surface. The disturbance of a surface wave parallel to the direction of wave
can be represented mathematically by propagation, for example when an
multiplying the amplitude of the wave by ultrasonic test is performed on a
a factor that causes the amplitude to cylindrical test object. When an ultrasonic
decay, or attenuate, in a direction transducer is placed on the end of the test
perpendicular to the surface (see the object, the wave is usually expected to
discussion of rayleigh waves elsewhere in propagate along the axis of the test object
this volume). (as always happens for isotropic materials)
and to have little or no interaction with
Propagation in Anisotropic the side wall of the test object. For an
Materials anisotropic material, the deviation of the
group velocity from the direction of
All metals and alloys are crystalline propagation, which will still be down the
materials, that is, their atoms or molecules axis of the material, can cause reflection
occupy a symmetric, defined position in from the side wall.
space under thermodynamic equilibrium
conditions. Most such materials used in Mathematical study of wave
engineering structures are elastically propagation in anisotropic materials leads
isotropic because they are composed of a to many interesting observations but is
very large number of randomly oriented, beyond the scope of this chapter.
small crystals. Elastically isotropic means
that their elastic properties are the same
in all directions. Some modern day
engineering structural materials, such as
advanced composites or specially prepared
single-crystal alloys, are especially
designed and used in engineering
structures to take advantage of their
elastic anisotropic properties. For
example, an advanced composite material
composed of unidirectional single fibers is
much stiffer elastically in the direction
parallel to the fibers than it is
perpendicular to the fiber direction. The
stiffness parallel to the fiber direction can
be five to fifteen times greater than that
perpendicular to the fiber direction. The
difference in mechanical properties as a
function of direction has a dramatic effect
on wave propagation. In anisotropic
materials, three distinct wave modes can
propagate. These modes are distinct in
that each of the three propagates with a
different phase speed. Each of these waves
also has a distinct direction of particle
vibration motion. Normally, one of these
waves will have a vibration disturbance
that is nearly parallel to the direction of
propagation (and therefore called the
quasilongitudinal mode) whereas the other
two modes have particle vibration
directions that are nearly perpendicular to
the direction of propagation (and are
called the quasi transverse modes). The
three directions of particle displacements
are always mutually perpendicular to each
other, just as the three coordinate axes of
a three-dimensional coordinate system are
mutually perpendicular. The
quasilongitudinal mode travels with a
phase speed that is almost always the
fastest of the three modes.
Another interesting effect of anisotropy
on wave propagation is that the group
velocity of the wave will travel in a
direction different than the direction of
propagation. This can lead to reflection
from a surface that one might think is
40 Ultrasonic Testing
PART 2. Wave Propagation in Isotropic Materials
Plane Body Waves (18) ∂σxz + ∂σyz + ∂σzz = ρ ∂2 uz
∂x ∂y ∂z ∂2 t
An infinite medium containing a linear,
elastic, homogeneous isotropic material where ux, uy and uz are the components of
has a constitutive equation (a relation the particle displacement vector and
between stresses and strains in the
material): where body forces are neglected.
( )( )(8) σxx = λ + 2μ ⑀xx + λ ⑀yy + ⑀zz Consider a plane wave of the form:
( )(19) ux = Ax cos kxx + kyy + kzz − ωt
(9) σyy = (λ + 2μ) ⑀yy + λ (⑀xx + ⑀zz ) ( )(20) uy = Ay cos kxx + kyy + kzz − ωt
( )( )(10) σzz = λ + 2μ ⑀zz + λ ⑀xx + ⑀yy ( )(21) uz = Az cos kxx + kyy + kzz − ωt
(11) σyz = σzy = 2μ⑀yz = 2μ⑀zy where Ax, Ay and Az are the components
of the amplitude of the displacement; kx,
(12) σzx = σx z = 2μ⑀zx = 2μ⑀xz ky and kz are the components of the wave
vector; and ω is the angular frequency
(13) σx y = σyx = 2μ⑀xy = 2μ⑀yx
(radian per second).
where positive ⑀ij represents components
of strains, σij represents components of These parameters have the same
stresses and λ and μ are Lamé’s constants.
physical and mathematical meaning as
This form is commonly called Hooke’s
law. Lamé’s constants are materials the corresponding terms for the
constants related to Young’s modulus E
and Poisson’s ratio ν by the relations: one-dimensional case discussed in the
introduction. The wave vector, in fact, has
a magnitude equal to the wave number k:
(22) k = kx2 + ky2 + kz2
where the wave number and the
frequency are related to the phase speed
of the wave by the relation:
(14) λ = Eν v = k
ω
(1 + ν) (1 − 2ν) (23)
(15) μ = E Other physical and mathematical
importance is attached to the wave vector.
2(1 + ν) The components (kx, ky, kz) are the
components of a vector to the plane of
The governing differential equation is the equal phase points. The wave travels
equation of motion for a continuum: through space so that the plane of equal
phase points moves so as to remain
(16) ∂σxx + ∂σxy + ∂σxz = ρ ∂2 ux parallel to itself. For this reason, the wave
∂x ∂y ∂z ∂2 t vector describes the direction of wave
propagation. Often, a unit vector ν is
defined by the relation:
(17) ∂σxy + ∂σyy + ∂σyz = ρ ∂2 uy (24) k = kv
∂x ∂y ∂z ∂2 t
Ultrasonic Wave Propagation 41
Here, ν is the direction of wave to obtain expressions for the stress
propagation and the magnitude of the k components. Finally, these expressions for
is the wave number, as stated above. the stress components are substituted into
the differential equations of motion for
With some mathematical the material (Eqs. 16 to 18). The resulting
manipulation, the relations between equations are algebraic and are known as
strain and displacements (Eqs. 25 to 30) Christoffel’s equations.
can be used to obtain the
three-dimensional wave equation:
( )(36) ⎤
(25) ⑀xx = ∂ux ⎡⎣⎢(λ + μ)vxvx + μ − ρv2 ⎥⎦ Ax +
∂x
(λ + μ) vx vy Ay + (λ + μ) vx vz Az = 0
(26) ⑀yy = ∂uy (37) (λ + μ)vy vx Ax +⎡⎣(λ + μ) vyvy +
∂y
(27) ⑀zz = ∂uz ( )μ − ρv2 Ay ⎤ + (λ + μ) vyvz Az = 0
∂z ⎦⎥
1 ⎛ ∂ux ∂uy ⎞ (38) (λ )+ μ vz vx Ax + (λ + μ) vzvy Ay +
2 ⎝⎜⎜ ∂y ∂x ⎟⎠⎟
(28) ⑀xy = + ( )⎣⎡⎢(λ + μ) vz vz ⎤
+ μ − ρv2 Az ⎥⎦ = 0
(29) ⑀yz = 1 ⎛ ∂uy + ∂uz ⎞ These equations are homogeneous in the
2 ⎜⎝⎜ ∂z ∂y ⎟⎠⎟ variable v2. If there is a nontrivial
(nonzero) solution for v2, then the
(30) ⑀zx = 1 ⎛ ∂uz + ∂ux ⎞ determinant of the coefficients of these
⎜ ⎟ equations must be zero. The determinant
2 ⎝ ∂x ∂z ⎠ shown as Eq. 39 (see below) provides the
values of wave speeds that can propagate
The three-dimensional wave equation in the given material. Mathematically and
governs wave propagation through the physically, it can be shown that Eq. 39
prescribed linear, elastic, homogeneous, always has real and positive roots, so that
isotropic material: there are always three real values of phase
speed that are solutions to Christoffel’s
equations.
(31) (λ + μ) ∂Δ + μ∇2ux = ρ ∂2 ux ( )Lvx vx + μ−ρ v2 Lvx vy Lvx vz
∂x ∂t 2
(39) Lvy vx ( )Lvy vy + μ−ρ v2 Lvy vz =0
∂2 uy Lvz vx Lvz vy ( )Lvz vz + μ−ρ v2
∂t2
(32) (λ + μ) ∂Δ + μ∇2uy = ρ where L is the sum of Lamé’s constants λ
∂y and µ.
(33) (λ + μ) ∂Δ + μ∇2uz = ρ ∂2 uz For isotropic materials, two of the roots
∂z ∂t2 of Eq. 39 are always equal so that there
are only two distinct values of wave
where Δ is the volume dilation of the speeds that can propagate in an isotropic
displacements given by the sum of the medium. The values of these roots are:
normal strains:
(40) vL = λ + 2μ
ρ
(34) Δ = ⑀xx + ⑀yy + ⑀zz (41) vT = μ
ρ
and ∇2 is the laplacian operator, defined
by the following set of derivatives:
(35) ∇2 = ∂2 + ∂2 + ∂2 If these roots are substituted for phase
∂x2 ∂y2 ∂z2 speed v in the set of homogeneous
equations (Eqs. 36 to 38), the
The assumed particle displacements corresponding directions of particle
displacement vectors that can propagate
(Eqs. 19 to 21) are substituted into Eqs. 31 with the two distinct wave speeds are
found. The first root has particle
to 33. Then, the resulting values of ⑀ij are displacements always parallel to the wave
substituted into Hooke’s law (Eqs. 8 to 13)
42 Ultrasonic Testing
vector. This mode is called longitudinal The longitudinal wave always
(because of the parallel relation between propagates at a speed faster than the
the particle displacement and the transverse mode because both λ and µ are
direction of propagation), dilational always positive. Typical wave speed values
(because it is associated with the dilation are given in Table 1.
or volume change that occurs locally as
the wave passes through a region) or Surface Waves
irrotational (because the displacement field
has no rotation field). Rayleigh Waves
The second root is the double root of It has been observed experimentally in a
the determinant equation. The variety of applications that large
corresponding particle displacement amplitude waves propagate in solid
vector is a solution to Eq. 35 for this value materials along the bounding surfaces.
of phase speed and can be shown to lie in These waves are constrained to lie near
any direction in the plane perpendicular the surface and hence expand in only two
to the wave vector. For this reason, this dimensions. Because of this fact, the effect
mode is called the transverse mode. of these waves can be felt at greater
Transverse mode is also referred to as shear distances from the wave source than the
(because the strain field associated with it three-dimensional body plane waves in
is pure shear), distortional (for the same the discussion of wave properties, above.
reason) or equivoluminal (because there is
no local volume change as the wave
propagates through a region).
TABLE 1. Acoustic parameters of typical materials.
Material ___V_e_l_o_c_it_y__(_k_m_·_s_–_1)___ Longitudinal Wavelength for Density
VL VT Acoustic Longitudinal (103 kg·m–3)
Metals
Aluminum uranium alloy Impedance Wave at 10 MHz
Aluminum, galvanized
Beryllium (106 kg·m–2·s) (mm)
Brass (naval)
Bronze, phosphor (5 percent) 6.35 3.10 17.2 0.635 2.71
Copper 6.25 3.10 17.5 0.625 2.80
Lead, pure 12.80 8.71 23.3 1.28 1.82
Lead, antimony (6 percent) 4.43 2.12 36.1 0.443 8.1
Magnesium 3.53 2.23 31.2 0.353 8.86
Mercury 4.66 2.26 41.8 0.466 8.9
Molybdenum 2.16 0.70 24.6 0.216 11.4
Nickel 2.16 0.81 23.6 0.216 10.90
Nickel chromium alloy (wrought) 5.79 3.10 10.1 0.579 1.74
Molybdenum alloy (wrought) 1.42 — 18.5 0.142 13.00
Silver nickel (18 percent) 6.29 3.35 63.5 0.629 10.09
Steel 5.63 2.96 49.5 0.563 8.8
Stainless steel, austenitic 7.82 3.02 64.5 0.782 8.25
Stainless steel, martensitic 6.02 2.72 53.1 0.602 8.83
Titanium 4.62 2.32 40.3 0.462 8.75
Tungsten 5.85 3.23 45.6 0.585 7.8
5.66 3.12 45.5 0.566 8.03
Nonmetals 7.39 2.99 56.7 0.739 7.67
Acrylic resin 6.10 3.12 27.7 0.610 4.54
Air 5.18 2.87 99.8 0.518 19.25
Fused quartz
Ice 2.67 1.12 3.2 0.264 1.18
Oil (transformer) 0.33 — 0.00033 0.033 0.001
Plate glass 5.93 3.75 13.0 0.593 2.20
Heat resistant glass 3.98 1.99 4.0 0.398 1
Quartz (natural) 1.38 — 1.27 0.138 0.92
Water 5.77 3.43 14.5 0.577 2.51
5.57 3.44 12.4 0.557 2.23
5.73 — 15.2 0.573 2.65
1.49 — 1.49 0.149 1.00
Ultrasonic Wave Propagation 43
As an example of two-dimensional (46a) α1 = k 1− v2
spreading, it is often observed that the vL2
major damage following an earthquake is
caused by waves that propagate at and:
velocities slightly slower than the phase
speed of the transverse body wave. To (46b) Az = ia1
model these waves mathematically, Ay k
Rayleigh suggested that they be
represented by the following equations: and:
(42) ux = (A1 e−αz cos ky − ωt )
(43) uy = A2 (e−αz cos ky − ωt ) (46c) Ax = 0
For mode 2:
(44) uz = (A3 e−αz cos ky − ωt )
(47a) α2 = k 1− v2
where A1, A2 and A3 are the amplitudes of vT2
the displacement field associated with the
wave and α is an attenuation factor. and:
The attenuation factor causes the wave (47b) Az = ik
amplitude to decay as the observer moves Ay α2
away from the boundary into the interior
of the body (that is, in the positive and:
Z direction) and the wave is assumed to be
propagating along the boundary of an (47c) Ax = not set
isotropic material in the direction of the
Y axis. The value of alpha must be positive When a wave with these characteristics
for the wave to be constrained to lie near is used to attempt to satisfy the stress free
the surface of the material; otherwise, the boundary equations, it is found that the
wave amplitude increases as the wave equations cannot be identically equal to
enters the material. The meanings of k zero unless the amplitude of the wave
and ω are the same as for the body waves itself is zero. Because experimental
in the discussion of plane body waves, observation says that such waves do
above. indeed exist, the mathematical modeling
must be adjusted to find an equation that
The wave having displacements given predicts more of the wave characteristics,
in Eqs. 42 to 44 must satisfy the equations particularly the phase speed and particle
of motion (Eqs. 31 to 33) and must satisfy displacement direction of the wave.
the boundary conditions along the surface
of the material. If the surface is a free Rayleigh suggested that both mode 1
surface, a good approximation for and mode 2 must be present
ultrasonic work in the laboratory, the simultaneously. When they are assumed
stresses on the surface must be zero. At to be so, the net displacement field is the
z = 0: sum of the two surface wave modes.
Then, if the boundary conditions of zero
(45) σx z = σyz = σzz = 0 stress are to be satisfied, the following
additional characteristics must be true for
At ultrasonic frequencies in the megahertz the surface (rayleigh) waves. That is, on
region and above, sound waves attenuate substituting the sum of the particle
rapidly in air. Ultrasonic waves traveling displacements of the two modes into the
in solid material almost totally reflect at a boundary equations, the following
boundary with air. For all practical conditions must be true on the boundary:
purposes, the wave is constrained to (1) A1 must be zero for mode 2 and (2) the
remain in the solid and the boundary is phase speed vR of the rayleigh wave must
considered to be in free space. Thus, the satisfy the equation:
assumption of zero stresses on the
boundary is very reasonable. (48) ⎡ vR2 ⎤3 ⎡ vR2 ⎤2
⎢ ⎥− 8⎢ ⎥+
When the assumed surface wave ⎢⎣ v 2 ⎥⎦ v 2 ⎦⎥
displacements (Eqs. 42 to 44) are T ⎣⎢ T
substituted into the equation of motion,
it is found that two possible modes may ( ) ( )8 ⎡ vR2 ⎤
propagate with the following wave 3 − 2γ2 ⎢ ⎥ − 16 1− γ2 =0
characteristics. For mode 1: ⎦⎥
⎣⎢ vT2
44 Ultrasonic Testing
where: FIGURE 5. Normalized longitudinal Uˆ and
transverse Wˆ displacements as function of
(49) γ= vT = 1 − 2v depth for plane rayleigh surface wave. Solid
vL curves indicate poisson ratio of 0.34; dashed
2(1 − v) curves indicate poisson ratio of 0.25.
It can be shown that Eq. 48 must have 1.0
a solution for nR·νT–1 that lies between
zero and one. Thus, nR < νT < νL and the Displacement (normalized) 0.8
attenuation coefficients for both modes Transverse wave
are therefore real and positive. Hence, the 0.6
surface wave has the desired 0.4
characteristics observed experimentally: 0.2
(1) the phase speed must be less than the 0 Longitudinal wave
transverse body wave speed νT and (2) the
attenuation coefficient is positive, thus –0.2
constraining the propagating wave to lie 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
in the vicinity of the bounding surface of z·λ–1
the solid. Legend
For a material with Poisson’s ratio ν
WˆWUˆˆDRRR = longitudinal displacement of rayleigh wave
having a value of 0.25, the phase speed of = transverse displacement of rayleigh wave
= transverse depth
the rayleigh surface is 0.91940 nT and the
corresponding values of α are:
(50) α1 = 0.84754 k z = depth in Z direction
λ = wavelength
(51) α2 = 0.3933 k
Note that the higher the frequency of the
surface wave, the larger is the value of the
wave number k (the wave number varies
inversely with the wavelength) and the
greater is the value of the attenuation
coefficients. Physically, this means that
higher frequency waves are constrained to
lie closer to the surface of the solid than
lower frequency waves. Thus, by varying
the frequency of the surface wave used in
an ultrasonic test, material properties may
be determined at varying depths in the
material.
