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5-17-2015
Demonstrations in Ultrasound Imaging and Beam
Steering Using Phased Arrays
Katherine Anne Roy
Dickinson College
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Roy, Katherine Anne, "Demonstrations in Ultrasound Imaging and Beam Steering Using Phased Arrays" (2015). Dickinson College
Honors Theses. Paper 197.
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Demonstrations in Ultrasound
Imaging and Beam Steering Using
Phased Arrays
Submitted in partial fulfillment of honors requirements
for the Department of Physics and Astronomy, Dickinson College,
by
Katie Roy
Research Advisor: Professor Brett Pearson
Reader: Professor David Reed
Reader: Professor Catrina Hamilton-Drager
Reader: Professor Lars Q. English
Reader: Professor David Jackson
Reader: Professor David Mertens
Carlisle, PA
May 10, 2015
Abstract
Ultrasound imaging is one of the most ubiquitous forms of sonography today.
It ranks among the cheapest and safest forms of medical imaging techniques
available, and it has a wide range of applications in medicine; it is gentle enough
to image a developing fetus, yet can also be powerful enough to destroy a kidney
stone. We have developed a series of demonstrations that examine several
physical properties of ultrasound, specifically as they pertain to imaging. We
intend these demonstrations to be used as a learning aid for students who are
studying the physics behind medical imaging. These demonstrations utilize
single-source transceivers from the maker Parsonics, and they cover the basic
properties of ultrasound, the remote sensing capabilities of sound, and more
advanced ultrasound theory. Many applications of ultrasound imaging require
instead an array of multiple sources. Following techniques used in radar and
loudspeaker implementation, we also explore analytically and computationally
the steering and shaping of ultrasound waves using source arrays. Operating in
the small-source, far-field limit, we vary the time delays introduced into each
source element to allow for changes in directionality. Following the principle of
superposition, we are able to focus the outgoing radiation at a particular point
in the far field. Applications of these techniques are most notably within the
medical field and include both ultrasound imaging and ultrasound therapies.
ii
Acknowledgements
I would like to thank my research advisor, Professor Brett Pearson, for all of his advice
and encouragement in this project, and for his constant enthusiasm in learning more
about each senior project; Jonathan Barrick, whose incredible construction ingenuity
brought our demonstrations to life, and without whose patient assistance we could
not have completed any of our experiments; Professors David Reed and Lars English
for their assistance and guiding ideas over the course of the semester; and Professors
Catrina Hamilton-Drager and Hans Pfister for their support and feedback both in
this project and outside of it.
To our peers and friends who listened diligently to multiple research update presentations and provided feedback on our work this whole year – we cannot thank you
enough and wish you the best of luck after graduation.
Finally, I am so grateful to have had as wonderful a research partner and co-author
for the demonstrations portion of this project as Andrew Chen. Thank you for being
such a supportive and inspiring collaborator over the course of this entire year.
iii
Contents
Abstract
ii
1 Introduction
1
2 Theory
2.1 Basics of Ultrasound Transducers
2.2 Ultrasound Wave Propagation . .
2.3 Imaging Methods . . . . . . . . .
2.4 Arrays of Sources . . . . . . . . .
2.5 Linearly Phased Arrays . . . . . .
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3
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3 Experimental Methods
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4 Demonstrations and Results
4.1 Speed of Sound Demonstration
4.2 Pulse-Echo Demonstration . . .
4.3 Wind Speed Demonstration . .
4.4 Intensity Mapping . . . . . . .
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5 Phased Array Simulations and Results
27
6 Conclusions and Further Research
30
A Demonstrations User’s Manual
A.1 Introduction . . . . . . . . . . . . . . .
A.2 Theory . . . . . . . . . . . . . . . . . .
A.3 Equipment . . . . . . . . . . . . . . . .
A.4 Experimental Procedure . . . . . . . .
A.4.1 Speed of Sound Demonstration
A.4.2 Pulse-Echo Demonstration . . .
A.4.3 Wind Speed Demonstration . .
A.4.4 Intensity Mapping . . . . . . .
31
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List of Figures
1
2
3
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5
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9
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11
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(Left) Incoming voltage potential deforms piezoelectric material in a
transducer and produces a sound wave. (Right) Incoming sound waves
cause the piezoelectric material to vibrate and produce a voltage potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interface between two media with Z1 << Z2 . Qualitative relative
initial, reflected, and transmitted intensities (I0 , IR , IT ) represented by
arrow width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Planar cross section of the radiation pattern of a single point source
with field point r. Semicircles represent crests of multiple wavefronts.
Acoustic radiation from a linear array of sources. Individual wavefronts
sum to a plane wave along this front. . . . . . . . . . . . . . . . . . .
Pathlength difference x between r0 and r1 . Due to the far-field assumption, r1 = r0 − x and the angle θ measured from vertical is equal
to the angle labeled above. . . . . . . . . . . . . . . . . . . . . . . . .
Matlab plot of |H(θ)|2 with varying N . Curves with greatest to least
spread represent N = 5, N = 10, and N = 100, respectively. Corresponding FWHM measured to be 0.74, 0.36, and and 0.04. Fixed
2π
m−1 and d = .001 m. . . . . . . . . . . . . . . . .
parameters k = .004
Matlab plot of |H(θ)|2 with varying λ. Curves with greatest to least
spread represent λ = 0.03 m, λ = 0.01 m, and λ = 0.004 m, respectively. Corresponding FWHM measured to be 0.92, 0.30, and and 0.12.
Fixed parameters N = 30 and d = .001 m. . . . . . . . . . . . . . . .
Acoustic radiation pattern from a linear array of sources with an effective linear time delay. Individual wavefronts sum to a plane wave with
angle θ0 from the vertical. . . . . . . . . . . . . . . . . . . . . . . . .
2π
Matlab plot of |H(θ)|2 with N = 30, d = 0.001 m, and k = 0.004
m−1 .
Left peak created when θ0 = 0, and peaks shifted right created when
θ0 = π6 and θ0 = π3 . Corresponding FWHM measured in Matlab to be
0.12, 0.14, and 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polar representation of the directivity function in Figure 9. Radial
length to the curve along some angle represents the magnitude of the
directivity function at that angle. (Left) Directivity for θ0 = 0. (Right)
Directivity for θ0 = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . .
Convention for angle offsets for our experiments. . . . . . . . . . . . .
Transceivers with custom mounts on optical bench. . . . . . . . . . .
Tektronix TDS 1012B oscilloscope. . . . . . . . . . . . . . . . . . . .
v
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6
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12
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Screenshots of oscilloscope data. The signal from Channel 2 (top signal) represents the signal received directly from the function generator,
while the sound was received by Channel 1 (bottom received signal)
and traveled through the transceivers. Note the difference in voltage
scale between the two images. (Left) At a separation distance of 26 cm
in air, we used the cursors to measure the time delay between when
the function generator released the pulse to when it was first detected
by the receiver: ∆t = 800 µs. (Right) Again, at a separation distance
of 26 cm, the time delay through the plexiglass cylinder was measured
to be ∆t = 210 µs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transceivers with metal plate. . . . . . . . . . . . . . . . . . . . . . .
Diagram of the Pulse Echo Demonstration. . . . . . . . . . . . . . . .
Schematic diagram of the wind speed demonstration. . . . . . . . . .
An overhead shot of the wind speed demo. The PVC pipe in the upper
left is what directs the airflow. . . . . . . . . . . . . . . . . . . . . . .
Wind speed data measured with anemometer. . . . . . . . . . . . . .
Method of data acquisition for intensity map. . . . . . . . . . . . . .
Graph showing amplitude as a function of distance for 0◦ , 5◦ and 10◦
off axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decibel map in relative dB units. . . . . . . . . . . . . . . . . . . . .
Decibel intensity level from RMS voltage data. . . . . . . . . . . . . .
Acoustic radiation pattern from array with arbitrary delay. Individual
wavefronts sum to an envelope wavefront leading out from the array. .
|H(θ)|2 for step function with N = 30, d = 0.001 m, and λ = 0.004
m. Second half of the array elements activated at a time mπ
after the
kc
first, where m ∈ R. When m = 1, we induce a π-phase delay (darker
curve). When m = 2, we induce a 2π-phase delay, which has no effect
in the far field (lighter curve). . . . . . . . . . . . . . . . . . . . . . .
