width, the analysis becomes more
difficult. The basic technique involved in these complicated deriva-
tions is that one has to integrate over the surface of the radiator
assuming each surface element dS acts as a spherical radiator.(8) As
a result of this integration, the expressions which are formed are given
below. Here the results for a finite circular element are considered.

In the far field where the patch difference can be approximated by

d SIN(O) where d is the spacing between elements andiais the angle of

observation.

24

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Ammmcmwvv mp+

 

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x x a
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o — I s a. a
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I _ I N o x
a . v ~ . s
I _ I . A. ~ .. x
I _ I m OI \ - \
I .\
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+

25
Furthermore, the distance r of observation from the transducer is
assumed much larger than 'a' the diameter of the piston type transducer.
This approximation basically separates the near zone and far zones of
observation.

The pressure distribution may be found to be given by (Reference 78)

 

14) P = p * c * k * n * a2 V (2J1 (k * a * SIN(e)
2(n)r * k * a * SIN(O)
p = Density of the medium
c = Propagation velocity
k = Wave Constant
. V = Velocity amplitude of source
J1= Bessels Function

And the intensity is defined as I = P2/2 *;3* c
In the 'NEAR ZONE'(8) where the point of observation is of a dis-
tance which is comparable to the diameter of the piston transducer and

path difference == d SIN(e) does not hold so (78)

i * fl * R (—N)/A/R(-N)+.. 21 * fl * R(n)/A/R(N))

p = Re (P(o)(e2 . + e
where R(N) is simply a notational symbol being given as
15) R(N) = (r2 + N2d2) SQRT. - r

N being the element number

d being the separation between elements

r is the distance of observation from the axis of the transducer

vertically downwards

A is the wave length of ultrasonics
2.5 Expressions for a Rectangular Elements Array

Here the source pattern is described as a function of x and y coordi-

nate positions rather than 9 as shown in Section One.

26
Let xo, yo be the position of the point where the source field pat-

tern is being measured and '8' indicates the appropriate phase shift
provided to each element in comparison to the last element. So, the

(5)

source field pattern for the far field is given as

”M 9 .Y0) =
o o 2
esz eJkr°/22(x SINC (§%5) * S comb (sxow/Az) SINC (xow/Az)) * y SINC(yoy/Az)

where 5 comb (sxo/Az) = Sum all terms 6 (xo/Az - n/s)

A = Lambda, Wave Length
n = Number of Elements in the Array
k=21r/A

All other terms are referred to Figure 8.
In a similar way focusing field source patterns may be analyzed as

well.(5)

27

 

 

 

 

 

 

 

 

—<

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_JL_

 

(x0. yo)
Figure 8 Rectangular Element Array

CHAPTER III ULTRASONIC PHASED ARRAY FABRICATION

This chapter deals with the fabrication technique developed during
the course of this research project. The details of fabrication are
mentioned below in a stepwise order as follows:

1) Normally the piezoelectric materials are available in large
circular discs with silver top and bottom electrodes, so the
first step is to cut the disc in small pieces of required di-
mensions using a diamond wheel saw; the dimension of the piece
depends upon how many elements one has to have in the array and
the dimensions of the individual elements. Cutting of the
piezoelectrical material is a very delicate process and it in-
volves considerable care with the saw and a special process.
Before cutting the piezoelectric disc, it is fastened on the
base plate of the saw using Apiezon Wax. This should be done
very carefully to avoid exceeding the Curie temperature of the
piezoelectric; for lead zirconate, the material used for fab-
rication of the array, the Curie temperature is 160 to 170
degrees centigrade.

Once the disc is firmly held on the base plate, it is ready for
cutting. The width of the diamond blade used in the diamond
saw depends upon the dimensions of the array; the blade used
has a width of 0.25 mm.

First, cut the large disc in small pieces, in this case, the

small rectangular piece has the dimension 1.4 cms by 0.7 cms.

28

29
These dimensions were selected in order to accomodate 11

elements in the array.*

2) Mount the base plate carefully on the saw and mark on the ver-
tical control handle of the saw the required depth of each cut.
For the array fabricated in this study, it was 3/4 of the total
depth of the piece.

Then the saw is started and its autohandle locked. Using the
automatic feed, after 5 minutes, the first out should be com-
plete with a speed setting of "40".

It is important to note that autofeed should not be started

until after the saw rotation is started. After the cut is fin-
ished, move the blade upwards so that it does not touch the

piece of piezoelectric, turn off the automatic feed and then the
saw. Now, move the horizontal position control handle by an
amount such that the spacing between the first and the second

cut is the desired amount, which was 1 mm in this case. Then,
lower the blade again to the same position as for the first cut
and start the saw and autofeed again. In a similar way 12 cuts
are made resulting in an 11 element array each 0.6 mm wide, if
the cut to cut separation is 1 mm as shown in the Figure 9. It
is to be noted that the speed of the saw, if not properly matched
with the elasticity of the material to be cut, can result in
breaking of the saw blade. A marked speed of 40 units is normal-
ly employed for all cutting.

3) Once the cutting was complete, a thin wire (32 ohms/cm) was sol-

dered to each element and then the base plate, to which the

* Only 7 elements were used in actual testing due to problems with
the end elements.

30

 

minim Foil

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

f 7 ’T‘ Water Proofing
4 5
Whea.Transducer
' , . Array
. V .i .i, -: i fCopper Strips
~ ' ' ‘ ~Wooden Frame
IE I \fionnecting
r‘) Wires
Base
iiifi??? Fiji §§_- Water Proofing
Output
Leads

 

 

Figure 9 Fabrication Details of Transducer

4)

5)

6)

31
rectangular piezoelectric piece was glued, was dipped in an

acetone filled beaker to remove the wax, in this way the rec-
tangular piece is removed from the base plate and made ready
for mounting. 32 ohms/cm thin wire is, infact, not desirable
because it causes a certain excitation power drop, but was

used as it was the only thin wire available. In the future,
wires with such a high resistance should be avoided.

It is worth noting that soldering is the most delicate part of
this fabrication technique and should be done very carefully.
It is also recommended to use silver solder to make connections
in order to increase the strength of the connection.

A circular balsa wood (insulator) frame was made to fit in a
hollow steel cylinderical case as shown in the Figure 9. Then,
a rectangular hollow cup was made in its center, with dimensions
1.8 mm by 0.9 mm of depth 3.0 m.mts.

