NDT.net • May 2006 • Vol. 11 No.5 
"Bicurved" Ultrasonic TransducersAlex Karpelson, Kinectrics Inc.,800 Kipling Avenue, Toronto, Ontario M8Z 6C4 Canada Email: alex.karpelson@kinectrics.com
1 IntroductionUltrasonic Testing (UT) of different tubes is a commonly used inspection method. Various techniques are employed for tube examination in order to detect, characterize and size such "abnormal" areas as flaws, laminations, pores, inclusions, material microstructure imperfections, inhomogeneities, grain size variations, material morphology changes, etc. However, sometimes the results of the inspection are not satisfactory due to insufficient sensitivity. One of the main reasons of low sensitivity is a poor focusing of acoustic beams. Transducers used for tube testing should be focused differently in the circumferential and axial directions of the tube because of the additional focusing provided by the cylindrically concave inner surface of the tube, which works actually as a second focusing acoustic lens. This is applicable to all normal beam (NB) and angle probes used for tube inspection.Typically the pulseecho (PE) and pitchcatch (PC) techniques are used for tube testing to detect, characterize and size flaws located within the tube wall, on the inside surface or outside surface. (Sometimes terms inside diameter (ID) and outside diameter (OD) are used in context of inside and outside surfaces, respectively). NB longitudinal waves and angle beam shear waves propagated in the circumferential and axial directions are commonly employed. Each technique has its advantages and deficiencies. For simplicity we assume that tube is filled with water, access is possible only from within the tube, and all transducers are positioned inside the tube. 2 Normal beam bifocused probesThe most probable reason for low sensitivity is that all standard NB probes are spherically focused, but tube has a cylindrical shape. As a result, only one component, the transducer acoustic lens, provides focusing in the axial direction of the tube (of course, the refraction at the interface water/tube decreases the focal length). But there are two focusing components in the circumferential direction of the tube: the inner cylindrically concave tube surface and the lens (of course again, the refraction at the interface water/tube decreases the focal length).Note that ID concave surface of the metal tube filled with water, can be considered in the Kirchhoff approximation (i.e. for paraxial rays and for all dimensions larger than a wavelength) as a focusing lens with focal length Ftube where R is the tube inside radius of curvature, C_{w} is the speed of UT waves in water, and C_{m} is the speed of UT waves in metal tube. For illustration, circumferential/radial crosssection of the acoustic field of a standard NB spherically focused transducer, used e.g. for inspection of metal tube with ID=100mm and wall thickness WT=4mm, is shown in Figure 1. This crosssection represents the probe acoustic field in water and within the tube wall for longitudinal waves. It was obtained by using special "Imagine3D" software developed in UTEX Scientific Instruments Inc. (Mississauga, ON) for acoustic beamtracing simulation. The shortening of probe focal length inside the metal tube occurs because of two reasons: usual refraction at the interface water/metal for oblique acoustic rays and additional focusing by the inner cylindrically concave tube surface.
In order to provide a good focusing and, subsequently, high inspection sensitivity, the NB transducers used for tube testing should be focused differently in the circumferential and axial directions of the tube. (Note that some types of the bifocused transducers are well known and were used for a long time but for different purposes: cylindrically focused probes, paintbrush transducers, etc). The required bifocused probe for tube inspection should contain an acoustic lens cylindrically focused in both lateral directions  azimuthal and elevation  with different curvatures. The lens curvature (and subsequently the focal length) in transducer elevation lateral direction, corresponding to the axial direction of the tube, should be calculated keeping in mind that there is only one component performing focusing in this direction, and this component is the lens itself. The lens curvature (and subsequently the focal length) in transducer azimuthal lateral direction, corresponding to the circumferential direction of the tube, should be calculated taking into account that there are two components performing focusing in this direction: the lens itself and cylindrically concave inner surface of the tube, which works as a second focusing acoustic lens. The required approximate computations can be done using the geometrical acoustics approximation, Snell's law, and simple formula for a focused probe and flat interface:
where F_{total} is the total focal length of transducer in water and metal, F is the nominal focal length of probe in water, WP is the waterpath. Novel 10MHz bifocused transducers were designed, manufactured in UTX Inc. (Holmes, NY), and then tested using computerbased scanning UT system, which included Winspect data acquisition software, SONIX STR8100 digitizer card, and UTEX UT340 pulserreceiver. The probes are shown in Fig. 2. The main parameters of these new transducers (acoustic fields, frequency responses and sensitivities) were measured on the scanning rig in a water tank. The acoustic fields of a novel bifocused probe in axiallateral (elevation) crosssection and axiallateral (azimuthal) crosssection are presented in Figs. 3a and 3b.
