Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 16.3
For a wave incident at a water-honey interface, determine the reflection factor at 45
and at 50
.
The sound speeds and impedances for water (medium 1) and honey (medium 2) are 1.48 km/s and
2.05 km/s and 1.48 MegaRayls and 2.89 MegaRayls, given in Table 16.1.
Solution
From Snell's law, Eq. (16.16b),
78.4
. Then from Eq. (16.19a),
y
T
¼
arcsin[(2.05/1.48) sin 45]
¼
2
:
89
cos 45
1
:
48
cos 78
:
4
RF
¼
4
¼
0
:
746
2
:
89
cos 45
þ
1
:
48
cos 78
:
Before finding the result for 50
, it is worth finding the critical angle at which the incident wave
is directed along the boundary. This angle, found by setting
90
y
T
¼
in Snell's law, is known as
46.22
in this case. For an incident angle of 50
that is past the crit-
ical angle, the wave is just reflected and does not get transmitted, or
the critical angle, which is
y
I
¼
RF
¼
1.
16.2.3 Transducer Basics
The essential part of an ultrasound system is a means to generate and receive acoustic
waves. This function is performed by the transducer, which can convert electrical signals
to acoustic pressure waves and vice versa. Inside the transducer is a piezoelectric crystal or
ceramic that deforms when a voltage is applied and transmits acoustic waves. A reciprocal
piezoelectric effect, in which electrical charge is created by the mechanical deformation of a
crystal, allows the transducer to convert returning acoustic waves back to electrical signals.
The piezoelectric effect was discovered by the Curie brothers in 1880.
A simple model is presented that describes the basic acoustic and electrical characteris-
tics of a piezoelectric transducer. Consider a transducer to consist of a rectangular piece
of piezoelectric material with electrodes on the sides, as shown in Figure 16.8. Each side
has a cross-sectional area,
A
, with a top and bottom that are much longer (
>
10X) than the
thickness,
. Because piezoelectric material is dielectric, it is essentially a capacitor with a
clamped capacitance
d
e
S
A
=
d
C
¼
ð
16
:
20
Þ
0
e
S
is a clamped dielectric constant under the condition of zero deformation.
Because the crystal is a solid rather than a liquid, elastic waves are created internally.
The elastic counterpart of pressure is stress, or force per unit area, represented by the
symbol
in which
, the relative change in length of
the crystal divided by its original length. Both stress and strain vary with direction in a
piezoelectric material, but for the single direction of wave propagation, it is possible to
relate the stress to the strain as
T
. Another important elastic variable is strain,
S
T
¼
C
D
S
hD
ð
16
:
21
Þ
in which
is a piezoelectric constant. This equation is known as a modified Hooke's law, in
which the first term relates the stress to the strain and the second term relates stress to the
h