N 2
|H(θ)|2 for parabolic time delay of τn = kc(mπ
N 2 (n − 2 ) with N = 30,
)
2
d = 0.001 m, and λ = 0.004 m. Furthest outside to furthest inside
curves correspond to m = 1, m = 0.75, m = 0.5, and m = 0.25,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transducer, optical track and hard interface. . . . . . . . . . . . . . .
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1
Introduction
The history of sonogrophy in medicine dates as far back as 1761 when Leopold Auenbrugger published a book entitled On Percussion of the Chest [1]. Methods of exploiting the body’s susceptibility to percussion, such as during a routine exam with
a stethoscope, are still used today in clinical settings, though the applications of
sonography in medicine have grown and expanded since Auenbrugger first published
his findings. Beginning as early as the 19th century, physicians have used sound in
medicine to gather information about the condition of their patients, most predominantly in the fields of obstetrics and cardiology.
In 1880 Pierre and Jacques Curie discovered the phenomenon known as the piezoelectric effect while experimenting with crystal structures [2]. The piezoelectric effect
is characterized by the deformation of the structure of a material due to an applied
voltage potential [2]. Conversely, a piezoelectric material also has the ability to produce an electric charge if it is subject to some compression or tension force. The first
application of a piezoelectric crystal in any form of sonography – the use of appropriate instruments to construct images from received sound waves – was in the field
of submarine sonar [2]. Since their origin, ceramic piezoelectric materials have been
used extensively throughout the consumer product market as actuators in devices, as
well as in the medical field for the production of ultrasound. Sound vibrating at frequencies higher than those of the audible range, generally above 20kHz, is classified as
ultrasound, and these high frequencies and corresponding small wavelengths are what
make ultrasound so suitable for imaging within the human body. Presently, ultrasound is used to image organs, tissues, blood and even to produce three-dimensional
fetal images.
One specific use of ultrasound imaging is fetal imaging. Since ultrasound techniques are safer than x-rays and other procedures utilizing ionizing radiation, and
because they are more cost effective than MRI, x-ray, and CT scans, they are the
preferred methods for imaging a developing fetus [1, 3, 4, 5, 6, 7, 8]. Frequency shifts
via the Doppler effect are also used in ultrasound imaging in order to locate and
amplify a fetal heartbeat and in the observation of blood flow through a patient’s
cardiovascular system. The pervasiveness of ultrasound in the medical field makes it
an essential imaging technique of modern health care.
This paper will discuss the basic physical concepts upon which these particular
applications of ultrasound are based, beginning with acoustic wave theory. Because
sound is a longitudinal wave that is transported through media by particle vibrations,
properties of host media will determine how the sound wave propagates. For example,
the speed of sound in air at room temperature is roughly 340 m/s, but its velocity
1
through more elastic materials – such as liquids and solids – is much higher.
The high frequencies of ultrasound are desirable particularly in medical imaging
because of the inverse relationship between frequency and wavelength at a given
propagation speed; objects smaller than one wavelength or objects separated by less
than one wavelength of the incoming sound pulse will not register as distinct entities
in an image, and so the higher ultrasound frequencies allow for greater resolution,
both laterally and axially [9].
The degree to which a sound pulse propagates, attenuates, or reflects back at an
interface between different media is determined by the relative acoustic impedances
of those media [10]. Acoustic impedance is essentially a measure of the elasticity of a
medium, and the fraction of the incident pulse that is reflected back is proportional to
the square of the difference in acoustic impedance between the two materials. Producing an actual image from ultrasound requires hardware that can interpret and convert
the returning pulse echoes and sound backscatter into markers for boundaries between
different fluids and tissues. The echoes from successive reflections are registered by a
receiver, and the magnitudes and time delays of the sound pulses are quantized and
converted into data that is analyzed and processed in order to reconstruct the area
over which the scans have been taken.
Using transducers from the maker Parsonics 1 , we seek to create laboratory demonstrations displaying a variety of ultrasound properties to students at the undergraduate level. We will test the spread of our ultrasound beam and will demonstrate the
effects of attenuation through measurements showing how received voltages drop off
as a function of distance and angle, and we will use mixed media and interfaces to
display other topics in ultrasound, including propagation velocity, transmission, and
reflection.
The transceivers used for these demonstrations are single-source elements that
send out pulses specified by the driving function generator. Historically, these are
similar to the early transceivers used in the field of medicine, which required mechanical scanning both in the linear and angular direction to achieve the desired image [9].
However, many imaging applications now utilize instead a combination, or an array,
of sources. Phased arrays are used in a wide variety of fields across physics, most
notably in radio astronomy, loudspeaker and radar implementation, and – of course
– medical ultrasound imaging for beam focusing and steering.
Because multiple sources make it possible to steer and focus an outgoing sound
wave to specific depths and angles, arrays allow for greater control than single aperture
sources and are useful in handling complicated geometries that a single source cannot
1
Parsonics Ultrasound Transducers, 935 Dieckman Rd., Woodstock, IL 60098.
2
resolve well [9]. Huygens’ Principle and the concepts of constructive and destructive
interference guide conceptual understanding of sound propagation. Operating within
the small-source, far-field limit, we explore radiation patterns and are able to sum the
total pressure released by multiple sources in an array. We will examine analytically
the directivity of a line of sources with no time delay between source activation, with
a linear time delay, and we will develop computationally the directivity for an array
with arbitrary time delays.
2
Theory
This section will cover a basic investigation of ultrasound and the properties that
make its use so effective in the field of medicine, in addition to relevant theory and
derivations concerning phased arrays.
2.1
Basics of Ultrasound Transducers
Ultrasound transducers work by means of piezoelectric material. Applying a voltage
across a piezoelectric crystal or ceramic material causes it to deform physically, producing sound. Conversely, these materials produce an oscillating voltage potential
when deformed by an incoming sound pulse, as seen in Figure 1.
Figure 1: (Left) Incoming voltage potential deforms piezoelectric material in a transducer and produces a sound wave. (Right) Incoming sound waves cause the piezoelectric material to vibrate and produce a voltage potential.
When an AC voltage is applied across a piezoelectric material, such as those
found in medical transducers, the alternating voltage drives the oscillation of the
3
piezoelectric material. As such, the frequency of the electrical energy in the current
dictates the frequency of the vibrational energy and of the sound released [2]. Through
the use of a function or pulse generator, it is possible produce sound waves of a
specified frequency.
2.2
Ultrasound Wave Propagation
One of the simplest properties of sound waves and of waves in general is the relationship between the propagation velocity, the frequency, and the wavelength:
v = λf,
(1)
where λ is the wavelength of the sound, f is its frequency, and v is the velocity.
Unlike light, sound requires a medium through which vibrations can occur. Because sound is a longitudinal wave, it propagates through media via compressions and
rarefactions. The speed of a sound wave through a medium varies with the density
and elasticity of the host medium:
r
κ
,
(2)
v=
ρ
where ρ is the density of the material in kilograms per cubic meter and κ is the elastic
kg
modulus, a measure of the medium’s resistance to compression in pascals or m·s
2 [10].
The speed of sound in air is approximately 340 m/s, while in water it propagates at
approximately 1500 m/s [10].
The energy of an ultrasound wave is proportional to the square of the pressure of
the wave [10]. As the wave propagates, there are two main factors that contribute to
a decrease in intensity with distance. The first is beam spreading, a phenomenon that
occurs as an ultrasound beam diverges. For a point source of spherical radiation, this
decrease is related to the inverse of the square of the distance traveled. The second
occurs when the media through which a wave travels absorbs energy as the sound
passes through. Mathematically, we can represent the decrease in amplitude due to
absorption as a function of distance [1]. This is given as:
A = A0 e−αx ,
(3)
where x is the distance traveled by the sound, A0 is some original amplitude, and α is
the absorption coefficient of the material, usually given in inverse centimeters, cm−1 .
Some typical values of the absorption coefficient for ultrasound with a frequency of 1
MHz are: α = 0.13 cm−1 in muscle, α = 0.11 cm−1 in the brain, and α = 2.5 × 10−4
cm−1 in water [1].
4
We are often more interested in the intensity of a sound wave than its pressure.