The particle displacement associated
with the rayleigh wave has components
in both the Y and the Z directions. Hence,
rayleigh waves cannot be classified as
longitudinal or transverse. Rather,
rayleigh waves have a combination of the
motions associated with longitudinal and
transverse waves. Particle displacement on
the surface of the test object moves
through an elliptical path as one complete
cycle of the wave passes a point on the
surface (Fig. 4). This motion is similar to
the path followed by a buoy floating on
the surface of a lake as a water surface
wave passes. The normalized longitudinal
and transverse displacements for a plane
rayleigh surface wave are shown as a
function of depth in the material in
Fig. 5.
Ultrasonic Wave Propagation 45
PART 3. Extensions to Other Types of Surface
Waves
Leaky Rayleigh Waves (polymethyl methacrylate, for instance),
the rayleigh wave speed lies below the
Planar Fluid-to-Solid Geometry acoustic velocity of water. In this case, no
radiation damping of a rayleigh wave on
An important extension of the simple the plastic surface occurs and the
geometry analyzed above is the case of a radiation angle calculated from the
solid half space mechanically coupled to a expression above is purely imaginary.
fluid half space. In this circumstance, the Although there is particle displacement in
boundary conditions of Eq. 45 must be the fluid near the interface, the incipient
modified to account for continuity of the wave is evanescent and carries no energy
normal stress component σzz. In addition, into the fluid.
a wave potential must be introduced for
the acoustic wave in the fluid. The text The typical behavior observed in such
below details the physical behavior of a studies is indicated schematically in Fig. 6.
surface wave under these conditions. An incident beam of ultrasound strikes
the fluid-to-solid interface at the rayleigh
A plane rayleigh surface wave angle. The ensuing reflected field is
propagating on a half space in contact characterized by (1) a displacement of the
with a fluid couples its vertical beam weight center along the propagation
displacements to the fluid medium direction and (2) a redistribution of
through the normal stress boundary acoustic energy into two main lobes
condition. This periodic particle motion (Fig. 6). These lobes are separated by a
in the fluid can lead, under favorable null zone resulting from a phase
circumstances, to the excitation of an difference between portions of the
acoustic wave in the fluid leaving the reflected field. Beyond this region is a
surface at an angle determined by Snell’s trailing field in which the amplitude
law. That is, the X components of the two decreases exponentially with propagation
wave vectors (surface and acoustic wave) distance. For comparison, a specular
are identical. Another way to visualize the reflection (ignoring diffraction) is shown
situation is to imagine the surface wave as dashed lines in Fig. 6.
crests to be associated with a particular
phase point on the acoustic wave. At One approach to solving this problem3
some instant of time, the phase difference assumes that the main contribution to the
between successive points along the reflected field arises from a simple pole of
acoustic wave vector is 2π, if the angle at the reflection coefficient lying in the
which this wave leaves the solid surface is complex wave vector plane and
given by sin–1 (νf·νR–1), where νf is the
fluid wave speed and the angle is FIGURE 6. Typical geometry for leaky wave
measured from the surface normal. studies: transducer position or frequency
may be varied. Dashed lines indicate
Time reversal invariance implies the specular reflection. When a leaky wave is
existence of the inverse phenomenon, present, sound energy is concentrated in
namely the generation of rayleigh surface shaded regions with trailing exponential
waves by means of an acoustic beam decay. The null zone is a region of phase
incident on the fluid-to-solid interface at cancellation.
the angle indicated above, called the
rayleigh angle. The two effects combine to Transmitter Receiver
produce an apparent displacement and
distortion of the reflected beam, Fluid
characteristic of this type of acoustic Layer
surface wave interaction. First reported in
1950,2 this effect has since been explained Solid
in full theoretical detail.3 Numerous
experimental studies have also been Null zone
carried out.4-8 Leaky wave field
The surface wave is radiation damped
by the leakage of energy into the acoustic
mode — hence the name leaky waves. For
elastically soft materials such as plastics
46 Ultrasonic Testing
corresponding to the solution of Eq. 48 As observed10 and modeled,11 this
generalized to include the fluid. By effect was given a full explanation in 1973
integrating analytically over the product in an analysis of acoustic beam reflection
of reflection coefficient and incident from fluid-to-solid media.3 In the lossless
beam profile, an expression is obtained case, the rayleigh pole zero pair (complex
for the reflected field in terms of rational wave vector values where the reflection
and tabulated functions: coefficient approaches infinity or zero,
respectively) sits across the real axis in the
(52) F = FSP + FLW complex plane with the pole in the first
quadrant and the zero in the fourth, each
FSP(x,z) = − Γ0 ⎛ −x2 ⎞ equidistant from the real axis. Rising
⎜ ar2 ⎟ absorptive losses in the solid cause the
⎠ pair to migrate to higher imaginary values
e⎝ until the zero eventually crosses the real
(53) axis from below. At this point, the plane
η ar cosθ wave acoustic reflection amplitude at the
rayleigh angle vanishes and all wave
× ei kf (x sin θ−z cos θ) energy leaks into the solid. The calculated
magnitude of the reflection coefficient11
(54) FLW (x,z) = − 2FSP ⎣⎡1 − η ar for a water-to-stainless steel interface is
shown in Fig. 7. For losses higher than the
× e γ2 er f c ( γ) ⎤ critical value, the acoustic reflection
⎥ increases but the distortion and lateral
Δs ⎦⎥ shift of the beam no longer occur.
(55) γ = ar − x Layered Half Space
Δs ar
Another complicating feature on the basic
where ar is a factor related to the incident fluid-to-solid geometry is the addition to
beam width, erfc(γ) is the complex the half space of a solid isotropic layer.
complementary error function,9 FLW is the The layer differs from the half space in its
leaky wave portion of the field, FSP is the elastic properties and may be of any
specular portion of the field, kf is the fluid thickness (Fig. 8). In this case, the surface
wave vector, Γ0 is a wave potential wave speed becomes dispersive or
amplitude and ΔS is a parameter frequency dependent. At low sound
depending only on material constants and frequency, the wavelength of the rayleigh
relating to the degree of coupling between FIGURE 7. Magnitude of reflection coefficient
for a steel to water interface with varying
acoustic and surface waves. values of transverse wave attenuation in
units of nepers per wavelength (Np·λ–1). At
The coordinates X and Z have their critical value of 0.073 Np·λ–1, reflected
amplitude falls sharply to zero. Higher or
origin at the intersection of the solid lower attenuation results in partial
reflection.11
surface and incident beam center.
0.0003
Although complicated, Eqs. 52 to 55 show 1.0
the only analytical expression calculated 1.0
so far for the reflected field of a leaky 0.8
rayleigh wave. In addition, the result of 0.3
0.6
those equations requires the assumption
0.073
of an incident gaussian beam profile for 0.4
its derivation.
Absorptive Losses in Solid Reflection coefficient 0.2 0.04
(normalized magnitude)
Several additional features may now be 0
added to the planar fluid-to-solid 0.4 0.5 0.6
geometry to increase its generality. The (25) (30) (35)
first feature is absorptive losses in the
solid. As the attenuation in the solid Angle of incidence,
medium increases with frequency, the rad (deg)
behavior of the reflected field increasingly
deviates from that expected on the basis
of lossless theory. At angles of incidence
higher than the transverse critical angle,
all energy is reflected if losses are ignored.
As the attenuation in the solid approaches
0.073 Np·λr–1, there is a rapid onset of
energy leakage into the solid,
accompanied by a consequent reduction
in the acoustic reflected field amplitude.
Ultrasonic Wave Propagation 47
wave is much larger than the layer substrate and the velocity dispersion takes
thickness. The sound wave speed is the form shown in Fig. 9, calculated for
thereby controlled primarily by the elastic silicon on a zinc oxide substrate. At zero
properties of the substrate and the layer layer thickness, the surface wave speed is
may be viewed as a perturbation to the indeed that of the substrate and as the
elastic environment of the surface. thickness or frequency increases, the
effective wave speed approaches the value
At the other frequency extreme, the appropriate for the layer. However, near
wavelength is much smaller than the kd = 0.8, the mode ceases to propagate
layer thickness and the surface wave is when the wave speed reaches the
hardly affected by the presence of the substrate transverse velocity. This effect is
substrate. Instead, the layer properties connected to the fact that, near the
essentially determine the speed of the cutoff, the vertical displacement
wave. At intermediate values of the amplitude decays very slowly away from
wavelength-to-layer thickness ratio, a the surface. If the wave does not become
proportionate mixing of the elastic leaky into the substrate, it must cease to
constants could be expected. This simple propagate.
physical representation is not completely
accurate, as discussed below. When the layer wave speed is lower
than the substrate’s, the dispersion
A complete theoretical exposition of follows the intuitive picture constructed
various effects involving the layered half above. These results for a zinc oxide layer
space in vacuum is given in the on silicon are shown in Fig. 10. Here, the
literature.12 The elastic wave propagation two limiting values of the curve are the
problem is formulated and solved for a rayleigh wave speed of the substrate and
wide variety of cases, including layer, as expected. This loading case is
anisotropic and piezoelectric materials. marked by the existence of higher order
Details of the phenomena are presented modes, not shown in Fig. 10. The first of
graphically and the reader is referred to these is called the sezawa wave, with the
this and other sources for thorough additional excitations simply being
treatments of the factors complicating numbered.13 The higher order modes
surface wave propagation.3,12-16 differ from the rayleigh mode in several
respects, including their asymptotic
For the layered half space, the wave approach to the layer transverse wave
displacements of Eqs. 42 to 44 and the speed at large kd. To derive these curves,
boundary conditions of Eq. 45 must be zeros of complicated transcendental
generalized to the two elastic media of the equations must generally be sought, a
layer and half space. Expressing Eq. 45 in delicate problem in nonlinear
terms of wave displacements leads to a set optimization best left to a high speed
of simultaneous linear equations in the computer.
wave potentials. For there to be solutions
of these homogeneous equations, the FIGURE 9. Velocity dispersion of rayleigh
determinant of the coefficients must wave on stiffened half space as function of
vanish. The wave vector values that satisfy kd. Phase velocity is bounded by the
this requirement are then roots of the substrate Cr and transverse wave speed.
suitably generalized secular equation, Materials are silicon on zinc oxide.12
similar to Eq. 48. In this geometry, the
roots are once again real (because no fluid Substrate Group
is present) and may be inverted to give transverse velocity
the phase velocity of the surface waves. velocity
There are two distinguishable cases: the
layer rayleigh wave speed can be either
higher or lower than that of the substrate.
A higher layer wave speed stiffens the
FIGURE 8. Layered half space geometry. Velocity (km·s–1) 2.85
2.80
X 2.75 Phase velocity
2.70
Fluid Z 2.65
Isotropic Substrate rayleigh velocity
layer
Substrate 0 0.2 0.4 0.6 0.8
Product of wave number k
and plate thickness d
48 Ultrasonic Testing
Fluid Coupled Layered Half Δs = 2
Space Im ξp
( )(56)
To a layered half space geometry, an
additional feature may be added — a fluid Examples of experimental data,
in place of the vacuum. With appropriate accompanied by theoretical predictions,
modification of the boundary conditions, for acoustic beam reflection from layered
the wave displacement technique now half spaces are shown in Figs. 10 and 11.
yields a 7 × 7 matrix containing all The material system in both cases is
mechanical motion of the layer, half copper on stainless steel. The products of
space and fluid. Leaky rayleigh waves frequency and layer thickness are nearly
have been studied in both loading and equal. In Fig. 11, the nonspecular nature
stiffening layers and beam profiles have of the reflected field is evident whereas
been measured and elicited.17-21 As in the the trailing leaky wave field can be seen as
case of the simple fluid coupled half a slower decay to zero amplitude on the
space, the fluid has little influence on the right side. At an incident angle only
wave speed dispersion but does cause the 3 mrad (0.2 deg) different from the
wave vector to become complex because rayleigh angle in Fig. 12, the deep null
the surface wave radiates energy into the expected at this value of frequency and
fluid as it propagates. layer thickness has nearly disappeared. In
both of these examples, the theoretical
Following the procedures of earlier model19-21 extending the original
studies,3 it has been demonstrated that calculation3 is in good agreement with the
the influence of the layer on finite beam measurements.
reflection in the lossless case is isolated in
the dispersive rayleigh wave pole.22 This Transverse Horizontal
result implies that the analytical Waves
expression of Eqs. 52 to 55 can be
generalized to the layered half space by Another topic deserves mention because
substituting the appropriate incident of its applicability to electromagnetic
angle and generalized, frequency acoustic transducers (EMATs).23,24 By
dependent values of Δs. If the incident restricting particle displacements to the
angle is different from the rayleigh angle, Y axis, a different type of wave motion is
Δs decreases rapidly and specular observed. In this case, Eqs. 42 to 44 are
reflection is soon restored. The analytical written as:
connection between Δs and the simple
half space calculation is straightforward.3 FIGURE 11. Reflected field amplitude as
This parameter is known as the schoch function of coordinate for copper layered
displacement2 and is related to the steel half space at rayleigh angle incidence.
complex rayleigh wave pole through: Experimental measurements are plotted as
discrete points. Solid curve is model
FIGURE 10. Velocity dispersion of calculation. Product of frequency times
fundamental rayleigh mode on loaded half thickness is 0.34.20
space as function of kd. Phase velocity is
bounded by substrate and layer Cr. Higher 1.0
order modes are not shown. Materials are
zinc oxide on silicon.12 0.8
Substrate transverse velocity
5 Rayleigh velocity
4
Phase velocity (km·s–1) 0.6
Amplitude (volts)
0.4
3 Transverse velocity
Rayleigh velocity
Layer 0.2
0 12 34 0 –20 0 20 40 60 80
–40
Product of wave number k
and plate thickness d Position (mm)
Ultrasonic Wave Propagation 49
(57) ux = uz = 0 development similar to that for the
and:
rayleigh wave and by noting that the
surface tractions σyz vanish at the top and
bottom of the plate of thickness d:
(58) uy = Ay e−αz cos (ky − ωt ) k2 ⎛ ω ⎞2 ⎛ πn ⎞2
⎜⎜⎝ vT ⎠⎟⎟ ⎝⎜ d ⎠⎟
(60) = −
where there is only a single nonzero where d is the thickness (meter) of a free
displacement amplitude. The appropriate plate in which these waves are
stress tensor element is: propagating (meter); n is a positive integer
and nT is the infinite medium transverse
(59) σy z = π ⎛ ∂uy + ∂uz ⎞ wave speed (meter per second);
⎜⎝⎜ ∂z ∂y ⎠⎟⎟
Traveling wave modes for this
The wave implied by the above conditions excitation exist for all values of n, such
is a horizontally polarized transverse that the right hand side of Eq. 60 is
wave, also known as a transverse horizontal greater than zero. Otherwise, the resulting
wave. Propagation at the transverse wave imaginary wave vector k corresponds to a
speed in the bulk or parallel to the surface standing wave solution. Depending on
of a half space is possible because no whether the number of particle
displacements are normal to the surface. displacement modes in the plate is even
The wave has convenient properties in or odd, the transverse horizontal modes
reflection at interfaces, making it will be symmetric or antisymmetric,
attractive in certain applications. The respectively. From Eq. 60, it can be seen
addition of a fluid to the solid half space that in the case of the lowest order
geometry is irrelevant in this case, where symmetric mode (N = 0) there is
the absence of vertical surface dispersionless propagation at the
displacements means there will be no transverse wave speed. All other modes
coupling between transverse horizontal are dispersive. The group velocity, which
waves and acoustic waves in the fluid. can be calculated from Eq. 60, is shown
for several transverse horizontal plate
As a preliminary step to the treatment waves in Fig. 13.
of a related wave type on a layered half
space, note how this wave behaves in a If the free plate in the above example is
free plate. An analytical dispersion in welded contact to a half space of
relation25,26 for this type of wave can be differing elastic properties, its wave
determined by following a line of characteristics, under the assumption of
Y axis displacements only, change
FIGURE 12. Reflected field amplitude considerably. The plate still functions as a
distribution for copper on steel with wave guide, as in the case of transverse
fd = 0.41, where f is frequency and d is plate
thickness. Open circles are data and solid FIGURE 13. Dispersion of group velocity for
curve is theory. Incident angle is 3 mrad several transverse horizontal plate waves.
(0.2 deg) different from rayleigh angle, Fundamental mode is nondispersive.
giving rise to much weaker null zone.19
1 n=0
1.0 n=1
n=2
0.8
n=3
0.6 0.8 n = 4
Amplitude (V) 0.6
Cg·(VT)–1
0.4
0.4 0.2
0.2 01 2 3 4 5
0 0 20 40 60 80 k·d·π –1
–40 –20
Legend
Position (mm) Cg = group velocity
d = distance
k = wave number
vT = transverse wave speed
50 Ultrasonic Testing
horizontal plate waves, but the cutoff behavior that occur when the material
behavior noted for the free plate is properties are anisotropic. Various
replaced by transitions to leaky waves for phenomena occur in piezoelectric
modes that exceed critical wavelengths. materials combined with conductors or
Furthermore, this wave type can insulators. New wave types are observed
propagate only if the layer transverse and familiar modes exhibit unusual
wave speed is less than that of the characteristics.
substrate. A dispersion relation for these
waves has been derived:14 There are also further possible solutions
in the leaky wave case where a slow
⎡ ⎛ ω ⎞2 ⎤ μ k2 − ⎛ ω ⎞2 guided mode at the fluid-to-solid interface
tan ⎢⎢d ⎜⎜⎝ vˆT ⎟⎠⎟ ⎥ μˆ ⎜ vˆT ⎟ (known as the scholte wave) can exist.28
(61) − k2 ⎥ = ⎝ ⎠ These and other surface wave types are
⎣⎢ ⎦⎥ covered in the literature.25,26,29-33
⎛ ω ⎞2 − k2
⎜ vˆT ⎟
⎝ ⎠
where k is the wave vector of the
disturbance, μ is the transverse modulus
(ratio of shearing stress to shearing strain)
and ω is the circular frequency (radian per
second).
