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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5814_Библиотеки_им_академика_М_И_Перельмана
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47Chapter three: Ultrasonic transducers and arrays
EC
= 0.50
3
(a) (b)(c)
Table3.1 Properties of Important Piezoelectric Materials
Property PVDF
–12
d33 (10
c/n) 15 2.31 583 100
Quartz
(X-Cut) PZT-5H
Lead
Niobate
g33 (10–2 v-m/n) 14 5.78 191 430
ECC
Kκ (10
t
–11
F/m)
0.11 0.14 0.55 0.34
9.7 3.98 3010 3054
c (m/s) 2070 5740 3970 3100
ρ (kg/m3)
Curie temp (°C)
ρ and c denote density and sound speed, respectively.
1760 2650 7450 5900
100 573 190 500
electromechanical coupling coefcient is not necessarily 100% because
some of the energy is stored as mechanical energy and the rest is stored
dielectrically in the form of electrical potential energy. Since only the
stored mechanical energy is useful, the electromechanical coupling coefcient is a measure of the performance of a material as a transducer. The
piezoelectric constants for a few important piezoelectric materials are
listed in Table3.1.
There are a few important material geometries that are frequently
encountered in ultrasonic imaging. These are shown in Figure3.7 using
the effect of geometry on an electromechanical coupling coefcient as an
example. For a disc of PZT-5H with the diameter much larger than the
thickness (c), the thickness mode electromechanical coupling coefcient
For PZT-5H
C33 = 0.75
Figure 3.7 The effect of a piezoelectric material affects its piezoelectric properties. This is illustrated by way of the electrical mechanical coupling coefcient. (a)
A tall element of square cross section frequently seen in composite materials. (b)
A tall element frequently seen in linear arrays. (c) A circular disc frequently seen
in single-element transducers.
2
1
ECC
t
ECC33' = 0.69
48 Diagnostic ultrasound: imaging and blood ow measurements
ECCt is used. For a long tall element with the sides of the square cross section much smaller than the thickness (a) and a tall element with rectangular
cross section (b), the electromechanical coupling coefcients are denoted
as ECC33 and ECC33’, respectively. The geometry shown in Figure3.7(b) is
the most important in that it is often used in linear arrays. It is interesting
to note that the electromechanical coupling coefcient is highest in the bar
mode shown in (a), attributable to the fact that more energy is irradiated
in the long-axis direction because the bar is surrounded by air, which has a
much lower acoustic impedance than PZT. The effect of material shape on
other measured piezoelectric properties can be found in Table3.2. There
is an additional denition for ECC, the planar ECC or ECCp, which is
dened as the strain or displacement generated in the 1 or 2 direction of a
thin disc for an electric eld applied in the 3 direction.
Ferroelectric ceramics are often classied as soft or hard by their rigidity. In general, soft ceramics have larger piezoelectric constants, dielectric
constants, ECC, and loss, but poorer linearity. They are also easier to pole.
In addition to PZT, piezoelectric polymers have also been found to be
useful in a number of applications (Brown, 2000). One of these polymers
is polyvinylidence diuoride (PVDF), which is semicrystalline. After processes like polymerization, stretching, and poling, a thin sheet of PVDF
with a thickness in the order of 6 to 50 μm can be used as a transducer
material. The advantages of this material are that it is wideband, exible,
and inexpensive. The disadvantages are that it has a very low transmitting constant, its dielectric loss is large, and the dielectric constant is low.
Although PVDF is not an ideal transmitting material, it does possess a
fairly high receiving constant. Miniature PVDF hydrophones are commercially available. P(VDF-TrFE) co-polymers have been shown to have a
higher electromechanical coupling coefcient.
One of the most promising frontiers in transducer technology is
the development of piezoelectric composite materials (Smith, 1989).
Innovation in fabrication technology has allowed the preparation of PZT
polymer 1-3 composites for applications in the 1 to 7.5 MHz range. These
1-3 composites, consisting of small PZT rods embedded in a low-density
polymer, illustrated in Figure3.8, where the dark and light regions indicate the piezoelectric ceramic phase and the polymer phase, respectively,
Table3.2 Material Properties of PZT-5H of Different Geometries
Property Bar mode Plate mode Element mode
Velocity (m/s) 3850 4580 4000
Acoustic impedance
(Mrayl)
Electromechanical
coupling coefcient
28.9 34.3 30.0
0.75 0.51 0.73
49Chapter three: Ultrasonic transducers and arrays
Bulk
2-2 Composite
1–3 Composite
tric
Polymer
(a) (b) (c)
Piezoelec
Figure 3.8 Congurations of piezoelectric composites: (a) Bulk, (b) 2-2 composites, and (c) 1-3 composites.
have been used for low-frequency underwater applications for many
years. The notation 1-3, 2-2, etc., has been coined by Newnham et al. (1978).