A damping material was developed in the laboratory by mixing
very fine powdered glass with Elmer's Epoxy in a proportion of
2 to 1. This material has the property of damping the ringing
oscillations of the elements since its acoustic impedance is
higher than balsa wood. Ideally, the damping impedance should
be even higher. The compound used represents a compromise be-
tween high electrical insulation and high acoustic impedance.
0n the bottom of the balsa wood frame, 11 square pieces of thin
copper sheet (0.2 cm by 0.2 cm) were attached using Elmer's
Epoxy (Resin and Hardner), each piece completely isolated from
the other. These copper pieces were used as a connecting

board between the thin wires coming from the transducer elements

and the external connecting wires which were to go out of the

7)

8)

32

assembly.

When the copper pieces were attached, we filled the rectang-
ular hollow cup in the balsa wood frame with the damping
material developed and then mounted the transducer on top,

as shown in Figure 9, such that its flat surface was in level
with the top surface of the balsa wood frame. This assembly
rested for 24 hours so that the damping material dried out.
Next, connect each of the thin wires coming from each element
of the array to each copper piece and also connect the exter-
nal wires, one with each copper piece.

Fix the balsa wood frame inside the steel cylinder such that
the top surface of the frame is in level with the top side

of the cylinderical case. Apply a light coat of silver con-
ducting paint on the tap face of the transducer and the edges
of the hollow steel case and then attach over the top surface
a thin tin foil. Seal the edges of the foil with Elmer's
Epoxy to make it waterproof. Fill the bottom of the assembly
also with the same water repellant epoxy after connecting a
small screw from the outside, to make it waterproof from the
bottom also. Attach to the screw a common ground wire for the
assembly and solder it to the body of the steel case. This

completes the fabrication of the array.

CHAPTER IV EXPERIMENTAL RESULTS

After the fabrication of the transducer, a series of experiments
were performed on this transducer in order to check its charaacteristics.
The experiments ranged from simply observing the field strength pattern
of the array and then checking for the field strength pattern of each
element in the array, to the more complex observations of the steering
and focusing characteristics of the array.

Some of these results, along with the block diagrams of the control-
ling electronics, are presented in this chapter. The experiments are
divided into two groups and the explanation and observations are stated
in the following sections of this Chapter. A discussion of the results

presented in this Chapter is in Chapter 6.
4.1 Group 1 Experiments

The Group 1 Experiments consist of the two experiments done to ob-
serve the field strength pattern of the array and of the elements individ—
ually using a Tone Burst Excitation. The block diagram of the controlling
electronics is shown in Figure 10.

In order to generate a tone burst consisting of a INHz sinosoidal
signal, one needs a pulse generator (Data Pulse Model #106); the output
of which can be applied as the trigger input to the tone burst generator
(Wave Tek Model #134) operating in gated mode. By adjusting the frequency
of the sinosoidal signal of the wave tek generator, one can obtain the

desired tone burst. This tone burst is normally of small magnitudes, on

33

34

Tone Burst Generator Power Amplifier

Wave Tek }———————7

 

 

 

 

 

 

 

 

 

 

 

Data PulseJ Scope 514

[‘6h. 2

 

 

 

 

Pulse GEnerator Ch. 1

 

 

 

 

     

 

| Transducer
Panometric
5050 51—<a Probe

Receiver Amplifier

Figure 10 Controlling Electronics of Group 1 Experiments

35
the order of 1 volts peak to peak which cannot excite the transducer;

the transducer needs large voltage for excitation. So, a power amplifi-
cation is needed and a power amplifier can be connected at the output

of the wave tek. From the output of the power amplifier the transducer
or the individual elements can be excited. The data obtained in Experi-

ments 1 and 2 is stated below.
Experiment 1

This experiment indicates the variation of field strength of the
array, all seven elements excited at the same time versus the axial
displacement 'x' about the center of the transducer array. (See Figure
11 for plot.)

Separation between transmitter and receiver = 8 cms

Number of sinosoidal signal cycles in tone burst = 4

Repetition rate of receiving amplifier (panametric 5050) = maXimum

36

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I x
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I x
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01
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37
Table 5 Output Voltage versus Axial Displacement of an Array

 

Axial Displacement Peak to Peak Signal Output
(cms) (volts)
1) 8 0.30
2) 7 0.51
3) 6 0.95
4) 5 0.80
5) 4 1.00
6) 3 1.80
7) 2 2.50
8) 1 4.00
9) 0 6.00
10) -1 3.81
11) -2 2.60
12) -3 1.70
13) -4 1.25
14) -5 0.82
15) -6 0.90
16) -7 0.42
17) -8 0.35

Experiment 2

This Experiment shows the field strength pattern of the central
element of the array versus the axial displacement 'x' about the cen-
ter of the transducer array. (See Figure 12 for plot.)

Separation between transmitter and receiver = 8 cms

Number of sinosoidal signal cycles in tone burst = 4

Repetition rate of receiver amplifier (panometric 5050) maximum

38

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mEo w u .xm new .xh cmmzumn cowpmcmamm

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m N. o m c m N .H o H: N: m: e: mu on mu m.

d ‘ d 1 d ‘ . Q

HmHHo>V
Hzo>

 

H

39
Table 6 Output Voltage versus Axial Displacement of an Array Element

 

 

Axial Displacement Peak to Peak Signal Output
(cms) __Q ,(volts)
1) 8 0.12
2) 7 0.18
3) 6 0.20
4) 5 0.30
5) 4 0.54
6) 3 0.60
7) 2 1.10
8) 1 1.50
9) O 2.00
10) ~1 1.52
11) -2 1.00
12) -3 0.70
13) -4 0.56
14) -5 0.40
15) ~6 0.25
16) -7 0.15
17) ~8 0.10

4.2 Group 2 Experiments

In this group of experiments, six experiments were performed using
an excitation pulse of 0.3 microsecond duration. The last four experi-
ments were performed for steering the beam electronically. For achieving
this objective,i1programmable delay line network was needed. One such
network was developed using digital design techniques during the course
of this research. The details of this circuit can be seen in the Appen-
dix A at the end of this thesis. The general block diagram of the support-
ing electronics is shown in the Figure 13. The programmable delay line
gives us ten delayed pulses, the delay between pulses is dependent upon

the comparison between the clock rate and the rate at which the DECX

Programmable

40

Delay Line‘ Gates

 

clock I
IDECX
or

Manual

Power Amplifiers

 

 

H %
T

+5 volts

H

l

 

 

 

 

 

 

 

 

 