The intensity of sound is measured in watts per square meter, and it is proportional
to the square of the amplitude of the pressure wave [1]. Thus, the similar expression
for the drop in intensity of a sound wave through a medium due to absorption is
shown to be:
I = I0 e−2αx ,
(4)
where I0 is an initial intensity and α and x are defined as above. Through comparison
of Equation 3 and Equation 4, we can see that intensity will taper off much more
quickly than amplitude when traveling through a medium since I ∝ A2 .
One common way that sound intensity is measured is through the decibel (dB)
scale. This scale is always defined relative to some reference intensity. As shown
in Equation 5, the equation for calculating the intensity of sound in dB given the
reference intensity I0 is:
dB = 10 log10
I
.
I0
(5)
For audible sound, I0 is typically chosen to be 10−12 W/m2 , the threshold of hearing.
This intensity is the lowest intensity audible to the human ear.
Since sound intensity is related to pressure by:
2
I
p
=
,
(6)
I0
p0
we can combine Equations 5 and 6 to find the decibel conversion equation:
p
dB = 20 log10
,
p0
(7)
where p is the pressure of the wave and p0 is the reference pressure.
Sound intensity undergoes a change when a sound wave encounters an interface of
two materials with different acoustic impedances. Acoustic impedance is a measure
of a medium’s elasticity and is expressed as [10]:
Z = ρv =
√
κρ,
(8)
where ρ is the density of the material, κ its elasticity, and v the speed of wave
propagation. At the interface, a portion of the sound is reflected back towards the
source while a percentage is also transmitted.
The extent to which an ultrasound pulse is reflected or transmitted is a direct
result of the square of the difference between the acoustic impedances of two media
5
at an interface. Assuming the wave approaches the interface perpendicularly, the
ratio of the reflected intensity to the initial intensity [10]:
IR
(Z1 − Z2 )2
=
.
I0
(Z1 + Z2 )2
(9)
Therefore, the transmitted intensity is:
IT = 100% −
IR
,
I0
(10)
where I0 , IR , and IT are the incident, reflected, and transmitted intensities, respectively, and Z1 , Z2 are the acoustic impedances of the first and the second media.
Figure 2: Interface between two media with Z1 << Z2 . Qualitative relative initial,
reflected, and transmitted intensities (I0 , IR , IT ) represented by arrow width.
One of the problems with ultrasonic imaging is that if too much of the sound wave
is lost due to absorption by a medium or to internal reflections, one can no longer
obtain any useful information.
Additionally, surfaces in reality are rough and will scatter the sound at various
angles. Beam spreading causes incoming sound to return at different angles, and the
transceivers receiving the pulse echoes must be designed to compensate for and to
interpret this scattering. When the backscatter echoes travel back towards a receiver,
a series of reflections are registered sequentially in time. Additional weightings and
time delays are usually introduced to emphasize the sound coming from the desired
area and “dim” the presence of backscatter from other areas. Each wave is converted
into a scan line, and within each line is stored information in time about threedimensional backscatter returning from different angles and depths [9]. These lines
are compiled and interpolated together to form the image [9]. In the next section we
discuss one of the imaging techniques in which the above method is implemented.
6
2.3
Imaging Methods
One of the most commonly used imaging methods is the pulse-echo technique. In
this system, a transducer sends out a single pulse of ultrasound. This pulse partially
reflects at different inerfaces and thus creates echoes that travel back towards their
origin. A receiver registers the echo pulses and translates them into electrical pulses
that can be used to form an image. One can calculate the distance of an object away
from the transducer through known media using this distance equation:
1
1
d = v∆t = λf ∆t
2
2
(11)
where d is the distance traveled by the sound and ∆t is time difference from when
the pulse was emitted to when its echo was received. Note that to obtain an accurate
measurement for the distance between a transducer and an object using Equation 11,
we must know the velocity of sound in the medium through which it propagates. This
method for calculating distance in scans is used extensively in the field of imaging.
Often only a single transceiver is used in the pulse-echo technique, and for this it
is important to avoid interference between the pulses created and the echoes received.
As such, it is standard to wait until the echoes from the first pulse are registered to
send out the second [10]. We define a pulse repetition rate, Rrep , in inverse time to
be:
v
(12)
Rrep = .
2d
Thus, those using the ultrasound machine can determine just how far they plan
to image, and, as a result, the minimum length of time to wait in between pulse
generations in order to avoid competing signals in one single transducer.
2.4
Arrays of Sources
Ultrasound arrays are an ubiquitous method of imaging today, and their creation
has greatly enhanced their applications in the medical field [9]. It is possible to
derive analytically several results concerning arrays of sources while operating under
a few given assumptions. Firstly we consider a straight line of sources, though other
geometries – such as curved or convex arrays – are also used in practice [9]. We only
study the case of small, assumed point sources such that the radius of the source, a, is
much smaller than the wavelength of the sound emitted: a << λ, or ka << 1, where
k is the wavenumber of the sound [11]. This allows us to assume that the radiation of
sound from each source is omnidirectional; that is, the wavefronts of outgoing sound
will spread spherically outwards from the source.
7
Expressions representing the sound waves emitted from a single source can be
found via solutions to the acoustic wave equation:
∇2 p =
1 ∂ 2p
,
c2 ∂t2
(13)
where p represents the pressure of the wave, c is the speed of sound, and t is time.
Solving this differential equation yields the familiar sinusoidal or complex exponential equations for wave propagation, and we can represent the spherical wavefront
leaving a simple source – a two-dimensional cross section of which can be seen in
Figure 3 – with the pressure expression:
p0 (r, t) =
A i(ωt−kr)
e
r
(14)
where A is an initial amplitude, r is the radial distance from a source located at the
origin, and ω is the angular frequency of the wave [11]. As the field point r approaches
the origin of a source, the pressure at the point grows considerably. There exists some
minimum radius (as r will never be zero) at which the pressure will maximize, and
the constant A is calibrated such that the magnitude of the expression for pressure
in Equation 14 equals the measured pressure at that minimum radius.
Figure 3: Planar cross section of the radiation pattern of a single point source with
field point r. Semicircles represent crests of multiple wavefronts.
To consider the total pressure at a field point r due to multiple point sources, we
must have an understanding of how sound pulses interfere. Sound waves follow the
principle of superposition and demonstrate constructive and destructive interference,
where the total pressure at a given point is equal to the sum of all of the localized
8
component pressures. Additionally, the overarching “envelope” connecting the wavefronts of each individual source represents the overall pressure wavefront (see Figure
4), an acoustic analog to Huygens’ Principle on the wavefronts of light.
Figure 4: Acoustic radiation from a linear array of sources. Individual wavefronts
sum to a plane wave along this front.
Specifically, we consider an array of N sources aligned as in Figure 4, with the
leftmost source positioned over the origin. We index these sources from 0 to N − 1.
The pressure wave from the nth source will thus be
pn (r, t) =
An i(ωn t−kn rn )
e
,
rn
(15)
according to Equation 14, where An , rn , ωn , and kn are the nth source’s corresponding initial amplitude, distance to the field point in question, angular frequency, and
wavenumber.
However, we choose the amplitude and frequency to be the same for each source,
and so A, ω, and k will be identical for each array element and independent of
index. As discussed above, we are also assuming r = r0 to be in the far-field; that
is, r >> (N − 1)d, where d represents the distance between each individual source.
In other words, the distance to the field point is much larger than the total spatial
extent of the array.
One important result of this assumption is that, in terms of amplitude attenuation, the distance from each source to the field point can be taken as the same r.
Because the distance between the array and the field point is by assumption so large,
the difference in distance from one element (to the field point) and from a second element (also to the field point) will create a negligible difference between the sources’
attenuation, and so we disregard these small differences. It follows that the 1r factor
in Equation 15 is independent of index. However, it is key to note that the far-field
assumption does not allow us to equate the phases in the argument of the exponential.
Though d is very small and r is much larger, the wavelength of ultrasound is also
quite small, and thus the seemingly small differences between each rn will actually
significantly impact the phase of the individual waves at the point r. As such, the
sum of all the pressure waves at r will be
9
N −1
ptot (r, t) =
A iωt X −ikrn
e
.
e
r
n=0
(16)
Figure 5: Pathlength difference x between r0 and r1 . Due to the far-field assumption,
r1 = r0 −x and the angle θ measured from vertical is equal to the angle labeled above.
