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McKeighen 99

6/7/1999 4:55:00 PM
Design Guidelines for Medical Ultrasonic Arrays
( SPIE International Symposium on Medical Imaging, Feb. 25, 1998, San Diego, CA )
(Invited Paper / Keynote Presentation)
Ronald E McKeighen, Ph.D.
Acoustic Imaging Transducers Inc.
10027 South 51 St. Street, Phoenix, AZ 85044
The basics of ultrasonic transducer array design in the frequency range useful for medical imaging ( 1- 10 MHz)
are discussed. Performance parameters of importance in transducer design are considered, including sensitivity,
coupling constant, band width, frequency downshift, pulse duration, beam focusing properties, and electrical
matching. Two and three dimensional effects must also be taken into account. The advantages of computer
modeling in three dimensions with finite element analysis code are highlighted. The principles of multi-element
array transducers useful for two dimensional real-time scanning are reviewed. Two dimensional arrays provide
the opportunity of focusing the elevation beam dimension in the short axis, or make possible full three
dimensional scanning volumes.
Keywords: ultrasonic arrays, transducer, broadband, beamforming, system, two dimensional arrays
1. Introduction / Basic Principles
A one dimensional array, when coupled with a beamformer, allows broad band ultrasonic pulses to be generated
and swept through a body for medical diagnostic imaging. The design of the array, including number of elements,
pitch, and frequency, are arrived at with the particular clinical application or target organ in mind. Image
resolution increases with frequency, however attenuation of energy by body tissue at the rate of 0.5 dB to 0.7 dB /
cm / MHz dictates a compromise in final choice of nominal transducer frequency.
The basic construction of a one dimensional array is illustrated in figure 1. Illustrated is the crossection of three
major array elements, with each major element made up of two sub-diced elements, to maintain desirable aspect
ratios. The basic principles(1-3) involved in constructing a transducer to generate a short pressure pulse can be
understood by considering a simple diagram illustrating the four basic pressure pulses (figure 2) generated when a
piezoelectric ceramic is shock excited by an impulse voltage of approximately 100 V. At each of the major
surfaces of the ceramic, two pressure waves are generated traveling in opposite directions, one a positive pressure
wave, and one a negative (rarefaction) pressure wave . If the ceramic is surrounded by air or water, most of the
wave amplitude will be reflected at the boundaries of the ceramic, causing the pulse to "ring" for a very long time.
The fraction of the pulse amplitude reflected at each interface is proportional to the difference in acoustic
impedance between the two materials on either side of the boundary. One way to make a broad band ( short pulse)
transducer is to attach a backing material to the back side of the ceramic with an acoustic impedance close to that
of the ceramic. Then the internal reverberations of the ceramic are eliminated , allowing a very short pulse to be
Second Matching Layer
First Matching Layer
Front Matching Layer
Figure 1. Sub-diced Linear Array
Figure 2. Four basic pressure waves generated by voltage impulse.
(+) Compression wave; (-) Rarefaction wave
The acoustic impedance is defined as the product of the density of the material and the longitudinal sound
velocity. Non destructive testing transducers (NDT) generate very short pulses by this principle of attaching a
very dense backing material(4,14) . The backing material chosen should ideally not only closely match the
ceramic acoustic impedance but also be very absorptive or "lossy" so that no energy is left to be reflected from the
non ceramic end of the backing and create an artifact echo. Although adequate for NDT, this ceramic construction
does not have enough sensitivity for medical applications, since so much of the energy applied in creating the
pulse is lost into the backing. The acoustic impedance of PZT ceramic is about 33.0 ( units of 106 Kg/M2/sec or
mega Rayl). The acoustic intensity in a material may be expressed by equation 1, where I is the intensity in
watts/cm2, Z is the acoustic impedance, u is the particle velocity, and P is the pressure:
I = 1/2 (Zu2) = 1/2 (P2/Z)
As an example, assume that the backing impedance is 15.0 , and that the front side of the ceramic is coupled into
water with an impedance of 1.5 . It can be seen that 10 times more power goes into the backing than goes forward
into the water. To increase the relative amount of energy transmitted into the front medium, one need a means to
make the water or tissue in front of the ceramic (the effective load) appear to have a high impedance which more
closely matches that of the crystal. The situation is analogous to an electrical circuit, where maximum power is
transferred from a generator to a load when the electrical impedance of the load equals that of the generator. The
mechanism for doing this is the impedance transformer. The basic physics is borrowed from transmission line
theory, where n stages of transmission line sections of varying impedances, each a quarter wavelength long, are
used to match the generator to a load. In the acoustic analog, the concept of acoustic impedance is used, and the
transmission line sections become matching layers of intermediate impedance values between those of ceramic
and water or body tissue, and of thickness dimensions nominally one-quarter of a wavelength at the center
frequency. Thus, for medical ultrasonic applications, an optimum combination of broad band width, short pulse,
and good sensitivity can be achieved by using a light impedance backing ( usually 3 to 5 mega Rayls), with two
quarterwave matching layers on the front of the piezoelectric element. One measure of the sensitivity of a
transducer is insertion loss. Insertion loss is defined as the ratio of the power delivered to that part of the system
following the transducer, before insertion of the transducer, to the power delivered to that same part of the system
after insertion of the transducer. The term transducer loss is defined as the ratio of the available power of the
specified source to the power that the transducer delivers to the specified load under specified operation
conditions. The available power is the quotient of the mean square of the open-circuit terminal voltage of the
source divided by four times the resistive component of the impedance of the source. To measure insertion loss
one must account for beam spreading diffraction losses, electrical impedance mismatches, etc. This usually
requires having the target reflector very close to the face, or a special shaped target to return all the pulse echo
energy to the transducer. For medical ultrasonic transducers, insertion loss numbers fall in the range of -6 dB to 10 dB typically. More practically, for comparison purposes in production, when testing the individual elements of
a multi-element array, one sets the target a fixed distance close to the elevation axis focal point, and defines a
more empirical term called "loop sensitivity". This is defined as the ratio of the peak to peak pulser voltage, to
the received (two way) pulse echo voltage peak to peak amplitude. Values for loop sensitivity are typically -25 to
-45 dB.