A notation of 1-3 means that one phase of the composite is connected
only in one direction, whereas the second phase is connected in all three
directions. A notation of 2-2 means that both phases are connected in two
directions, as illustrated in Figure3.8. These composites, typically in volume concentration of 20 to 70% PZT, have a lower acoustic impedance (4 to
25 MRayl) than conventional PZT (34 MRayl), which better matches the
acoustic impedance of human skin. The composite material can be made
exible with a free dielectric constant, Kκ, from 44 ⋅ 10
–11
to 2213 ⋅ 10
–11
F/m ,
and with an ECCt higher than that of PZT. The higher coupling coefcient
and better impedance matching can lead to higher transducer sensitivity
and improved image resolution. An electron micrograph of a 2-2 composite is shown in Figure3.9 with 28 μm PZT-5H piezoelectric ceramics and
5 μm polymer ller. In designing composites, care must be taken to avoid
spurious lateral resonance modes. The height of a square pillar should be
5 µm wide
28 µm wide
PZT-5H
piezoelectric
Figure 3.9 An electron micrograph of a 2-2 composite consisting of PZT-5H piezoelectrics separated by polymer.
polymer
filled kerf
50 Diagnostic ultrasound: imaging and blood ow measurements
Table3.3 Piezoelectric Properties of Single-Crystal Piezoelectric Materials
Property PZN-PT PMN-PT PIN-PMN-PT
ECC
33
Tc, °C
ε
K
Z, MRayl 26 30 30
0.93 0.94 0.94
140 155 160
294 800 700
much larger than the width of the sides and the width of the kerf, which
is the gap among the pillars. The pillar and the kerf width should all be
much smaller than the wavelength. The lateral resonance modes are also
determined by the ller material, which must be carefully chosen. EpoTech 301-2 is a popular ller material that has a longitudinal velocity of
2650 m/s, a shear velocity of 1230 m/s, and a density of 1150 kg/m3. It also
has fairly large attenuation coefcients for longitudinal (9.5 dB/mm at
30 MHz) and shear (3.6 dB/mm at 30 MHz) waves.
One of the problems associated with composite materials is the higher
fabrication cost. Typical fabrication approaches include (1) dice and ll
where PZT is rst diced and subsequently the gaps are lled with a polymer, and (2) injection molding. The performances of composite annular
and linear arrays with frequencies from 3 to 7.5 MHz have been found to
be superior to those of similar PZT devices. A substantial number of commercial high-performance arrays are made from composites.
Conventional PZT has a grain size in the order of 3 to 5 μm, which
is not particularly suited for high-frequency applications. Fine-grain PZT
and single-crystal ferroelectric materials like Pb(Zn
(PZN-PT), Pb(Mg
Pb(Mg
1/3Nb2/3)O3
1/3Nb2/3)O3
-PbTiO3 (PIN-PMN-PT), which have been shown to
-PbTiO3 (PMN-PT), and Pb(In
1/3Nb2/3)O3
-PbTiO3
1/2Nb1/2
)-
have a higher dielectric constant and electromechanical coupling coefcient than conventional PZT, are potentially promising piezoelectric materials for high-frequency applications (Shrout and Fielding, 1990; Zipparo
et al., 1997; Tian et al., 2007). These materials possess extremely high ECC.
Their ECC33 was found to be as high as 0.9. Table3.3 lists piezoelectric
properties of these single-crystal materials. It is known that a number of
commercial clinical scanner probes at frequencies from 3 to 7 MHz now
are made from single-crystal piezoelectric materials, and they exhibit
superior bandwidth and sensitivity, measures of transducer performance
that will be discussed below.