As one can see from Figs. 3a3b, the bifocused probe has smaller focal length in lateral elevation direction F^{elev}=31mm than in lateral azimuthal direction F^{azim}=35mm. As a result, taking into account the additional focusing due to tube curvature in circumferential direction, corresponding to the transducer lateral azimuthal direction, this probe at WP=21mm will be focused approximately in the middle of tube wall (for tube with ID=100mm and WT=4mm) in both circumferential and axial directions. Note that rationale was to develop a probe with F^{elev}=33mm and F^{azim}=42mm. Such a probe would be really focused in the middle of tube wall in both directions. However the obtained experimental results for acoustic field (see Figs. 3a3b) differed substantially from theoretical calculations. The reason is that the approximation of geometrical acoustics, used for computations, is probably not valid for the bicurved transducers because they transmit nonparaxial rays. Due to a bicurved surface, the acoustic rays are transmitted in various directions and at different angles. Therefore, the condition of geometrical acoustics, that the transmitted rays should be the paraxial ones, is not met. Finally, the sensitivity of these bifocused transducers was measured by detecting calibration slots made on the tube ID and OD. The results showed that the bifocused probes had the same absolute sensitivity but were a little bit more uniformly balanced regarding responses from the ID and OD flaws in comparison with the most appropriate standard spherically focused transducers (f=10MHz, FL=33mm, D=9.5mm, WP=21mm) typically used for these tubes inspection. 3 Shear wave angle elliptical bifocused probesThe additional focusing performed by cylindrically concave inner surface of the tube, is particularly important for angle shear wave transducers, because their acoustic fields are significantly distorted after refraction at the surface water/tube.Standard spherically focused angle transducers do not have the focal points after refraction on flat, cylindrical or spherical surfaces, because such surfaces substantially distort the oblique converging wave front transmitted by angle transducer. See beamtracing simulation in Figs. 4 and 5.
In order to obtain the focal points for angle probes after refraction on flat, cylindrical or spherical surfaces, the working surface of the transducer should have a more complicated curved shape. For example, to create the focal point after angle refraction at the flat surface, transducer should have an elliptical focusing surface in transducer lateral elevation direction and spherical focusing surface in transducer lateral azimuthal direction. Fig. 6 schematically shows acoustic fields created respectively by spherically and elliptically focused immersion angle probes in transducer lateral elevation direction (in Figure plane along the refraction surface) after angle refraction at the surface water/metal. Note that probes should be spherically focused in transducer lateral azimuthal direction (perpendicular to Figure plane). It is clear that in order to create a focal point after angle refraction the incident acoustic beams should impinge the refraction surface at different specified angles, which depend on the transducer position, orientation and surface shape. Moreover, acoustic waves from different directions should come to the focal point inphase in order to create the constructive interference. This factor also should be taken into account during calculation of probe position and orientation. All these parameters can be determined applying rather simple algebraic equations based on the geometrical acoustics, Snell's law, and wave phase consideration.
Computation of position and shape of the probe creating the required acoustic field is a synthesis problem. It can be solved by using system of algebraic equations, based on the geometrical acoustics and refraction law for beam trajectories, and also timeofflight and wave phase calculations for different acoustic beams. This system of equations can be finally reduced to the following system of two algebraic equations, which determine the transducer shape and location: where T is the total timeofflight from any point on the probe surface to the focal point, b=b(x) is the abscissa of the incident beam, d is the ordinate of the focal point, quantities d and x  b(x) are shown in Fig. 6b, y(x) is the function describing the probe shape. To solve system (3) one can determine function b=b(x) from the second equation using constants C_{m}, C_{w}, d, and T, then substitute b(x) into the first equation, and finally find function y=y(x), i.e. probe shape and location. The required probe should contain a rectangular piezoelement with acoustic lens cylindrically focused in both lateral directions  azimuthal and elevation  with different curvatures. The lens in transducer elevation lateral direction, corresponding to Xaxis in Fig. 6 plane, should be elliptical. The lens in transducer azimuthal lateral direction, perpendicular to Fig. 6 plane, should be spherical. For curved refraction surfaces (see Fig. 7) the probe's profile will be more complicated than the elliptical one shown in Fig. 6b.
The computation of probe curved surface and transducer position inside the tube can be done using the system of algebraic equations, based on the geometrical acoustics and refraction law for beam trajectories, and also timeofflight and wave phase calculations for different acoustic beams. This system of equations can be finally reduced to the following system of two algebraic equations, which determine the transducer shape and location:
Solving system (4), one can determine refraction angle b = b(j) and function r = r(j), describing the probe shape and location in polar coordinates r and j. Using the same method of computation, the equations analogous to (4) can be easily derived for a convex spherical (cylindrical) interface water/metal shown in Fig. 8. This case occurs when probe with a curved surface creating a focal point inside the tube wall, should be located outside the tube.