2. Design of Acoustic Stack
A variety of references discuss the design and construction of ultrasonic transducers(1-11) . What determines the
frequency output of a medical ultrasonic transducer? At what frequency does the transducer resonate ? The
frequency output is related to the thickness of the ceramic material from which the transducer is constructed.
However, if one attaches leads to the electrodes on a piece of ceramic in air, and measures the electrical
impedance as a function of frequency, one observes not one resonance, but two (figure 3). The resonance at
which the impedance reaches a minimum is called fr or the resonance frequency, and the higher frequency
resonance where the impedance reaches a peak is called fa, the anti resonance. In the literature, these two
resonances are also variously referred to as the following:
Frequency Definitions
Fr=resonant frequency=frequency of zero
Fa =Anti-resonant frequency
=frequency of zero reactance
2) Fs=Series resonant frequency=frequency
Fp=Parallel resonant frequency
of maximum conductance
= frequency of maximum resistance
3) Fm =Frequency of minimum impedance
Fn=Frequency of maximum impedance
The following relationships apply:
for the lossless case:
for the lossy case:
So, which resonance should one build a transducer around ? A little known phenomena is that the position and
shape of the fr resonance depends dramatically on the length and width dimensions, L & W of the ceramic slab,
as well as the thickness Tk . There is a strong influence due to modal coupling, of the other dimensions on fr. For
example if a piece of ceramic used to dice a one dimensional array has dimensions of approximately 15 mm x 60
mm, the resonance valley near fr will appear washed out, and broad. However, if a smaller piece of dimensions
about 15mm x 6 mm is broken off of that same slab, the resonance will be more pronounced and closer to the true
resonance behavior because of much lower modal coupling to the later dimensions. If one had observed the anti
resonant peak, fa in both of these measurements, one would note that it is very well defined, and repeatable ! This
is because the anti-resonance (or parallel resonance) frequency is fundamentally related only to the ceramic
thickness, and not influenced by modal coupling. Also, as the amount of poling is increased in a given material, fa
will remain nearly constant, while fr will steadily decrease until the material is completely poled. Thus, fa
depends on the velocity and thickness, while fr depends on the motional reactance, which varies with the poling
state of the material, and cross coupling to other modes. It for these reasons that a more predictable way to specify
and construct ultrasonic transducers is to specify ceramic dimensions, and matching layer thickness, related to fa,
rather than fr .
Figure 3. Fundamental resonance and antiresonance of slab of PZT ceramic
Starting with a ceramic that resonates at fa, a
broadband transducer design usually has a
nominal center frequency much lower than
fa. This is because of several downshifting
factors, as will be explained below. The final
center frequency of an ultrasonic array
transducer may be expressed as:
fc = Ds * fa = ds1 * ds2 * ds3 * ds4 * fa
where, fc = transducer design center
frequency, and fa = bulk ceramic antiresonant frequency.
Ds = total downshift factor
ds1 = downshift factor #1 = lower frequency resonance caused by dicing ceramic into sub-elements, i.e. the
acoustic velocity is lower because of side wall boundary conditions (wave guide).
= 0.82 ( when kerfs are later filled with polymer resin, ala construction for linear or convex arrays)
ds2 = downshift factor #2 = mass loading effect of acoustic stack (matching layers and backing) on ceramic resonance
≅ 0.95
ds3 = downshift factor #3 = due to frequency dependent attenuation of RTV lens or face material
≅ 0.93
ds4 = downshift factor #4 = this phenomenon is due to the fact that pulser circuits usually have very low
impedance (5 to 10 ohms). Thus, energy is more easily coupled to the low impedance resonance peak fr,
rather than the higher impedance resonance fa.
≅ 0.78
Therefore, the total transducer frequency downshift factor can be
DS = ds1*ds2*ds3*ds4 = 0.56
Thus, fc = 0.56 * fa . Therefore, to start a transducer design (linear or convex array), one wants to order ceramic
with bulk resonance fa (anti-resonance) in the slab, of frequency approximately 1.77 times the nominal transducer
design center frequency. For example, for a 3.5 MHz linear array, one would order ceramic with fa = 6.10 MHz.
The final transducer center frequency and bandwidth, can also be heavily influenced by matching layer thickness,
and the method and values of electrical tuning chosen.
Design of Acoustic Matching to Load;
Quarter Wave Matching Layers for Broadband Response:
For short impulse response and broad bandwidth, ultrasonic transducers are designed with one or two matching
layers on the face. The front matching layers are analogous to anti-reflection coatings on optical lenses. The
acoustic impedances of these layers are usually chosen based on transmission line theory. A new way to calculate
the optimum quarter wave matching layer impedance values will be shown.
The challenge in transducer design is to match the high acoustic impedance of about 33 mega-Rayls typical of
PZT ceramics to the relative low impedance 1.5 mega-Rayls characteristic of a water or tissue load on the face of
the transducer. Classical transmission line theory may be used to design a multi-stage quarter wave section
impedance transformer(12,13). Using the classical approach of Collins 12 for a Butterworth type of filter response,
one would calculate the requisite impedances of double quarter wave layers to be applied to the front acoustic
port of the ceramic as:
(where we assume the ceramic Zc = 33.2, and the front load impedance is water with ZL = 1.50 )
One section
Two section
Table I. Quarter Wave Matching Sections predicted by transmission line theory.
Example values to match PZT ceramic
Z1 (first layer)
Z2 (second layer)
Zc • ZL
Zc3/4 ZL1/4
Matching layers are typically fabricated using epoxy resins loaded with dense powders such as tungsten or
alumina14. However, it is very difficult to achieve dense enough loading to produce acoustic impedances in the
range of 8 to 15 mega-Rayls. Solid materials such as glass and graphite have been considered as matching layers,
but there is the classical problem of bonding on pre-fabricated matching layers without the bond line destroying
the effectiveness of the matching layer15 .
