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The Transducer Influence on the Detection of a Transient Ultrasonic Field Scattered by a Rigid Point Target

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The Transducer Influence on the Detection of a Transient Ultrasonic Field Scattered by a Rigid Point Target

H. Khelladi

*

and H. Djelouah

**

*

Institut d’Electronique,

**

Institut de Physique USTHB, B.P. 32, El Alia, 16111, Alger

Abstract – The use of ultrasound in the detection of defects improves the methods of Non Destructive Testing (NDT). Among the principal processes of detection, the echo-mode is the most used. Because of the importance of this later in NDT, we have analysed the effects, on the shape of the transient ultrasonic field, of the defect position and of some parameters constituting the detection line. As the transducer plays leading role in the detection of defects, the influence of its characteristics (damping size and nominal frequency) has been investigated in the present study.

Résumé – L’utilisation des ultrasons dans le domaine de la détection des défauts a pour but d’améliorer les méthodes de contrôle. Parmi les principaux procédés de détection, l’écho-mode est le plus utilisé. En raison de l’importance de ce dernier en contrôle non destructif des matériaux, nous avons analysé l’influence de la position et de la variation de certains paramètres de la chaîne de la chaîne de sondage sur la forme du champ ultrasonore impulsionnel détecté par le capteur récepteur. Dans ce présent travail, nous nous sommes spécialement intéressés à l’influence des caractéristiques du transducteur (amortissement, taille, et fréquence nominale) sachant qu’il joue un rôle déterminant dans la détection des défauts.

Mots clés: Transducteur plan – Ultraons – Echo-mode – Diffraction – Simulation numérique.

1. INTRODUCTION

In Non Destructive Testing of materials, all the information is included in the signal detected by the transducer. This later transmits and receives the ultrasonic waves which propagate in the medium to analyze.

This, it is important to take into accounts its influence on the shape of the detected signal.

The quality of the NDT analyses (detection sensitivity, precision in the defect localization, resolution .... ) is conditioned by each element of the detected line.

A good interpretation of the different detected signals necessitates a control on how each element of the detected line operates. Therefore, the experimenter needs to know the influence of each element that constitutes the detection line. Moreover, a physical phenomenon such as the scattering is generally present and can lead to an erroneous diagnosis.

In the present work, the effects of the radial and axial defect position on the average pressure detected by the receiving transducer are investigated. Thereafter the influence of the transducer characteristics (size, nominal frequency, factor of damping…) is examined in our analysis.

2. DESCRIPTION OF THE DETECTION LINE As shown in figure 1, the ultrasonic equipment in the NDT is simple enough [1}.

Fig. 1: Synopsis diagram of the detection line

The voltage generated by the electrical pulser stimulates a piezoelectric transducer that produces pulses which propagate in the medium to be analyzed (propagation medium); these ultrasonic waves undergo a

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reflection in the presence of any discontinuity in the propagation medium. A receiving transducer allows the detection of these reflected waves. An analysis of these waves permits the localization of the defect and the evaluation of its dimensions.

3. TRANSIENT ULTRASONIC FIELD SCATTERED BY AN INSONIFIED POINT TARGET

The ultrasonic waves emitted by the transmitting transducer undergo a reflection in the presence of any discontinuity in the propagation medium. This discontinuity can be represented by a point target, having an acoustic impedance much larger than that of the propagation medium. So the reflection coefficient at the interface fluid/target can be considered as equal to unity (total reflection). A simple configuration is considered, where the surface Sc of the target is parallel to the surface ST of the transducer.

In the case of an isotropic, homogeneous and lossless fluid, and in the hypothesis of linear acoustic, it is easy to prove that near the axis, the average pressure detected by the receiving transducer is given by [2] - [5] :

) t c, r iER( 0 t

) t ( v ) t c, r iER( P ) t ( v ) t (

p r φ r

∂ ρ ∂

=

= (1)

where ⊗ denotes a time convolution, ρ0 is the density of the propagation medium at rest, rrc is the vector position of the point target; PiER is the transmit-receive impulse response for the pressure; φiER is the transmit- receive impulse response for the acoustical potential.