To determine the shape of probe curved surface and transducer position outside the tube, one can use equations, based on geometrical acoustics, Snell's law, and timeofflight calculations for different acoustic beams. This system of equations can be finally reduced to the following system of two algebraic equations, which determine the transducer shape and location:
Solving system (5), one can determine refraction angle b = b(j) and function r = r(j), describing the probe shape and location in polar coordinates r and j. 4 Probes with bicurved logarithmic lensesWhen the depth position of a flaw within the tube wall is unknown, the wall should be insonified in radial direction using a narrow weakly diverging acoustic beam. Standard focused probe, creating a narrow beam only at the focal point, cannot do it. Only transducer, forming a narrow weakly diverging acoustic beam within the required range, can provide high detectability and accuracy of the inspection. Piezotransducers with "logarithmic" acoustic lenses [1] have a stretched focal zone, i.e. they form a narrow weakly diverging UT beam within the required insonification range, thus providing a high lateral resolution.The required probe should contain a rectangular piezoelement with a logarithmic acoustic lens cylindrically focused in both lateral directions  azimuthal and elevation  with different curvatures. The lens curvature (and subsequently the stretched focal zone) in transducer elevation lateral direction, corresponding to the axial direction of the tube, should be calculated keeping in mind that there is only one component performing focusing in this direction, and this component is the lens itself. The lens curvature (and subsequently the stretched focal zone) in transducer azimuthal lateral direction, corresponding to the circumferential direction of the tube, should be calculated taking into account that there are two components performing focusing in this direction: the lens itself and cylindrically concave inner surface of the tube, which works as a second focusing acoustic lens. The required computations can be done using e.g. geometrical acoustics approximation and method of imaginary (virtual) acoustic source (probe). Computation of the position and shape of the probe, creating the required acoustic field, is a synthesis problem. It can be solved in two stages applying beam trajectories and refraction law. At the first stage, using the geometrical acoustics approximation, one should determine position and shape of the "imaginary" probe, replacing the real one, located within the metal and creating the desired acoustic field. At the second stage, using the Snell's law and geometrical acoustics, the position and shape of the real transducer, located in water and creating the required field should be determined. Sometimes it is more convenient to use another computation method, which allows determining directly the shape of the probe by solving the system of algebraic equations. These equations are based on the geometrical acoustics and refraction law for beam trajectories, and also timeofflight and wave phase calculations for different acoustic beams. Note that real acoustic field of a bicurved transducer may differ significantly from a theoretical one, calculated on the basis of geometrical acoustics approximation. The reason is that such an approximation is valid only for the paraxial acoustic rays, while transducer with a bicurved surface transmits the UT waves in the various directions. (Recall that paraxial rays are the rays propagating with small height and inclination angle in comparison with the aperture radius and aperture angle of the transducer, respectively). For accurate computation of the acoustic field of a bicurved probe one can use e.g. rather complex but rigorous theory, based on the surface integral calculation, where the integration should be carried out over the bicurved surface. 5 Probes with bicurved mirrorsTill now only transducers with the bicurved acoustic lenses were analyzed. However, it is not easy to design and manufacture such probes. This procedure is a rather complicated one, and often leads to the disappointing results. Another way to achieve the same goal is to make transducer with a bicurved piezoelement. But to make such a piezoelement is probably even more difficult than manufacture a probe with a bicurved acoustic lens. The third and probably the simplest way to solve the problem, is to employ a bifocused mirror in the transducer assembly.Application of mirror allows changing the beam propagation direction and focusing/defocusing of the acoustic field. Performing UT inspection of a tube with probe located inside it, one can use the defocusing acoustic mirror cylindrically convex in the circumferential direction. Such a mirror will compensate the focusing, which occurs due to tube inner concave surface in the circumferential direction. Fig. 9 shows the schematic of the longfocused NB probe with a convex mirror.
As one can see, usage of the mirror allows, first of all, employing the longfocused transducer even for small tubes inspection by positioning this probe axially. Secondly, mirror, convex in the circumferential direction, defocuses UT beam and compensates focusing on the tube concave inner surface. As a result, the focal points in the circumferential and axial directions coincide (see Figs. 9a and 9b). It should increase the sensitivity of inspection and measurement accuracy. 6 Conclusions

© NDT.net  Top 