The circumflex (^) denotes the layer
properties. By way of illustration, Fig. 14
shows transverse displacements for the
fundamental wave, known as a love
mode,27 for several values of kd. As kd
approaches zero, the phase velocity tends
to the transverse wave speed of the
substrate. At large kd values, the
excitation is confined to the layer and
propagates at a phase velocity near b for
the layer. The presence of the tangent
function in Eq. 61 leads to the existence
of higher order modes, all of whose phase
velocities are bounded by b and b^.
Other Elastic Excitations
There are many more elastic excitations
connected with surface wave propagation,
including interesting variations in
FIGURE 14. Transverse displacement as function of depth for
fundamental love mode at several values of kd, where k is
wave number and d is plate thickness. Materials are gold on
fused quartz.12
1 Layer Substrate
Amplitude (V) 0.5 kd = 0.5
kd = 20 1.4
2.4
6.0 3.5
1.0 0.5 0 –0.5 –1.0 –1.5
Transverse displacement z·d–1, where z
is distance in Z direction and d is depth
Ultrasonic Wave Propagation 51
PART 4. Reflection at Plane Boundary in Stress
Free Media
The simplest problem that can be treated mathematical statement leads to the
for an ultrasonic wave incident at a important physical principle known as
material surface is that of a plane wave Snell’s law for reflection or refraction. The
incident at a free, plane boundary in an condition relates the phase speeds of the
isotropic material. Because the material is incident and reflected (or refracted waves
isotropic, any convenient coordinate in the situation where the boundary
system may be used. For the discussion surface separates two different media) to
below, the YZ plane is considered the the angles of incidence and reflection, as
plane of incidence. The incident plane shown below by Eq. 66.
(see Fig. 15 for the coordinate system)
contains the normal n to the plane Incident Longitudinal
boundary and the wave vector of the Wave
incident wave k.
An incident longitudinal wave has particle
The problem can be considered in displacements parallel to the direction of
three parts: (1) an incident longitudinal the wave vector k. Direction of k is given
wave (called a pressure wave in relative to the normal n of the plane
geophysics); (2) an incident transverse boundary by the incident angle θI. Hence,
wave with particle displacements parallel the particle displacements of the incident
to the boundary surface (a transverse wave may be written as:
horizontal wave); and (3) an incident
transverse wave with particle (62) uI = ⎣⎡j sin θI + k cos θI⎤⎦ AI
displacements lying in the incident plane
(a transverse vertical mode). Any incident × eωt −ky y+kzz
transverse mode can be resolved into a
component parallel and a component where j is unit vector parallel to the
perpendicular to the plane of incidence. Y axis, k is unit vector parallel to the
Z axis, ky = kI sin θI and kz = kI cos θ.
Any incident transverse wave also may
be resolved into such components and This displacement field must satisfy the
each component can be treated separately equations of motion (Eqs. 31 to 33) and
(see below). All equations that follow are must simultaneously satisfy the stress free
linear so the complete displacement field boundary conditions on the surface of the
is the sum of the displacements found in material. A longitudinal wave with
the incident transverse horizontal mode displacements in the form of Eq. 59
and the incident transverse vertical mode. satisfies the motion as long as the velocity
of the wave is given by Eq. 40. However,
Whatever type of wave is incident on this displacement field cannot by itself
the boundary, consideration of the stress satisfy the stress free boundary conditions.
free boundary conditions requires that the
phase of each plane wave (both incident To satisfy the conditions of no stress on
and reflected) be equal at all positions the reflection boundary, it is necessary to
along the boundary and for all time. This have the reflection of two independent
waves because there are two independent
FIGURE 15. Definition of terms for boundary conditions — the stress
longitudinal wave incident on plane stress component σzz and szy must
free boundary. simultaneously be zero. In fact, the third
stress component σzx on the surface must
π, Y kT also be zero. It can be shown that if the
incident wave has a wave normal in the
kI θT kL YZ plane as assumed, szx will be
θI θL identically zero and this condition does
not offer any further constraints on the
Z problem solution. Because two types of
body waves can satisfy the equation of
motion for an isotropic material, consider
the possibility that two reflected waves
result from the incident longitudinal wave
(recall that two independent reflected
52 Ultrasonic Testing
waves are needed to satisfy the boundary (67) R = sin 2θI sin 2θT − κ cos2 2θT
conditions). The reflected waves are then sin 2θI sin 2θT + κ cos2 2θT
a longitudinal wave at angle θL:
(68) T = 2κ sin 2θI cos2 2θT
(63) uL = ⎣⎡j sin θL + k cosθL ⎦⎤ AL sin 2θI sin 2θT + κ cos2 2θT
× eωt −kPy +kPz
where R is the reflection coefficient for
and a reflected transverse wave S at the the reflected longitudinal wave, T is the
angle θT (see Fig. 15): reflection coefficient for the reflected
transverse wave, κ is the ratio of the
(64) uT = ⎣⎡j sin θT + k cosθT ⎤⎦ AT longitudinal wave speed to transverse
× eωt −kPy +kPz wave speed and ν is Poisson’s ratio for the
material.
The quantities involved have the same
physical and mathematical meanings as (69) κ = vL2 = 2 (1− v)
those given for Eq. 67. The subscripts v 2 1−v
have been changed to indicate different T
waves of interest. The boundary
conditions can now be written as in Figure 16 provides a typical solution
Eq. 65 at z = 0: for the reflection coefficients of an
incident longitudinal wave in a material
(65a) σzz = 0 having a Poisson’s ratio of 0.25.
and:
(65b) σzy = 0 Incident Transverse
Horizontal Mode
When substituting into these equations
using the constitutive equations (Eqs. 8 to An incident transverse mode with particle
13), the strain displacement relations displacements parallel to the reflecting
(Eqs. 25 to 30), the displacement fields for boundary is considered a transverse
the incident longitudinal wave (Eq. 62), horizontal mode. The reflection problem
the reflected longitudinal wave (Eq. 63) for this mode is comparatively simple.
and the reflected S wave (Eq. 64), a system Write the displacement field as:
of two algebraic equations is obtained.
These equations provide two important (70) uSH = i ASH eωt −kyy+kzz
mathematical and physical conclusions
concerning the nature of the reflected Assume that the reflected waves are a
waves. First, in order that the equations longitudinal wave of the type in Eq. 63
be satisfied for all time and position along and a transverse wave of the type in
the boundary, the following set of Eq. 64 with potentially an additional
equations must be satisfied: component in the direction of the X axis:
(66) vI = vL = vT
sin θI sin θL sin θT
These equations are the mathematical FIGURE 16. Reflection coefficients for longitudinal wave
expression for Snell’s law. They state that incident on plane stress-free boundary for material with
the incident and reflected waves must Poisson’s ratio of 0.25.
propagate along the boundary at the same
phase speed. Note that the equations state Reflection coefficient magnitude 1.20 Transverse T
that the component of the phase speed of
each wave parallel to the plane boundary 0.56
is equal.
–0.08 Longitudinal R
The second important property –0.72
obtained from these algebraic equations
provides information on the relative size –1.36
of the amplitude of each reflected wave
compared to the incident wave. The –2.00
amplitude ratios of the reflected 0 17 33 50 66 83 100 116 132 150
longitudinal wave (AL·AI–1) and the
reflected transverse wave (AT·AI–1) (0.95)(1.9) (2.8) (3.8) (4.7) (5.7) (6.6) (7.6) (8.5)
compared to the incident wave amplitude
are called the reflection coefficients: Incident angle θI, mrad (deg)
Ultrasonic Wave Propagation 53
( )(71) uT = i AT1 + j sin θT + k cosθT AT2 Because vL is always greater than vT, the
ratio of the phase speed on the right hand
× eωt −kTy +kTz
side of Eq. 75 is always greater than 1.
After substitution into the same stress free
boundary conditions, it is found that AL There is a real solution for θL only for
and AT2 must be identically zero, that AT1 values of sin θT in which the product of
must be identically equal to ASH and that sin θT and the ratio of the phase speeds
the angle of the reflected wave must equal vL·vT–1 is less than 1. In fact, there is a
the angle of the incident wave. That is, an single value of the incident angle θT that
incident transverse horizontal mode is makes the right hand side of Eq. 75
reflected as a transverse horizontal mode,
no mode conversion occurs and the exactly 1:
amplitude of the reflected mode equals
the amplitude of the incident mode. (76) θ TC = sin−1 vT
vL
Incident Transverse For all incident angles less than θTC,
Vertical Mode there are real solutions for θL from Eq. 75.
For all incident angles greater than θTC,
This is the most complicated of the three no real solution for θL exists and there can
incident wave problems. The problem be no reflected longitudinal body wave.
must be further categorized into the The angle θTC is called the critical angle
ranges of incident angles smaller than a reflection because it separates the reflection
specific angle called the critical angle and
incident angles greater than the critical problem for an incident transverse vertical
angle. Assume an incident transverse
wave with particle displacements lying mode into two regions depending on the
parallel to the incident plane:
incident angle. There is a region where
both longitudinal and transverse (shear
vertical) are reflected and a region where a
transverse vertical mode and a surface
mode are reflected (for incident angles
below and above the critical angle,
respectively).
(72) uI = ( j sin θI + k cosθI ) AT2 Incident Angles Less than
Critical Angle Reflection
× eωt −kIy +kIz
For a transverse vertical wave incident at
The critical angle for reflection (and all angles below the critical angle, both a
refraction) of a wave is defined by Snell’s reflected longitudinal mode and a
law as the angle of incidence for which reflected transverse vertical mode
the angle of the reflected longitudinal generally occur. Depending on the value
mode becomes 0.5 π. For any incidence of Poisson’s ratio, there are for some
angles greater than the critical angle, no materials specific values of incident angles
reflected longitudinal mode can occur. when the amplitude of the reflected
The boundary conditions, still numbering transverse vertical mode is zero. In all
two, cannot be satisfied by the reflection cases, there is always a reflected
of the two types of body waves and it is longitudinal mode for this range of
necessary, both physically and incident angles. With stress free boundary
mathematically, for a nonbody wave to be conditions, there is a set of reflection
reflected. The only other possible type of coefficients for this problem as follows:
wave is a surface wave similar to the
rayleigh wave. (77) R = 2κ sin 2θT sin 2θT
By Snell’s law (Eq. 66), the relation 2sin 2θL sin 2θT + κ cos2 2θT
between reflected angles, incident angles
and phase speeds is:
(73) vI = vL = vT 2sin 2θL sin 2θT − κ cos2 2θT
sin θI sin θL sin θT 2sin 2θL sin 2θT + κ cos2 2θT
(78) T =
When the incident wave is a transverse All parameters have the same definitions
wave, this relation becomes: as given above for the reflected
longitudinal mode. This problem, with
(74) vT = vL the exception of the appearance of the
sin θT sin θL critical angle phenomenon, is symmetric
in results to the reflected longitudinal
or: mode discussed above.
(75) sin θL = sin θT vL First, there is mode conversion on
vT reflection of a longitudinal mode. Second,
54 Ultrasonic Testing
note the reciprocity between the values of The phase angle ζ is given by:
the reflection coefficients (compare
Eqs. 77 and 78 with Eqs. 67 and 68). (80) ζ = tan−1 − cos2 2T
Finally, note that Snell’s law is identical
with Eq. 66 as long as the appropriate ÷ ⎛ sin2 θT − 1 ⎞
wave speed and incident angle are used ⎝⎜⎜2sin θT sin 2θT k ⎟⎠⎟
for the incident mode.
One of the major differences between
Incident Angles Greater the reflected surface wave and the
than Critical Angle rayleigh wave is the fact that the phase
Reflection velocity of the reflected surface wave is
not less than that of the transverse mode
For a transverse vertical wave incident on in the material. In fact, the phase velocity
a stress free boundary at incident angles of the reflected wave must vary with the
greater than the critical angle given by angle of incidence as given by the
Eq. 76, a reflected longitudinal mode equation:
cannot occur. Because there are still two
independent, not identically zero, (81) vS = vT
boundary conditions on the stress sin θT
components that must be satisfied, it is
both physically and mathematically The amplitude of the surface wave
necessary to postulate the reflection of a
second mode, in addition to a reflected decreases exponentially with distance
transverse vertical mode. The governing
equation of motion allows only the away from the surface according to the
longitudinal wave and the transverse factor e–αz where the attenuation
wave to propagate in a linear elastic coefficient α is:
medium. There is no physically
distinguishable difference between the (82) α = k2 1− vS2
transverse vertical and transverse vT2
horizontal mode as far as the interior of
the body is concerned. Hence, the only Finally, the wave number of the reflected
possible second reflected mode is a mode surface wave is found from the relation:
that cannot propagate into the interior
but must be confined to propagation (83) k2 = k sin θT
along or near the boundary surface.
The values of α and the wave number k2
By using the displacement fields are also distinctively different from the
assumed for the incident and reflected corresponding relations for the rayleigh
transverse vertical modes (Eqs. 63 and 64, wave. Note again that all of these
respectively), a surface wave with relations are mandated by the solution to
displacements of the form assumed by the equation of motion and the boundary
Rayleigh (Eqs. 42 to 44) in the equations conditions of stress on the reflecting
of motion (Eqs. 31 to 33) and in the boundary. Figure 17 presents sample
boundary conditions for stress (Eq. 45),
then a set of algebraic equations is found FIGURE 17. Reflection coefficients for transverse vertical wave
to characterize the exact forms of the incident on plane stress free boundary for material with
wave displacements. It is found, for Poisson’s ration of 0.25.
example, that the reflected transverse
vertical wave has a reflection coefficient 1.20Reflection coefficient magnitudeT S
equal to –1 in this case. On reflection, the 0.56 Phase angle (rad)R
wave also undergoes a change in phase –0.08 π
relative to the incident wave by an –0.72 ζ
amount 2ζ that depends on the angle of –1.36
incidence (see below). 0.5 π
The reflected surface wave is similar in –2.00
properties to the rayleigh wave but is 0 17 33 50 66 83 100 116 132 150
distinctly different from a rayleigh wave. (0.95) (1.9) (2.8) (3.8) (4.7) (5.7) (6.6) (7.6) (8.5)
The surface wave is also phase shifted
relative to the incident wave by the Incident angle θI, mrad (deg)
amount ζ. The reflection coefficient of the
reflected surface wave is:
(79) S = 2 sin ζ sin θT tan 2θT
Ultrasonic Wave Propagation 55
values for the reflection coefficients of
incident transverse vertical waves
throughout the entire range of incident
angles.
Figure 18 shows orientation of several
waves. In this schematic, the longitudinal
waves are horizontal and the transverse
waves are close to the normal. On top are
the reflected waves and on bottom are the
transmitted waves. The incident wave can
be either of two different cases: the
incident wave can be a longitudinal wave
or a transverse wave. The behavior of
these waves is described in standard
texts.34,35
FIGURE 18. Transmission and reflection of
ultrasonic waves.
T RS RL
Medium 1
Medium 2
TS TL
Legend
I = incident wave
RL = reflected longitudinal (or pressure) wave
RS = reflected transverse (or shear) wave
TL = transmitted longitudinal (or pressure) wave
TS = transmitted transverse (or shear) wave
56 Ultrasonic Testing
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20. Nayfeh, A. and D.E. Chimenti. p 50-54.
“Reflection of Finite Acoustic Beams
from Loaded and Stiffened 30. Nondestructive Testing Handbook, first
Half-Spaces.” Journal of the Acoustical edition. Columbus, OH: American
Society of America. Vol. 75, No. 5. Society for Nondestructive Testing
Melville, NY: American Institute of (1959).
Physics, for the Acoustical Society of
America (1984): p 1360. 31. Papadakis, E.P. “Absolute
Measurements of Ultrasonic
21. Chimenti, D.E. “Energy Leakage from Attenuation Using Damped
Rayleigh Waves on a Fluid-Loaded, Nondestructive Testing Transducers.”
Layered Half-Space.” Applied Physics Journal of Testing and Evaluation.
Letters. Vol. 43, No. 1. Melville, NY: Vol. 12, No. 5. West Conshohocken,
American Institute of Physics (1983): PA: ASTM International (1984):
p 46. p 273-279.
22. Nayfeh, A., D.E. Chimenti, L. Adler 32. Papadakis, E.P. “Diffraction of
and R. Crane. Journal of Applied Ultrasound Radiating into an
Physics. Vol. 53. Melville, NY: Elastically Anisotropic Medium.”
American Institute of Physics (1982): Journal of the Acoustical Society of
p 175. America. Vol. 36, No. 3. Melville, NY:
American Institute of Physics, for the
23. Fortunko, C., R. King and M. Tam. Acoustical Society of America (1964):
“Nondestructive Evaluation of Planar p 414-422.
Defects in Plates Using Low-Frequency
Shear Horizontal Waves.” Journal of 33. Papadakis, E.P. “Ultrasonic Diffraction
Applied Physics. Vol. 53, No. 5. Loss and Phase Change in Anisotropic
Melville, NY: American Institute of Materials.” Journal of the Acoustical
Physics (1982): p 3450. Society of America. Vol. 40, No. 4.
Melville, NY: American Institute of
24. Fortunko, C.M. and R.E. Schramm. Physics, for the Acoustical Society of
“An Analysis of Electromagnetic America (1966): p 863-876.
Acoustic Transducer Arrays for
Nondestructive Evaluation of Thick 34. Graff, K. Wave Motion in Elastic Solids.
Metal Sections and Weldments.” New York, NY: Dover (1963, 1991).
Review of Progress in Quantitative
Nondestructive Evaluation. Vol. 2A. 35. Kolsky, H. Stress Waves in Solids. New
New York, NY: Plenum (1983): York, NY: Dover (1975, 2003).
p 283-307.
25. Beaver, W.L. “Sonic Nearfields of a
Pulsed Piston Radiator.” Journal of the
Acoustical Society of America. Vol. 56,
No. 4. Melville, NY: American Institute
of Physics, for the Acoustical Society
of America (1974): p 1043-1048.