Transducer

Figure 13 Controlling Electronics of Group 2 Experiments

41
pulses are coming when operating in auto mode. In the manual mode, as

soon as the clock pulse count becomes equal to the preset count read-
ing, we get a pulse which is our output from the programmable delay
lines. The change of level of this pulse can be used to trigger a
monostable to generate pulses of duration of 0.3 microseconds. These
pulses are the input to the power amplifiers (see Appendix B for de-
tails) after passing them through the gates. There are ten outputs of
the programmable delay line so we need ten gates. The power amplifiers
are necessary to provide the high voltage pulses for the excitation of
the transducer elements. The delay time between the exciting pulse

for each element is controlled by switches attached to the programmable
delay line in the manual mode and by the DECX signal (coming from the
computer) in the auto mode. This delay time determines the angle
through which the main lobe (beam) is steered. It is to be noted that
the data obtained during this group of experiments will have some error
because one of the seven elements of the array stopped operating dur-
ing experimentation. The data obtained during the six experiments per-

formed is listed in the following few pages.
Experiment 1

This experiment indicates the field strength pattern of the array
with all six elements excited at the same time by a pulse of 0.3 micro-
seconds duration versus the axial displacement 'x' about the center of
the transducer array. (See Figure 14 for plot.)

Separation between transmitter and receiver = 7.5 cms

Exciting pulse amplitude = 104 volts

Noise level at the output = 0.03 volts

42

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43

Table 7 Output Voltage versus Axial Displacement of an Array

 

Axial Displacement Peak to Peak Signal Output
(cms) (volts)
1) 7.5 0.040
2) 6.5 0.045
3) 5.5 0.074
4) 4.5 0.060
5) 3.5 0.065
6) 2.5 0.100
7) 1.5 0.160
8) 0.5 0.180
9) O 0.200
10) -0.5 0.170
11) -1.5 0.160
12) ~2.5 0.090
13) ~3.5 0.060
14) -4.5 0.055
15) ~5.5 0.070
16) -6.5 0.040

Experiment 2

This experiment indicates the variation of field strength of the
central element alone, excited by a pulse of 0.3 microseconds duration
versus the axial displacement 'x' about the center of the transducer
array. (See Figure 15 for plot.)

Separation between transmitter and receiver = 7.5 cms

Exciting pulse amplitude = 104 volts

Noise level at the output = 0.03 volts

44

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(I)
E
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45
Table 8 Output Voltage versus Axial Displacement of an Array Element

 

Axial Displacement Peak to Peak Signal Output
(CNS) (volts)
1) 5.5 0.031
2) 4.5 0.032
3) 3.5 0.032
4) 2.5 0.035
5) 1.5 0.040
6) 0.5 0.053
7) 0 0.060
8) ~0.5 0.050
9) -1.5 0.045
10) -2.5 0.038
11) -3.5 0.035
12) -4.5 0.035
13) -5.5 0.033

Experiment 3

This experiment shows the variation of field strength on electron-
ically steering the beam using the programmable delay line. Each of
the excitation pulses are of 0.3 microsecond duration. The delay be-
tween each of the seven pulses is the deciding factor for the steering
angle of the main lobe of ultrasound being transmitted by the array.

In order to steer the beam through 30 degrees, one needs a 0.3 micro-
second delay between each of the exciting pulses, which, in other words,
corresponds to a clock frequency of 3.334 MHz. The programmable delay
line designed has the capability of steering the beam towards left or
right along the axial displacement axis 'x'. In this particular exper-
iment we want to steer the beam towards left by 30 degrees. The data
obtained is as follows: (See Figure 16 for plot.)

Separation between transmitter and receiver = 7.5 cms

46

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Amplitude of exciting pulses

47
= 104 volts

Noise level at the output = 0.03 volts

each

Table 9 Vout versus Axial Displacement for Left Side Beam Steering

Axial Displacement

Peak to Peak Voltage Output

 

#(cms) Vout - (volts)

1) 2.5 0.048

2) 1.5 0.052

3) 0.5 0.060

4) 0 0.065 (center of Tx.)
5) ~0.5 0.075

6) ~1.5 0.075

7) ~2.5 0.082

8) ~3.5 0.130

9) -4.5 0.180 (peak of central lobe)
10) ~5.5 0.120

11) ~6.5 0.085
12) ~7.5 0.074
13) ~8.5 0.070

14) ~9.5 0.065

15) -10.5 0.05

From this data the steering angle of the

being transmitted can be calculated.

main lobe of ultrasound

The shift in the position of central lobe, by applying a delay

of 0.3 microseconds between each exciting pulse observed, = 4.5 cms

Separation between transmitter and receiver = 7.5 cms

So, tan(e)= %~%-, 6 being the observed steering angle

0 = tan'1(%~%

) = 30.92 degrees

Steering angle expected theoretically = 30 degrees

Experimental error

0.92 degrees

48
Experiment 4

This experiment is very similar to Experiment 3. The only differ-
ence between the two being that we want to steer the ultrasound beam
by 30 degrees toward right rather than left along the axial axis 'x'.
This can be achieved simply by operating the shift left/shift right
switch on the programmable delay line to the shift right position. The
data obtained is as follows: (See Figure 17 for plot.)

Separation between transmitter and receiver = 7.5 cms

Amplitude of exciting pulses = 104 volts each

Noise level at the output = 0.03 volts

Table 10 Vout versus Axial Displacement for Right Side Beam Steering

 

Axial Displacement Peak to Peak Voltage Output
(ems) (V01 ts)

1) 10.5 0.038

2) 9.5 0.045

3) 8.5 0.058

4) 7.5 0.062

5) 6.5 0.070

6) 5.5 0.095

7) 4.5 0.165 (peak of central lobe)
8) 3.5 0.100

9) 2.5 0.082

10) 1.5 0.06

11) 0.5 0.06

12) 0 0.052 (center of Tx.)
13) ~0.5 0.045

14) ~1.5 0.035
15) ~2.5 0.032

The steering angle can be calculated from this data very easily.

The shift in the position of central lobe by applying a delay of

49

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4H

50

0.3 microseconds between each exciting pulse observed = 4.5 cms
Separation between transmitter and receiver = 7.5 cms so,

tan(0) = %~%-, 0 being the observed steering angle

0 = tan'1 04%} 30.92 degrees
Steering angle expected theoretically = 30 degrees

Experimental error = 0.92 degrees
Experiment 5

This experiment is very similar to Experiment 3. We want to steer
the ultrasound beam by 30 degrees towards left along the axial axis 'x'.
The only difference being that the distance between transducer array
transmitting ultrasound and the receiver has been increased to 10 centi-
meters so that we can have a better idea of how the field pattern varies
as this distance is increased. The data obtained is as follows: (See
Figure 18 for plot.)