Considering the case in which all sources oscillate in phase, we can calculate the
difference between each rn using a geometric pathlength difference, as shown in Figure
5.
Since r >> (N − 1)d, we approximate that each rn is parallel to every other rm .
From this approximation, the angle θ represents both the offset from the normal to
the line of the array, and the angle labeled in Figure 5. It follows that the pathlength
difference between two adjacent source elements is x = d sin(θ). Since we have taken
the distance from the source at the origin to the field point to be r0 = r, by geometry
the nth source will have a distance from that field point of roughly
rn ≈ r − nd sin(θ).
Substituting this back into Equation 16 yields [11]
N −1
X
A
ptot (r, t) = ei(ωt−kr)
einkd sin(θ) ,
r
n=0
which can be simplified further to
ptot (r, t) =
A i(ωt−kr) eN ikd sin(θ) − 1
e
r
eikd sin(θ) − 1
10
(17)
using the summation identity
N
−1
X
n=0
xn =
xN − 1
.
x−1
(18)
Factoring from both the top and bottom of the fraction yields
" N
#
N
ikd sin(θ)
ikd sin(θ)
−N
ikd sin(θ)
2
2
2
A
e
e
−e
ptot (r, t) = ei(ωt−kr) 1 ikd sin(θ)
,
1
1
ikd
sin(θ)
r
e2
e2
− e− 2 ikd sin(θ)
and simplifying after substituting cosines and sines in place of the complex exponentials results in
sin( N2 kd sin(θ))
A
ptot (r, t) = ei(ωt−kr+δ)
,
(19)
r
sin( 21 kd sin(θ))
N −1
where δ = e 2 ikd sin(θ) is a global phase shift that is completely independent of index
n. Additionally, often the more meaningful property of the wave is its intensity,
which varies with |p|2 , and so the complex exponential will itself convert to a total
magnitude of 1. The phase shift can therefore, for many purposes, be ignored.
This result of the pressure summation is a bit unwieldy, and it is difficult to understand what it represents physically in the system. Therefore, instead of analyzing
directly this equation, we convert total pressure into a directivity function, H(θ), that
provides information about the relative strength of the sound with respect to θ, at a
given distance from the source array. We divide total pressure by the maximum value
of the total pressure, here where θ = 0, to find the normalized directivity function.
We must use L’Hopital’s Rule to avoid dividing by zero, and we ignore any extra
factors that come about as a result of δ to determine the directivity of this system to
be
1 sin( N2 kd sin(θ))
H(θ) =
.
(20)
N sin( 12 kd sin(θ))
Visual representations of the square of this function, |H(θ)|2 , with different varying
parameters, can be found in Figures 6 and 7. These plots display the magnitudes of
the constructive interference for a varying angle θ at a fixed distance away from the
source array. Intuitively, the greatest constructive interference occurs at an angular
offset of 0◦ , as the wavefront in this system propagates parallel to the normal to the
line array.
The lateral resolution of the emitted wavefront is related to the angular spread
of the peak of the directivity function. Using a narrower beam will allow for better
resolution over a smaller range of angles. The spread of the peak of H(θ) is dependent
on the number of sources in the array, N , and the wavelength, λ, of the ultrasound
in use.
11
Figure 6: Matlab plot of |H(θ)|2 with varying N . Curves with greatest to least
spread represent N = 5, N = 10, and N = 100, respectively. Corresponding FWHM
2π
m−1 and d = .001
measured to be 0.74, 0.36, and and 0.04. Fixed parameters k = .004
m.
Employing a greater number of sources allows for better “fine tuning” capabilities
in terms of narrowing the peak of the outgoing sound, and so we can see in Figure 6
that the spike of narrower width corresponds to the larger value of N . It is also the
case that the wavelength of outgoing sound itself limits how small of an object the
pulse can resolve. As shown in Figure 7, the shorter the wavelength, the narrower
the region of constructive interference resulting from the phased array will be.
Figure 7: Matlab plot of |H(θ)|2 with varying λ. Curves with greatest to least spread
represent λ = 0.03 m, λ = 0.01 m, and λ = 0.004 m, respectively. Corresponding
FWHM measured to be 0.92, 0.30, and and 0.12. Fixed parameters N = 30 and
d = .001 m.
12
Calculating the full width at half maximum (FWHM) for each of these peaks
through Matlab demonstrates the nature of the relationship between differences in
spread and differences in both λ and N . In the case of λ, the widths differ by a factor
nearly identical to the factor of difference between the parameter values; for example,
in transitioning from λ = 0.004 m to λ = 0.01 m, a scale factor of 3, the width
also scales up by a factor of 3, implying something of a linear relationship between
these values. In the case of varying finite N , the widths differ by a factor of one
over the factor of difference between the parameter values: when transitioning from
N = 5 to N = 10, a factor of 2, the FWHM value instead halves, implying an inverse
relationship between the lateral width and the number of sources.
2.5
Linearly Phased Arrays
Just as we derived the directivity function for an array without any sort of time delay,
we can also find the directivity function for a “phased array” with a linear time delay.
We call these arrays “phased” because the time delay is converted to a distance delay
which is included in the phase of the wave and thus affects the interference pattern.
Suppose we induce a time delay, τn , to the activation of each element. As the
sources are activated at different times, the wavefronts of each source will interfere,
and the radiation pattern of the sum will rely on the delays introduced. In order to
avoid the addition of an indexed time and instead continue working with an indexed
distance, we convert the time delay into a corresponding distance delay by multiplying
by the speed of sound to find that
cτn = δn .
(21)
This distance delay δn is then added on to rn for every n from 0 to N − 1, yielding
rn ≈ r − nd sin(θ) + δn ,
(22)
which conforms to the case given above when the time delay τn = 0.
Since we are considering a linear delay in this section, we define an element-toelement – or adjacent source – delay, τ , that will be independent of index. We take
0)
, where θ0 is the angle offset from θ = 0 to
this defined time delay to be τ = d sin(θ
c
which the array will steer the outgoing waves [11]. This time and distance delay is
also calculated geometrically, almost identically to the path length difference derived
in Section 2.4.
In order to apply this delay linearly to the array, each element will contain its own
unique multiple of τ , according to index, as shown in the schematic in Figure 8. It
13
Figure 8: Acoustic radiation pattern from a linear array of sources with an effective
linear time delay. Individual wavefronts sum to a plane wave with angle θ0 from the
vertical.
0)
follows that τn = nd sin(θ
, and so the corresponding distance delay is δn = nd sin(θ0 ).
c
Substituting our new rn into Equation 16 yields
N −1
X
A
einkd(sin(θ)−sin(θ0 )) .
ptot (r, t) = eiωt
r
n=0
We can again use Equation 18, and the derivation proceeds exactly as that for
zero time delay, but with an additional term of −kd sin(θ0 ). The resulting directivity
function H(θ), normalized to the maximum when θ = θ0 , is thus
1 sin N2 kd(sin(θ) − sin(θ0 ))
.
H(θ) =
(23)
N sin 12 kd(sin(θ) − sin(θ0 ))
For a given angle θ0 , we have derived the time delay necessary to enact steering
to that angle. The directivity functions for systems containing linear time delays are
pictured in Figures 9 and 10. Note the shift in angle offset of the maximum peak
in Figure 9 corresponds exactly to the value of θ0 used, π6 . Figure 10 represents the
directivity function in a polar plot, in which the length of the radius to the curve
along some angle represents the magnitude of the directivity function at that angle.
The maximum radial component in Figure 10 also shifts to 30◦ when the necessary
time delay is applied.
The spread of the peaks in the previous section depended on the number of sources
N and the wavelength λ of the sound being used, and the peak width also depends
on the angle θ0 to which the array is steering the outgoing wave. As the desired angle
gets larger, the spread of the peak also increases.
We can treat the cases of no delay and linear delay analytically, and it is possible
to derive expressions to describe the directivity function for each system. However,
if we are to introduce an arbitrary time delay we cannot use the same mathematical
tools to eliminate the sum and dependence on index. Thus, we will discuss the effects
of an arbitrary time delay computationally in Section 5.