26. Seki, H., A. Granato and R. Truell.
“Diffraction Effects in the Ultrasonic
Field of a Piston Source and Their
Importance in the Accurate
Measurement of Attenuation.” Journal
of the Acoustical Society of America.
Vol. 28, No. 2. Melville, NY: American
Institute of Physics, for the Acoustical
Society of America (1956): p 230-238.
27. Love, A.H. Some Problems in
Geodynamics. London, United
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28. Scholte, J. “Geophysics.” Royal
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Supplement 5. Oxford, United
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Publications (1947): p 120.
58 Ultrasonic Testing
3
CHAPTER
Generation and Detection
of Ultrasound
Gary L. Workman, University of Alabama in Huntsville,
Huntsville, Alabama (Part 6)
George A. Alers, San Luis Obispo, California (Part 9)
Theodore L. Allen, Southwest Research Institute, San
Antonio, Texas
Yoseph Bar-Cohen, Jet Propulsion Laboratory,
Pasadena, California (Part 5)
Frederick A. Bruton, Southwest Research Institute, San
Antonio, Texas (Part 2)
Francis H. Chang, Fort Worth, Texas
Butrus Pierre T. Khuri-Yakub, Stanford University,
Stanford, California (Part 10)
Michael Moles, Olympus NDT Canada, Toronto,
Ontario, Canada (Part 4)
Jean-Pierre Monchalin, National Research Council
Canada, Québec (Part 8)
Joseph L. Rose, Pennsylvania State University,
University Park, Pennsylvania (Part 7)
Part 1 adapted from Piezoelectric Technology Data for Designers (Morgan Matroc, Vernitron Division)
PART 1. Piezoelectricity1,2
Piezoelectricity means “pressure electricity” Piezoelectric Actions
and is a property of certain crystals,
including quartz, rochelle salt, tourmaline The type of piezoelectric material
and barium titanate. As the term suggests, determines the degree of deformation
electricity results when pressure is applied resulting from application of an electric
to one of these crystals. The reverse effect field and, conversely, the nature of the
also is present — when an electric field is deforming forces needed to develop an
applied, the crystal rapidly changes shape. electric charge (Fig. 1).
Piezoelectric Materials Generally, at least two of these
deformations are present simultaneously.
Piezoelectric materials commonly used for In some cases, one type of expansion is
electromechanical transducers include accompanied by a contraction that
barium titanate and ceramics containing compensates for the expansion and
lead zirconate and lead titanate. These produces no net change in volume. For
ceramics are polycrystalline and do not example, the expansion of plate length
have piezoelectric properties in their may be compensated by an equal
original state. Piezoelectric behavior is contraction of width or thickness. In
induced by polarization or so-called poling some materials, the compensating effects
procedures. In addition to the traditional are not of equal magnitude and net
ceramic elements, organic materials such volume change does occur. In all cases,
as polyvinyllidene fluoride (PVDF) have the deformations are very small when
been demonstrated to be effective in amplification by mechanical resonance is
piezoelectric ultrasound transducers.3 not involved. The maximum
Polyvinyllidene fluoride transducers are displacements are on the order of
lightweight, low cost and broad band. micrometers.
Transducer manufacturers provide
polyvinyllidene fluoride transducers. Piezoelectric Actions from Various
Other, similar ferroelectric materials have Materials
been developed for ultrasonic transducers.
In piezoelectric ceramic materials, the
FIGURE 1. Basic deformations of piezoelectric directions of the electrical and mechanical
plates: (a) thickness shear; (b) face shear; axes depend on the direction of the
(c) thickness expansion. original direct current polarizing field.
During the poling process, a ceramic
(a) element undergoes a permanent increase
in thickness between poling electrodes
+– and a permanent decrease in length
parallel to the electrodes.
0
–+ When a direct current voltage of the
same polarity as the poling voltage but of
(b) smaller magnitude is subsequently applied
between the poling electrodes, the
+ – element experiences further but
0 + temporary expansion in the poling
direction and contraction parallel to the
– electrodes. Conversely, when direct
current of opposite polarity is applied, the
(c) element contracts in the poling direction
and expands parallel to the electrodes. In
+ either case, the element returns to the
– original poled dimensions when the
voltage is removed from the electrodes.
0
–+ These effects are shown greatly
exaggerated in Fig. 2. The thickness and
transverse effects are not of equal
magnitude and there is a small volume
change when voltage is applied to the
electrodes.
60 Ultrasonic Testing
When compressive force is applied in Mechanical and Acoustical
the poling direction or when tensile force Impedance Considerations
is applied parallel to the electrodes, the
voltage that results between electrodes has The mechanical impedance or acoustical
the same polarity as the original poling impedance of crystal plates and ceramics
voltage. Reversing the direction of the are on the same order of magnitude as
applied force reverses the polarity of the those of solids or liquids. For this reason,
resulting voltage between electrodes. such piezoelectric elements are well suited
for underwater sound applications and
Assume that the poling electrodes are those mechanical applications involving
removed from a ceramic element and the large forces with small displacements.
element is provided with signal electrodes
perpendicular to the poling direction. Because these impedances are several
When a voltage is applied, a deformation orders of magnitude greater than those of
takes place transversely, around an axis gases, the transfer of energy to air, for
perpendicular to both the poling and example, is poor. For acoustical
signal directions (Fig. 3). When transverse applications in air and mechanical
forces are applied to the element, applications involving small forces with
corresponding voltage appears between comparatively large displacements, a
the signal electrodes. system of levers is often used to obtain a
better impedance match.
FIGURE 2. Basic deformations of piezoelectric
ceramic plates: (a) poled but at rest; (b) top Resonant Devices
electrode positive, bottom negative; (c) top
electrode negative, bottom positive. To obtain optimum performance from a
piezoelectric device, the circuit to which it
(a) Poled along is connected must have certain
this axis characteristics in turn dictated by the
design of the device. In discussing this
0 subject, it is convenient to divide
piezoelectric devices into two broad
(b) categories: nonresonant devices and
resonant devices.
+
– Nonresonant devices are so named
because they are designed to operate well
(c) below resonance or over a relatively large
frequency range, usually several octaves.
– All or most of the operating frequency
+ range typically lies below the resonant
frequency of the device. However, in some
FIGURE 3. Shear action of ceramic plates. cases, the useful frequency range includes
the frequencies of one or more
Poled along this axis resonances. In such cases, heavy damping
is used. Nonresonant devices include
+– microphones, headphones,
0 accelerometers, high voltage sources and
some underwater receiving transducers.
–+
Resonant devices are designed to
operate at a single frequency (the
mechanical resonance frequency of the
device) or over a band of frequencies
usually less than an octave (the band
includes the resonance frequency of the
device). Resonant devices include
ultrasonic transducers and underwater
power transducers.
The electrical impedance of a
piezoelectric device is more complicated
than the simple capacitor representation
typically used in discussing nonresonant
devices. A more appropriate model is a
capacitor representing the static
capacitance of the piezoelectric element,
shunted by an impedance representing
the mechanical vibrating system. In most
nonresonant devices, the latter impedance
may be approximated by a capacitor.
Generation and Detection of Ultrasound 61
Under these circumstances, there is a these materials. Electric boundary
capacitor in parallel with a capacitor (a conditions are identified by indicating
single capacitor representation). locations and connections of electrodes.
In devices designed for operation at Axes
resonance, the impedance representing
the mechanical system may at resonance Piezoelectric materials are anisotropic and
become a resistance of relatively low their electrical, mechanical and
value, shunted by the same static electromechanical properties differ for
capacitance. electrical or mechanical excitation along
different directions. For systematic
The shunt static capacitance typically is tabulation of properties, a standardized
undesirable, whether the device is means for identifying directions is
designed for operation at resonance or for required. Where crystals are concerned,
broad band, below resonance operation. the orthogonal axes are referred to by
In electrically driven devices, shunt static numerals: 1 corresponds to the X axis, 2
capacitance shunts the driving amplifier corresponds to the Y axis and 3
or other signal source requiring that the corresponds to the Z axis.
source be capable of supplying extra
current. In the case of mechanically Piezoelectric ceramics are isotropic and
driven piezoelectric devices, the static are not piezoelectric before they are
capacitance acts as a load on the active polarized. Once they are polarized, they
part of the transducer, reducing the become anisotropic. The direction of the
electrical output. poling field is identified as direction 3. In
the plane perpendicular to axis 3, the
In nonresonant devices, not much can ceramics are nondirectional. Accordingly,
be done about the shunt capacitance, the 1 and 2 axes may be arbitrarily
except choosing a piezoelectric material located but must be perpendicular to each
with maximum activity. In resonant other.
devices, the static capacitance may be
neutralized by using a shunt or series Elastic Constants
inductor chosen to resonate with the
static capacitance at the operating To identify the directions of stress and
frequency (Fig. 4). strain, two numerical subscripts are added
to the symbol S for elastic compliance
Properties of Piezoelectric (strain and stress). The first numeral
Materials indicates the direction of stress or strain
and the second numeral indicates the
Figure 5 shows typical symbols used to direction of strain or stress. The symbol S
describe piezoelectric materials. These with appropriate subscripts is used to
symbols are used to identify properties of identify elastic behavior under the specific
materials and should not be used to condition that all external stresses not
describe piezoelectric elements made of embraced by the symbol remain constant.
Thus S13 is the symbol for the ratio of
FIGURE 4. Resonant piezoelectric device with strain in direction 3 to stress in direction
static capacitance neutralized by inductor: 1 provided there is no change in stress in
(a) for low impedance electrical source or directions 2 and 3. It also is the symbol
load; (b) for high impedance electrical for the ratio of strain in direction 1 to
source or load. stress in direction 3 provided there is no
change in stress in directions 1 and 2.
(a) 3
The restriction regarding stresses in
2 other directions needs emphasizing.
1 Suppose that a piezoelectric plate is
clamped in a vise that applies a load,
(b) causing a stress in direction 3. Now, apply
the load to cause stress to the plate in
2 direction 1 and calculate the resulting
41 strain in direction 1. If the plate were not
clamped in the vise, the stress along axis 1
Legend causes strain in direction 1 equal to S11
1. Impedance representing resonant mechanical times the stress. In addition, there is
strain in directions 2 and 3. The vise,
system. however, tends to prevent the strain in
2. Static capacitance. direction 3 and in so doing, causes stress
3. Series inductor. in direction 3. The development of this
4. Shunt inductor. stress violates the requirement imposed in
the definition that S11 cannot be used to
calculate the strain in direction 1.
62 Ultrasonic Testing
Consider a second example. This time Because piezoelectric materials
the plate is resting on the table and a interchange mechanical (elastic) and
weight is placed on top. The weight electrical energy, the elastic properties
causes a stress in direction 3. Now, apply depend on electric boundary conditions.
the load to cause a stress in direction 1 For example, when electrodes on a bar of
and calculate the resulting strain in piezoelectric material are connected
direction 1. As a result of the stress in together, the bar displays higher elastic
direction 1, the plate experiences strain in compliance than when the electrodes are
direction 3, lifting the weight slightly. not connected together. Thus, in defining
However, stress in direction 3 caused by elastic properties, the electric boundary
the weight has not changed (except conditions must be identified. This is
momentarily while the weight was being done by adding a superscript to the
lifted) and accordingly the symbol S11 is, symbol.
in this case, appropriate for calculating
the strain in direction 1 resulting from When the electric field across the
application of stress in direction 1. piezoelectric body is held constant, for
example by short circuiting the electrodes,
Shear stress or strain around axis 1 is the superscript E is used. When the
indicated by the subscript 4, around axis 2 electric charge density is held constant,
by subscript 5 and around axis 3 by for example by maintaining an open
subscript 6. Thus, S44 is the ratio of circuit at the electrodes, the superscript D
transverse strain around axis 1 to is used. Thus S3E3 is the symbol for the
transverse stress around axis 1. A ratio of strain to stress along axis 3 if all
restriction requiring that stresses not other external stresses are constant and
embraced by the symbol must remain the electric field is constant.
constant is theoretically applicable here
also but, because of symmetry conditions, Dielectric Constants
it is not applicable to transverse
compliances of ceramics. In piezoelectric materials, the dielectric
constant ε (dielectric displacement or
FIGURE 5. Typical symbols used to describe piezoelectric material properties.
SD Indicates that compliance is measured with electrode circuit open SE Indicates that compliance is measured with electrodes
11 36 connected together
Indicates that stress (or strain) is in direction 1a
Indicates that strain (or stress) is in direction 1 Indicates that stress (or strain) is in shear form around axis 3
Compliance = strain/stress Indicates that strain (or stress) is in direction 3
Compliance = strain/stress
KT Indicates that all stresses on material are constant KS Indicates that all strains in material are constant
1 Indicates that electrodes are perpendicular to axis 1 3 Indicates that electrodes are perpendicular to axis 3
Relative dielectric constant Relative dielectric constant
k15 Indicates that stress (or strain) is in shear form around axis 2 kp Subscript used only for ceramics indicates electrodes
Indicates that electrodes are perpendicular to axis 1 perpendicular to axis 3 and stress (or strain) equal in all
directions perpendicular to axis 3
Electromechanical coupling
Electromechanical coupling
d33 Indicates that the piezoelectrically induced strain (or the induced dh Indicates that stress is induced equally in directions 1, 2 and 3
g31 stress) is in direction 3b (hydrostatic stress) and that electrodes are perpendicular to axis
Indicates that electrodes are perpendicular to axis 3 3 for ceramics or axis 2 for lithium sulfate
Strain/induced field = short circuit charge/electrode g15 Short circuit charge/electrode area/induced stress
area/induced stress
Indicates that induced stress (or piezoelectrically induced strain)
Indicates that induced stress (or piezoelectrically induced strain) is in shear form around axis 2b
is in direction 1b Indicates that electrodes are perpendicular to axis 1
Indicates that electrodes are perpendicular to axis 3 Field/induced stress = strain/applied charge/electrode area
Field/induced stress = strain/applied charge electrode area
a. All stresses other than stress involved in one subscript are constant.
b. All stresses other than stress involved in second subscript are constant.
Generation and Detection of Ultrasound 63
charge density per electric field) depends Coupling
on the directions of field and dielectric Coupling is an expression for the ability
of a piezoelectric material to exchange
displacement. For this reason, subscripts electrical energy for mechanical energy or
vice versa. Coupling squared is equal to
are added to the symbol to indicate the the transformed energy divided by the
total energy input. The same constant is
directions. The first subscript denotes the applied for conversion from electrical to
mechanical energy and from mechanical
direction of the electric field or dielectric to electrical energy.
displacement. The second subscript Except in one special case noted below,
the coupling coefficients typically used
denotes the direction of the dielectric and those given here are for the cases
where all external stresses (except the
displacement or electric field. Thus ε33 is input stress considered in the energy
the ratio field applied in direction 3 to the transformation) are constant:
resulting dielectric displacement in
direction 3.
In most piezoelectric materials used in
ultrasonic transducers, a field along one
axis results in dielectric displacement only
along the same axis, so that the two
subscripts for these materials are always
the same. Accordingly, one subscript often a
b
is omitted: ε3 means the same as ε33. (2) K321 =
Because piezoelectric materials
interchange electrical and mechanical
energy, the electrical properties depend on where a is transformed electrical energy
causing mechanical strain in direction 1
mechanical boundary conditions. When a when all external stresses are constant
and b is electrical energy input to
piezoelectric body is completely free to electrodes on faces perpendicular to
axis 3. For example:
vibrate, the dielectric constant is higher
than when the body is mechanically
restrained. Accordingly, superscripts are
added to the symbol for dielectric
constant to indicate the mechanical (3) K321 = c
d
boundary conditions. Superscript T
denotes the condition of constant stress
(no mechanical restraint). Superscript S where c is transformed mechanical energy
causing an electrical charge to flow
denotes the condition of constant strain between connected electrodes on faces
perpendicular to axis 3 and d is the
(material completely restrained to prevent mechanical energy input accompanying
the stress in direction 1, with all other
any mechanical deformation when field is external stresses constant.
applied, a condition that can be A special case of considerable practical
importance involves using thickness
approached only under very special vibrations in ceramic plates or disks at
conditions). Thus ε1T1 is the dielectric frequencies above the resonant
constant for field and dielectric frequencies determined by the length and
width of the element. Under these
displacement in direction 1 under the conditions, the inertia of the piezoelectric
material effectively prevents lateral
condition of constant stress on the body vibrations. The effect is the same as
S though infinitely rigid clamps were
and ε 11 is the corresponding dielectric applied to the plate to prevent length and
width vibrations. Such theoretical clamps
constant under the condition of constant would cause opposing dynamic stresses as
the element tries to vibrate laterally. The
strain in the body. qualification that all external stresses are
constant is not met and accordingly k33
The relative dielectric constant, does not define electromechanical
coupling under such conditions. The
sometimes identified by the symbol K, is coupling in this special case is identified
by the symbol kt.
the ratio of the material dielectric
Another special case involves coupling
constant ε to the dielectric constant of a between electric field in direction 3 in
ceramics and mechanical action
vacuum ε0 (ε0 = 8.85 × 1012 F·m–1): simultaneously in the 1 and 2 directions.
This coupling is identified by the symbol
(1) K1⌻1 = ⑀1⌻1 kp (planar coupling). It is important
⑀0 because of the ease with which it may be
measured with high accuracy, yielding a
Piezoelectric Constants
The most common electromechanical
constants are coupling k, strain constant d
and stress constant g. For each of these,
the directions of field and stress or strain
are indicated by two subscripts.
The first subscript indicates the
direction of electric field. The second
subscript indicates the direction of stress
or strain. As in the case of elastic
constants, subscripts 4, 5 and 6 denote
stress or strain around axes 1, 2 and 3
respectively.
64 Ultrasonic Testing
simple measure of the effectiveness of Frequency Limitations
poling of ceramic components.