) t , r t ( ) t , r t ( S C ) S

t , r (

P iE c iE c

T 0

c c 0

iER r r φ r

⊗ ∂

∂ φ

− ρ

= (2)

)) t , r ( )

t , r ( t( S C ) S

t , r

( iE c iE c

T 0 c c

iER r φ r ⊗φ r

− ∂

=

φ (3)

C0 is the propagation velocity of the acoustic waves in the fluid, φiE(rrc,t) is the spatial impulse response.

The acoustic pressure, at any point in the fluid, can be determined after deriving the expression of φiE(rrc,t).

Fig. 2: Geometry for the determination of the field scattered by a rigid point target

As shown in figure 2, two regions are defined according to whether the projection of the point M is on the surface ST of the piston or on the surface SB of the baffle. The first area corresponds to the direct radiation zone.

It is formed by a straight cylinder perpendicular to the transducer plane, and whose basis is the surface Sr. The second area represents the shadow zone, and is constituted by the remainder of the half space. In the case of a circular piston with a radius a, the expression of φiE(rrc,t) is summarized in Table. 1.

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The transducer normal velocity v (t) has been simulated by the function defined by Funch/Mulier [6], and it represents a damped sinusoid of frequency ω, containing N cycles (N=2), and having a duration of τ=2πN/ω.

Table 1: Analytic expression of φiE(rrc,t) for any position ξ off-axis

Region I : ξ < a





=

=

− ξ

− + + ξ π

<

<

= =

= φ

max 2

max 2 2 1

2 c 2c 2 2 0 2

1 0 min 0

0 min

c iE

t t t 0

t t t t z

R 2

z R cos a

C Ar

t t t t

C t t t

0 ) t , r (r

Region I : ξ > a





=

>

=

− ξ

− + + ξ π

=

<

=

φ =

2 max

t2 tmax t min 2 1

2 c 2c 2 2 0 2

1 min c

iE

t t t 0

t t z

R 2

z R cos a

C Ar

t t t 0

) t , r (r

The normal velocity is given by :





  < <



 ω

 

 +



 

 ω + ν

=

elsewhere 0

r t 0 N t

1 sin N 1 N . N ) t ( sin . )

t ( v

0

L L L L L L L L L L L L L L L L L L L L L L L

L L L L L L L L L

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Using equation (4), the variation of the normalized normal velocity versus the time is plotted in figure 3.

Then, the convolution integral has been evaluated using the Fast Fourier Transform.

Fig. 3: Simulation of the velocity waveform for : N = 2, f = 2.25 MHz, vo=10-4 mm/µs 3.1 Influence of the target position on the detected pressure field

3.1.1 On axis case

As shown in figure 4a, for a point target on the axis, the transmit-receive impulse response for the potential )

t , z , 0

( c

φiER is made up of two rectangular functions. It begins at

0 0 Cc

z

t = 2 with a constant value that lasts

until

0 2 2c

1 c C

a z

t z + +

= (this interval represents the plane wave regime) and finishes at

0 2 2c

2 C

a z

t 2 +

= .

At a farther position of the target, the duration of the plane wave regime decreases (Fig. 4d).

The transmit-receive impulse response for the pressure PiER(0,zc,t) is consisting of 3 successive Dirac peaks (Fig. 4b). The first and the third peaks possess the same polarity, have a weight of 0 0

T

c C

S S ρ

− , while the

second one has an inverted polarity and a weight of 0 0

T

c C

S

2S ρ .

The pressure p(t) is consisting of three components having the same waveform than the transmitting transducer normal velocity (Fig. 4c ). This result can be interpreted by these equations.

[

(t t ) 2 (t t ) (t t )

]

) t ( v S C

) S t , z , 0 (

P 0 0 0 1 2

T

cc ρ ⊗ δ − − δ − +δ −

= (5)

[

v(t t ) 2v(t t ) v(t t )

]

S C ) S t , z , 0 (

P 0 0 0 1 2

T

cc ρ − − − + −

= (6)

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Fig. 4: Variation effect of the target position on φiER , PiER and p(t) for : a = 12,7 mm, f = 2.25 MHz, N = 2, ξ = 0 mm, b = 0.04 mm.

The delays of these three pulses correspond respectively to the propagation times t0, t1 and t2.

0 0 c

C z

t = 2 represents the time propagation from the transducer centre to the target, then after reflection back to the transducer centre.

0 2 c2

1 c C

a z

t z + +

=

corresponds to two different acoustic paths covered during the same duration. The first path is from the transducer centre to the target, after reflection to the edge of the transducer; the second path is front the edge of the transducer to the target, then after reflection to the transducer centre.

