NDT.net • May 2006 • Vol. 11 No.5

"Bi-curved" Ultrasonic Transducers

Alex Karpelson, Kinectrics Inc.,
800 Kipling Avenue, Toronto, Ontario M8Z 6C4 Canada
E-mail: alex.karpelson@kinectrics.com


1 Introduction

Ultrasonic 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 pulse-echo (PE) and pitch-catch (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 bi-focused probes

The 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

Ftube = R / (Cm / Cw - 1) , (1)

where R is the tube inside radius of curvature, Cw is the speed of UT waves in water, and Cm is the speed of UT waves in metal tube.

For illustration, circumferential/radial cross-section 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 cross-section represents the probe acoustic field in water and within the tube wall for longitudinal waves. It was obtained by using special "Imagine-3D" software developed in UTEX Scientific Instruments Inc. (Mississauga, ON) for acoustic beam-tracing 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.


Fig 1. "Imagine-3D" beam-tracing simulation for standard NB probe with focal length F=15mm, center frequency f=10MHz, aperture diameter D=4.5mm, and water-path WP=5mm, located inside the metal tube with ID=100mm and WT=4mm.

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 bi-focused transducers are well known and were used for a long time but for different purposes: cylindrically focused probes, paintbrush transducers, etc). The required bi-focused 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:


Fig 2. Novel bi-focused probes: focal length in elevation direction Felev=31mm, focal length in azimuthal direction Fazim=35mm, f=10MHz, aperture diameter D=9.5mm.

Ftotal = WP + (F - WP) ยท Cw / Cm , (2)

where Ftotal is the total focal length of transducer in water and metal, F is the nominal focal length of probe in water, WP is the water-path.

Novel 10MHz bi-focused transducers were designed, manufactured in UTX Inc. (Holmes, NY), and then tested using computer-based scanning UT system, which included Winspect data acquisition software, SONIX STR-8100 digitizer card, and UTEX UT-340 pulser-receiver. 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 bi-focused probe in axial-lateral (elevation) cross-section and axial-lateral (azimuthal) cross-section are presented in Figs. 3a and 3b.


Fig 3a. Transducer acoustic field in water in PE mode with 1.6mm diameter ball-reflector. Axial-lateral (elevation) cross-section, probe elevation direction corresponds to the axial direction of the tube. Probe: novel bi-focused transducer, focal length in elevation direction Felev=31mm, f=10MHz, D=9.5mm.


Fig 3b. Transducer acoustic field in water in PE mode with 1.6mm diameter ball-reflector. Axial-lateral (azimuthal) cross-section, probe azimuthal direction corresponds to the circumferential direction of the tube. Probe: novel bi-focused transducer, focal length in azimuthal direction Fazim=35mm, f=10MHz, D=9.5mm.

As one can see from Figs. 3a-3b, the bi-focused probe has smaller focal length in lateral elevation direction Felev=31mm than in lateral azimuthal direction Fazim=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 Felev=33mm and Fazim=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. 3a-3b) differed substantially from theoretical calculations. The reason is that the approximation of geometrical acoustics, used for computations, is probably not valid for the bi-curved transducers because they transmit non-paraxial rays. Due to a bi-curved 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 bi-focused transducers was measured by detecting calibration slots made on the tube ID and OD. The results showed that the bi-focused 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 bi-focused probes

The 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 beam-tracing simulation in Figs. 4 and 5.


Fig 4. "Imagine-3D" beam-tracing simulation for metal tube in water and standard shear wave angle probe with F=33mm, f=10MHz, D=9.5mm, WP=20.6mm, and incident angle a=250.


Fig 5. "Imagine-3D" beam-tracing simulation for metal plate in water and standard shear wave angle probe with F=33mm, f=10MHz, D=9.5mm, WP=20.6mm, and incident angle a=250.

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 in-phase 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.


Figure 6. Schematics of acoustic fields of spherically (A) and elliptically (B) focused immersion angle probes in transducer lateral elevation direction (in Figure plane along the refraction surface) after angle refraction at flat interface water/metal.

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 time-of-flight 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 time-of-flight 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 Cm, Cw, 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 piezo-element with acoustic lens cylindrically focused in both lateral directions - azimuthal and elevation - with different curvatures. The lens in transducer elevation lateral direction, corresponding to X-axis 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.


Fig 7. Schematic of acoustic field of focused immersion angle probe with curved surface in transducer lateral elevation direction (in Figure plane along the refraction surface) after angle refraction at spherical (cylindrical) concave interface water/metal.

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 time-of-flight 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.


Fig 8. Schematic of acoustic field of focused immersion angle probe with curved surface in transducer lateral elevation direction (in Figure plane along the refraction surface) after angle refraction at spherical (cylindrical) convex interface water/metal.

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 time-of-flight 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 bi-curved logarithmic lenses

When 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. Piezo-transducers 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 piezo-element 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 time-of-flight and wave phase calculations for different acoustic beams.

Note that real acoustic field of a bi-curved 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 bi-curved 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 bi-curved 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 bi-curved surface.

5 Probes with bi-curved mirrors

Till now only transducers with the bi-curved 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 bi-curved piezoelement. But to make such a piezoelement is probably even more difficult than manufacture a probe with a bi-curved acoustic lens. The third and probably the simplest way to solve the problem, is to employ a bi-focused 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 long-focused NB probe with a convex mirror.


Fig 9. Schematic of the long-focused axially positioned NB probe with a cylindrically convex mirror: a - axial-radial cross-section, b - circumferential-radial cross-section.

As one can see, usage of the mirror allows, first of all, employing the long-focused 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

  • It is shown that transducers used for tube testing should be focused differently in the circumferential and axial directions of the tube because of an 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 the all NB and angle probes used for tube inspection.
  • 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. The required bi-focused probe 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 computations can be done using simple formulae based on the geometrical acoustics approximation and Snell's law.
  • NB bi-focused transducer was designed, manufactured and tested. It has a longer focal length in the lateral elevation direction than in lateral azimuthal direction. As a result, taking into account the additional focusing due to tube curvature in the circumferential direction corresponding to transducer lateral azimuthal direction, this probe can be really focused approximately in the middle of the tube wall in both circumferential and axial directions. The experiments showed that the bi-focused probes are more uniformly balanced regarding responses from the ID and OD flaws than similar standard spherically focused transducers.
  • In order to obtain the focal points for angle probes after refraction on the flat, cylindrical or spherical surfaces, the working surface of the transducer should have a 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 probe lateral elevation direction and spherical focusing surface in transducer lateral azimuthal direction. The probe shape can be computed using formulae (3)-(5) based on the Snell's law, geometrical acoustics, and time-of-flight calculations for various acoustic beams.
  • Transducers with "logarithmic" acoustic lenses curved differently in two lateral directions can form a narrow weakly diverging UT beam within the required insonification range, thus providing a high lateral resolution in both directions, azimuthal and elevation. Such a lens can be computed using equations based on the geometrical acoustics and refraction law for beam trajectories, and also time-of-flight and wave phase calculations for different acoustic beams. Sometimes it is more convenient to use another computation technique based on the geometrical acoustics approximation and method of the "imaginary" probe.
  • The real acoustic field of a bi-curved 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 bi-curved surface transmits the UT waves in the various directions. For accurate rigorous computation of the acoustic field of a bi-curved probe one can calculate e.g. the surface integral, where the integration should be carried out over the bi-curved surface.
  • 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 the tube inner concave surface in the circumferential direction.

    References

    1. Karpelson A., Hatcher B. "Collimating and Wide-Band Ultrasonic Piezotransducers", The e-journal of NDT, vol. 9, No. 1, January 2004.

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