An important restriction must be observed
Piezoelectric d Constants in the constants discussed above when
calculating the behavior of actual bodies
The piezoelectric d constants express the of piezoelectric material. If the stress,
ratio of strain developed along or around strain, electric field and dielectric
a specified axis to the field applied parallel displacement involved in the constants
to a specified axis, when all external are uniform throughout the body, then
stresses are constant. The d constants also the constants may be used directly in
express the ratio of short circuit charge simple calculations. For example, when a
per unit area of electrode flowing between static or low frequency alternating field is
connected electrodes that are applied, the change in length of a
perpendicular to a specified axis to (1) the piezoelectric bar equals the product of the
stress along a specified axis or (2) the appropriate d constant, the applied field
stress around a specified axis, when all and the length.
other external stresses are constant. For
example, d31 denotes the ratio of strain in On the other hand, if the stress or field
direction 1 to the field applied in is not uniform throughout the body, then
direction 3 when the piezoelectric the behavior of the body as a whole may
material is mechanically free in all be determined only by integrating the
directions. It also denotes the ratio of behavior of all the incremental portions
charge (per unit area of electrode) that of the body. In this case, the constants
flows between electrodes perpendicular to discussed above are used to relate stress,
axis 3 and connected together to the strain, field and charge density in each
stress in direction 1 when the material is increment.
free of external stresses in all other
directions. The most common situation where
nonuniform distribution of stress is
A special case applies to piezoelectric encountered, preventing simple bulk
ceramics. These materials develop calculation, is when the frequency of the
substantial charge when subjected to electrical or mechanical excitation of the
uniform load along all three axes body is at or close to the mechanical
(hydrostatic pressure). The ratio of short resonance frequency of the body.
circuit charge per unit area of electrode to
the applied hydrostatic pressure is Frequency Constant
identified by the symbol dh, the
hydrostatic d constant. For ceramics, the The frequency constant N is the product
electrodes are understood to be of the mechanical resonant frequency
perpendicular to axis 3. For lithium under specified electrical boundary
sulfate, the electrodes are understood to conditions (short circuit or open circuit)
be perpendicular to the Y or 2 axis. A and the dimension that determines that
more descriptive symbol is d3h or d2h but resonant frequency. It is applicable only
such designations are not commonly to specific boundary conditions.
used.
For example, N1 for piezoelectric
Piezoelectric g Constants ceramics applies only to a long, thin,
narrow bar polarized perpendicular to the
The piezoelectric g constants express the length and measured with the polarizing
ratio of field developed along a specified electrodes connected together. The same
axis to the stress along or around a bar poled along the length has a different
specified axis when all other external frequency constant. In this case, the
stresses are constant. The g constants also assigned designation N3a applies when
express the ratio of strain developed along measurement is made with the electrodes
or around a specified axis to the electric open circuited.
charge per unit area of electrode applied
to electrodes perpendicular to a specified Electrical Losses
axis.
Some piezoelectric materials, including
For example, g33 denotes the ratio of quartz, are high quality dielectrics. Other
field developed in direction 3 to stress in piezoelectric materials, notably the
direction 3 when all other external piezoelectric ceramics, are relatively lossy.
stresses are zero. It also denotes the ratio The dielectric losses in ceramics may be
of strain developed in direction 3 to the the limiting factor in the power handling
charge per unit area of electrode applied capabilities of transducers. The losses are
to electrodes on faces perpendicular to expressed as a dissipation factor, the ratio
axis 3. of effective series resistance to effective
series reactance.
Generation and Detection of Ultrasound 65
Mechanical Losses In addition, at elevated temperatures,
the aging process is accelerated, electrical
When an elastic body is deformed, most losses increase and the maximum safe
of the mechanical energy applied in stress is reduced.
causing the deformation is stored as
elastic energy. However, a small part of Properties of Piezoelectric
the applied energy is dissipated as heat Elements
because of molecular friction.
Equivalent Circuits
In some applications of piezoelectric
materials, such mechanical losses may Equivalent circuit techniques have been
become important. Usually they are far used for many years to obtain solutions to
outweighed by mechanical losses in other electrical and mechanical problems. These
elements of the piezoelectric device — in same techniques are used to describe the
particular, in cement joints between behavior of piezoelectric elements.4,5
piezoelectric materials and driving or Table 1 shows the electrical and
driven members. Mechanical losses are mechanical units used in the construction
expressed in terms of mechanical Q, the of equivalent circuits for
ratio of mechanical stiffness reactance or electromechanical systems.
mass reactance at resonance to the
mechanical resistance. These equivalent circuits are shown in
Fig. 6. If the constants in these two
Aging circuits are suitably related, the circuits are
Most of the properties of piezoelectric TABLE 1. Analogous electrical and
ceramics change gradually. The changes mechanical units for electromechanical
tend to be logarithmic with time after the systems.
original polarization. For example, the
dielectric constant 1 h after poling may be Electrical Unit Mechanical Unit
1000 and 10 h after poling it may be 990.
At 100 h, it is about 980, at 1000 h about Voltage force
970 and at 10 000 h about 960. In this Current velocity
case, the dielectric constant is said to age Charge displacement
about 1 percent per time decade. The Capacitance compliance
aging of various properties depends on Inductance mass
the ceramic composition and on the way Impedance impedance
the ceramic is processed during
manufacture. Exact aging rates cannot be FIGURE 6. Basic equivalent circuits: (a) ratio
specified but it is typical to specify that of N to 1; (b) ratio of 1 to N´.
the aging of a given property is less than
some limiting rate. (a)
Because of aging, exact values for Ce N:1 M
various properties such as dielectric
constant, coupling and elastic modulus Electrical RL Cm Mechanical
may be specified only at a stated time
after poling. The longer the time period (b) 1:N C’m M
after poling, the more stable the material Mechanical
becomes. R’L C’e
Electrical
High Stress
Legend
Most of the properties of piezoelectric
ceramics vary with the level of electrical C = capacitor
or mechanical stress when such stresses M = mechanical inductance
are large. Data commonly presented for N = voltage output proportional to number of turns
piezoelectric ceramics are for stress levels R = resistor
low enough for the results to be
independent of stress.
Curie Point
For each piezoelectric material, there is a
characteristic temperature called the curie
point. When a ceramic element is heated
above the curie point, it suffers
permanent and complete loss of
piezoelectric activity. In practice, the
operating temperature for a piezoelectric
ceramic must be limited to some value
substantially below the curie point.
66 Ultrasonic Testing
equivalent at all frequencies. The measured resonance peak of the element
mechanical terminals represent the face or is finite because of mechanical losses. The
point of mechanical energy transfer to or equivalent circuit data include no
from the piezoelectric element. The mechanical loss element so that the
inductance symbol M represents the computed response is infinite at
effective vibrating mass of the element. resonance.
The transformer symbol represents an For most nonresonant
ideal electromechanical transformer, a electromechanical transducer design
device that transforms voltage to force problems, this omission is not significant.
and vice versa and current to velocity and Introduced losses (usually for control of
vice versa, without loss and without frequency response) greatly outweigh
energy storage. The transformation ratio internal losses.
N:1 in Fig. 6a is the ratio of voltage input
to force output of the ideal transformer
and also the ratio of velocity input to
current output. It is used in purely
electrical network calculations. The
transformation ratio 1:N′ in Fig. 6b has
similar significance. The capacitance
symbols on the electrical side represent
electrical capacitances. The capacitance
symbols on the mechanical side represent
mechanical compliances.
The choice of circuit for a particular
problem depends on the external circuit
elements connected to the piezoelectric
element. Because the two circuits are
equivalent, either may be used but often
one is more convenient than the other.
For example, if the electrical terminals are
connected to a constant current generator
and it is necessary to calculate the force
applied to a mechanical load connected to
the mechanical terminals of the
piezoelectric element, the problem is
simplified by selecting the circuit in
Fig. 6a to eliminate capacitance Ce from
the calculations.
These circuits are useful for most
design purposes. However, it should be
emphasized that they only approximate
the actual behavior of the piezoelectric
elements. The approximation is useful up
to and slightly above the first resonance
frequency of the mounted, unloaded
element. At higher frequencies, further
resonances typically may be observed in
the behavior of a piezoelectric element
because of the distributed nature of the
mass and compliance, for which only
lumped elements are used. Better
approximations of the actual performance
can be obtained by modifying the
equivalent circuits to include additional
reactance elements or by allowing some
elements of the simple circuits to vary
with frequency in a prescribed manner.
Because of the approximate nature of
equivalent circuits, the accuracy of
representing the piezoelectric element is
better at some frequencies than at others.
The most accurate frequency range
depends on the choice of values assigned
to the elements of the circuit.
The effective mass or moment of
inertia is chosen so that the equivalent
circuit has the same resonant frequency as
the actual crystal. The height of the actual
Generation and Detection of Ultrasound 67
PART 2. Transduction1
The approach to transducer design Transducer Characterization
presented below is aimed at engineered Station
design using computer modeling. Before
the design process is addressed, important A computerized system for measuring
performance parameters are identified and typical ultrasonic transducer parameters
techniques are selected to measure these has been developed. It can measure
parameters. The instrumentation used to waveform and spectrum as well as relative
characterize transducers must not modify
the transducer response. Some of the FIGURE 7. Comparison of performance
measurements should be performed under characteristics from commercial transducer
conditions that simulate the actual testing (see Fig. 8): (a) signal waveform with
application of the transducer. The damping and gating; (b) frequency
transducer‘s components must be spectrum; (c) data furnished by
characterized individually and then manufacturer (frequency and waveform).
incorporated into the transducer design.
(a)
A computer model has the advantage
of predicting the performance of a Amplitude
transducer, allowing fine tuning of the (relative scale)
design before committing resources and
materials to fabrication.
Instrumentation
Instrumentation and Transducer (b)Amplitude Time
Characterization (c) (relative scale) (relative scale)
Significant differences have been observed Amplitude Frequency
between transducer characterization data (relative scale) (relative scale)
from different sources. One investigation
revealed that differences in Time
instrumentation and measurement (relative scale)
technique could seriously affect the
apparent performance of a transducer
(Figs. 7 and 8). Figure 7c shows the
waveform and frequency data furnished
by a manufacturer with its 10 MHz
transducer. Figure 8 shows the data
acquired from the transducer when tested
at the buyer‘s facility. The waveforms and
the spectra are very different: the
manufacturer‘s spectrum peaks at about
10 MHz and the buyer‘s spectrum peaks at
about 6 MHz.
Research showed that the discrepancy
was largely a matter of technique. In an
effort to duplicate the manufacturer‘s
data, pulser damping and signal gating
were adjusted until a close match was
achieved (see Figs. 7a and 7b). This
exercise strongly emphasizes the need to
include the essential elements of test
procedures, directly or by specific
reference, within a transducer
specification.
68 Ultrasonic Testing
sensitivity, electrical impedance and the trigger to be used depends on the
sound field beam profiles. Transducer particular measurement being performed.
design and optimization software allows The computer oscilloscope is used for
for transducer modeling and predicted acquisition of waveform data and an
performance based on the transducer analog oscilloscope is used for setup and
model. Impedance matching software is monitoring purposes. A jet printer is used
included for selection of effective to produce hardcopy. A personal
impedance matching networks, including computer controls the entire process,
component value calculation. including the motor/drive unit used to
provide precise positioning capability for
System components and their sound field measurements.
individual functions are as follows. A
pulser is used to pulse the transmitting Differences in Transducer
transducer and initiate overall system Performance
timing, including repetition rate. A pulse
generator produces a variable width pulse Several factors contribute to differences
for delayed triggering and the between the measured and predicted
introduction of a time offset between the performance of ultrasonic transducers.
two pulsers. A second pulser is used for The observed differences outside the range
pulsing the transducer being evaluated. of instrument and test error and the
Lastly, a commercial receiver is used for contributing factors are discussed below.
amplification and attenuation of the test
transducer‘s received signal. Inductance and Resistance
The catch transducer‘s received signal Inductors used in the matching networks
is used as a trigger to produce a moving contain a resistive element that affects the
gate while the function trigger select circuit. However, it is customary to
switch box (Fig. 9) is used to differentiate assume that the effect is negligible. To
the pulse generator‘s output. It is also verify this assumption, measurements
used to switch the correct trigger signal to were taken on the inductive and resistive
both the first pulser and the computer components of inductors at 5 MHz. The
oscilloscope. Switching is needed because study consisted of measuring the
resistance and inductance of a group of
FIGURE 8. Data from research transducer five inductors at each of 34 nominal
developed to meet specific performance inductor values. Figure 10 shows an
criteria: (a) signal waveforms; (b) frequency example of inductive and resistive
spectrum. components over a frequency range of 1
to 10 MHz for one inductor.
(a)
Table 2 reports the measured averages
Amplitude for each inductor grouped by the nominal
(relative scale) inductor value. The inductors are specified
not to exceed ±10 percent of the nominal
FIGURE 9. Function trigger select switch box (50 Ω cable
used throughout, with shields tied to common chassis
ground).
Time 120 Ω 51 Ω
(relative scale) C
(b) 14 (–) out
11 I 510 pF 15 Receive
Frequency Y2 13 Sync
(relative scale)
Z
Amplitude Ext trig 12 C
(relative scale) I
Z
Legend
Z = impedance/pulser characterization
I = Immersion transducers
C = contact transducers
Generation and Detection of Ultrasound 69
FIGURE 10. Gain phase analyzer data of 15 µH (nominal) value. Figure 11 shows plots of inductance
inductor. versus resistance data taken as illustrated
in Fig. 10. This study of transducer
Inductance (µH) 15.6 Inductance 45 Resistance (Ω) fabrication incorporating matching
15.4 5.5 40 networks has been mostly in the 20 to
15.2 30 30 µH range with inductors that have
15.0 Resistance 20 resistive elements on the order of 8 to
14.9 10 15 Ω. The resistance of the inductors is
14.8 sufficient to affect transducer performance
0 and should be included in the calculation
1 10 of the series resistance for the matching
network. The series resistance value of the
Frequency (MHz) matching network is included as one of
the parameters in the transducer
modeling software.
TABLE 2. Measured average inductor FIGURE 11. Plots of measured inductance
values. versus measured resistance over range of
inductors: (a) small nominal inductance;
Nominal Average Average (b) medium nominal inductance; (c) large
Inductor Measured Measured nominal inductance.
Inductance Resistance
Value (a)
(µH) (µH) (Ω)
1.3
0.10 0.090 0.162
0.15 0.142 0.256 1.0
0.18 0.172 0.314 Resistance (Ω)
0.22 0.213 0.394 0.5
0.27 0.259 0.416
0.33 0.306 0.496 0.1 0.2 0.4 0.6 0.8
0.39 0.385 0.688 0 Inductance (µH)
0.47 0.451 0.836
0.56 0.546 0.990 (b) Resistance (Ω)
0.68 0.660 1.142
0.62 0.772 1.272 4.0 24 6 8
1.0 0.956 1.400 3.5 Inductance (µH)
1.5 1.502 1.148 3.0
1.8 1.750 1.434 2.5 Resistance (Ω)
2.2 2.198 1.910 2.0
2.7 2.764 1.946 1.5 20 40 60 80 100
3.3 3.116 1.512 1.0 Inductance (µH)
3.9 3.822 1.936
4.7 4.574 2.388 0
5.6 5.194 2.828
6.8 6.502 2.864 (c) 60
8.2 7.652 3.558
10.0 9.560 4.486 50
15.0 14.636 7.086 40
18.0 17.504 9.794 30
22.0 22.040 15.064 20
27.0 26.528 14.954 10
33.0 32.562 18.34 0
39.0 39.027 23.35
47.0 45.996 27.88 0
56.0 56.812 25.056
68.0 65.912 48.24
82.0 78.960 38.2
100.0 95. 480 57.22
70 Ultrasonic Testing
Piezoelectric Element The input capacitor Cg is charged to a
voltage Vg through resistors Rg and RL and
The piezoelectric element parameter then the switch is closed. Following the
values supplied by the manufacturer are
usually based on an average value derived closure, the voltage Vt across the
from sampling several batches of transducer rises too rapidly to be affected
piezoelectric materials. The nominal
values are not sufficiently precise for use by the mechanical nature of the
in modeling the performance of
transducers. A means of measuring the transducer (represented in the diagram by
piezoelectric coupling factor, the damped
capacitance, the dielectric constant and equivalent series circuit LCR) or by the
the free resonant frequency of
piezoelectric elements are a key to charging resistors Rg and RL. The
transducer modeling. magnitude of this voltage step is used to
To measure the damped capacitance C0 calculate the value of the damped
and the electromechanical coupling factor
k of piezoelectric transducer elements, a (constant strain) capacitance of the
technique consists of the sudden
placement of an electronic charge on the transducer by using the following
plated surfaces of the element and then
measuring the variation of the resulting equation:
voltage between these surfaces.6 What is
observed is a voltage step followed by a (4) C0 = Cg Vg − Vt 0
train of triangular waves as shown in Vt0
Fig. 12. Figure 13 is a simplified
illustration of this technique. This equation is derived from the
FIGURE 12. Wave train during measurement of damped equality of Eqs. 5 and 6. After the Cg is
capacitance. fully charged and before the switch is
Parasitic closed, the charge on Cg is given:
oscillation
(5) Q 0 = Vg Cg
T = —1
f After the switch is closed, that same
quantity of charge is shared by Cg and C0:
( )(6) Q0 = Vt0 Cg + C0
Before the switch is closed, the energy
stored in Cg is:
(7) Wg = Q 2
0
2Cg
V VV After the switch is closed and after the
high frequency transients have died out,
Time (relative scale) the stored energy is given by:
Legend
f = frequency = Q 2
T = wave cyclic interval ( )(8)Wa 0
V = signal amplitude
2 Cg + C0
FIGURE 13. Basic circuit for measuring damped capacitance Note that Wa is smaller than Wg by the
and electromechanical coupling factor of piezoelectric ratio Cg·(Cg + C0)–1. That is, by closing the
transducers. switch, energy is lost in the amount of
Wg – Wa. This energy is not recoverable. It
Rg Cg is lost either through lossy elements in
+ the circuit or by electromagnetic
radiation. Initially, this energy is stored
Power supply Switch L1 alternately in the parasitic inductances of
C0 C1 R the circuit and in the circuit capacitances.