Separation between transmitter and receiver = 10.00 cms

Amplitude of exciting pulses = 104 volts each

Noise level at the output = 0.03 volts

51

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mEo m m N m m w m N H o H- N- m- w- m- o- N- w-

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omH com
’ILV
L I
I
H
H
I \ili
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/
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er
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4‘

52
Table 11 Vout versus Axial Displacement for Left Side Beam Steering

 

Axial Displacement Peak to Peak Voltage Output
(cms) (volts)

1) 1.5 0.035

2) 0.5 0.040

3) O 0.050 (center of Tx.)
4) ~0.5 0.058

5) ~1.5 0.085

6) ~2.5 0.075

7) ~3.5 0.082

8) ~4.5 0.095

9) ~5.5 0.110

10) ~6.0 0.120 (peak of central lobe)
11) ~6.5 0.105

12) ~7.5 0.090

13) ~8.5 0.082

14) ~9.5 0.071

15) ~10.5 0.062

16) ~11.5 0.070

17) ~12.5 0.040

The steering angle can be calculated from this data as follows:

The shift in the position of central lobe by applying a delay of
0.3 microseconds between each exciting pulse observed = 6.0 cms

Separation between transmitter and receiver = 10.00 cms so,

tan(0) =lgf%%-; 0 being the observed steering angle

_ -1 6.00)
9 ‘ ta" (1_0.00

30.92 degrees
Steering angle expected theoretically = 30.00 degrees

Experimental error = 0.92 degrees

53

Experiment 6

This experiment is very similar to Experiment 5. We want to steer
the ultrasound beam by 30 degrees towards left along the axial axis 'x'.
The only difference being that the distance between transducer array
transmitting ultrasound and the receiver has been reduced to 5 centi-
meters so that we can have some idea of how field pattern of the steered
ultrasound beam looks like near the transducer array. The data obtained
is as follows: (See Figure 19 for plot.)

Separation between transmitter and receiver = 50 cms

Amplitude of exciting pulses = 104 volts each

Noise level at the output = 0.03 volts

Table 12 Vout versus Axial Displacement for Left Side Beam Steering

 

Axial Displacement Peak to Peak Voltage Output
(cms) (volts)

1) 1.5 0.050

2) 0.5 0.062

3) 0 0.068 (center of Tx.)
4) -0.5 0.075

5) ~1.0 0.080

6) ~2.0 0.120

7) ~2.5 0.180

8) ~3.0 0.220 (peak of central lobe)
9) ~3.5 0.170

10) ~4.0 0.105

11) ~5.0 0.080

12) -6.0 0.072
13) ~7.0 0.065
14) ~8.0 0.060
15) ~9.0 0.055
16) ~10.0 0.046
17) ~11.0 0.042

54

sH mesmHs

 

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msso m N o m e m N H o H- N- m- H7 m- o- N- N- m- S- 26
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omH oom
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55
The steering angle can be calculated from this data as follows:

The shift in the position of the central lobe by applying a delay
of 0.3 microsecond between each exciting pulse observed = 3.0 cms

Separation between transmitter and receiver = 5.0 cms so,
_ 3.00

tan(0) - 5—00'; 0 being the observed steering angle.
_ -1 3.00}
0 ~ tan (5500'

30.92 degrees
Steering angle expected theoretically = 30.00 degrees

Experimental error = 0.92 degrees

CHAPTER V CROSS COUPLING

Piezoelectric transducer arrays normally consist of closely pack-
ed radiating elements separated by a distance smaller than half the
wavelength of fundamental resonance frequency. This is a fundamental
requirement in the ultrasonic array design, but on the other hand, it
brings along with it the cross coupling problem.

Cross coupling is a phenomona which describes the influence of an
excited element on the neighboring elements due to induction. Cross
coupling may, in fact, have an appreciable effect on the field strength
pattern. In the near field this effect can be easily observed, but in
the far field it can also be seen. It will also arise from the fact
that the individual array elements reside on a common piezoelectric
base. These may be minimized by shielding the wires and minimizing
base thickness. However, even if these sources of cross coupling were
completely eliminated, appreciable cross coupling would still exist as
is seen in the analysis below which closely follows the analysis of
Wei Ming Wang.(1) This Chapter describes some theoretical aspects of
cross coupling.

The theoretical analysis in this Chapter is based on the assumption
that only one element is excited at a time and its effect, with regard
to coupling, is observed on the neighboring elements. Under this assump-
tion, cross coupling will have two componants, one is due to the direct
reception of the spherical waves at the receiving element being generated

56

57
by the excited element and the second is due to reflected wave componant

from the boundary of any object obstructing the propagation of ultrasonic
waves. The boundary conditions of the reflecting medium become excess-
ively important for this componant for the case where the medium is not
well defined (like a body). In order to deal with such situations, an
assumption is made that considers the reflected wave componant being
generated by a ficticious 'image source' formed at the same distance as
the actual source is from the reflecting boundary, as shown in the Fig-
ure 23. All elements are considered to be similar and resonate at the
same frequency.

Consider the induced voltage at an element due to a neighboring
excited element. The electro-mechanical equivalent circuit of each

)

element is shown in the Figure 20 (2 where

Z0 is output impedance

is the short circuit mechanical impedance

N

is the input electrical impedance

N

is the impedance of the voltage source

4 is the transformation ratio. This ratio basically describes the
conversion ratio of electrical excitation to mechanical stress
wave generated by the transducer.

From the equivalent circuit of Figure 20 we note that 20 impedance

is given as
- 2
Zo ~ (ZS Ze/Ze + 25) 0 + 2m and
V1 = (Ze/Ze + Z5) Es so

Voc (open Circuit) =<I>V1
For the above circuit a Norton's equivalent circuit can also be

drawn as shown in the Figure 21, where

58

 

 

11
If

 

 

 

 

4

Figure 20 Equivalent Circuit Model of an Array Element

 

 

#9

Figure 21 Norton's Equivalent Model of an Array Element

59

(Z Z ) 2
(Ze/zs + 29)):S 0/ s e o + Zm

ZS+Ze

I(norton)

Ze ES <b/(ZS + Ze)Zo

I(norton) is a current source in the circuit, but from the trans-
mission line anology it is in fact a velocity source because the trans-
ducer element is generating an ultrasonic stress wave traveling at a
certain velocity in the medium. So, I(norton) can be replaced by 'V0'
in the above expression; where VO represents the velocity of ultrasonic
stress wave, so

v0 = ZeZSo/(Ze + Z

)Z

S 0

Let 'V' be the induced velocity in the nth element due to the rad-
iating element. This induced velocity is due to the induced wave in the
secondary side of the receiver equivalent circuit as shown in the Figure
22. So, the induced voltage can be found out in the primary circuit.