DeSilets and Fraser 16 of Stanford University found that assuming the transducer was air backed, effectively lower
target values for the matching layer impedances could be derived, based on the KLM 17 equivalent circuit model,
which places a virtual node in the middle of the ceramic. Their analysis predicts :
One section
Two section
Table II. Quarter Wave Matching Sections predicted by KLM theory.
Example values to match PZT ceramic
Z1 (first layer)
Z2 (second layer)
Zc4/7 ZL3/7
The optimum impedances for the layers (see Table III.) can also be derived with minimal understanding of the
underlying physics, by the means of statistical design of experiment methodology (DOE) and computer
simulation models18 . The exact layer thicknesses can also be tailored as another design optimization parameter.
An excellent starting point for design optimization is to use the design rule that the layer thickness is 0.350
lambda, at the anti-resonance frequency of the diced ceramic resonance.
Table III. Quarter Wave Matching Sections predicted by DOE computer optimization .
Example values to match PZT ceramic
Z1 (first layer)
Z2 (second layer)
One section
Two section
computer optim..
computer optim..
Influence of back side on front side matching:
The formulas for matching layers derived by DeSilets16 et. al. assumed an air load on the back side of the
transducer ceramic. The transducer designer has more flexibility in optimizing the design trade off if the impact of
the back side loading and impedance on the ceramic are included in choosing the matching layer impedance
values applied to the front side of the transducer. One usually assumes that the acoustic impedance of a PZT
ceramic is about 33.2 mega-Rayls, and one calculates the requisite matching layer impedances as discussed
above. In a new approach 19 , one uses the KLM model or other transmission line model, to first calculate the
average acoustic impedance as a function of frequency, looking into the face of the ceramic. This value will be
impacted by the load on the back side of the ceramic and by the electrical tuning, because of the electricalacoustic coupling. An example of the acoustic impedance at the face of the ceramic as a function of frequency is
shown in figure 4, using a backing impedance of 3.0.
Figure 4. Effective acoustic impedance looking into ceramic,
as influenced by presence of backing.
One may calculate the average effective impedance of the face
of the ceramic over the bandwidth of interest, say 80 %, and
use this value to plug into the transmission line equations to
calculate the desired quarter wave matching layer values. In
this example, over a 70 % bandwidth, using an effective
average acoustic impedance of 18.54, the resultant matching
layer values would be:
First matching layer: Z1 = Zc4/7 ZL3/7 = 6.31
Second matching layer: Z2 = Zc1/7ZL6/7 = 2.15
Table IV. Quarter Wave Matching Sections predicted by Modified theory .
Example values to match PZT ceramic
Z1 (first layer)
Z2 (second layer)
One section
Two section
Zc4/7 ZL3/7
This approach has the added advantage that the matching layer impedances predicted are much lower than in the
classical approach, and thus much easier to fabricate. Figure 5 shows the pulse echo response (Piezocad) of a
transducer with these values for the two front matching layers, and a backing impedance of 3.0 . The design
yields a respectable bandwidth of 72 %, a -20 dB pulse length of 0.591 micro-seconds, with a center frequency
of 3.73 MHz.
Figure 5. Pulse Echo response with double matching layers of
Table IV.
For broad band width and short pulse ringdown , the thickness of
the matching layers should be ground to a tightly controlled
dimension. Since repeatable transducer designs are based on the
ceramic anti-resonant frequency fa, the matching layers should be
ground to a thickness of 0.350 lambda, where lambda is the
wavelength of sound in the layer at a frequency given by fa.
Optimal performance can usually be achieved with layer
thicknesses tailored to within +/- 15% of this design rule starting
point. This design rule also coincidentally translates to
approximately 0.25 lambda thickness at the net operating center frequency of the total acoustic stack, fc. Hence
the layers are effectively "quarter wave layers", and in effect behave as anti-reflection coatings on the face of the
transducer to eliminate reverberation of energy in the form of internal reflections that would make the pulse
ringdown lengthy. Ted Rhyne20 has analyzed broadband transducer design from the perspective of a filter design
problem , and has calculated that with a three matching section design, bandwidths up to 90 % are achievable.
3. Materials
Modern ultrasonic arrays for medical imaging are typically constructed from piezoelectric ceramic of the PZT-5H
type. PZT ceramic is available in various choices of dielectric constants. Because of the small array element sizes,
and subsequent low capacitance and high impedances, one typically chooses a ceramic with a very high relative
dielectric constant and high coupling factor. Various additives, such as chromium, iron, nickel, and lanthanum are
used to modify the dielectric constant and the piezoelectric properties of the material. Table V lists some typical
piezoelectric ceramics and their properties. The constant d33 is the strain developed per applied field and is a
measure of the effectiveness of the material as a driver or transmitter. The constant g33 is the electric field
generated per applied stress and is a measure of the effectiveness when used as a receiver. A useful figure of merit
when choosing materials is the product of d times g. The electromechanical coupling factor (k33) is not actually a
measure of intrinsic efficiency of transducer material as its name implies; but the square of k33 does give the
relative portion of stored input electrical energy converted to stored mechanical energy, and vice versa. Theory
indicates that materials with higher coupling coefficients k33 can give better band widths and pulse response in
medical applications. Note that the high dielectric loss of PVDF plastic film limits its signal to noise ratio,
achievable in a transducer.
Table V: Piezoelectric Material Properties
PZT-5A PZT-5H Motorola PZT-7A
3203 HD
Free Dielectric Constant
1700.00 3400.00 3800.00
Dielectric Loss Tangent
d33 (M/V x 10-12)
g33 (V*M/Newt x 10-3)
Acoustic Impedance Z
Mechanical Q
The acoustic velocity in materials, both in the ceramic and the matching layers, depends on the aspect ratio w/h of
the element geometry. Typically, to avoid spurious resonances, and to allow flexibility of conforming to a convex
geometry, the major pitch of a diced array is diced into sub-elements of width to height ratio w/h of between 0.4
to 0.6 . One advantage of this is that the piezoelectric coupling factor for this geometry is higher than for flat plate
ceramic slabs. While kp is only 0.5, k33 can be as high as 0.75 .