Vg V
– R1 During a short period after the switch
closure, the high frequency oscillations
Transducer are observed on the voltage waveform.
These oscillations die out as this
Legend unavailable energy is dissipated. The
oscillations do not significantly affect the
C = capacitor measurement of element characteristics
g = subscript designating input when relatively low frequency elements
R = resistor are being evaluated (that is, for elements
L = inductance with characteristic frequencies below
V = electric potential 5 MHz). But when element frequencies
Generation and Detection of Ultrasound 71
approach 10 MHz, these oscillations mechanical work. Conversely, k2 also
interfere with accurate measurement
because they are superimposed on many represents the fraction of mechanical
of the triangular wave‘s first period.
energy input to an element, transformed
In a typical circuit configuration, the
frequency of these parasitic oscillations is and available to do electrical work. In the
about 100 MHz. The test circuit is laid out
and fabricated with care to minimize the test device described above, very little
parasitic circuit inductance and ensure
that the frequencies of the parasitic energy is radiated as acoustic energy
oscillations are well above the frequencies
of the transducers to be tested by the because of the poor acoustic coupling to
device.
the element holding fixture and very little
Recall from the analysis above that
much energy is contained in the is dissipated in the resistors of the test
components controlling these oscillations.
Therefore, in addition to maximizing the device.
frequency of parasitic oscillations, a
means is needed for damping the Therefore, this available energy flows
oscillations so that they die out within a
few cycles. Placing resistors in series with back and forth between electrical storage
the switch provides one means of
damping the oscillations but this step is and mechanical storage for a long time as
not frequency selective and tends to
reduce the effectiveness of the circuit in evidenced by the long triangular wave
the frequency range of interest (above
5 MHz). In particular, the severely train that follows the first voltage step.
rounded peaks of the characteristic
triangular wave increase the likelihood of This fortunate circumstance allows precise
significant measurement error. measurements. The resistor R1′ (Fig. 13)
represents the dissipative mechanisms
Ferromagnetic ceramic beads have
provided an effective solution for this within the piezoelectric element. In high
problem. Two or three of these beads
threaded over conductors carrying the Q elements, R1 is very small compared to
high frequency currents provide adequate the reactance of either C1 or L1 at the
damping. The effects of the beads decrease element‘s resonant frequency. When the
with decreasing frequency and the effect
on the circuit at the operating frequencies element is loaded acoustically, the
of the transducers is negligible.
effective value of R1 is increased and the
A silicon controlled rectifier as the effective Q of the transducer is reduced.
switch makes rise times on the order of
10 ns possible, fast enough to excite the To determine the electrical energy
100 MHz parasitic oscillations discussed
above. The design of a piezoelectric input to the transducer by this special test
element test circuit is shown in Fig. 14.
circuit and the available mechanical
The electromechanical coupling factor
k, when squared, represents the fraction energy, two voltage measurements are
of the electrical energy input to a
piezoelectric element — the fraction made across the terminals of the
transformed and available to do
transducer: (1) the voltage Vt0 at the first
peak and (2) the voltage Vti at the first
valley of the triangular wave. As a first
step, assume that the input capacitor Cg is
very small compared to C0 and its effect is
not included in the following
calculations. At the end, a correction
factor is added to account for this
omission. The voltage at the first peak
provides the information needed to
compute the input energy Wt0:
( )(9) = 2 ⎛ C0 ⎞
Wt0 Vt0 ⎜ ⎟
⎝2⎠
but:
(10) Q0 = Vt0 C0
FIGURE 14. Diagram of piezoelectric element test circuit.
Ferrite bead
100 V 75 K 10 K 100 pF Element holder
Ferrite bead
0 001 5 pF 9 MΩ
100 25 pF Scope
Power supply 100 1__M__Ω_
and trigger 20 pF
generator
Test head
72 Ultrasonic Testing
therefore: according to their relative sizes, so that
the total residual energy is given by:
( )(11) = Q0 Q02
Wt 0 2 Vt 0 (16) Wt 0 = C C0
Next, the residual stored energy is After all oscillations have ceased, the
computed after all oscillations of the
triangular wave of the transducer voltage voltages across C0 and C1 are equal and Q0
have died out. Discounting the effects of is shared according to their relative sizes,
any direct current paths, the voltage
across the transducer settles to a level so that the total residual energy is given:
halfway between the peak and the valley
of the triangular wave. The residual = Q02
energy stored in the transducer is given 2 C0 + C1
by: ( )(17)Wtr
(12) Wtr = Q0 ⎜⎛ Vt 0 + Vt i ⎞ Again, the difference between Wt0 and
⎟ Wtr is the available mechanical energy
Wam and k2 is the ratio of Wam to Wt0:
2⎝ 2 ⎠ (18) k2 = Wam = Wt0 − Wtr
Now, the energy available for doing Wt0 Wt0
mechanical work is computed as the
difference between Eqs. 10 and 11: = C1
C0 + C1
(13) Wam = Wt0 − Wtr This equation may be confirmed by
comparing it with the standard on
= Q0 ⎛ − Vt 0 + Vt i ⎞ piezoelectricity.7 There, the effective
2 ⎝⎜Vt 0 2 ⎟ electromechanical coupling factor is
⎠ defined:
= Q0 ⎛⎜ Vt 0 + Vt i ⎞
⎟
2⎝ 2 ⎠
fp2 − fs2
(19) ke2ff =
The electromechanical coupling factor is f 2
determined by the ratio of Wam to Wt0: p
(14) k2 = Wam where fs is frequency of maximum
Wt0 conductance and fp is frequency of
maximum resistance. Therefore, keff can
= V2t0 − Vti be determined by measuring fp and fs with
2Vt0 an impedance analyzer. By multiplying
1 − Vti both top and bottom of the fraction by
= Vt0
(2π)2:
2
(20) ke2ff = ω2p − ω2s
This equation is valid when the input ω2p
capacitor Cg is very small compared to the
element‘s damped capacitance C0. When where ωp is radian frequency of maximum
this is not the case, the voltage resistance and ωs is radian frequency of
measurements across the element are maximum conductance.
made in exactly the same way but the
input voltage Vg must also be known and The characteristic radian frequencies of
used to compute a correction factor. With
the correction factor added, the the equivalent circuit shown in Fig. 13
expression for the electromechanical
coupling factor becomes: can be defined in the following ways:
(21) ω2a = 1
L1 C1
⎛ Vt i ⎞⎛ Vt 0 ⎞ and:
⎜1 Vt0 ⎟⎠⎜⎜⎝1 Vg ⎟⎟⎠
⎝ − + 1
L1 C0
(15) k2 = (22) ω20 =
2
It may be helpful to approach the and:
calculation of k2 in a different way. This is
(23) ω2p = ω2s + ω20
based on the assumption that the value of
C0 and C1 are known and Q0 is shared
Generation and Detection of Ultrasound 73
Substituting these values into Eq. 17, fired silver deposit or sputter deposit,
all of the L1 terms cancel: resulted in silver layers of uniform
thickness on the order of 2.5 µm
(24) ke2ff = C1 (0.0001 in.). However, electrode plating on
C1 + C0 some elements was found to vary from 25
to 75 µm. In this range of thickness, the
This is identical to Eq. 18 obtained for k2 plating thickness affects the performance
by a different analysis. of the transducer.
The period of the triangular wave Electrodes are shown in Figs. 15 and 16
excited by the test device described above (cross section microphotographs of four
gives a very precise measurement of the transducers). The transducers were
piezoelectric element‘s resonant manufactured using nominal frequency
frequency. Good results can be obtained elements from the same manufacturer.
by measuring period time on high quality The sampling consisted of two groups of
oscilloscopes. When the time calibration transducers: 9.7 mm (0.38 in.) round at
or linearity of the oscilloscope is not 5 MHz and 6.4 mm (0.25 in.) round at
adequate, a comparison technique may be 10 MHz. The transducers were
used (the output of a precision signal characterized, sectioned and
generator is adjusted to match the photographed at 100× magnification to
frequency of the triangular wave). The test sample plating thickness and to measure
devices discussed here measure three the acoustic impedance of the backing.
important parameters of piezoelectric The variation in electrode thickness is
elements quickly and with good precision. visible in the figures.
Those characteristics are the damped
capacitance C0, the electromechanical The effects of variations in electrode
coupling factor k and the resonant thickness on the performance of
frequency. Experience shows the device to transducers can be minimized by using
be an important tool for transducer the thinnest electrodes possible.
development and fabrication facilities. Piezoelectric elements are available with
gold alloy electrodes with thicknesses on
Electrode Thickness on the order of 5 nm (2 × 10–7 in.), several
Piezoelectric Elements orders of magnitude less than that of
silver plating. Consistency of thickness is
The effect of the electrode deposited on not as important when electrodes are less
the piezoelectric element was once than 2 µm.
considered to have little effect on
performance. It was assumed that the two Effects of Angle Beam Wedge
common methods of depositing silver,
With angle beam transducers, plastic
wedges are used between a longitudinal
FIGURE 15. Examples of two 9.7 mm (0.38 in.), 5 MHz FIGURE 16. Cross sections of 6.4 mm (0.25 in.)
transducers photographed at 100×: (a) first transducer; photographed at 100X: (a) 10 MHz transducer; (b) 5 MHz
(b) second transducer. transducer.
(a) Loaded epoxy (a) 0.25 mm Loaded epoxy face
face (0.01 in.) Piezoelectric element
Silver electrode Silver electrode Loaded epoxy
Piezoelectric backing
0.25 mm element (b)
(0.01 in.) Loaded epoxy face
Loaded Silver electrode Piezoelectric element
epoxy
backing
(b) Loaded epoxy Loaded epoxy
face backing
Silver electrode
Piezoelectric 0.25 mm
0.25 mm element (0.01 in.)
(0.01 in.)
Loaded
epoxy
backing
74 Ultrasonic Testing
transducer and a test object to generate Table 3 gives the mechanical properties
mode converted transverse waves. The of an acrylic and a polystyrene wedge
plastics commonly used for the wedges material. The attenuation values in Table 4
are acrylics such as methyl methacrylate are from the literature.
and polymers such as polystyrene. These
materials have a strong effect on the Figure 17 is a diagram of experimental
frequency response of the pulse emerging equipment used for measuring
from the wedge. A 5 MHz center attenuation at frequencies between 1 MHz
frequency transducer transmitting a and 10 MHz. Attenuation measurements
longitudinal wave through the plastic are made at several discrete frequencies
wedge in the angle beam transducer over the range of interest (2.25, 5.0, 7.5
assembly, for example, produced a 3.5 to and 10 MHz) using narrow band, tuned
4.0 MHz peak frequency, transverse beam and undamped transducers. Consistently
in the test object coupled to the wedge. uniform coupling between each
transducer and the test material is
The effect becomes more acute for important in making all such attenuation
frequencies around 10 MHz, when the measurements. One good procedure is to
peak frequency in the transverse beam is apply light machine oil between a
shifted to lower values of 6 to 7 MHz. The transducer and a block of the chosen
path length of the ultrasonic wave varies plastic material, wringing the transducer
from front to rear as it travels from the well onto the block to remove air bubbles.
transducer through the wedge to the A loading device can be designed and
surface in contact with the test object. used for applying uniform pressure to
Different frequencies present in the ensure consistent coupling throughout
incident beam suffer different amounts of the measurement.
attenuation. (Attenuation increases with
increasing frequency.) The effect is more Plastics have high ultrasonic
pronounced at high frequencies for the attenuation coefficients compared to
incident pulse of the broader band width. those of metals. In most metals,
Therefore, it is important to know the
transmission characteristics of ultrasonic TABLE 3. Mechanical properties of two arbitrarily chosen
pulses through the plastics commonly plastic wedge materials — results will vary with other
used for wedges. specimens. The acrylic is preshrunk and absorbs
ultraviolet radiation.
Reliable wedge materials are
manufactured as acrylics (polymethyl Property Acrylic Polystyrene
methacrylate) and polymers (polystyrene).
Acrylic thermoplastics are products of the Ultrasonic wave velocity (mm·s–1) 2.73 × 106 2.36 × 106
homopolymerization of acrylic ester Longitudinal 1.43 × 106 1.45 × 106
monomers, principally methyl Transverse
methacrylate. The chemical formula is 0.273 0.236
[-CH2(CH3)(COOCH3)] and is typically Longitudinal wavelength (mm) 0.546 0.472
amorphous. Many acrylics are copolymers 10 MHz 2.730 2.360
of two monomers. Some of them are 5 MHz
linear whereas others are crosslinked. Cast 1 MHz 0.143 0.145
acrylic sheets made of methyl 0.286 0.286
methacrylate monomer are available in Transverse wavelength (mm) 1.430 1.450
linear and crosslinked compositions. 10 MHz
5 MHz 1.180 1.050
Widely used polystyrene brands are 1 MHz 4.15 × 109 3.2 × 109
thermoset crosslinked styrene copolymer. 3.20 × 105 2.47 × 105
Its chemical formula is [–CH2CH(C6H5)–] Other M 85 M 65
and it is typically amorphous. Density (g·cm–3)
Commercial polystyrene typically Young’s modulus (N·m–2)
contains up to 2000 styrene units in the Acoustic impedance (g·cm–2·s)
polymer chain. General purpose Hardness (rockwell)
polystyrene is essentially pure polystyrene
with relative molecular mass Mr between TABLE 4. Ultrasonic attenuation factor for two plastic
50 000 and 60 000. The relative molecular materials at various frequencies.
mass (formerly known as molecular weight)
is the ratio of the average mass per Material Ultrasonic Attenuation Factor (dB·mm–1)
molecule to 12–1 of the mass of an atom 1 MHz 2.5 MHz 5 MHz 10 MHz
of the nuclide carbon-12. Monomers
frequently used for copolymerization with Acrylic 1.5 3.5 7 ___
styrene are acrylonitril, alphamethyl
styrene butadiene, maleic anhydride and Polystyrene 0.8 1.6 3.5 ___
methyl methacrylate. Polystyrene is
semilinear in structure and amorphous in
nature. Impact strength is improved for
impact grades by blending with rubbers
such as polybutadiene.
Generation and Detection of Ultrasound 75
attenuation measurements may be made attenuation in decibels based on the first
over a wide range of frequencies (using a two echoes in the pulse echo train,
block of constant thickness and the pulse according to Eq. 25:
echo technique) by observing the decay in
amplitude of successive echoes. This same (25) α = 20 log V1
technique may be used to measure V2
ultrasonic attenuation in plastics.
However, to cover the frequency range of Equation 25 is based on the
interest, it is usually necessary to use assumption that no energy is lost at the
blocks of different thicknesses for front and back interfaces, by beam spread
different frequencies. These different or by nonparallel reflecting surfaces.
thicknesses may be provided by steps in a Under such ideal conditions, the pulse
single block. echo train produces:
A block of dimensions of 100 × 50 × (26) V1 = V2 = V3
50 mm (4 × 2 × 2 in.) with four steps of V2 V3 V4
heights 13, 25, 38 and 50 mm (0.5, 1.0,
1.5 and 2 in.), for example, is a useful where Vx is the amplitude in volts of the
design for measuring attenuation in first through the fourth echoes in the
polystyrene and methyl methacrylate at train. However, the sources of error
frequencies ranging from 1 to 10 MHz. mentioned above do cause Eq. 25 to
The top and bottom surfaces must be indicate values that are higher than the
highly finished (typically 0.8 µm) and true attenuation attributable to internal
parallel to each other within 17 mrad loss mechanisms. A portion of ultrasonic
(1 deg). For reliable attenuation energy is lost in the bond between the
measurements, the number of echoes transducer and the test object on
obtained in a given material for a given successive round trips of the ultrasonic
frequency must exceed two. pulse. Effects of these factors must be
minimized or accounted for.
In Fig. 17, the functions of the pulse
modulator and receiver are to generate The effect of divergence of the sound
and receive radiofrequency pulses (tone beam can be minimized by making
burst or wave train). The pulse width, the measurements with all multiple echoes
pulse repetition rate, the pulse amplitude appearing in the near field. In this respect,
and the receiver gain must be carefully the stepped blocks are useful for shifting
adjusted to maintain the narrow band measurements to the step of the most
width of the wave train pulse and to practical height. Tables 5 and 6 show the
avoid saturation of the receiver. It is data needed to make such a shift. When
advisable to obtain as many equally proper care is taken to minimize losses
spaced pulse echoes of exponentially from beam divergence, surface finish and
decreasing heights as possible on the nonparallelism of the step faces, the
oscilloscope screen. The attenuation measured attenuation can be attributed
recorder provides a measure of the total primarily to internal losses such as
scattering and absorption.
FIGURE 17. Diagram of test setup for attenuation
measurement. Figure 18 shows the attenuation versus
frequency for methyl methacrylate and
Pulse modulator and receiver styrene polymer. The vertical bar at each
point indicates the spread (maximum to
Radiofrequency minimum) in measured values. At low
module frequency, the spread is about 0.05 dB,
rising to 0.1 dB at high frequency. The
Video Sync Test object overall accuracy of the measurement is 5
out out to 7 percent. The ultrasonic attenuation
coefficient plotted in Fig. 18 was obtained
Video Sync Z axis from total measured attenuation divided
in in Channel 1 by twice the height of the step chosen in
the test object.