E ind. = II (ZeZs/Zs + Ze)
where I1 is the current in the primary circuit due to induced current
12 so

E ind. = (o 12)(ZeZS/ZS + Ze)
or in terms of the acoustic analogy, we can replace induced current I2
by the induced velocity 'v'. So,

E ind. = (4 v)(ZeZs/Ze + ZS)

The expression for the induced velocity can now be derived under
certain assumptions. The most important of all being that although the
generated and received waves are spherical or semi-spherical in nature
we can treat them as plane waves for the reason that the reflecting plane
is at a number of wavelengths away from the source (far field approxima-

tion).

60

 

 

1:0
I I
Ililll

Al

 

._HA

 

*

Figure 22 Receiving Element Equivalent Circuit

61

So, under this assumption the stress exerted by the transmitted

wave on the receiving element is given as

F = P * A * D where

P = Acoustic pressure at the center of the receiving element
A = Face area of the element

0 = Directivity function of the element face

As the dimension of the elements are smaller than the wave length
we can approximate the expressions of 'P' and 'D' by the standard ex-
pression of P and D for a circular transducer. Using these standard
expressions, a standard expression for induced stress can be found and
using the concept that the total stress wave reaching the receiver ele-
ment is the sum of direct plane wave plus the reflected plane wave from
the reflecting medium, the second part of received wave can be approxi-
mated by a plane wave generated by a fictitious source,as shown in Fig-
ure 23,to meet the non-uniform reflecting conditions of the obstructing
medium; one can derive the expression for the induced velocity,(3) the
result stated below for the induced velocity for a receiving element
when only one element in the whole array is excited.

V = VoA202(-iwp eikrF(p))/2nr12O where

Vo = Velocity of ultrasound waves in the medium under consideration

A = Face area of the element

0 = Directivity function of the receiver element face
w = Frequency of the ultrasound waves

p = Numerical distance

ikr2(v0 + Bo)2/(1 + yo Bo)2 where
k and r2 represent distances shown in the Figure 23

cos (Angle of incidence)

Y0

cos(eo)

yo

 

62

 

 

 

 

 

‘1‘“?

 

 

 

 

 

 

 

 

 

 

 

I Reflecting Surface

 

Image Elements

 

 

Figure 23 Plane Wave Model of a

 

 

 

 

 

 

Transmitting/Receiving Element of an Array

63

80 = acoustic admittance

H."
A
'0
v
N

error function introduced to compensate for plane wave

assumption

(1 - I n p e0 erf cI p where

(1)

erf C(IFE_) is the complementary error function and
r1 = the distance between transmitting and receiving element.
So, the expression for induced voltage in an element is

E ind. = <l> V0 A2 o2 [}i-- peikr i(p)] [zezs/ze + ZS] /21rr1 z0 .

If more than one element is radiating then this induced voltage
will be the summation of induced voltage componants due to each element.

This is a generalized expression for a case where the elements are
separated from each other. For the case where the elements are joined
by a common substrate of piezoelectric material, another componant due
to mutual induction between the elements should be added to find out
the induced velocity.

The conclusion, which can be drawn from this mathematical analy-
sis, is that the induced voltage is inversely proportional to the center
to center distance between the transmitting and the receiving elements.
The effect of this cross coupling is considerable in the near field in
comparison to the far field. As most of the observations are normally
taken in the far field, these cross coupling effects are given a second-
ary importance in the design. If, in fact, appreciable cross coupling

is observed in the far field, it may be due to a "non-ideal" form of

cross coupling, such as the common base or intra-wire coupling.
5.1 Experimental Results

In order to study the effect of cross coupling on the transducer

array fabricated during this research, three similar experiments were

64
performed. In all these experiments one of the elements of the array

is excited by a voltage pulse of the same amplitude as was used in all
six experiments of Group 2 in Chapter 4, and the induced voltage is

measured at the terminals of other elements.
Experiment 1

In this experiment the central element '4' (see Figure 9, Chapter
3 for element numbering) of the array was excited by a 0.3 microsecond
pulse and the induced voltages on other elements of the array were ob.-
served. The data obtained is listed below:

Amplitude of the exciting pulse = 104 volts

Element excited = No. 4

Table 13 Induced Voltage for Different Elements of the Array

 

Element Number Induced Voltage
(volts)
5.20

7.0

8.0

7.90

6.75

5.00

Vacuum-H

Experiment 2

In this experiment, the first element '1' (see Figure 9, Chapter 3
for element numbering) of the array was excited by a 0.3 microsecond'
pulse and the induced voltages on other elements of the array were ob-
served. The data obtained is listed below:

Amplitude of the exciting pulse = 104 volts

Element excited = No. 1

65
Table 14 Induced Voltage for Different Elements of the Array

 

Element Number Induced Voltage

,(volts)
10.00
7.20
5.10
3.50
3.00
2.80

\immbwm

Experiment 3

In this experiment, the last element '7' (see Figure 9, Chapter
3 for element numbering) of the array was excited by a 0.3 microsecond
pulse and the induced voltages on other elements of the array were ob-
served. The data obtained is listed below:

Amplitude of the exciting pulse = 104 volts

Element excited = No. 7

Table 15 Induced Voltage for Different Elements of the Array

 

Element Number Induced Voltage

(volts)
2.9
3.05
3.50
4.95
6.85
9.50

05011::me

CHAPTER VI CONCLUSIONS

The experimental results obtained in general for the field strength
pattern of the array and the steering data obtained agrees well in some
aspects with the theoretical predictions of Chapter 2. However, there
are important differences too. Following are some conclusions which
can be drawn by comparison of theoretical results and experimental data:

1) The experimental field strength pattern of the array with all

elements excited at the same time, shown in Figure 14, seems
to be agreeing in general with the theoretical field strength
pattern of Figure 3; both have a main lobe and two major side
lobes. However, a close study indicates marked differences
in the ratio of main lobe height to side lobe height, the
width of main lobes, the position of side lobes and a missing
pair of first side lobes. The theoretical ratio of main lobe
to side lobe height was 1.37 to the observed ratio of 3.07.
The theoretical width of the main lobe was 25 degrees to the
observed width of 58 degrees and the theoretical position of
the first side lobe was 20 degrees to the observed first side
lobe of 36 degrees.