The “material” coupling factor (squared) is: ( Jaffe and Belincourt 21 ):
k t = (π / 2 )( Fs / Fp ) tan{(π / 2 )( Fp − Fs ) / Fp }
The “effective” coupling constant may be given as:
(it is this coupling factor that determines filter or transducer bandwidths !)
keff2 = ( Fn − Fm ) / Fn
The square of the coupling factor is proportional to the partitioning of the energy available in the electrical
domain over that in the mechanical domain, or vice versa. Also, theoretical analysis has shown that the ultimate
bandwidth available from a transducer design is proportional to keff squared 16 .
The acoustic materials used to fabricate a transducers front and back layers are typically made from polymer
resins. These are heavily loaded with powders to achieve the desired acoustic properties of acoustic impedance,
velocity, density, and attenuation. Desired acoustic impedances for backing the transducer range from 3.0 to 7.0
mega-Rayl. Backing materials are typically epoxy, loaded with powders of aluminum oxide, tungsten and its
oxides, and additives to increase the absorption of acoustic energy in the backing. A typical formulation has 6 to 9
components. The material used in the backing must be attenuative enough so that any backing echo is at least 90
to 100 dB down from the main bang pulse amplitude. Broadband transducer designs are usually made with double
matching layers on the front. The first layer usually needs an acoustic impedance of about 8 mega-Rayls (14,16) and
may be made from epoxy loaded with tungsten or aluminum oxide (Kossoff 14). Acoustic impedance is an
especially important quality of transducer material that serves as a measure of the pressure required per unit
particle velocity, and thus of the stiffness or resistance to passage of pressure waves. The second matching layer
materials are usually unloaded epoxy or urethane, with an impedance between 2.2 and 2.9 mega-Rayl. For
fabricating matching layers on a transducer, there are two schools of thought . One proposes to grind the precured layers to precise dimensions first and then apply them to the ceramic with thin bond techniques. The other
approach is to cast the materials in place on the ceramic, let the thermoset polymer resins cure in place without an
intrinsic bond line, and then machine to final dimensions. Casting and grinding requires formulations that do not
have significant settling or gradients formed during curing. The grind and glue approach requires that the thin
bond lines be less than 1/200 of an acoustic wavelength, in order to be invisible acoustically 15 . The practical
problems of doing repeatable ultra thin bond lines in high volume production can be formidable.
4. Thermal Management
Another practical issue in good transducer design is to avoid overheating the face of the probe, causing patient
discomfort. The temperature at the face of the probe must not exceed 41 degrees centigrade. Phased array probes
used for cardiology scanning tend to overheat more so than linear arrays, because the same elements are used to
generate every beam firing. Analysis shows that a lot of the heat generated in the probe is caused by self
absorption in the materials of the acoustic waves generated. Good thermal management techniques include
inserting low profile conductive heat pipes or fins in the backing, and loading the backing and front layer
polymers with powders of excellent thermal conductivity, such as aluminum nitride, or industrial grade diamonds.
A metallic RF/EMI shield grid placed over the face of the transducer can also serve to quickly carry away the
thermal energy.
5. Elevation Focusing:
Focusing of the short axis, elevation dimension determines the effective slice thickness of the image plane. There
are two generic approaches to focusing the beam elevation in the short axis, internal focusing (by shape of
ceramic) , or external focusing (see figure 6.) with a RTV silicone lens.
Figure 6. Two alternate geometries to focus short axis
For use with flat ceramic geometry (in short axis), an external
convex lens made with material of a lower velocity than water
Match Layer
are used. These materials are in the RTV silicone family, and
are available form GE and Dow Corning. RTV typically has
Face Lens/Filler
Urethane face
RTV Lens
and acoustic impedance between 1.3 and 1.5 mega-Rayl and
Internal Focus
External Focus
an acoustic loss of about 7 dB/cm/MHz. A non-linear curve
for choosing soft or sharp beam focusing has been given by
Kossoff 22. The general purpose curves are normalized by the far field transition distance T.
T = D2 / 4*lambda
with D = the active aperture diameter
A formula for determining the requisite radius of curvature for a RTV lens may be derived from optics:
ROC = Reff { (Vmedium - Vlens) / Vlens } ~ 0.525 Reff
where, Reff is the effective geometric focal point desired, for tissue, Vmedium = 1.540 Km/sec, and for typical
RTV silicones, Vlens = 1.010 Km/sec
Alternatively, for internally focused array designs, with cylindrical shaped ceramic, it is desirable to have a
neutral density face filler with impedance close to tissue or water. These requirements are met by certain families
of castable urethane organic polymers. Some moderate level of acoustic absorption is actually desirable in the
face material, because it helps attenuate multi-path reverberation from the tissue, which creates near field clutter
in the image.
To minimize clutter in the image the dimensions of the beam in the elevation focus dimension (out of the
electronic focal plane) should also be carefully specified. Note that diffraction theory shows that a lens cannot
focus an aperture beyond its farfield Fraunhofer transition distance T, given above. For typical arrays, the short
axis (elevation) dimension width is about 30 lambda, and the short axis lens is usually set with a focal distance of
about T, for soft focusing with long depth of field.
While beam widths are classically quoted as the - 3dB points in optical applications, for ultrasonic imaging,
experience has shown that the effective beam thickness in the elevation dimension that influences image quality, is
proportional to the -10 dB beam width (one way ) (or -20 dB send / receive beam). The depth of focus of the elevation
beam is determined by the mechanical focusing properties of the acoustic lens . Kino gives from optics23 :
DOF = 7.1 (f#)2 * lambda
(where f# = f number)
This expression is the depth of field for -3 dB rolloff in the axial intensity, from the peak value at the focus. For
ultrasonic imaging a more useful value is the depth of field for a broadband pulse, for only a 1.5 dB rolloff in
sensitivity. This acoustic depth of field would be given by:
DOF = 4.7 (f#)2 * lambda
Extended Elevation Focusing:
The depth of field of the beam can be improved passively by borrowing from the physics of non-diffracting beams, and
building apodization into the transducer structure (24-30) .This may be done by one of three ways, 1) patterning of the
electrode on the ceramic, 2) gradient attenuation in the lens material, or 3) varying the effective polarization strength of the
piezoelectric ceramic across its elevation dimension. The beam may be made more collimated and uniform with a
Hamming apodization function applied across the aperture. Figure 7 is a comparative plot of just the -10 dB isocontour
beamwidths for the apodized case versus the classical flat aperture case . Experience has shown that the -10 dB one way
beamwidth (-20 dB two way), is more representative of the actual effective slice thickness (elevation dimension) seen in the
image, than the -3 dB contour classically quoted in optics.