Automatic Strobe Oscilloscope
gain Attenuation coefficients in the acrylic
are higher than those of polystyrene at all
control ultrasonic frequencies between 1 and
10 MHz. At high frequencies, the
Attenuation recorder differences in ultrasonic attenuation
coefficients for two plastics is higher than
that for low frequencies. This behavior
can be explained by those characteristics
of polymeric structure important for
determining its mechanical response to
the elastic stress. These characteristics are
76 Ultrasonic Testing
TABLE 5. Calculated values of near field length and half-angle of main lobe to point of
zero energy for commercial acrylic, having longitudinal velocity of 2.73 × 106 mm·s–1
(1.07 × 105 in.·s–1), and for ultrasonic transducer, having diameter of 13 mm (0.5 in.).
Transducer Diameter ____N__e_a_r_F_i_e_l_d____ ____H__a_lf_A__n_g_l_e____
Frequency Wavelength to Wavelength mm (in.)
(MHz) (mm) (ratio) mrad (deg)
10.0 0.27 47 148 (5.84) 26 (1.5)
7.5 0.36
5.0 0.54 35 111 (4.37) 35 (2.0)
2.25 1.09
1.0 2.72 23 74 (2.92) 52 (3.0)
11 33 (1.31) 117 (6.7)
5 15 (0.58) 264 (15.13)
TABLE 6. Calculated values of near field length and half angle of main lobe to point of
zero energy for polystyrene, having longitudinal velocity of 2.39 × 106 mm·s–1
(9.4 × 104 in.·s–1), and for ultrasonic transducer, having diameter of 13 mm (0.5 in.).
Transducer Diameter ____N__e_a_r_F_i_e_l_d____ ____H__a_lf_A__n_g_l_e____
Frequency Wavelength to Wavelength mm (in.)
(MHz) (mm) (ratio) mrad (deg)
10.0 0.24 54 171 (6.72) 23 (1.30)
7.5 0.31
5.0 0.47 40 128 (5.04) 30 (1.73)
2.25 1.01
1.0 2.36 27 85 (3.36) 45 (2.60)
12 38 (1.51) 101 (5.78)
5 17 (0.67) 229 (13.12)
FIGURE 18. Ultrasonic attenuation versus frequency for two its relative molecular mass, crystallinity,
plastics: (a) methyl methacrylate (acrylic) and (b) styrene crosslinking and chain stiffening.
polymer (polystyrene).
The polystyrene shown in Fig. 18 is the
Attenuation coefficient 1.2 (30) Acrylic thermoset crosslinked styrene polymer.
dB·mm–1 (dB·in.–1) 1.1 (28) The crosslinking involves the formation
1.0 (26) Polystyrene of strong covalent bonds between
0.9 (24) individual polymer chains. The
0.85 (22) 1 2 3 4 5 6 7 8 9 10 crosslinking increases the mechanical
0.8 (20) Frequency (MHz) strength and decreases the ability of
0.7 (18) individual chains to slide past one
0.6 (16) another. This, in turn, enhances the
0.55 (14) elastic response and reduces the viscous
0.5 (12) response of the polymer to the induced
0.4 (10) stress. Polystyrene, crosslinked, therefore
0.3 (8) shows less ultrasonic attenuation than
0.25 (6) does acrylic, not crosslinked.
0.2 (4)
0.1 (2)
Generation and Detection of Ultrasound 77
PART 3. Generation and Reception of Ultrasound8
For optimum operation, it is important to The influence of transmitter parameters
understand the effect of the front panel on the shape of the ultrasonic signal has
controls on the internal functions of an been extensively explored experimentally9
ultrasonic testing instrument. Described and theoretically.10 The effects of
below are the principles of operation for transducer parameters on the shape of the
key components in a typical ultrasonic emitted ultrasonic signal are also well
test instrument. understood.11 Figure 19 shows the two
waveforms most often used in ultrasonic
Transducer Excitation test instruments: spike and square wave
pulses. The less popular bipolar tone burst
Most transducers used for ultrasonic and step waveforms are also shown.
testing incorporate a thin plate of
piezoelectric material to convert electrical Bipolar waveforms that have higher
energy, typically stored in a capacitor, into energy content are preferred in certain
an ultrasonic signal that is radiated away. specialized testing applications,
In most discontinuity detection and particularly to penetrate thick, highly
thickness gaging, it is advantageous to attenuating materials. The use of bipolar
generate a compact ultrasonic waveform. signals to excite the transducer can result
This is best accomplished by exciting the in significant improvement in signal
transducer with a short, unipolar voltage amplitude but at the expense of a
waveform whose rise time is shorter than reduction in resolution. This tradeoff is
the time required for an ultrasonic often acceptable, particularly in
impulse to move through the piezoelectric through-transmission testing.
plate.
An alternative method of exciting
FIGURE 19. Waveforms used by ultrasonic ultrasonic transducers uses the step pulse.
testing systems: (a) spike pulse; (b) square Under certain circumstances, this pulse
wave pulse; (c) bipolar tone burst; (d) step shape is preferred, because it can cause a
pulse. piezoelectric transducer to emit a
compact, unipolar ultrasonic waveform.12
(a) Unipolar waveforms generated with step
pulse excitation are sometimes used in
(b) thickness gages and high resolution pulse
echo discontinuity detectors. They can
(c) also yield information about gradual
changes in material properties. However,
(d) because of diffraction effects, which cause
the transmitted ultrasonic waveforms to
become bipolar at relatively short
distances from the transducer, step pulsers
are not used in general purpose ultrasonic
test instrumentation.13
Spike Pulsers
Spike pulsers are among the earliest
electronic circuits used to excite
piezoelectric transducers. Efficient spike
pulsers are relatively simple to construct.
The essential components of a spike
pulser are shown in Fig. 20a and
associated pulse shapes are shown in
Fig. 20b.
The spike pulser operates as follows.
First, the charging capacitor is charged to
a high voltage (typically 250 to 400 V)
through the charging resistor and the
damping resistor. In most portable
instruments, the value of the charging
capacitor is 1 to 4 nF while the charging
resistor and the damping resistor seldom
78 Ultrasonic Testing
exceed 200 and 10 kΩ, respectively. These voltages. Silicon controlled rectifiers were
values permit charging the capacitor to used in many general purpose
the full value of the direct current power instruments but their switching is not fast
supply in less than 1 ms. Thus, pulse enough for high resolution. In general,
repetition frequencies of 1000 Hz can be avalanche transistor circuits should not be
sustained. used to generate pulse voltages in excess
of 200 V. Pulses of 1 kV can be achieved
After applying voltage, the switch is using silicon controlled rectifier switches
abruptly closed, causing the voltage of the while thyratrons can control pulses with
fully charged capacitor to appear across voltages on the order of 10 kV. Thyratrons
the terminals of the transducer. The and silicon controlled rectifiers exhibit
abrupt voltage change causes the fundamental limitations as fast switching
piezoelectric material of the transducer to devices.
respond in the form of an emitted
ultrasonic wave. The exciting voltage then Ultrasonic transducers are typically
rapidly decays because of the damping connected to the pulser with a length of
resistor, connected in parallel with the coaxial cable whose capacitance increases
transducer. The value of the damping at a rate of about 100 pF·m–1. Therefore,
resistor is typically 10 to 100 Ω. This value the capacitance of several meters of cable
can be adjusted by the operator to can easily equal or exceed the capacitance
accommodate different transducer of many transducers. In such cases, a
impedances. Proper adjustment of the significant portion of the pulse energy can
damping resistor is important because it be shunted away from the transducer.
directly determines transducer ringdown
times and the resulting near surface The efficiency of excitation can also be
resolution. degraded by the series inductances and
other parasitic impedances. Series
Because the acoustic pressure at the inductances tend to increase rise times
front face of the transducer is directly and prevent the high frequency portion
proportional to the time derivative of the of the pulse energy from reaching the
applied voltage dV·dt–1, it is important to transducer. These effects may severely
minimize the rise time of the applied affect the ability of a spike pulser
pulse. The rise time is affected primarily efficiently to excite thin film transducers,
by the speed at which the switch can be typically 50 mm (2 in.) thick and
fully closed and by the presence of exhibiting capacitances of only a few
parasitic inductances in series with the picofarads.
capacitor, switch and transducer (parasitic
inductances are not shown in Fig. 20). Because the electrical operating points
of devices used as switches in spike
Historically, fast switching avalanche pulsers cannot generally be controlled,
transistors have been used in very fast impedance matching networks are not
circuits. Spike pulsers, using avalanche recommended. Square wave and tone
transistors, are still frequently used in burst pulsers are better suited for this
thickness gages and high resolution purpose.
discontinuity detectors.
Square Wave Pulsers
In the past, gas filled thermionic tubes,
principally thyratrons, have been favored Semiconductor switching technology has
in applications requiring very high pulse led to the proliferation of square wave
pulsers in ultrasonic instruments. The
FIGURE 20. Spike pulser: (a) circuit diagram; (b) pulse shape. principal disadvantages of square wave
pulsers (high component count and
(a) Charging Capacitor appreciable power consumption) are offset
resistor by important operational advantages. In
Direct current particular, the use of a square wave pulse
power supply increases the ability of the operator to
control and stabilize important test
Transducer parameters, including the harmonic
content (spectrum) of the transmitted
Avalanche transistor or Switch Damping ultrasonic pulses. In addition, the use of a
silicon controlled rectifier resistor square wave pulse may result in higher
pulse amplitudes.
(b)
Figure 21a shows the operation of a
Time square wave pulser and Figs. 21b and 21c
show its typical pulse shapes. Although
the circuit in Fig. 21a is topologically
identical to that in Fig. 20, note that the
shape of the pulse used to excite the
transducer is significantly different. This
difference arises from the use of a
switching device known as the metal
Generation and Detection of Ultrasound 79
oxide superconductor field effect waveform is observed. If the pulse is too
transducer, or mosfet. short, then the ultrasonic pulse amplitude
is significantly smaller than that achieved
High power mosfet devices are widely with an equivalent spike pulser. In
used in square wave pulsers intended to practice, the pulse duration is adjusted
operate between 0.1 and 10 MHz.14 empirically by the operator from the front
Mosfet transistor switches permit the panel or an external computer.
application of 1000 V excitations in less
than 10 ns. In addition, they can safely A properly adjusted square wave pulser
handle pulse currents of 30 A and higher. can generate twice as much signal voltage
Consequently, square wave pulsers are as a spike pulser charged to the same
well suited for driving large, low voltage. This effect is illustrated in Figs. 22
frequency transducers, which frequently and 23 and using 2.25 and 5 MHz broad
exhibit high capacitances. band transducers.15 Even larger
improvements in signal strength are
Initially, the square wave pulser possible when a suitable impedance
operates as a spike pulser. The sharp matching device is interposed between
transition associated with the closing of the transducer and the pulser.
the switch causes the generation of an
ultrasonic signal by the transducer. The theoretical and practical
However, because the charging capacitor advantages of square wave pulsers are well
C is much larger than that used in the understood.16 Except for specialized
spike pulser (typically 1 mF compared to applications, such as thickness gages and
1 nF), the pulse voltage is not allowed to high resolution discontinuity detectors,
decay while the switch remains in the square wave pulsers offer better
closed position. When the switch is performance than spike pulsers. However,
finally restored to its original open to optimize the performance of a square
position, the pulse voltage is returned to wave pulser, the damping resistor and
zero. This second abrupt transition also pulse duration must be adjusted
causes the generation of an ultrasonic independently for each transducer. In the
signal. Because the second transition is
opposite to the first, the second excited FIGURE 22. Comparison of ultrasonic signal
signal is inverted with respect to the first voltages with 2.25 MHz broad band
excited signal. transducer: (a) spike pulser; (b) properly
adjusted square wave pulser.
The time duration of the square wave
pulse must be carefully adjusted to (a)
produce a positive interference between
the ultrasonic signals excited by the
positive going and negative going
transitions of the transducer. If the pulse
is too long, then a distorted ultrasonic
FIGURE 21. Square wave pulser: (a) circuit diagram; (b) open Amplitude
switch pulse shape; (c) transducer voltage pulse shape. (relative scale)
(a) Charging Large
resistor capacitor
Direct current
power supply
Transducer Time
(relative scale)
Metal oxide Switch Damping (b)
semiconductor field resistor +6 dB
effect transistor
(b) On Amplitude
(c) Off (relative scale)
80 Ultrasonic Testing Time Time
(relative scale)
spike pulser, only the value of damping tone burst signals are used in many
resistor is operator adjustable. ultrasonic interferometers for material
velocity measurement.18
Tone Burst Pulsers
Step Pulsers
Tone burst operation can be achieved by
repetitively closing the switch S of the The excitation of ultrasonic transducers
square wave pulser shown in Fig. 21a. The with step pulses requires circuits that are
main advantage of operating the square topologically more complex than those
wave pulser in this mode is that it allows discussed previously. Figure 24a shows a
the operator to maximize the energy of circuit that can impose a step shaped
the transmitted signal at a specific excitation on a piezoelectric transducer.
frequency. The spike pulser and square wave pulser
use one switching device but the step
Tone burst operation can also be pulser requires two separate switching
achieved when a spike pulser is used to devices.
drive an inductively tuned transducer. In
this case, however, frequency control can Figure 24b shows the timing diagram
only be realized by altering the value of for the step pulser in Fig. 24a. First,
the tuning inductor. switch 1 is closed to allow the transducer
to charge to a high voltage. Next, switch 1
Tone burst pulsers are often designed is restored to the open position and
for compatibility with impedance switch 2 causes the transducer voltage to
matching networks required to maximize decay rapidly to zero. This rapid transition
the output of unconventional transducers: causes the generation of the unipolar
electromagnetic acoustic transducers, air ultrasonic waveform.
coupled elements, dry coupled and roller
probes. Pulsers capable of generating Figure 25 shows the effect of pulse
200 A, 450 V tone bursts at frequencies of shape on waveforms observed at the front
several megahertz are available. Tone face of a broad band, thin film
burst excitation is often used in special ferroelectric polymer transducer. In this
instruments, including acoustic case, the same transducer is excited in
microscopes, where frequencies of several turn by different step and spike pulsers.19
gigahertz have been demonstrated.17 Also, The unipolar pulse is more compact than
the bipolar pulse produced by a spike
FIGURE 23. Comparison of signal voltage pulser. In this case, an external damping
with 5 MHz broad band transducer: resistor was not required because a
(a) spike pulser; (b) square wave pulser. transducer with high internal damping
was used.
(a)
Auxiliary Devices
The capabilities of many instruments can
sometimes be significantly extended by
Amplitude FIGURE 24. Step pulser: (a) circuit; (b) timing
(relative scale) diagram.
(a) Switch 1
Transducer
Direct current
power supply
Time Switch 2
(relative scale)
Amplitude(b) (b) Timing circuit
(relative scale)
+5 dB Switch 1 On
Switch 2 Off
Time Transducer voltage
(relative scale)
Generation and Detection of Ultrasound 81
auxiliary devices. These are typically Multiplexers
connected between the transducer and
the instrument and include diplexers External multiplexers are generally used to
(transmit/receive switches), multiplexers, permit connection of multiple transducers
impedance matching networks, external to general purpose, portable or laboratory
power amplifiers and low noise instruments. The capabilities of most
preamplifiers. With the exception of modular equipment can be expanded by
multiplexers, which allow the use of adding specialized modules. Multiplexing
multiple transducers, auxiliary devices are modules are also available for such
needed to improve electrical compatibility instruments. Typically, 2:1, 4:1, 8:1 and
between special purpose transducers and higher signal multiplexing is possible.
general purpose test instruments. However, care must be used to ensure that
transducer mechanical scanning rates are
Diplexers and Transmit/Receive compatible with the instrument pulse
Switches repetition frequency setting. Figure 27
shows a multiplexer setup with a general
All pulse echo instruments provide an purpose ultrasonic instrument. Possible
internal diplexer or transmit/receive applications of this configuration include
switch function. This function can be plate, pipe and laminate testing.
activated by a front panel switch. With
the switch disabled, the instrument is set FIGURE 26. Typical configurations of external transmit/receive
to operate in the pitch catch mode. switch and general purpose ultrasonic test instrument:
(a) output signal amplification; (b) preamplified input signal.
Transmit/receive switches may be used
to protect sensitive internal receiver (a) Transmit/receive External Ultrasonic
circuitry from the effects of high voltage switch power instrumentation
transmitter pulses. External switches are Transducer amplifier
needed mainly to facilitate the use of Transmitter
external power amplifiers or low noise
preamplifiers on portable or laboratory Receiver
instruments.
(b) Transmit/receive Ultrasonic
To operate a system with an external switch instrumentation
transmit/receive switch, the internal Transducer
diplexer must first be disabled to permit Transmitter
the instrument to operate in the pitch
catch mode. The output of the switch can External Receiver
then be connected directly to the receiver low noise
input. Typical uses of an external switch preamplifier
with a general purpose instrument are
shown in Fig. 26.
FIGURE 25. Ultrasonic waveforms observed
at front face of broad band, thin film
ferroelectric polymer transducer: (a) unipolar
pulse; (b) bipolar pulse.
(a)
Amplitude (relative scale) FIGURE 27. Test setup for multiplexer with
general purpose ultrasonic test instrument.
(b) Time Transducer Ultrasonic
resolution array instrumentation
Transmitter
Multiplexer
Receiver
Time (relative scale)
82 Ultrasonic Testing
Impedance Matching Devices dominated by cable inductance and
capacitance and must be excited using
Impedance matching networks are used sharp spike pulses. External power
primarily to improve the ratio of signal to amplifiers are also required to drive most
noise and to facilitate special purpose low frequency transducers, air coupled
transducers. Ultrasonic test equipment is transducers and most electromagnetic
generally designed for compatibility with acoustic transducers.
transducers operating between 1 and
10 MHz. However, poor signal-to-noise Such transducers generally exhibit low
performance can result when an attempt electrical impedances and often require
is made to use piezoelectric transducers square wave and tone burst pulsers that
operating at lower frequencies or can supply peak currents above 100 A.
unconventional transducers such as The efficiency of square wave and tone
electromagnetic acoustic transducers, air
coupled transducers and many dry FIGURE 28. Typical impedance matching
coupled probes. networks: (a) series inductor; (b) tapped
inductor or autotransformer (step down);
Generally, it is difficult to improve the (c) tapped inductor or autotransformer (step
performance of spike pulsers by using up); (d) transformer; (e) low pass; (f) high
impedance matching devices. On the pass.
other hand, such devices can greatly (a) From pulser
improve the power output of many square
wave pulsers and most tone burst pulser (b)
designs. On reception, significant
improvements in signal to noise can also (c)
be achieved by using such networks to
match the impedances of the transducer (d)
and the input preamplifier.20
(e)
Because the electrical impedances of
piezoelectric transducers are dominated (f)
by a large static capacitance, inductors, air
cored tapped inductors8,21 and broad band
transformers22 can be used for matching.