2) The experimental field strength pattern of the central element
of the array, shown in Figures 12 and 15, does not closely agree
with the theoretical field strength pattern of Figure 4. The
difference is that we expected the field strength pattern to
be much flatter than what we observed experimentally.

66

4)

67
One reason for deviation between theory and experiment may be
fabrication flaws, such as non-uniform solder connections be-
tween the elements of the array and the external connecting
wires. Also we could probably get better results by reducing
the common substrate coupling during fabrication.
The observed angle of beam steering agrees quite closely with
that predicted theoretically as may be seen from a study of
the beam steering Experiments 3, 4, 5, and 6 of Group 2 in
Chapter 4. The experimental results show that by exciting the
elements by sharp pulses of 0.3 microsecond duration, each
being delayed by 0.3 microseconds from its proceeding pulse,
we can steer the beam left or right along the axial axis
through an angle of 30.92 degrees (see calculations for Exper-
iments 3, 4, 5, and 6, Group 2, Chapter 4) whereas the theo-
retical steering angle expected is 30.00 degrees. The experi-
mental error in the steering angle, which is the difference
between experimental and theoretical steering angles, may be
due to the fact that one of the central elements of the array
went out of operation during testing or may be due to dissym-
metries in the elements of the fabricated array. Again, the
ratio of side lobe height to main lobe height is higher than
the theoretically predicted value.
The programmable delay line developed during the course of this
research is capable of steering the ultrasound beam through
discrete angles between 0 degrees and 445 degrees. Using the
programmable delay line in automode (DECX control), gates and
power amplifiers, the ultrasound beam should be steerable through

angles between 0 and 445 degrees in steps of 2 degrees under the

5)

6)

68
computer control. However, computer control was not actually

exercised because a key high speed IC chip was not yet in

hand at the time of taking data. Finer angular resolution is
obtainable by extension of the design to higher clock frequen-
cies.

The peak to peak output voltage obtained during Experiments

3, 5, and 6 of Group 2 leads to the conclusion that the field
strength varies as R'Z, where R is the distance between trans-
mitter and the receiver. However, the main lobe to side lobe
height ratio does not remain constant as expected theoretical-
ly; it also decreases with field strength as the distance be-
tween transmitter and receiver is increased. This may be due
to the fact that the field strength pattern of each element is
not as flat as expected. At present, this remains an unanswer-
ed question. Also, a R'1 dependence was expected rather than

a R'2 dependence observed.

The data obtained during Experiments 1, 2, and 3 of Chapter 5
(Cross Coupling) leads to the conclusion that the induced volt-

d-0o67

age varies somewhere between (1’1 to , where d is the axial

distance between transmission and reception points; whereas, a

d-l

dependence was predicted in Chapter 5. However, an exact
agreement should not be expected due to the presence of other
components of cross coupling. Since the excitation pulse was
quite short (0.3 microseconds) in duration, it seems like the re-
flection contribution was quite small and it was probably due

to the reflection near the transducer face such as transducer-

medium boundary. The other components contributing towards

69

cross coupling were probably the common substrate induction and

the intra-wire coupling componants.

CHAPTER VII RECOMMENDATIONS

Keeping in view the conclusions, the following recommendations

are proposed for future research.

1)

2)

4)

The fabrication process proposed in Chapter 3 can be improved
by applying deeper cuts for separating the elements. This
would reduce the common substrate induction, subsequently
reducing the cross coupling. A better technique should be
developed for solder connections between the array elements
and the external wires. The external wires must also be
shielded throughout to reduce intra-wire coupling.

In order to improve dynamic range (ratio of main lobe height
to side lobe height) and lateral resolution in the far field,
arrays with a larger number of elements should be fabricated.
Commercial units have on the order of 20 elements.

The reduction of main lobe height to side lobe height with the
increase of distnce between transmitter and receiver is also
one of the aspects to be studied in detail.

The central element of the transmitting array could also be
used as a receiver. More ideally, if all elements receive with
the same directionality as they transmit better dynamic range
is achievable. This is done in current commercial units by

Grumman and Varian, for example.

70

REFERENCES

CHAPTER I
1)

2) A.
3) 0.
4) H.
CHAPTER 11
1) A.
2) B.
3) 0.
4) G.
5)

6) M.
7)

8)

REFERENCES

Harry E. Thomas, "Handbook of Biomedical Instrumentation and

Measurement," Prentice-Hall Company, 1975, pp. 2-5.

. Oliner and G. H. Knittel, ”Phased Array Antennas," Artec

House, Inc., 1970- pp. 1-7.

. Gooberman, "Ultrasonic Theory and Applications,” English

Universities Press LTD., 1968, pp. 52-55.

. Wells, ”Ultrasound Applications," Plenum Press, 1965,

pp. 17-24.

. Oliner and G. H. Knittel, "Phased Array Antennas," Artec

House, Inc., 1970, pp. 4-10

. Hildebrand and B. B. Breden, "An Introduction to Acousti-

cal Holography," Plenum Press, 1972, pp. 16-19.

. Bobikov, "Ultrasonics and its Industrial Applications,"

Consultants Bureau NY, 1960, pp. 56-64.

. Gooberman, "Ultrasonic Theory and Applications," English

Universities Press LTD, 1968, pp. 54-59.

Glen Wade, "Acoustical Imaging," Plenum Press NY, 1975, pp.

111-126.

. Skolnik, "Introduction to Radar Systems," McGraw-Hill

Company, 1962, pp. 101-106.

"Ultrasonics in Medicine, Conference Papers," Munchen, 1975,

pp. 75-80.

Yoshimitsu Kikuchi, "Ultrasonic Transducer," Corona Publishing

Company, LTD, 1969, pp. 240-244.

71

72

CHAPTER V

1) Wei Ming Wang, "Mutual Coupling through Radiation Transducer
Elements in an Infinite Array," IEEE Sonics, 1972, pp. 28-31

2) Charles Sherman, "Analysis of Acoustic Interactions in the Trans-
ducer Arrays," IEEE Sonics, 1966, pp. 9-15.

3) Uno Ingard, "0n the Reflection of a Spherical Sound Wave from
an Infinite Plane Wave,” Journal of the Acoustical Society,
1951, pp. 329-335.

APPENDIX B

1) James F. Gibbons, "Physics of Semiconductor Devices,” McGraw-Hill

Company, 1965, pp. 275-325.