Figure 7. Additional depth of focus achieved by apodizing the short
axis aperture.
6. System Considerations
There must be tight coupling between system design and
transducer design, to achieve the best image quality in
clinical applications(31-34) .One of the first questions to ask
when designing a new ultrasonic array, is what should the
major element pitch be ? The number of sub-elements is
determined by keeping the sub-element aspect ratio in the
range of 0.4 to 0.6 for W/H. A guideline to make a first
estimate of a required array pitch, is to decide how many
channels are going to be used in beamforming.
The lateral image resolution is given by :
resol = 1.22 * f# * lambda
(8) where f# = Focus depth / Aperture = "f number", and Aperture = N * Pitch ;
where N is the number of channels used in beamforming.
and lambda = wavelength of sound at image frequency
In order to adequately display this resolution on the system monitor, the resolution cell must be sampled 2 to 4
times. This means the number of acoustic lines fired per expected resolution cell in the lateral dimension must be
2 to 4. The density of acoustic lines formed, can be increased by stepping the active aperture across the array in
alternate odd then even element groupings. Let's consider designing a convex geometry array, to scan a fan
shaped image sector.
In terms of angular resolution of beamforming from the face of the array:
Θres = arctan( resol / Z) ≅ (resol / Z )
(for small angles)
= 1.22 • f# • lambda / Z
whereas, the image sampling rate for display, or sampling of fired acoustic lines, is given by the angle subtended
by stepping the aperture on ceramic element boundaries and is:
Θtxd = P / R
where P = the array element pitch ; and R = the convex radius of the face of the array.
For adequate sampling of the image resolution and to avoid aliasing in the image, we want:
Number Samples per resolution = Θres / Θtxd ≥ 2
Θres / Θtxd = (R/P) • (1.22 • f# • lambda/Z) ≥ 2
Pitch=P ≤ =0.6 • (R • f# • lambda/Z)
and since
f#=Z/(N • P)
we may also write:
P ≤ = 0.6( R • lambda / N )
Figure 8 plots this curve for the cases of a 64, 96 , and 128 channel beamformer for a 3.5 MHz convex array with
a 40 mm radius. This pitch value can be used as a starting value in array design, and is adjusted depending on the
clinical application and details of the system beamformer .
Figure 8: The first estimate of a desired array element
pitch for a 3.5 MHz convex array, as function of the
convex ROC and number of beamformer channels.
Typically to maintain image resolution with depth, the
number of beamformer channels active is expanded with
depth, up to the maximum number. If the aperture is
expanded too slowly, image resolution is sacrificed ;
however, if the aperture is expanded too rapidly, acoustic
clutter increases and hurts the systems ability to maintain
high dynamic range and contrast resolution. This is because
of the finite angular acceptance sensitivity of individual
array elements.
Antenna Angular Sensitivity:
The theoretical angular sensitivity of a array element of ceramic width w ( Pitch minus kerf), is given by :
P/Po= sinc (π ⋅ x ) • cos( Θ ) = [(sin π ⋅ x ) / π ⋅ x ] • cos( Θ )
Convex Array Element Pitch as function of Convex radius ROC
and Number of Beamformer Channels
Array major element Pitch (mm)
Convex Radius ROC (mm)
where P = pressure at angle Θ , Po = pressure on axis Θ = 0, x = (w/lambda)*sin( Θ )
(this equation was derived assuming non-rigid baffle boundary conditions 35 ).
Many times a system engineer or transducer designer needs a good estimate of what the element sensitivity
function will be before empirical data is available. A useful formula has been derived to give the rolloff angle at
various dB criteria. This is a transcendental, iterative relationship, and will give accurate results after the 3rd or
4th iteration. ( formulas for ith iteration estimate)
I ) -3 dB One Way Sensitivity Angle (-6 dB two way):
Θi = arcsin(0.442 * lambda/w)
( this is the seed value (first estimate of Element (-3db 1way) Sensitivity Angle )
Θ( i + 1) = arctan((1.414/ π )*(lambda/w)*sin( π *(w/lambda)*sin Θi )) [Radians] (17)
( this will converge in 3 to 4 iterations)
II ) -6 dB One Way Sensitivity Angle (-12 dB two way):
Θi = arcsin(0.60 * lambda/w)
(Seed value (first estimate of Element (-6db 1way) Sensitivity Angle )
Θ( i + 1) = arctan((2/ π )*(lambda/w)*sin( π *(w/lambda)*sin Θi )) [Radians]
( this will converge in 3 to 4 iterations)
Figure 9 shows the angular sensitivity pattern (directivity ) as calculated by the Pzflex FEA code for a 3.5 MHz
phased array element. Comparison is made between air maintained in the kerfs, versus the response if RTV
material is allowed to wick into the kerfs, or alternately epoxy resin as a kerf filler.
Angular Sensitivity & Kerf FIller
Figure 9. Plot of angular sensitivity of individual
array element, and influence of boundary conditions
and kerf fillers (from FEA simulations)
Amplitude (Linear)
Air Kerf
Undiced Layers
RTV Kerfs
Epoxy Kerf
Stiffened Baffle
Cos Theory
Experience has shown that a good balance between
resolution requirements and keeping down unwanted
clutter acceptance , is given by expanding the
beamformer aperture at a rate allowed by the element
angle rolloff at the -12 dB ( two way send/receive)
point. An example of the suggested expanding aperture
rate for a 3.5 MHz, convex array with 40 mm radius of
curvature is given in figure 11. One can see that having
larger number of beamformer channels is only useful at deeper depths.