As pressures add, the amplitude of the second pulse is twice that of the first one.

0 2 2c

2 C

a z

t 2 +

=

Corresponds to the propagation time from the edge of the transducer to the target then after reflection from the target, back to the edge of the transducer.

Thus, at any instant t < t0 the pressure is null on the receiver, since it has not been reached by any perturbation. The finite dimension of the transmitting-receiving transducer has no influence between the instant time interval is called the plane wave regime. In this case, the pressure is equal to the product of the coefficient by the particle velocity delayed by t0.

However, from the instant t1, the pressure field includes in addition the waves stemming from the periphery of the piston. These waves, called edge waves, express the diffraction by the boundary of the piston and the point target.

Referring to figure 5, for a point target on the axis, the pressure detected by the receiving transducer is made up of three pulses whose delays depend on the target position zc. For a target close to the transducer (Fig. 5a, 5b, 5c, 5d and 5e), the three pulses are separated in the time. On the other hand, at a farther position, the delay between the three pulses becomes smaller, causing their overlap as can be seen in (Fig. 5f).

3.1.2 Off-axis case

For a point target off-axis, the situation is slightly more complicated (Fig. 6). In this case, different instants corresponding to the arrival of the waves are defined.

0

0 C c

z t = 2

corresponds to the acoustic path from point H of the source to the target then back to the same point H of the source. The point H represents the orthogonal projection of the point M on the transducer (Fig. 2).

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Fig. 5: Influence of the target position on the detected pressure for : a = 12.7 mm, f = 2.25 MHz, N = 2, ξ = 0 mm, b = 0.04 mm

0

2 2c

1 c C

) a ( z

t z + + −ξ

=

corresponds to two different acoustic paths covered during the same period of time. The first path is from point H of the source to the target, then to the closest edge of the source; the second path is from the closest edge of the source to the target, then to the point H of the source.

0 2 2c

2 C

) a ( z

t 2 + −ξ

=

corresponds to the acoustic path from the closest edge of the source to the target then to the closest edge of the source.

0

2 2c

3 c C

) a ( z

t z + + ξ+

=

corresponds to two different acoustic paths covered during the same period of time. The first path is from point H of the source to the target then to the farthest edge of the source; the second path is from the farthest edge of the source to the target then to the point H of the source.

0

2 c2

2 2c

4 C

) a ( z )

a (

t z + ξ+ + + −ξ

=

corresponds to the acoustic path from the farthest edge of the source to the target then to the closest edge of the source.

0 2 2c

5 C

) a ( z

t 2 + ξ+

=

corresponds to the acoustic path from the farthest edge of the source to the target then to the farthest edge of the source.

In this case the transmit-receive impulse response for the potential φiER(rrc,t) begins at t0 with a constant value that lasts until t1, (Fig. 6a); this interval represents the plane wave regime. For a given position of the target, the duration of the plane wave decreases when ξ increases (Fig. 6d).

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Fig. 6: Variation effect of the target position on φiER, PiER011 and p(t) for : a = 12.7 mm, f = 2.25 MHz, N = 2, Zc = 15 mm, b = 0.04 mm

It is important to notice that off-axis, the transmit-receive impulse response for the pressure PiER(rrc,t) presents an important number of discontinuities (Fig. 6b). PiER(rrc,t) is null everywhere, except at the instants t0 (represented by a Dirac peak with a weight of 0 0

r

c C

S

S ρ ), tl, t2, t3, t4 and t5 where the discontinuity occurs (Fig. 6b).

Fig. 7: Influence of the target position on the detected pressure for : a = 12.7 mm, f = 2.25 MHz, N = 2, Zc = 20 mm, b = 0.04 mm

The pressure field is constituted by the plane wave which has the same waveform than the transmitting transducer normal velocity, followed by the edge waves staggered in the time.

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Let us point out that the pulse amplitude relative to the instants t1 and t3 is more important than that of instants t2, t4 and t5. For the first series of pulses, each instant corresponds to two different acoustic paths;

therefore the contribution is twice as much. On the other hand, for the second series of pulses, each instant corresponds to one acoustic path (Fig. 6f).