Reactive transformers23 and ladder
networks of inductors and capacitors are
also effective. Generally, computer
modeling is required to achieve optimum
results.24 In broad band matching, a
tradeoff between band width and
mismatch loss must be accepted.25
Examples of useful impedance matching
networks are shown in Fig. 28.
Figure 29 shows a broad band step-up
transformer used to increase the outputs
of a high power pulser capable of
operating in the square wave and tone
burst modes. In the case of the square
wave pulser, a 22 dB gain in output power
was observed relative to that produced by
a spike pulser charged to the same
voltage. The low output impedance of the
pulser (2 to 3Ω) improved the quality of
the match.
Figure 30 shows a simple series
inductor used as an impedance matching
element. Otherwise, this experimental
configuration is identical to that used in
Fig. 29 and a 23 dB gain was observed for
single pulse excitation. Use of the tone
burst signal resulted in a 29 dB
improvement in the ratio of signal to
noise.
External Power Amplifiers and
Pulsers
External power amplifiers are generally
used to drive unconventional transducers.
For example, thin film transducers exhibit
low capacitances that can be easily
Generation and Detection of Ultrasound 83
FIGURE 29. Broad band, step up transformer burst amplifiers can be significantly
to increase power outputs of pulser capable increased by impedance matching.
of operating in square wave and tone burst
modes: (a) system configuration; (b) square External Receiver Preamplifiers
wave input; (c) square wave output;
(d) tone burst input; (e) tone burst output. External preamplifiers are often justified
to improve signal to noise in critical
(a) Step up 100:1 situations, particularly when operating in
a through-transmission or pitch catch
transformer Oscilloscope mode. In such cases, a properly selected
preamplifier can increase the received
Pulser 1:2 signal to noise by 30 dB or more.
However, considerable care must be
(b) 1500Signal (V) FIGURE 30. Simple series inductor used as
impedance matching element: (a) system
1000 configuration; (b) single-pulse input;
500 (c) single-pulse output; (d) tone burst input;
0 (e) tone burst output.
–500
0 (a) 100:1
(c) 4 60 µHz Oscilloscope
Pulser
2
0 5 10
–2 Time (µs)
–4 (b)
10 1000
Signal (V) 0
Signal (dB) –1000 10 20
0 Time (µs)
20 30 (c) 8Signal (dB)
Time (µs)
4
(d) 1500Signal (V)
0
1000
500 –4
0
–500 –8
0 10 20 30
(e) 4 Time (µs)
(d)
2
0 1000
–2 Signal (V)
–4 5 10 0
Time (µs)
10 –1000
0 10 20
Time (µs)
Signal (dB) (e) 8
Signal (dB) 4
0
–4
20 30 –8 20 30
Time (µs) 10 Time (µs)
84 Ultrasonic Testing
exercised when selecting a preamplifier require input signals with amplitudes
for a particular application. between 1 and 10 V, most received signals
must be amplified. To establish the best
Generally, bipolar transistor ratio of signal to noise, they must also be
preamplifiers are preferred in applications preamplified immediately after reception
where the magnitude of the transducer and then filtered. These fundamental
input impedance is less than 500 Ω. For signal transformations are performed in
input impedance between 500 Ω and the instrument’s receiver.
10 kΩ, preamplifiers with either bipolar
transistor or junction field effect The block diagram in Fig. 31 shows the
transistors are recommended. For arrangement of the four signal
transducers with exceptionally high input conditioning stages in the receiver section
impedances, the preamplifier should use of an ultrasonic discontinuity detector or
either a junction field effect transistor or thickness gage. After reception by the
another field effect transistor as the input transducer, the signals are successively
device.26 filtered and preamplified, attenuated and
again amplified and filtered. The band
Preamplifiers using bipolar transistors width and gain parameters of the receiver
as input devices can generally be operated are adjusted by the operator using front
in the 218 to 398 K (–55 to 25 °C; –67 to panel controls. An inconsistent selection
+257 °F) temperature range. Preamplifiers of signal band width and gain parameters
equipped with field effect transistor front can lead to unrepeatable test results.
ends can be operated at low temperatures
but should not be operated at The repeatability of the ultrasonic
temperatures higher than 373 K (212 °F). testing process is strongly influenced by
the frequency dependent gain
Because broad band film transducers characteristics of the receiver.28 For this
often exhibit low capacitances, the reason, ultrasonic test procedures that
preamplifier input capacitance should be prescribe initial instrument adjustments
about 1 pF while the input resistance generally stipulate ways to adjust the gain
must be in excess of 4 MΩ. To maximize and frequency response of the instrument.
the signal-to-noise ratio of signals received
from low frequency piezoelectric The best band width and gain settings
transducers and electromagnetic acoustic are also influenced by transducer,
transducers, one preamplifier design uses discontinuity and pulser frequency
a low noise bipolar transistor as the input response characteristics.29,30 The wide
device. Such devices generally achieve a
higher ratio of signal to noise when used FIGURE 31. Signal conditioning stages in
with low impedance transducers between receiver section of ultrasonic discontinuity
0.1 and 5.0 MHz. As in the previous detector or thickness gage.
example, this design also incorporates a
line driver circuit that suits it for driving Signal from
long cable lengths. transducers
Generally, the band width of the Input protection circuits and
preamplifier has little effect on the final preamplifier
signal-to-noise ratio, except in the case of
thin film ferroelectric polymer Attenuator
transducers.27 Because the signal band
width is generally adjustable within the Radiofrequency amplifier
receiver, most preamplifiers are not
equipped with frequency controls. The Radiofrequency filter banks
voltage gains of most preamplifiers are in
the 20 to 40 dB range. These gain levels
are generally sufficient to override the
internal noise of most receivers and
compensate for cable transmission losses.
Signal Reception and
Conditioning
Ultrasonic transducers are typically
excited with pulse amplitudes from 100 to
1000 V. The voltages of the received
signals can range from microvolts to
several volts. The received signals may
also exhibit frequency characteristics
much different from the pulses used to
excite the transmitting transducers.
Because most signal processing devices
(signal gates, video detectors, level
comparators, analog-to-digital converters)
Generation and Detection of Ultrasound 85
variability of the pulser, receiver and In addition, the input circuits may
transducer frequency responses is incorporate features such as special high
illustrated in Fig. 32. Because the pass frequency filters for eliminating
characteristics of the individual undesirable responses of some transducers
components are so widely variable, and reducing power line interferences.
appropriately damped ultrasonic reference Properly designed input filters can
standards are needed to ensure the significantly reduce the effects of
repeatability of test results. transducer radial modes and improve
recovery times.
Input Circuits
In some modern instrument designs,
Typical input circuits for discontinuity input circuits incorporate features that
detectors and thickness gages provide two facilitate automatic recognition of the
essential functions: (1) rapid recovery transducer type. In such instruments, the
from the transmitter pulses (input gain and filter settings are established
protection circuits) and (2) establishment automatically by an internal
of the signal-to-noise performance of the microprocessor. This feature can improve
instrument (preamplifiers). test reliability.
FIGURE 32. Variability of pulser, receiver and Input preamplifiers are typically broad
transducer frequency responses: (a) square band, low gain devices that can linearly
wave pulser, broad band receiver spectral amplify the full amplitude range of the
response; (b) tuned pulser, narrow band ultrasonic signals. Preamplifiers are
receiver spectral response; (c) broad band needed for boosting the signals to levels
pulser, narrow band receiver spectral high enough to overcome the noise levels
response. of ensuing amplification and filtering.
Their band width and gain characteristics
(a) Receiver response are generally not adjustable. Some
laboratory and modular instruments allow
10 for the selection of different preamplifiers
Amplitude (relative unit) to accommodate low and high impedance
8 transducers.
6 Transmitter The band widths of the receiver
spectrum sections of most portable and modular
instruments usually extend from 1 to
4 into 50 Ω load 15 MHz. In some cases, the frequency
responses of such instruments can be
2 extended down to 100 kHz by external
broad band preamplifiers with sufficient
0 gain to overcome the low frequency
0 1 2 3 4 5 6 7 8 9 10 filtering of the input protection circuits.
Frequency (MHz) Attenuators
(b) Amplitude (relative unit) Transmitter Attenuators are broad band calibrated
spectrum devices used for two reasons. First, they
10 into 50 Ω load are needed to reduce the amplitudes of
8 strong signals to prevent them from
saturating the receiver gain stages that
6 Receiver response follow. Second, they are needed to
4 compare accurately the amplitudes of
signals with a signal from a known
2 reflector or calibration block.
0 Attenuators are typically calibrated in
0 1 2 3 4 5 6 7 8 9 10 decibels to allow quantitative
measurements of signal amplitudes over
Frequency (MHz) the entire dynamic range of an ultrasonic
instrument. Generally, coarse (10 dB) and
(c) Amplitude (relative unit) Transmitter spectrum fine (1 or 2 dB) front panel attenuator
into 50 Ω load controls are provided. In many
10 instruments, the attenuators can also be
remotely controlled using a digital signal
8 generated by external computers.
6 Typically, attenuators permit gain
adjustments in the 30 to 50 dB range.
4 This range generally exceeds the linear
range of the following analog
2 Receiver radiofrequency amplifier. In some designs
response using nonlinear radiofrequency
amplification stages, a much larger
0 attenuator adjustment is provided.
0 1 2 3 4 5 6 7 8 9 10
Frequency (MHz)
86 Ultrasonic Testing
Radiofrequency Amplifiers Signal Processing
Radiofrequency amplifiers are used to In traditional discontinuity detectors,
raise the amplitudes of ultrasonic signals signal processing was accomplished with
to a level that permits signal processing simple analog circuits. The time domain
circuits to operate properly. For example, amplitudes of the signals were first
the correct gain of the radiofrequency recovered from the radiofrequency signals
amplification stages is needed to directly using a video detection process and then
display the received signals on a cathode were further amplified using low
ray tube or similar display device. The frequency (video) amplifiers. Typically,
internal design of the instrument then the resulting signals were then time gated
ensures the proper operation of the signal
processing circuits, including signal gates, FIGURE 33. Block diagrams for signal
video detectors, alarm level comparators processing: (a) analog; (b) digital, where
and analog-to-digital converters. accept/reject decisions are made by external
computer.
Generally, radiofrequency amplifiers (a)
are arranged as two or three blocks of
fixed gain linear amplification that can be 1
switched in or out using front panel
controls. As a rule, the band widths of 2
radiofrequency gain blocks are not
individually adjustable using front panel 3 5
controls. Typical radiofrequency gain 4
blocks provide amplification in 20 dB 97
steps. Because the gain of typical
preamplifiers is 20 dB and an additional 86
30 to 50 dB gain adjustment is possible
using attenuators, the instruments can (b)
typically be adjusted to process ultrasonic
signals over a 100 dB range. 1
Special purpose instruments may use a 2
logarithmic rather than linear
radiofrequency amplifier. The use of such 3
amplifiers allows processing of signals 10 4
with large dynamic range. The dynamic
range of instruments using linear 11 10
amplification is normally less than 30 dB; 12
the dynamic range of instruments using
logarithmic amplification can exceed 14 13
100 dB.
To computer 9. Threshold level
Frequency Filters 10. Wideband amplifier
Legend 11. Video amplifier
The frequency characteristics of the 1. Transducer 12. Flash analog-to-digital
signals are established by filters after 2. Receiver
radiofrequency amplification. Ultrasonic 3. Signal gate converter
discontinuity detectors must be 4. Video detector 13. Cathode ray tube
compatible with a variety of piezoelectric 5. Video amplifier 14. Buffer
transducers. Typically, transducers with 6. Cathode ray tube
nominal frequencies of 1.0, 2.25, 5.0 MHz 7. Comparator
and higher must be accommodated. 8. Alarm
Transducer sizes are also widely variable.
To ensure correct operation over the
maximum range of transducer parameters,
it is desirable to limit the band width of
the received signals after radiofrequency
amplification. This limit helps maximize
the ratio of signal to noise. Bandpass
radiofrequency filter banks are provided
for this purpose.
The center frequency of radiofrequency
filter banks is selectable using front panel
controls. Typically, the center frequencies
of the filter banks are established to
correspond with the nominal frequencies
of standard transducers: 1.0, 2.25 and
5.0 MHz. Often, a wideband setting is also
selected.
Generation and Detection of Ultrasound 87
and compared against threshold levels to approach offers practical advantages at
permit accept/reject decisions. the expense of the signal phase
information.
Advances in digital signal processing
make it possible to apply processing Figure 33 illustrates the essential
techniques directly to the radiofrequency differences between the analog and digital
signals. However, the signals must first be approaches to signal processing.
converted to a digital format using an Figure 33a shows the traditional analog
analog-to-digital converter. Alternatively, approach. The approach in Figure 33b
signals can be digitized following peak allows accept/reject decisions to be made
detection and video amplification. This by an internal microprocessor or an
external digital computer.
FIGURE 34. Transformation of radiofrequency
signal as it passes through video detector: Signal Gates
(a) input waveform; (b) after positive
half-wave rectification; (c) after negative Ultrasonic discontinuity detectors are
half-wave rectification; (d) full-wave usually equipped with one or more signal
rectification. gates. The signal gates are used to isolate a
time domain region of the received
(a) ultrasonic pulse train. The selected region
can then be processed by the following
Pulse amplitude + signal processing stages. The signal gates
(relative scale) 0 are designed to process the selected region
– of the signal train and reject the other
regions.
Time (relative scale)
(b) The width and position of the signal
gates can usually be adjusted by the
Pulse amplitude + operator. In addition, the timing of each
(relative scale) 0 gate can either be synchronized with the
– transmitted pulse or with the arrival of a
selected ultrasonic signal such as an
Time (relative scale) interface echo. The latter feature allows
automatic tracking of the gate with a
(c) Pulse amplitude particular portion of the signal train.
(relative scale)
+ Video and Peak Detectors
0
– The video detector is essentially a rectifier
circuit with low pass filtering that
Time (relative scale) eliminates signals at twice the highest
frequency of the ultrasonic signal. The
(d) Pulse amplitude band width of the video detector is
(relative scale) adjustable using controls that adjust the
+ resistive capacitive time constant.
0 Figure 34 shows the transformation of the
– radiofrequency signal as it is passed
through the video detector. The effects of
Time (relative scale) positive and negative half-wave
rectification on the same input waveform
(three half cycles) are clearly shown in
Figs. 34b and 34c.
Most discontinuity detectors use
full-wave rectification (see Fig. 34d).
Consequently, the operator cannot select
the shorter of the two rectified
waveforms. However, some laboratory and
production instruments allow the
selection of either positive, negative or
full-wave rectification. The ability to
choose the polarity of the rectification is
only important for broad band
transducers.
A peak detector is basically a half-wave
rectifier with a narrow band width. This
effect is achieved by choosing a long time
constant. Peak detectors are often used
with signal gates in discontinuity alarm
circuits. The signal gate is needed to
88 Ultrasonic Testing
permit processing of later signals. The plotters. They can also be used to compare
associated waveforms are shown in Fig. 35. the amplitudes of different portions of the
same ultrasonic signal.
Video Amplifiers
Analog-to-Digital Converters
Video amplifiers are usually needed to
boost the amplitude of the signal after Analog-to-digital converters are circuits
video detection so that they are large that typically allow the conversion of a
enough to drive the vertical plates of a voltage signal to a digital word. In the
cathode ray tube. Video amplifiers are also case of direct conversion of
used as buffer amplifiers to drive radiofrequency signals to digital format,
sample-and-hold circuits. flash analog-to-digital converters are
available. Currently, waveforms with
Generally, the gains of video amplifiers frequencies up to 50 MHz can be sampled
in portable ultrasonic test equipment with eight-bit precision. Peak detected
cannot be adjusted using front panel waveforms do not contain high frequency
controls. However, video amplifiers can be components. Currently, such waveforms
followed by low pass filter circuits that can be converted using the more precise
allow the operator to change the 12-bit converters.
appearance of the demodulated ultrasonic
signals by using front panel controls. FIGURE 36. Sample-and-hold circuit:
(a) circuit diagram; (b) input signal;
Sample and Hold Circuits (c) output signal.
Figure 36a shows the principle of a (a) Output
sample-and-hold circuit. The basic circuit buffer
uses two switches. When closed, the input Input amplifier
switch allows the storage capacitor to buffer Sampling
charge to the output voltage of the peak amplifier gate
detector. This voltage can then be
sampled by the analog-to-digital Capacitor Reset switch
converter. The capacitor is then fully
discharged using the second switch. The (b)
associated waveforms are shown in
Figs. 36b and 36c.
Sample-and-hold circuits are often used
for interfacing to slow analog-to-digital
circuits and external display devices,
including strip chart recorders and
FIGURE 35. Peak detector waveforms: + Pulse amplitude
(a) input pulse; (b) output pulse. 0 (relative scale)
(a) –
Pulse amplitude Time (relative scale)
(relative scale)+ (c)
0 Pulse amplitude
– + (relative scale)
0
Time (relative scale) –
(b)
Pulse amplitude + Sample
(relative scale) time
0
– Time (relative scale)
Signal
gate
Time (relative scale)
Generation and Detection of Ultrasound 89
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