Angle - degrees
Figure 10: Geometry for determining desirable expanding aperture
rate of number of beamformer channels turned on at a given
depth, as dictated by individual array element polar sensitivity angle.
The aperture subtended by "ni " active elements across a convex array
is given by the chord:
A = 2 R sin (θ )
The depth at which an given channel ni may first be turned on is
determined by its sensitivity angle α . The relation between a given
target depth Z, and the allowed aperture and number of channels
activated (related to angle θ ), may be shown to be (figure 10):
Z = [R sin θ / tan( α - θ )] - R(1-cos θ )
This may be plotted in graphical form as the expanding aperture rate ( number of channels ) Vs image depth Z
(mm) . (see figure 11 )
Allowed Expanding Aperture as function of Acceptance Angle
6.5 MHz Convex Array, ROC=10 mm
60% Theoretical
80% Theoretical
Depth Z
100% Theoretical
Number of Channels Allowed
Figure 11 . Plot of expanding aperture of
Figure 12. Desired turn on rate of beamformer channels
beamformer channels Vs depth.
as function of individual element angular sensitivity
Another example for a tight radius (10 mm 6.5 MHz) convex array is shown in figure 12. One can see that there
is a significant benefit in utilization of available beamformer channels, as the angular sensitivity of individual
array elements is improved. The criteria used here is that of the target falling within the -12 dB two-way
acceptance angle of the outer array elements included in the aperture.
7. Tuning and Electrical Impedance Matching
Another important aspect of relating an array transducer design to the system requirements is electrical impedance
matching. Because a multi-element array may have 100 to 200 elements, the individual elements are very small, and
subsequently have high electrical impedances ( 100 to 600 ohms). For efficiency of transferring energy, the array
elements should be matched to the pulser during transmit, and the pre-amp impedance on receive. For maximum
energy transfer between the transducer and pulser or receiver, the load impedance of the transducer should be the
complex conjugate of the electrical circuit coupled to it. This is a compromise, since pulser impedances, being
voltage sources, have low output impedance of 5 to 10 ohms, whereas pre-amp impedances are usually 50 to 300
ohms. To match the impedances of array elements to the coax cable and system impedances, "tuning" elements(36-43)
are typically placed in the probe connector between the coax cable and the system. The equivalent circuits to model
send/receive in the finite element code Pzflex are illustrated in figure 13. To solve a consistent set of equations
between switching the send and receive circuits in place, each circuit must have the identical number of
components, however, their values can be adjusted to emulate the real system. One broad bandwidth tuning
component would be a transformer with the proper turns ratio to step the array element impedance down to the
system load, and with an intrinsic inductance wound in, to cancel reactive loads. Canceling reactive effects
(capacitive or inductive) makes the effective load of the transducer array and cable look primarily resistive to the
system pulser / receiver, for more efficient transfer of energy and broad band performance . Because the array
contains hundreds of elements, transformer tuning can occupy a large volume of space, and be expensive. Because
of space and cost constraints, a more cost effective way of tuning a probe is to use surface mounted miniature
inductors. As a design guideline, series inductors will drop the effective element impedances down to 40 to 60 % of
their initial value. On the other hand, parallel mounted inductors, will raise the effective net impedance of the
elements a factor of approximately 1.5 to 2.0 . Parallel tuning will also help roll off low frequency content in the
pulse ( high pass filter), which can sometimes degrade image quality. The choice of series or parallel tuning depends
on the specifics of system requirements and the behavior of the particular pulsing circuit. In general, proper tuning
of the array elements yields higher sensitivity , aids in broad band response over a range of frequencies, and
influences the net center frequency of the transducer design. If an array element with coax attached has an
impedance and phase angle at the nominal center frequency of Z ∠ ( Θ ), then a nominal series tuning component
can be chosen as: Ls = |Z| * sin ( Θ ) / 2 π Fc , where Fc is the center frequency. A parallel tuning component
may be chosen as: Lp = |Z| / (sin( Θ ) * 2 π Fc) . The broadest frequency response can actually be achieved with a
distributed tuning network (figure 13) that includes a series inductor at the transducer end and, and a series/shunt
inductor at the connector/system end of the coax. Even more broad band frequency response can usually be
achieved by skewing the final values of tuning components away from their nominal value. Tuning should be
chosen not only to cancel out coax cable reactance, but also the parasitic capacitances of interconnecting system
boards, and the inductances of transmit/receive switching circuits.
Circuit Models for Transmit / Receive
for finite element modeling code PZFLEX
date: 2/25/97
PCB + T/R switch + Transmitter
System side
interface )
Ls2= 10.0 uH
Ls1= 15.0 uH
Rtxm= 4 ohm
Lp = 10.0 uH
Coax Transmission Line
Rp = 47 ohm
Figure 13: Equivalent circuit schematics used in finite element
code (PZFLEX) to model coax cable, tuning, and effects of
system reactance loads.
RL=10,000 ohm
Ls2= 10.0 uH
Ls1= 15.0 uH
interface )
Lp = 10.0 uH
Coax Transmission Line
Rp = 47 ohm
RL=640 ohm
Cap=140pf Lt/r=100uh
RL=640 ohm
By optimizing the tuning of a new, innovative phased array
architecture, bandwidths of over 90 % have been achieved [43],
with a two matching layer design. ( see figure 14 )
PCB + T/R switch + Receiver
System side
PL 20 = 1.04 usec
BW = 95.2 %
Figure 14 : Experimental results of new transducer architecture for phased array construction,
with series-shunt tuning. Bandwidth BW = 95.2 % ; -20 dB pulse length PL20 = 1.04 microsecond.
8. Computer Simulation / Modeling
Modal Resonances in Array Structures
The achieving of broad band performance can also be limited by modal coupling and spurious resonances in the
matching layer materials when the geometry has undesirable aspect ratios. Studies of modal coupling lead to a
handy rule in transducer design; one should try to maintain aspect ratios in material crossections of approximately
1:2 whether it be for the ceramic pillars, or the matching layer crossection. Therefore, as a general design rule
when designing acoustic transducers, it is usually desirable to keep the width to height aspect ratios
approximately 1:2 tall and thin, or greater than 3:1 flat and wide. The dynamic behavior of the elements of an
array acoustic stack are illustrated in the following figures. The basic geometry was modeled as a mesh and
analyzed with the finite element analysis (FEA) code PZFLEX 54, as illustrated in figure 15. The crossection
shows the expansion of the center element (two sub-elements) when activated by a pulse voltage.