A small variation of the parameter ξ leads to a noticeable change comparatively to the on axis pressure. On the axis, the pressure is constituted by three pulses and with amplitude ratios given by 1: 2: 1 (Fig. 5). While, for a variation equal to 0.05 of the parameter ξ/a, the pressure is composed of six pulses (Fig. 7c). For a higher value of the parameter ξ, the amplitude of the first pulse decreases, this is due to the effect of the second pulse which possess an inverted polarity (Fig. 7f).

3.2 Transducer characteristics influence on the detected pressure field 3.2.1 Nominal frequency

As file transducer nominal frequency increases, the three pulses detected by the receiving transducer are separated in the time (Fig. 8a, 8b, 8c and 8d). For a better detection of the diffraction phenomena, a transducer with a high nominal frequency must be used.

3.2.2 Damping

The duration of the three pulses detected by the receiving transducer depends on the number of cycles in the transducer normal velocity v (t). In practice, the number of cycles depends on

Fig. 8: Influence of the transducer nominal frequency on the detected pressure for : a = 12.7 mm, N = 2, Zc = 20 mm, ξ = 0 mm, b = 0.04 mm

whole the electroacoustic line, especially the damping of the transducer. For this reason, it is advisable to use a wide-band transducer. Referring to (Fig. 9a, 9b and 9c), it appears that increasing the number of cycles increases the duration of the three pulses, causing an overlap of the different pulses, and leading to a difficulty in the interpretation of the detected pressure (Fig. 9d).

So for a good interpretation of the detected pressure, we must use not only short pulses to excite a wide-band transducer having a high nominal frequency but also a receiver having a suitable band-pass.

3.2.3 Size

The transducer radius affects the amplitude of the detected signal. More the radius of the transducer decreases, more the amplitude of the signal increases (Fig. 10a). This can be explained by the tact that the amplitude of the signal is proportional to the ratio

r c

S S .

Furthermore, we observe that for a transducer possessing an important size, the detected pulses are resolved in time. Since in this case the acoustic path relative to t1 and t2 is increased, the second and third pulses are delayed (Fig. 10d). In general, the diffraction phenomena are preponderant for a small aperture.

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Fig. 9: Influence of the number of the cycles on the detected pressure for:

a = 12.7 mm, f = 2,25 MHz, Zc = 20 mm, ξ = 0 mm, b = 0.04 mm 4. CONCLUSION

The influence of the transducer characteristics on the ultrasonic field scattered by a rigid point target has been investigated. It has been noticed that the detected pressure is closely related to the characteristics presented by the transducer. For a better signal analyses, it is advisable to use a strongly damped transducer, possessing a high nominal frequency and having an important size, in order to avoid the overlapping of the pulses even for targets far from the transducer. The generalization of this model, by taking into account the effect of the target size on the detected pressure field, will be the subject of a next study.

Fig. 10: Influence of the transducer size on the detected pressure for : f = 2.25 MHz, N = 2, Z = 20 mm, b = 0.04 mm, ξ = 0 mm

REFERENCES

[1] J.L. Pe1letier, ‘La Pratique du Contrôle Industriel par Ultrasons’, Communications actives, ENSAM, Paris.

[2] H. Khelladi et H. Djelouah, ‘Modélisation en Echo-mode du Champ Ultrasonore Impulsionnel Diffracté par une Cible Ponctuelle’, Conférence Maghrébine sur le Contrôle Non-Destructif, Alger, 28-30 Juin 1997.

[3] A.J. Hayman and J.P. Weight, ‘Observations of Propagation of Very Short Ultrasonic Pulses and their Reflection by Small Targets’, J.

Acoust. Soc. Am., 63 (2), pp. 396-404, 1978.

[4] M. Ueda and H. Ihikawa, ‘Analysis of an Echo Signal Reflected from a Weakly Scattering Volume by a discrete Model of the Medium’, J.

Acoust. Soc. Am., 70 (6), pp. 1768-1775, 1981.

[5] S. Mc. Laren and J.P. Weight, ‘Transmit-receive Mode Responses from Finite Sized Targets in Fluid Media’, J. Acoust, Soc. Am., 82, (6), pp. 2102-2112, 1987.

[6] A. Ilan and J.P. Weight, ‘The Propagation of the Short Pulses of Ultrasound from a Circular Source Coupled to an Isotropic Solid’, J.

Acoust. Soc. Am., 88, (2), 1990.

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