3.3 Ultrasonic transducers
The simplest ultrasonic transducer is a single-element piston transducer
shown in Figure3.10, where (a) and (b) show, respectively, a photo and
the internal construction of a single-element ultrasonic transducer. The most
51Chapter three: Ultrasonic transducers and arrays
(a)
Lithium Niobate Transducers
(b)
Lens-focused
Epoxy
Lens
SMA
Connector
Brass
Housing
Conductive
Backing
Insulating Epoxy
with
LiNbO
Cr/Au Electrodes
3
Silver Epoxy
Matching Layer
Press-focused
Parylene
Layer
Figure 3.10 (a) Photo and (b) detailed construction of two single-element ultrasonic transducers with one or two matching layers and the backing material. The
transducer on the left has a lens, whereas the one on the right is self-focused.
important component of such a device is the piezoelectric element. A
number of factors are involved in choosing a proper piezoelectric material
for transmitting or receiving the ultrasonic wave. They include stability,
piezoelectric properties, and the strength of the material. The surfaces of
the element are electroded with red-on silver or sputtered chrome-gold.
The outside electrode is usually grounded to protect the patients from
electrical shock. The housing is metallic or plastic. An acoustic isolating
52 Diagnostic ultrasound: imaging and blood ow measurements
L
2
acoustic
Ba
acoustic
po
port
L
ck
rt
Electrical
Figure 3.11 A piezoceramic disc can be treated as an electromechanical device
with two acoustic ports representing the front and rear interfaces between the
ceramic and the surrounding media and an electrical port.
Front
A
port
material is placed between the piezoelectric element and the housing to
prevent ringing of the housing that follows the vibration of the piezoelectric element itself.
By considering the two surfaces of the piezoelectric element as two
independent vibrators, as illustrated in Figure3.11, one can easily see that
the resonant frequencies for such a transducer are given by
nc
p
=2f
o
(3.10)
with the lowest resonant frequency being n = 1, where cp is the acoustic
wave velocity in the transducer material, L is the thickness of the piezoelectric material, and n is an odd integer. In other words, resonance occurs
when L is equal to odd multiples of one-half wavelength, or
λ
=
Lnp (3.11)
where λp is the wavelength in the piezoelectric material.
The transducer can be treated as a three-port network, as shown
in Figure3.12, two being mechanical ports representing the front and
back surfaces of the piezoelectric crystal and one being an electrical
port representing the electrical connection of the piezoelectric material to the electrical generator (Kino, 1987). Various sophisticated onedimensional (1D) circuit models exist to model the behavior of the
transducer. The most well known are the Mason model, the Redwood
53Chapter three: Ultrasonic transducers and arrays
Mechanical port I
Me
Electrical port
Mechanical port IMechanical port II
port
chanical port II
F2 and u
2
Transducer
I1 and V
1
F1 and u
1
Figure 3.12 A system model for a single-element transducer.
model, and the KLM model. Commercial software based on these models is available.
The Mason model can be derived by considering the three-port conguration shown in Figure3.13 for a circular disc with area A and thickness L, where I, V, F, and u denote current, voltage, force, and medium
velocity, respectively.
Using the transmission line theory, the constitutive equations relating
mechanical properties to electrical properties, and boundary conditions
that u and F must be continuous across interfaces, the following simulta-
neous equations can be obtained:
cotcos
ZkLZ eckL
cp cp
F
1
coscot
Zeck LZkL
j
=−
F
2
V
3
cp cp
e
εε
K
ωωω
e
KC
e
ε
K
ω
e
ε
K
ω
1
0
u
1
u
I
(3.12)
2
3
where Zc = Z0A is called the radiation impedance and Z0 is the acoustic
impedance of the piezoelectric element, e is the piezoelectric stress constant,
kp is the wave number in the piezoelectric material, and C0 = Kε(A/L), the
Z
– Z
11
12
F2 and u
2
Figure 3.13 The Mason model or equivalent circuit for a single-element transducer.
Z
– Z
11
12
F1 and u
1
Z
12
n:1
–C
0
I
3
+
C
Electrical
V
0
3
–
54 Diagnostic ultrasound: imaging and blood ow measurements
Zk
cp
clamped capacitance. These simultaneous equations can be represented by
an electrical equivalent network, shown in Figure3.8, where Z11 = –jZccot(kpL),
Z12 = –jZccosec(kpL), Z11 – Z
= jZctan(kpL/2), and n = e(A/L).
12
Assuming that the loading media have acoustic impedances Z1 and
Z2, respectively, at ports I and II, the input electrical impedance Z3 of the
transducer at the electrical port can be readily calculated using this equivalent network.