For a linear array, for wider angular response, often the first matching layer is severed. Even with a wider one
lambda pitch, maintaining a flat 2:1 aspect ratio still does not guarantee a crossection free from modal coupling.
Modal distortions can be set up as seen in figure 15, depending on the relevant material properties and element
pitch. In essence, the thickness dimension is working as a quarter wave resonator, whereas the lateral extent on
the diced pitch boundary can work as a half wave resonator, and the lateral dimension can thus couple strongly to
the thickness dimension and cause undesirable spurious resonances. So aspect ratios greater than 2:1 are desired
in this architecture. For this reason, it is sometimes desirable to avoid notching the matching layers when
constructing a convex or linear array. However, the design tradeoff dilemma is that the angular sensitivity is
improved if the layers are severed.
Figure 15. Crossection of expansion and contraction of center element of three
element array, as modeled by finite element code PZFLEX.
In a phased array transducer, which is usually diced from the front side (matching layer), sub-dicing the matching
layers can keep a desirable tall 1:2 aspect ratio. The net effective acoustic velocity in matching layers with a tall ,
thin aspect ratio is different from the bulk measured value, and must be taken into account when specifying
matching layer dimensions to achieve broad band matching. The velocity downshift is due to wave guide and plate
mode effects on the sound velocity. Kino 23 gives a graph of downshifted velocity of a "strip guide mode" mode
velocity over the bulk longitudinal sound velocity, Vg/Vl as a function of the material’s Poisson ratio. Dispersion
curves that give the matching layer velocity downshift as a function of the height to width aspect ratio H/W are
based on the coupling mode theory of Onoe and Tiersten44 . Ayter 45 has expressed this in a useful form requiring
only knowledge of a materials shear and bulk longitudinal velocities (value of Poisson ratio not needed). The
expression is :
(Vg/ Vl )2 = ( ( 2 • f • H / Vl ) = 0.5(1 + x −
x = (H/W)
(1 + x ) 2 − 4 • x • k )
and k = 1 - [1-2 •(Vs / Vl ) ]2
(26) ; with Vs = shear velocity, Vg=longitudinal
velocity in strip guide, and Vl = longitudinal sound velocity (bulk),
Recently available computer simulation code PZFLEX 54 based on time domain finite element analysis of two
dimensional coupling effects, can also used to study these modal resonances46.
For a sector phased array probe, to achieve wide angular directivity, it is beneficial to sever the matching layers and
dice into the backing (see figure 16 ). However, at a pitch of 0.5 lambda, this leaves a very undesirable aspect ratio
for the front matching layers. One solution is to come back and sub-dice the matching layers to achieve the 1:2 tall
aspect ratio guide rule. However, plate modes and Lamb waves can still be set up as illustrated in the PZFLEX
simulation plotted in figure 16.
Figure 16. Modal behavior of the acoustic stack of the
center element of a phased array transducer being excited by
pulse voltage.
The three dimensional FEA code is also useful for evaluating
material effects , such as sound speed in the backing and its
influence on crosstalk and acoustic coupling . Crosstalk can
degrade the ability of the beamformer to focus with precision on
the target objects with out additional degradation effects.
Crosstalk can occur acoustically in the transducer acoustic stack,
and surrounding structures and housing, or electrically due to
capacitances and inductances; or crosstalk in the interconnect
paths leading through the system to the input points of the preamps. Another form of cross talk is through the backing.
Simulations show that when the velocity of the material in the
backing is high ( 2.8 KM/sec), the wave emanated into the backing from two sub-elements, can propagate side ways
and excite crosstalk energy on nearest neighbor elements, before the pulse has fully left the front of the array. The
use of backing material with slower velocity (1.6 KM/sec) can greatly reduce the progression of the wave into the
backing relative to the front going waves. The influence of kerf fillers on crosstalk can also be studied. A
propagation delay in the occurrence of the crosstalk components indicates, that in this case it is predominately
acoustic, rather than electrical. Typical crosstalk isolation for array elements is -28 to -32 dB. A level of -30 dB is
quite adequate for most imaging situations. For Doppler shift detection in continuous wave mode (CW), a higher
crosstalk isolation of -40 to -50 dB is desired. This can be achieved by leaving a buffer of a few unused element in
the array between the groups of elements transmitting versus receiving. Crosstalk can also seriously degrade the
achievable wide angle response (element acceptance angle theta) of array elements needed for wide angle beam
steering. This can be very important for phased array type transducers, or for linear arrays that need to be steered for
improved Doppler shift detection of blood flow.
9. Two Dimensional Arrays
A current very active area is the development of two dimensional array. Adding individually phased elements in the
orthogonal dimension (elevation) gives the opportunity for focusing the beam for better image slice thickness,
and/or arbitrarily steering the beam for volumetric scanning in three dimensions (47-50) . Furthermore, having
addressable elements in the elevation dimension will enhance the possiblity of getting aberration correction schemes
to work successfully .
The design and fabrication of 2-D arrays brings its own formidable challenges. These include the task of a multitude
of miniature interconnects, and building an acoustic stack that is well behaved. For good angular sensitivity, the
matching layers are typically severed, but can lead to spurious mode coupling, and difficulty in connecting the
signal ground return path. For 2-D arrays, new design rules for element aspect ratios, kerf widths, and accounting
for long range coupling/composite effects will have to be developed.