APPENDICES

APPENDIX A
PROGRAMMABLE DELAY LINE

APPENDIX A

Programmable Delay Line

The programmable delay line can be used to produce a number of de-
lay outputs upon initiation by an excitaion pulse. The time delay be-
tween the output can be controlled in a predetermined fashion. A pro-
grammable delay line was designed and developed during the course of
this research. The initial design was originated by Mr. Philip Chimento,
and was completed by the author with consultation by Mr. Mark Funk. The
details of that are given below and shown in Figure 24.

There are two modes of operation of this programmable delay line,
namely the manual and auto mode. In manual mode of operation, the delay
provided between the output pulses is controlled by six single pull
double throw switches and in auto mode the computer controls the delays
through its control signal called DECX. The delay line is initiated by
the inverted DECX signal which allows the AND gate 1 to open and allows
the upper counter to count the clock pulses. The frequency of the clock
determines the minimum time delay between two consecutive outputs of
the delay line. The outputs of the upper counter are applied to an
equality gate (combination of XOR's and AND gate) along with the other
outputs coming from the presettable counter, shown in the bottom part
of the Figure. The outputs of this counter can be set to a predeter-
mined value by setting the six single pull double throw switches in

manual mode and operating the count/load momentary switch, or it can
73

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2.: >28 mHseHEeHmoHs HHN 953s

            
               

m+

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unmvz umvzm
LO

umoH uw—gm

  

sousaou

 

NaHvN
Housaou

mpznsH
mouea
op

concN

msuumHmom
Hstm

    
        
   

«Heeeeumees

muHos m+ NchN

xumo

  

guarxm

seeHzm
eeeH

HessexNOua< suqum ueoa

75
be set to count the DECX pulses in the auto mode. In the eqUality gate

these two outputs of the top counter, counting clock pulses, and the
presettable counter are compared and it will give an output pulse when-
ever the two counts become equal. This output is applied as a clock
pulse to shift registers. In this way the output of the equality gate
is used to shift one's throw shift registers. Before this pulse appears
the shift register outputs are zero's. When the first pulse appears
the first output bit of shift registers 90 high, on second pulse from
equality gate the second output bit goes high and so on. It is obvious
from this description that the time interval between two consecutive
output bits depends upon the time interval between two consecutive
equality gate output pulses.

The shift registers are dual type and can shift the data right or
left. The mode of operation of shift registers (left or right shift)
'depends upon its inputs S0 and $1. Shifting the data left means we
want to steer the beam towards left side from the control position of
the beam. The inputs S0 and S1 are controlled by the outputs from a
flip flop, whose operation is controlled by an external single pull
double throw switch. When this external switch is in position '1' the
outputs S0 and S1 are at levels '0' and '1' respectively and so the
data in the shift registers starts moving left. When this switch is in
position '2' the outputs of the flip flop SO and S1 are at levels '1'
and '0' respectively and in this case the data will be moved towards
the right. Therefore, by controlling the position of this switch the
outputs of the shift registers can be obtained either from least sig-
nificant bit to most significant bit (shift left) or from most signif-

icant bit to least significant bit (shift right). Each output bit of

76
shift registers controls the initiation of excitation of one element of

the transducer array. Once the data is shifted from right to left or
from left to right completely, meaning that all output bits have '1's

at their terminals, no more data can be shifted before clearing the
shift registers. The same DECX pulse which starts the operation of this
delay line is used to clear the shift registers so that by the time the
equality gate starts giving output the shift registers are ready to
accept new data from it. The programmable delay line developed has a
maximum capability of 10 output bits, which means that an ultrasonic
transducer array with 10 elements can be excited by this delay line.

The limitations of this circuit is that in auto mode it can steer
the beam only by 1’40 degrees from the central position of the main lobe.
In order to do so the 4th and 6th bits of the presettable counter is
connected to the AND gate '2' as shown in the Figure. The output of
this AND gate resets the presettable counters and also changes the
state of the flip flop controlling the shift left shift right operation
of the shift registers. In this way, after steering the beam in one
direction by 40 degrees, this delay line starts steering the beam in
the other direction till it reaches 40 degrees beam shift and then the

whole cycle is repeated again.

APPENDIX B
GATES AND POWER AMPLIFIERS

APPENDIX B

Gates and Power Amplifiers

The outputs of the programmable delay line are levels rather than
pulses in the manual mode and are pulses of low frequency in case of
the auto mode, so one needs some kind of short pulse generating circuit
to operate at the beginning of each level shift ('0' to '1') of the out-
put bits of shift registers. A monostable circuit can be used for gen-
erating sharp pulses at the beginning of each level shift. However,
the monostable circuit has a low fan out and is not capable of supplying
enough current to the power amplifiers without considerably loading it-
self. Digital multiplexing gates can be used to solve this problem of
loading with their continuous input being connected to +5 volts and the
control input to the monostable output. The output of the gates can
not be connected to the inputs of the power amplifiers.

The power amplifiers are needed to amplify the low voltage pulse
generated by the monostables. This is necessary because one needs a
high power pulse to excite the transducer elements. For this purpose
a cascaded power amplifier with two stages was designed. (See Figure
26.) The coupling capacitor and 470 0'5 resistance forms a filtering
network which eliminates the small leakage spikes coming from the pre-
settable counter. The first stage is a common emitter configuration
to give the necessary voltage gain to the input signal. The second

stage is a direct coupled common collector stage to provide the necessary
77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“‘LJ‘

10 Power Amlifiers

h-(E—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 Gates

 

 

 

+4:
i-l:
-—[:

To Transducer
Array

 

 

 

 

 

 

r77“
|__1

10 Monostables.

re +"

,1 T;—
L:

H: F—L_
L__l.

s r:—
L.

H I71.
L...-

 

 

 

+5'volts

Figure 25 Block Diagram of Gates and

 

Input

[H

from

Programmable
Delay Line

Power Amplifier

79

high current gain so that the amplifier can supply large currents to
the load (array elements) without loading the common emitter stage.
During the course of this research, seven such cascaded amplifiers (one
for each element) were designed using the standard formulae (1) and the

circuit diagram of one of such amplifiers is shown in Figure 26.

 

80

I-FIZO volts

 

 

 

 

1000 pF nggn
1. IL H
' j1
Vin
470 0 J0. 0220F
1 k 15
' 12

 

0. 047uF

k

 

 

 

c—J
-

Figure 26 Circuit Diagram of Power Amplifier

Transducer
Element

APPENDIX C
COMPUTER PROGRAMS

 

APPENDIX C

Computer Programs

This appendix includes the computer programs developed for simulat-

ing the steering and focusing effects of the ultrasonic transducer array.