14
2π
Figure 9: Matlab plot of |H(θ)|2 with N = 30, d = 0.001 m, and k = 0.004
m−1 . Left
peak created when θ0 = 0, and peaks shifted right created when θ0 = π6 and θ0 = π3 .
Corresponding FWHM measured in Matlab to be 0.12, 0.14, and 0.25.
Figure 10: Polar representation of the directivity function in Figure 9. Radial length
to the curve along some angle represents the magnitude of the directivity function at
that angle. (Left) Directivity for θ0 = 0. (Right) Directivity for θ0 = 30◦ .
3
Experimental Methods
To create and observe demonstrations of the principles of ultrasound imaging, the
most crucial tool is the transceiver shown in Figure 12. We used the Parsonics
Corporation 4012 Series transceivers with teflon housing. The transceivers are rated
to work best at an operating frequency of 40 kHz ±4%. As shown in Figure 12,
we attached our transceivers to custom aluminum mounts that could be attached to
Cenco Physics optical mounts. These mounts pivoted on an axis perpendicular to
the optical bench with a needle attached to the pivoting portion. This guiding needle
15
was directly above a stable protractor. This allowed us to point our transceivers’
faces down the length of the track precisely. The mounts also allowed us to offset
our transceiver faces accurately to a chosen degree value. From now on, we will
refer to a 0◦ offset as the position where the needle is pointing to the 90◦ mark on
the protractor. We feel this is more intuitive since a 0◦ offset on both transceivers
means they are facing down the axis of the optical bench towards one another. The
convention described can be seen in Figure 11.
Figure 11: Convention for angle offsets for our experiments.
Figure 12: Transceivers with custom mounts on optical bench.
To generate a driving voltage for our transceivers, we used an Agilent 22330A wave
form generator with the following settings: a pulsed sine wave with an amplitude of
5 V Pk-Pk, a period of 25.641 µs, and a frequency of 39 kHz. To find the optimal
frequency of our transceivers, we faced them toward each other and sent a signal into
one. We then scanned the frequency while keeping the driving voltage the same and
noting where received voltage was the highest. We found our particular transceivers
operated optimally at a frequency of 39 kHz. We also used the function generator
to generate bursts or pulses, and we had the burst setting at 6 cycles (periods) per
burst, a start phase of 0◦ and a burst period of 20ms. To sync the oscilloscope to the
16
generated pulse we triggered the oscilloscope using the sync output of the function
generator. In all cases we kept the sync setting on the function generator set to use
the internal sync, not manual.
We were not operating any single transceiver as a transmitter and receiver at the
same time, as is done with medical equipment. Instead we used one as a transmitter
and another as a receiver. In conjunction with this setup we utilized both channels on
our Tektronix TDS 1012B oscilloscope. One channel was used as a trigger channel and
was connected to the sync output of our function generator. The other channel was
connected to the transceiver that was operating as the receiver for our experiments.
All of the electrical components were connected with BNC cables. Our oscilloscope
had a USB interface which, in conjunction with a memory stick, is how we extracted
data and images.
Figure 13: Tektronix TDS 1012B oscilloscope.
4
4.1
Demonstrations and Results
Speed of Sound Demonstration
One of the simplest demonstrations one can use to demonstrate the properties of
sound is to use our system to display how the speed of sound changes when propagating through different media. Using measured distances between the transceivers, we
were able to demonstrate not only the difference in the propagation speed of sound
through different media, but also the impact of acoustic impedance differences on the
transmission and reception of sound at interfaces.
This demo clearly demonstrates how one can obtain a speed of sound for the
transducers through different media. The hardware and settings of the function generator were prepared as described above. Both transceivers were set at 0◦ (facing each
17
other). While the optical track allows for accurate distance measurement, it should
be noted that the actual separation of the transceivers will be less than the indicated
distance since the piezoelectric elements in the transceivers are not centered directly
over the center of the optical mounts. The speed of sound can be calculated using the
time cursors on the oscilloscope to determine the time delay of the pulse and received
signal; with that time delay, along with the transceiver separation, we calculate the
speed of sound. Once the speed of sound is determined, Equation 1 can be used to
calculate the wavelength of the ultrasonic pulse.
This demo can also be used to show the variability of the speed of sound in different
media. Additional materials required for this portion of the demonstration are indexmatching ultrasound gel and a solid cylinder of material. We used plexiglass as our
new medium. The length of the material should be measured and the transceivers
placed at a separation of that distance. The cursors can be used at this separation
to determine the time delay of the received signal. First insert the cylinder between
the transceivers with no ultrasound gel applied. Since the interface between air and
plexiglass borders materials with very different acoustic impedances, essentially no
received signal will be seen. This step is important to the students understanding
how much of an effect the difference in acoustic impedance has on transmission of
sound. Seeing this also serves to amplify the effect that the gel has when applied to
the transducers. Next we apply index-matching gel to the ends of the cylinder, and
the cylinder is placed back in between the transceivers. Now that the air to solid
interface has been eliminated, a notable increase in the received voltage should be
seen on the oscilloscope, as well as a shorter time delay between the pulse and the
received signal (than when using air as a medium).
As seen in Figure 14, the time delays for the pulses to travel the same distance,
26 cm, through air and plexiglass were quite different. We were able to calculate
a speed of 325 m/s through air and a speed of 1238 m/s through the plexiglass.
The accepted value for the speed of sound through air at standard temperature and
pressure is roughly 340 m/s, and so our calculated speed appeared to be low; however,
the piezoelectric crystals are not located at the face of the transceivers. In fact, we
know there to be a width of matching layer between the crystal and the face, and so
the distance with which we calculated the speed was too small. Similarly, the speed
we calculated through the plexiglass was also too slow; the accepted speed of sound
through Lucite is 1800 m/s [12].
In order to calculate a closer estimate for the speeds, we manipulated the velocity
equation to solve instead for the distance between the transducers, using the time difference measured and the accepted speed of sound through air. Through this process,
18
we found a distance 12 mm longer than that which we measured, implying that the
piezoelectric materials must be roughly 6 mm beneath the surface of the transceivers.
This is plausible, given their width of 4 cm. Armed with this information, we were
able to substitute the new distance in to solve for the speed of the pulse through glass,
and found that to be about 1295 m/s, which is significantly larger than our original
value and strikingly larger than the speed of sound in air, as we desired to show.
Figure 14: Screenshots of oscilloscope data. The signal from Channel 2 (top signal) represents the signal received directly from the function generator, while the
sound was received by Channel 1 (bottom received signal) and traveled through the
transceivers. Note the difference in voltage scale between the two images. (Left) At
a separation distance of 26 cm in air, we used the cursors to measure the time delay
between when the function generator released the pulse to when it was first detected
by the receiver: ∆t = 800 µs. (Right) Again, at a separation distance of 26 cm, the
time delay through the plexiglass cylinder was measured to be ∆t = 210 µs.
Taking 39 kHz to be the driving frequency of the pulse and substituting our
calculated speeds into the equation, we found the wavelengths to be roughly 8.7 mm
and 33.2 mm for the sound in air and in plexiglass, respectively.
It was also apparent from these images that the voltages, and thus intensities, of
the transmitted and reflected components of the pulse depended very much on the
matching – or lack thereof – of acoustic impedances at interfaces. As seen in Figure
14, there is a much larger signal received through the plexiglass than through air.
Because the interface between air and Teflon is a very hard boundary – the difference in acoustic impedances of air and a solid is very high – a fraction of the incoming
pulse is not transmitted through to the piezoelectric material, but reflected from the
surface of the transducer face and again from the receiver face. As such the received
intensity is relatively low, around 13.6 mV at its highest.
19
When we initially placed just the plexiglass cylinder in between the two transducers, we again experienced difficulties due to the harsh interface. In fact, the signal
essentially disappeared because instead of just one of these interfaces, the pulse was
required to travel through four of them – two air-transceiver and two plexiglass-air interfaces – in order to reach the receiver, resulting in a negligible transmitted intensity.
However, after applying Spectra 360 Electrode Gel from Parker to each end of the
cylinder, creating a smoother transition from transducer to glass and back again, the
received signal increased more than tenfold. With the gel in place, our received signal
was instead 296 mV, demonstrating two effects of the glass cylinder: the presence of
the glass likely acted as a focusing agent and decreased beamspreading, and the presence of the gel created interfaces with much lower differences in acoustic impedances
and allowing for greater transmission through these interfaces.