The electrical matching problem for 2-D arrays is even more challenging, because of the very large electrical
impedance of very small elements. This creates some interesting phenomena when running computer simulations
with modeling codes, such as PZFLEX. A curious baseline drift, or sagging of the pulse echo is sometimes observed
(figure 17.) For very large receiver impedances, the effect is even more exacerbated (figure 18 ). In effect, the
capacitance of the coax cable swamps the capacitance of very small elements, and terminated with a pre-amp of
high input impedance, the coax cable acts as a capacitor being charge up by the returning pulse. The effect is
particularly noticeable for pulse echo ringdown shapes that are not symmetrical above and below baseline. The RC
time constant of the equivalent circuit creates a low frequency response super imposed on the pulse. In the FEA
code (PZFLEX) this phenomena can be compensated for by use of a pass band filter. However the software
algorithm used by the code, itself imposes a slight DC base line shift to the pulse, and will give erroneous results for
the -20 dB and -40 dB pulse ringdown, unless compensated for by an additional correction for DC offset. When
processed in this manner, the strange looking pulse of figure 18, can be filtered and adjusted to look like the more
normal pulse echo shown in figure 19 ( along with its frequency spectrum ).
Figure 17. Sag and baseline drift of
returning pulse/echo, R=1000
Figure 18. Sag and base line drift due to
Figure 19. Restoration of integrity of pulse
charging of coax, with Rcvx=14,000 ohms
shape by band pass filtering & DC offset
10. Issues / Challenges
A particular challenge is to be able to fabricate arrays that achieve the predicted performance, and the ability to
reproduce them in high volumes, with consistent, repeatable performance specifications. One must be able to create
in the laboratory, acoustic materials with the requisite properties called for by the optimized design. Conversely, for
accurate simulation results, one needs careful characterization of the material properties to be called upon by the
computer simulation. Typical material properties that need to be carefully measured are: density, longitudinal
velocity, Poisson’s ratio or shear wave velocity, acoustic impedance, and the frequency dependent attenuation of an
acoustic wave passing through the sample. There are at least three ways to model the frequency dependent
attenuation of materials within the PZFLEX code. The accuracy of the simulations depends on using the best fit to
the empirical data. Excellent agreement can be obtained between empirical results and the predictions of FEA
computer simulations.
The fabrication yield of good arrays with very fine pitch in production, depends on the fundamental strength of
materials, particularly in the piezoelectric ceramic. Pinhole defects, porosity, and incipient fault lines or stress
fracture can wreck havoc on good yields. The recent development of ceramics with very fine grain structure
promises better transducer results and the ability to fabricate transducers of much higher frequencies 51
Another important area of active research, is the growth of single crystal forms of the relaxor-lead titanate (PT)
materials, in pieces large enough to fabricate arrays. These new materials have the performance advantage of much
larger coupling constants, which should facilitate the fabrication of high sensitivity arrays, having useable bandwidths(52)
over 100 % .
A very important, practical challenge in transducer development is cost reduction, concurrent with performance
improvement. The signal micro-interconnect challenge for arrays with large numbers of elements is significant. The
majority of the cost of materials in a probe is in the coax cable and system connectors ! So innovations here will be
11. Summary:
The principles of designing and fabricating high sensitivity, broadband, short impulse response array transducers for
medical ultrasonic imaging have been reviewed. The most commonly used ceramic piezoelectric for multi-element
arrays is the ceramic PZT-5H (or equivalent). Design rules for selection and optimization of front matching layer
impedance values and backing were reviewed. New design rules taking into account the influence of the backing on
the effective front impedance of the ceramic, reveal that impedances for the matching layers much lower than
previously estimated can be effective. Design rules for a good place to a start a broadband design with two front
matching layers are:
First matching layer ( acoustic impedance):
Z1 ≅ 7.0
Second matching layer:
Z2 ≅ 2.2
Layer thickness:
Thk ≅ 0.350 ( Vlayer/ Fa )
where Vlayer is the layer longitudinal acoustic velocity (downshifted) , and Fa is the diced ceramic element anti-resonant
For selecting the pitch of the array elements, typical design rules are:
Linear/Convex array:
Pitch = P ≅ 1.00 to 1.50 lambda
Sector Phased array:
Pitch = P ≅ 0.50 to 0.70 lambda
Principles for making broad band transducer designs, well matched to the system requirements were discussed.
Since a transducer operates in both mechanical and electrical domains, electrical tuning or impedance matching of
the probe is as important as the clever design of acoustic matching layers and backing, on achieving short impulse
response, over a broad range of frequencies, with good sensitivity.
Recently available computer simulation tools such as Pzflex 54 greatly assist in studying and understanding two and
three dimensional effects on array performance. This code is based on finite element analysis in the time domain,
and thus allows modeling of broadband pulses with computational efficiency 10 to 100 faster than traditional FEA
codes that operate in the frequency domain. Subtle transient effects that would be missed by a frequency domain
code can be predicted accurately by the time domain FEA code43. The two dimensional and 3-D effects, if not
controlled can cause unwanted mode coupling and spurious responses that seriously degrade pulse length and
frequency bandshape. These phenomena cannot be modeled directly in simpler computer simulation models such as
Piezocad53, based on the KLM model of a transducer 17 . Also, the KLM model is essentially a steady state
frequency domain type of calculation, and cannot predict all the effects of transient response and electrical tuning
that are accurately modeled by a time domain FEA code such as Pzflex (43,54) .
Image quality is a multi-parameter quantity, and depends on both the array transducer and the system performance,
and each should be designed and optimized with the other in mind. Advances in array technology and system
electronics are making possible ultrasonic real time imaging with significantly improved image quality. Image
quality is ultimately determined by and limited by transducer performance. Fundamental determinates of image
quality are axial and lateral resolution, frequency content, and signal processing. Clutter in beam sidelobes and
grating lobes can limit dynamic range and contrast resolution capability. Careful matching of the transducer array
element electrical impedances can optimize sensitivities and ultimate band widths achievable, so that 80 to 90 %
bandwidths are realizable. Recent progress in growing larger single crystals of relaxor-lead titanate with higher
coupling constants should make bandwidths over 100% realizable. Broad band, well behaved response in the
transducer array opens up many signal processing opportunities in the system. Examples are speckle reduction
through frequency compounding, image improvement via tracking filters, and image enhancement and clutter
reduction by use of harmonic imaging techniques.
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