The Program 1 listed below is for steering and Program 2 is for fo-

cusing effect simulation.

Program 1 (VORTXII FTN IV(G) SAEED)

\OCDVO‘U‘I-hwm

NNNHHi—Iv—Ii—ti—IHHo—bi—o
NHOkOmNOiUi-wai-JO

C DESIGN OF PHASED ARRAY FOR RECONSTRUCTIVE TOMOGRAPHY
C DATA VARIABLES ARE AS FOLLOWS

C D = SPACING OF ELEMENTS

C V = VELOCITY OF ULTRASONICS IN THE MEDIUM UNDER CONSIDERATION
C F = FREQUENCY OF ULTRASONICS

C N = NUMBER OF ARRAY ELEMENTS

C A = WIDTH OF ONE ARRAY ELEMENT

C T = TIME DELAY OF EXCITATION

C THETA = ANGLE OF MAIN LOBE WITH NORMAL

REAL LAMBDA
DIMENSION P(150), IP(2), 0(2), 10(2)
DATA (ID(I), I = 1, 2)/1 HP, 2 HPD/
READ (4, 10) B, V, F, N, A, T
WRITE (5, 10) B, V, F, N, A, T

10 FORMAT (F10.0, 2E10.3, I5, 5X, 2F10.0)

IP (1) = 1
IP(2)=1
D (1) = 1
D(2)=1
THETA = ~60
PHI = 3.14592

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LAMBDA = (V/F)
NI = N
00 20 I = 1,121
TETA = (THETA * PHI/180.)
X = SIN (TETA)
THERD = (PHI* SIN (TETA)/LAMBDA)
U = (THERD * B)
UP = (THERD * A)

UF = (PHI * T * F)

UT = (U ~ UF)

US = FLOAT (NI) * UT
UR = (SIN (US))

P1 = SIN (UP)/UP

P2 = SIN (US)/SIN(UT)

IF (P1. GT. 1) 00 TO 100

IF (P2. GT. NI) GO TO 200
GO TO 80

P1 = 1

IF (P2. GT. NI) GO TO 150
P2 = SIN (US)/SIN (UT)

P(I) = P1 * P2

GO TO 85

P2 = N1

P1 = SIN (UF)/UP

P(I) = P1 * P2

GO TO 85

P1 = 1

P2 = NI

P(I) = P1 * P2

GO TO 85

P(I) = P1 * P2

P08 = (20.0 * ALOG 10 (ABS (P(I)/NI)))
WRITE (5, 50) THETA, P(I), PDB, P1, P2
FORMAT (5(F10.6, 5X))

THETA = THETA + 1.0

WRITE (8) THETA, P (I), P08
CONTINUE

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60 75 I = 121

61 ENDFILE 8

62 CALL PRPLOT (2, I, 8, 5, IP, 0, ID, 0)
63 END

 

84
Program 2 VORTXII FIN IV(G) SAEED

2 c FOCUSING EFFECT SIMULATION FOR SECOND PHASED ARRAY
3 C 2 = FOCAL POINT

4 C D = ELEMENT T0 ELEMENT DISTANCE

5 C Y = DEPTH 0F MEASUREMENT

6 REAL LAMBDA

7 DIMENSION P(100), IP(2), 00(2), ID(2), L(4), M(4), 1(4), PHE(4)
8 DATA (ID(I), I = 1, 2)/1HP, 2HPD/

9 READ (4, 10) O, 2, v, F, Y

10 WRITE (5, 10) 0, 2, v, F, Y

11 10 FORMAT (2F10.0, 2E10 3, F10.0)

12 IP (1) =1

13 IP (2) = 1

14 DD (1) = 1

15 DD (2) = 1

16 A = 0.4

17 N = 7

18 x = -100

19 PHI = 3.141592
20 LAMBDA = (V/F)
21 L (I) = (3.0 * D)

22 L (2) = (2.0 * D)

23 L (3) = D
24 DO 20 I = 1, 3

25 M (I) = L (I) * SIN (ATAN (L(I)/Z))

26 T (I) = M(I)/v
27 PHE (I) = (T (I) * 2 * PHI 8 F)
28 20 CONTINUE
29 DO 30 J = 1, 41

30 R = ((x * * 2.0 + Y * * 2.0) * * 0.5)

31 R1= ((((x+3.0*0) **2.0) +Y**2.0) ** 0.5)
32 R2 = ((((x + 2.0 * D) * * 2.0) + Y * * 2.0) * * 0.5)
33 R3= ((((x+D) **2.0) +Y**2.0) **0.5)

34 R4 = ((((x - D) * * 2.0) + Y * * 2.0) * * 0.5)

35 R5 = ((((x - 2.0 * D) * * 2.0) + Y * * 2.0) * * 0.5)
36 R6= ((((x- 3.0*D)**2.0)+Y**2.0) **O.5)

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E1 = ((2.0 * PHI * (RI - R))/LAMBDA

E2 = PHE (1) + ((2 O * PHI * (R2 - R))/LAMBDA
E3 = PHE (2) + ((2.0 * PHI * (R3 - R))/LAMBDA
E4 = PHE (3)

E5 = PHE (2) + ((2.0 * PHI * (R4 — R))/LAMBDA
E6 = PHE (1) + ((2.0 * PHI * (R5 - R))/LAMBDA
E7 = ((2.0 * PHI * (R6 - R))/LAMBDA

P1 = (COS (El/R1) + COS (E2/R2) + COS (E3/R3) + COS (E4/R))
P2 = (COS (Es/R4) + COS (E6/R5) + COS (E7/R6))
P4 = P1 + P2

IF (x. E0. 0) TH = PHI/2.0

IF (x. NE. 0) TH = ATAN (Y/X)

THERD = PHI * SIN (TH)/LAMBDA
UP = THERD * A

U = THERD
US= FLOAT (N) * THERD
UR= SIN (US)

IF (U. E0. 0) GO To 30
P3 = SIN (US)/SIN (U)
P(J) = P3 * P4
P08 = (20.0 * ALOGIO (ABS (P(J)/N)))
WRITE (5, 50) x, P(J), PDB, P3, P4
FORMAT (1x, SFIO.6)
x = x + 5.0
WRITE (8) x, P(J), PDB
CONTINUE
I = 21
ENDFILE
CALL PRPLOT (2, I, 8, 5, IP, 00, ID, 0)
END