4.2
Pulse-Echo Demonstration
The following demonstration showed the remote sensing capabilities of ultrasound
transceivers. Much like in the speed of sound and wavelength demo, we used the time
delay to calculate a distance between objects. However, for this demo, we needed a
second optical bench and an additional hard metal interface. For the interface we used
an aluminum plate affixed to a stand designed for the optical bench. We assumed the
effective Zal of the aluminum to be much larger than the effective Zai . Referencing
Equation 9 we can see that:
2
Zal
(Zai − Zal )2
Ir
≈
= 1.
=
2
I0
(Zai + Zal )2
Zal
The two optical benches were placed in a “T” shape as shown in Figures 15 and 16.
The plate was then placed on the spine of the T, 22 cm from the center of the optical
bench that held the transceivers. The two receivers were placed equidistant from the
intersection of the optical benches at the top of the T with a separation distance of
26 cm. We then positioned our transceivers at angular offsets of +75◦ and −75◦ . We
once again observed a time delay between the output signal and the received pulse.
We could then calculate the distance from the face of one transceiver to the plate
using this time delay and the speed of sound in air. Note that this distance needed
to be halved since we actually calculated the distance from the transducer face to the
plate, and then from the plate to the receiver face. This value could be compared
to a theoretical distance using the value of x2 (half the separation of the transducers)
and y (the distance of the plate away from the T junction) shown in Figure 16.
We again assumed the speed of sound to be 340 m/s, and we measured a time
delay of ∆t =1.310 ms. Using Equation 11 we could calculate the distance of the
20
Figure 15: Transceivers with metal plate.
Figure 16: Diagram of the Pulse Echo Demonstration.
object from the transducer:
1
d = (340 m/s)(0.00131 s) = 0.223 m.
2
We could also easily find the distance of half of the path length covered by the
sound, as seen in Figure 16 using the Pythagorean Theorem. We first took half our
x separation to be 7 cm and then estimated the y distance to be 21 cm. With these
two values, we could calculate the approximate distance from the transducer to the
plate along the path traveled by the sound using the Pythagorean theorem:
p
d = (7 cm)2 + (21 cm)2 = 22.1 cm,
which is comparable to the above value.
21
4.3
Wind Speed Demonstration
While there seem to be many similarities between light and sound, especially in the
reflection or echo of sound off of a surface, an important distinction to make between
sound and light is that sound requires a medium to propagate. One thing that
greatly affects the propagation of sound is properties of the medium, as seen with the
speed of sound in Subsection 4.1. In this demonstration, we investigated how sound
propagation changes in a moving medium.
We positioned the transceivers and electrical equipment, at least initially, the
same way as in the speed of sound demo. Some additional equipment was required to
produce effective wind. These materials included a fan, a polyethylene garbage bag,
and a length of PVC piping. Ours was 40 cm long with a 10 cm diameter opening.
The polyethylene bag was firmly attached to the fan and an opening was made in
the bottom of the bag where the PVC was also attached such that the wind was
collimated through the PVC.
Figure 17: Schematic diagram of the wind speed demonstration.
For this demonstration, we first measured the time delay with the fan turned off,
without any wind. We then turned the fan on to its highest setting and directed it at
a shallow angle across the path of the sound propagation, the general idea of which
can be seen in Figure 17. When using the wind we made sure not to position the air
stream directly on or around the transceiver face to avoid causing any unnecessary
turbulence. While holding the PVC steady, we made note of the new time delay.
22
The procedure described can be seen in Figure 18. To measure the speed of sound
propagating against the windspeed, we merely switched the positions of the transducer
and receiver and found the delay to be of the same magnitude, but in the opposite
direction of the trial in which the sound traveled with the wind.
Figure 18: An overhead shot of the wind speed demo. The PVC pipe in the upper
left is what directs the airflow.
When we performed the experiment we were able to see that the time delay of the
sound pulse at a 60 cm separation was 1.82 ms without the fan applied. Using this
time delay and separation, we found our calculated speed of sound to be
vs =
0.6 m
= 329.67 m/s.
0.00182 s
Then, when we turned the fan on, this caused a shift in the time of the received sound
pulse to be 1.81 ms. We found our new sound velocity of sound to be
vw =
0.6 m
= 331.49 m/s.
0.00181 s
We could then calculate the velocity contribution to the sound due to wind. We called
this velocity contribution component va
va = vw − vs
= 331.49 m/s − 329.67 m/s
= 1.82 m/s.
23
Above is our calculation for the velocity contribution component due to the average wind speed across the field of sound propagation. To check this value, we used
an anemometer to measure wind speed values in the direction of sound propagation
along the optical track on which the transceivers were mounted. The results of these
measurements can be seen below in Figure 19. When we found the average value
Figure 19: Wind speed data measured with anemometer.
of our data set, we discovered that the value of the wind velocity component was
vt = 2.3 m/s. We concluded that this value was a reasonable confirmation of our
theoretical value va . Since our anemometer measured wind speed with a fan located
inside a bulky housing, this created uncertainty in the accuracy of our anemometer measurements. We felt that the discrepancy between our calculated va and the
measured average wind speed was allowable within this uncertainty.
4.4
Intensity Mapping
The ultrasound imaging process requires an understanding of how the transceivers
operate. Something that may prove particularly useful is the spatial output of sound
intensity from the transceivers. In an attempt to better acquaint ourselves with the
Parsonics transceivers, we collected data to create a spatial sound intensity map for
our equipment.
We used the function generator settings described above with a pulse length of
156 µs and pulse separation of 20 ms to ensure that there would be no interference
24
between consecutive pulses. Positioning the faces of the transceivers at a 0◦ offset,
we kept the transducer stationary while stepping the receiver back in 1 cm intervals.
We started to measure at a separation of 10 cm to avoid interference of the pulse
with its own reflections, formed by the high difference in acoustic impedance between
air and the transceiver faces, and we stopped measuring at a separation distance of
60 cm. Once we completed the measurements for the 0◦ offset, we then moved our
transducer to offsets of 2.5◦ , 5◦ , 7.5◦ , and 10◦ and repeated our measurements. We did
not explore larger angles as the voltages measured at 10◦ fell off to levels equivalent
to background signal after just a few steps back. This set up is shown in Figure 20.
Figure 20: Method of data acquisition for intensity map.
The data we took was in the form of the Pk-Pk voltage displayed by the oscilloscope. It would be more meaningful for us to record the data as a RMS (root mean
squared) voltage. Taking data with RMS values will allow us to complete our dB
calculations. We can convert our data to RMS simply by dividing our data set by
√
2. Our results for some select angle offsets are shown below in Figure 21.
Figure 21: Graph showing amplitude as a function of distance for 0◦ , 5◦ and 10◦ off
axis.
Using the above data and Equation 7, we can create a two dimensional decibel
intensity map. Since the voltage signal is generated by the pressure deformation in
the piezoelectric crystal, we take the RMS voltage readings to be proportional to the
25
incident sound pressure wave. Thus, we substitute voltage in for pressure in Equation
7 to find the conversion:
dB = 20 log10
V
V0
,
(24)
where V is the recorded RMS voltage and V0 is the reference voltage.
We chose our highest voltage reading to be the reference voltage required for
Equation 24. This means that our dB scale values range from 0 to the negative
numbers. Using Matlab, we plotted our results as shown in Figure 22. It is important
to note that this plot appears to be perfectly symmetric. This is because we assumed
our transceivers to be approximately symmetric, and mirrored our measurements
from just one side to generate the final figure.
Figure 22: Decibel map in relative dB units.
We are able to gain a better intuition of how our transceivers work from the
decibel map. Our understanding of how the sound waves attenuate is also made
clearer when we take into consideration Figure 23, which shows the data from Figure
21 plotted with the log function from Equation 24. This graph shows the same dB
values that were used in our intensity map, but we can clearly see in Figure 23 that
the decrease in dB intensity is linear with propagation distance. This indicates that
our original RMS voltage data experiences an exponential decrease, which confirms
our hypothesis that the majority of the sound loss is due to attenuation through air
26
and not through beam spreading. The 10◦ data in Figure 23 does not appear to be
linear throughout the entire decay, but this can be attributed to attenuation and the
fact that the signal at the end is entirely background.