Z
V
3
==
3
IjC
30
+−−
()sin2(1 cos)
1
+
1
ω
ECC
jZ ZZ kL
2
t
[( )sin ()cos]
12
2
ZZZkLjZZZkLkL
+−+
12 12
cp cpp
cp
2
L
(3.13)
For the special case where Z1 and Z2 = 0, this means a disc loaded by air on
both sides. Z3 becomes
1
=
Z
3
ω
jC
+
Z
(3.14)
a
0
where Za is given by
2
tan( /2)
ECC
=−
Z
a
t
ω
jCkLkL
0
p
(3.15)
/2
p
Equation (3.14) can be represented by an equivalent network consisting of an impedance Za in series with a capacitor C0. It can be shown
easily that Za → –ECC
2
/(jωC0) when ω → 0. Therefore, at low frequen-
t
cies a circular disc resonating in air basically behaves like a free capacitor
C
= Za + C0 = Kκ(A/L). As ω → ∞, Equation (3.14) yields Z3 → 1/(jωC0).
0κ
It behaves like a clamped capacitor, C0 = Kε(A/L). This phenomenon
has been used as a method for measuring Kε and Kκ of a piezoelectric
material.
A more careful examination of Equation (3.13) shows that the input
impedance has a minimum and maximum as a function of frequency
for a transducer irradiating into a medium with acoustic impedance Z1
with a backing medium of acoustic impedance Z2. These frequencies are
called, respectively, the series and parallel resonant frequencies. At these
frequencies the transducer can be represented by two-port networks consisting of a capacitor and a resistor. At series resonance or simply resonance, ω = ωr, Z3 is minimal, and the phase angle changes from –90° to 90°.
At parallel or antiresonance, ω = ωa, Z3 is maximal, and the phase angle
changes from 90° to –90°. These two networks are shown in Figure3.14(a)
55Chapter three: Ultrasonic transducers and arrays
()
2
12
ZZ
(
)
12
0
(a)
50 MHz LiNbO3 Transducer (6 mm Diameter)
(c)
Magnitude (Ohms)
Phase Angle (Degrees)
C
R
–40
–50
–60
–70
–80
–90
a
C
0
At series resonance
28
26
24
22
20
18
16
14
12
10
8
40 60 80 100
R
r
f
a
f
r
Frequency (MHz)
At parallel resonance
(b)
Magnitude
Phase
Figure 3.14 Equivalent electrical network for a single-element transducer near
resonance: (a) at series resonance and (b) at parallel resonance. (c) The magnitude
and phase of the input electrical impedance of a circular disc resonating in air as
a function of frequency.
and (b), where Ra and Rr are, respectively, the radiation resistances at parallel and series resonances and are given by
4
ECCZ
=
R
a
=
R
r
πω +
4
tc
C
a
0
π+
ZZ
ECCC
ω
Z
t2r0
(3.16)
(3.17)
c
A plot of the magnitude and phase of a circular disc resonating in air
as a function of frequency is shown in Figure3.14(c), where the resonance
and antiresonance frequencies and the change in phase near these frequencies are clearly seen. A more popular 1D transducer mode is the KLM
model, which is shown in Figure3.15. This model divides a piezoelectric
56 Diagnostic ultrasound: imaging and blood ow measurements
acoustic
Ele
po
Transmission lineTransmission line
–15.000
–65.000
10.000
–5.000
Pulse-Echo (Two-way) Impulse Response - V(Rx)/V(Tx)
1.000
Frequency in MHz
sinc
2ω
Front
port
ω
a
Back
acoustic
port
ctrical
rt
Z
, cp, L/2
c
C´
φ = k
ECC
t
C
0
t
C
0
φ: 1
C' = –
ωa C0 Z
sinc
2
Z
π
, cp, L/2
c
C
ω
ω
a
1/2
–1
ωa = anti-resonant frequency
Figure 3.15 The KLM model for a single-element transducer.
element into two halves, each represented by a transmission line. It is more
physically intuitive. The effects of matching layers and backing material
can be readily included.
A typical response for a PZT-5A single-element transducer of 1 cm
diameter, 5.5 MHz resonant frequency, loaded by water, and air backed,
obtained with the KLM model, is given in Figure3.16. Both the pulse-echo
5.000
V
o/Vi
mV/V
0.000
Figure 3.16 The pulse-echo waveform and its spectrum of a single-element transducer with no matching and backing obtained by the KLM model.
Time in Usec
3.320
V
dB
o/Vi
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