Figure 23: Decibel intensity level from RMS voltage data.
5
Phased Array Simulations and Results
We spent most of Sections 2.4 and 2.5 in the Theory portion discussing and deriving
the directivity function for a variety of arrays. We examined H(θ) for an array with
no delay, and an array with a linear time delay where the nth element had a delay of
nτ , and τ was determined independently of index. Because of geometric properties
of the sums in the derivations for both of these cases, our results were relatively easy
to handle analytically.
When we introduce an arbitrary time (or distance) delay into the system, perhaps
something like in Figure 24, the helpful properties and equalities from past sections,
in particular Equation 18, are no longer useful. As such, in place of analytical exploration, we instead developed a numerical summation to be entered into Matlab, and
we examined several different types of delays computationally.
The results presented here are parabolas of differing coefficients and a “step”
function where the left half of the elements are activated at one time, and the right
half are activated some time later, but for the purposes of the derivation, we consider
the nth element in the array to have some general time delay τn .
27
Figure 24: Acoustic radiation pattern from array with arbitrary delay. Individual
wavefronts sum to an envelope wavefront leading out from the array.
We substitute the corresponding distance delay into Equation 16 to find that the
total pressure at field point r resulting from the array as a whole is
N −1
ptot (r, t) =
A iωt X −ikrn
e
e
r
n=0
(25)
N −1
=
A i(ωt−kr) X inkd sin(θ)−ikδn
e
e
,
r
n=0
(26)
where δn = cτn is the total distance delay with respect to the first element located at
the origin (n = 0).
We defined our H(θ) for the array with no delay to be the total pressure divided by
the total pressure along the θ = 0 axis, and though this will not necessarily normalize
the directivity function in this case, we follow the same procedure in order to cancel
out the factor Ar ei(ωt−kr) . Thus,
PN −1
H(θ) =
n=0
einkd sin(θ)−ikδn
PN −1
n=0
e−ikδn
.
(27)
For the purposes of plotting, Matlab also allows us to divide this whole function
by its maximum in order to normalize it to 1.
Shown in Figures 25 and 26 are the directivity functions for a step and a parabolic
delay, respectively. Each trial was run with N = 30, d = 0.001 m, and λ = 0.004 m.
We can see that constructive and destructive interference relies upon the relationship between the time delay and the parameter values, and that even delays of
the same type can yield differently shaped directivities depending on the relationship
between coefficients of the time delay and the inherent physical parameters of the
system.
Since kδn = kcτn , in order to introduce a step function with difference in phase
π
of π, we must introduce a time delay of τ = kc
. The second half of the sources are
activated a time τ after the first. As seen in Figure 25, this phase delay causes the
peak of the directivity function to split into two. When inducing a 2π-phase delay
28
Figure 25: |H(θ)|2 for step function with N = 30, d = 0.001 m, and λ = 0.004 m.
Second half of the array elements activated at a time mπ
after the first, where m ∈ R.
kc
When m = 1, we induce a π-phase delay (darker curve). When m = 2, we induce a
2π-phase delay, which has no effect in the far field (lighter curve).
we use τ = 2π
, and because we are operating under far-field assumptions, our model
kc
displays no effects from this phase delay, as expected. Were we considering the near
field, there would exist some transient behavior before the second half of the sources
were activated, but after some time the pulses would once again propagate in phase,
as though there were no delay.
N 2
Figure 26: |H(θ)|2 for parabolic time delay of τn = kc(mπ
with N = 30,
N 2 (n − 2 )
)
2
d = 0.001 m, and λ = 0.004 m. Furthest outside to furthest inside curves correspond
to m = 1, m = 0.75, m = 0.5, and m = 0.25, respectively.
29
In order to enact a parabolic delay, we set τn =
mπ
(n
kc( N
)2
2
−
N 2
).
2
In the limiting
2
case as m approaches zero and the time delay grows smaller, |H(θ)| tends towards
the directivity curve representative of the total pressure of an array with no induced
time delays, as physically expected.
6
Conclusions and Further Research
A large part of this project was intended to create a series of demonstrations for
students to gain a better grasp of the physics of ultrasound. Through these demonstrations, students will be able to observe these processes and properties right before
their eyes, as opposed to through the completion of numerical calculations – which
are not always very intuitive.
We hope that these demonstrations will be used for years to come in the aid
of students learning about the physics of ultrasound imaging. There are further
areas of ultrasound imaging to explore should other students wish to take up the
task. Demonstrations in ultrasound lateral resolution would prove helpful in students’
understanding of ultrasound imaging, as well as demonstrations in Doppler ultrasound
imaging.
Similarly, there are a great many new directions in which the study of phased
arrays could be taken. While we have theoretically derived the directivity for two
particular cases of time delays and have shown several examples of results for arbitrary
delays, further examination into the relationship between the directivity and the
parameter values could prove quite fruitful.
30
A
A.1
Demonstrations User’s Manual
Introduction
This series of demos is designed to display to students the basic physical principles
that are at work in medical ultrasound imaging. Ultrasound imaging techniques
utilize several inherent properties of sound to produce an image. Medical ultrasound
is one of the most ubiquitous and cheapest forms of imaging for the body, and as such
it is important to understand the physical properties behind the technique.
A.2
Theory
A basic knowledge of the theory behind ultrasound properties is essential to understanding key concepts in the demonstrations. The first principle to understand is the
relation of the sound wavelength to its velocity and frequency: v = λf . This equation is all that is needed for the speed of sound demonstration. For the pulse-echo
demonstration – which shows the remote sensing capability of ultrasound – we use a
distance calculation equation given as d = λf ∆t. For the pulse-echo demo, we also
(Z1 −Z2 )2
IR
make use of the relation IIR0 = (Z
is the ratio of reflected
2 , where the ratio I
1 +Z2 )
0
sound intensity to incident sound intensity. When utilizing this equation to describe
an interface between media with drastically different acoustic impedances Z1 and Z2 ,
we see that effectively 100% of the intensity is reflected.
As a tool to help the students better understand ultrasound transducer functionality, we plotted the sound intensity radiation pattern of our transceivers over a 10◦
offset to either side. To complete this demo, one will need to use a modified version
of the equation to convert measured voltages to the decibel scale, dB = 20 log10 VV0 ,
where V is a root mean squared voltage value calculated from the peak-to-peak voltage measurements on the oscilloscope, and V0 is the reference (maximum) voltage
measured.
A.3
Equipment
Some of the required materials that we have used for our demonstrations are pictured
in Figure 27, including two Parsonics Corporation 4012 Series ultrasound transceivers
(only one of which is shown in Figure 27), a hard interface for the pulse-echo experiment, and the optical track upon which all of our demos are mounted. Note that our
custom mounts were made in the Dickinson College metal shop and fit perfectly the
optical mounts. They each were fitted with a protractor as well, in order to measure
offset angles, and the mount for the aluminum plate was also made in the shop on
31
campus. To generate the sound pulses for our transducers, we used an Agilent 22330A
function pulse generator. To measure and collect readings from our transceivers, we
used a Tektronix TDS 1012B oscilloscope. All of the necessary electrical components
were connected with BNC cables.
Figure 27: Transducer, optical track and hard interface.
A.4
Experimental Procedure
For each of the experiments you will need the same basic configuration of the equipment detailed above. First, connect the Agilent 22330A Waveform Generator directly
to the Tektronix TDS 1012B Oscilloscope using a BNC cable. Make this connection
from the “Sync” channel on the waveform generator to Channel 2 on the oscilloscope,
inserting a 50 Ω coaxial impedance matching connector into the oscilloscope channel.
Connect the “Output” channel on the waveform generator to the transceiver you are
planning to use as the source transducer, making sure to use a BNC cable with breakout wires connected to test clips (we used Pomona “Minigrabbers”) and to connect
the ground cable to the thicker of the two wires located at the end of the transceiver.
Next connect the oscilloscope’s Channel 1 – through another coaxial connector and
BNC cable – to the transceiver you plan to use as the receiver, once again connecting
the ground clip to the thicker wire at the end of the transceiver cable. Electronically,
this is all you need.
32
You will also use the same waveform generator settings for each experiment, as
they are listed here:
• Sine wave (not Square, Ramp, etc.)
• Amplitude of 5 V
• Frequency of 39 kHz
• Burst on
• 6 cycles per burst
• Start phase of 0◦
• Burst period of 20 ms
• Trigger internal sync (keeping the actual “Trigger” button off)
• Output on
Use a USB drive or memory stick to take pictures and data directly from the
oscilloscope, using the “Print” button to save the information needed.
Attach the transducers to mounts of equal height with the capability of resting on
an optical track. It would also be useful for demos relying on varying angles to attach
a protractor and a needle to each mount to make easier the determination of angular
position of each transceiver; position the protractor such that the needle (and thus
face of the transceiver) rests above the 90◦ mark when it is facing parallel down the
optical bench.
A.4.1
Speed of Sound Demonstration
1. Prepare as above.
2. Extra materials include ultrasound gel and a plexiglass or other material cylinder (we used one 26 cm in length and with a diameter of roughly 7 cm).
3. Set both transceivers at angular positions of 90◦ (0◦ offset).
4. Place the transceivers at a separation distance equal to the length of the cylinder.
5. Measure the maximum Pk-Pk voltage received for the sake of later comparison.
33
6. Measure the resulting time delay due to propagation through air on the oscilloscope using the vertical cursors, placing the first at the first side of the trigger
pulse and the second at the earliest sign of the received ultrasound pulse signal.
7. Complete calculations for velocity using this time delay, and compare to the
accepted value for speed of sound. Be sure to remind students that the piezoelectric crystals are not located immediately behind the faces of the transceivers.
8. Using the accepted speed of sound and measured time delay, calculate the actual
separation distance between the piezoelectric crystals.
9. Repeat this process, but with the plexiglass cylinder held between the transceivers.
Note that the signal all but disappears due to impedance mismatch.
10. Repeat this process with a layer of gel spread between the cylinder and the face
of each transceiver.
11. Once again measure the resulting time delay to determine the speed of sound
through glass (making sure to add the additional distance from transceiver face
to piezoelectric material).
12. Measure the maximum Pk-Pk voltage received.
13. Compare speeds and maximum voltages.
14. Discuss with the class the following: what properties of the cylinder and air
result in such differences in propagation speed, and the high probability of the
presence of a kind of “ultrasound fiber optic” effect occurring via total internal
reflection within the cylinder.
A.4.2
Pulse-Echo Demonstration
1. Prepare as above.
2. Additional materials include a second optical track and an aluminum plate with
a stand to be mounted on the optical bench.
3. Place the optical tracks perpendicular to one another, in a “T.”
4. Position the aluminum plate a measured distance down the spine of the T (we
set it 22 cm from the tracks’ intersection).
34
5. Place each transceiver a given distance from the center of the T on either side
of the spine, with angular offsets that are equal in magnitude and direct the
propagation path from the transceiver, to the plate, and then to the receiver.
Our transducers were each 7 cm from the center of the T and had offsets of
+75◦ and −75◦ , angled in towards one another.
6. Measure the time delay using vertical cursors on the oscilloscope. Using the
accepted speed of sound, calculate the distance traveled by the pulse.
7. Using the Pythagorean theorem, calculate the distance traveled by the pulse,
dictated by geometry.
8. Note that in the case where the transducer and receiver are the same instrument,
the distance between the instruments imaging and the object being imaged is
roughly half of this calculated distance.
9. Discuss with the class the following: once again the probable location of the
the piezoelectric material within each transceiver, and how repetition rate when
imaging with multiple pulses and one single transceiver (as is often the case in
medical applications) is key; the time between pulses must be, at the very least,
slightly longer than the time it takes the initial pulse sent out to return to the
receiver in order to avoid undesired interference.
A.4.3
Wind Speed Demonstration
1. Prepare as above.
2. Additional materials include a wind anemometer, a fan, and materials with
which to collimate the outgoing wind. We secured a large polyethylene bag to
the outer rim of the fan, and, attaching a PVC tube with length 40 cm and
diameter 10 cm, created a funnel to align the airflow.
3. Place the transducers a given distance apart, and keep them at this distance
for the rest of the demonstration.
4. Stabilize the fan and cone to ensure the airflow is consistent through all parts
of the demo. Do not place the fan directly behind one of the transducers, as
this will create unwanted additional turbulence. Instead, position the piping
such that the stream of wind will cross the sound propagation path at a shallow
angle while avoiding crossing the transducers directly.
35
5. Use the time delay without any airflow present to measure the speed of sound
in (approximately) stationary air.
6. Turn the fan on such that the airflow is in the direction of the sound propagation.
Measure the time delay and calculate the speed of sound with wind. Note that
the time delay on the oscilloscope screen shifts left.
7. Switch the positions of the transceiver and receiver to examine the effects of
the airflow acting against the sound propagation direction. Again, measure the
time delay, note that it shifts towards the right, and calculate the velocity of
sound propagating “upstream” against the wind.
8. Calculate the difference in the velocities calculated with and without wind
speed.
9. Next, using an anemometer, take measurements of wind speed from the face of
one transducer to the other, in intervals of 5 or 10 cm.
10. Average the values obtained and compare the result to the difference in velocities
calculated in (8).
11. Discuss with the class the following: why the velocity taken from the average
wind speed across the propagation path and the velocity calculated from the
difference in time delays may differ (mention turbulence around the anemometer
if this is significant), and that we do not see a Doppler shift in frequency as a
result of the flowing medium (with no moving source or observer/object).
A.4.4
Intensity Mapping
1. Prepare as above.
2. No additional materials required.
3. Position both transceivers at 0◦ offset.
4. Place the source transducer at 0 cm and the receiver at a distance of 10 cm
from the transducer on the optical bench.
5. Measure the Pk-Pk voltage and record, ensuring the scale allows the entire
height of the pulse to be read on the oscilloscope screen.
6. Walk the receiver back along the track in increments of 1 cm, measuring and
recording the voltage at each separation distance, until 60 cm separation.
36
7. Repeat this process with the transducer set to four more angles: offsets of 2.5◦ ,
5◦ , 7.5◦ , and 10◦ .
8. Convert the Pk-Pk voltage measurements to root mean squared voltages by
multiplying by √12 .
9. Plot these voltages vs. separation distance to visualize the exponential decrease
due to attenuation.
10. Use the decibel equation located in the Theory section of this manual to convert
each measurement to a sound decibel reading.
11. Use Matlab (or another similar program) to create a plot representing the radiation field of the transceiver, utilizing mirroring to construct the negative angles
as well.
12. Discuss with the class the following: the transceivers are relatively collimated
and for that reason attenuation due to beam spreading (usually an inverse
radial dependence for pressure) is negligible, and why the assumption that the
measured voltage is directly proportional to pressure is valid.
References
[1] J. Cameron and J. G. Skofronick, Medical Physics, John Wiley & Sons, New
York, Chester, Brisbane, Toronto, 1978.
[2] X. Zhu, Piezoelectric Materials: Structure, Properties and Applications, Nova
Science Publishers Inc., New York, NY, 2010.
[3] P. L. Carson and A. Fenster, Medical Physics (36), 411 (2009).
[4] E. M. L. Jr, J. A. Seibert, J. T. Bushberg, and J. M. Boone, The Essential
Physics of Medical Imaging, Williams & Wilkins, Philadelphia, 1993.
[5] M. Patterson, Ultrasound Imaging Lecture Notes,
http://www.science.mcmaster.ca/medphys/images/files/courses/4T03/4T3
ultrasound.pdf, September 29, 2014.
[6] L. George and J. Lai, AACA Pre-conference Workshop perioperative ultrasound,
http://www.bats.ac.nz/resources/pdfs/physics.pdf, September 29, 2014.
[7] T. A. Stiles, American Journal of Physics (82), 490 (2014).
37
[8] M. Se-yuen, Physics Education (38), 441 (2003).
[9] T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, Elsevier Inc., New
York, NY, 2014.
[10] S. A. Kane, Introduction to Physics in Modern Medicine, Taylor & Francis
Group, Boca Raton, Florida, 2009.
[11] B. E. Anderson, B. Moser, and K. L. Gee, Journal of the Acoustical Society of
America (131), 2394 (2012).
[12] D. R. Raichel, The Science and Applications of Acoustics,
ence+Business Media, Inc., New York, NY, 2006.
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Springer Sci-