On the suitability of induction coils for crack detection and sizing in metals by the surface magnetic field measurement technique

NDT&E International 34 (2001) 493±504

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On the suitability of induction coils for crack detection and sizing in metals by the surface magnetic ®eld measurement technique S.H.H. Sadeghi*, B. Toosi, R. Moini Department of Electrical Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran Received 30 June 2000; revised 5 December 2000; accepted 11 December 2000

Abstract The surface magnetic ®eld measurement (SMFM) technique has proved to be an accurate means for crack detection and sizing cracks in ferrous metals. The technique involves the use of two U-shaped current-carrying wires of suf®ciently high frequency while measuring the discontinuity in the resultant magnetic ®eld at the crack edge with an appropriate magnetic ®eld sensor. In this work, we describe a mathematical algorithm to obtain the crack signal from the output of an induction coil used in a SMFM probe. We also discuss the measurement errors due to the coil size and shape. To reduce the measurement errors, we present an algorithm in which the crack signal is recovered by appropriate deconvolution of the coil output signal and its spatial transfer function. The algorithm is then used to recover crack signals for various coil shapes and sizes. The study of the results demonstrates the effectiveness of the algorithm in the case of large coils. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Crack; Surface magnetic ®eld; Induction coil; Deconvolution

1. Introduction Since the early 1950s, one of the most commonly used methods for detection of ¯aws in metals has been the eddycurrent technique. In spite of recent advances in the technique, the eddy-current quantitative measurements of cracks usually require a complicated calibration procedure and often involve the use of calibration standards [1±4]. Moreover, magnetic materials generally present special problems in eddy-current testing. In fact, real defect signals are not usually distinguishable from those due to normal permeability variation. Factors that can in¯uence permeability include variations in heat treatment, degree of cold work, minor alloy composition differences and presence of residual stresses. In the recent years, examination of the magnetic ®eld phenomenon at the regions of ¯aws and cracks in ferrometallic components has shown an increasing potential and success as a means for nondestructive detection and evaluation of defects. Sadeghi and Mirshekar-Syahkal [5± 10] have laid the foundations for the development of a new technique for detection and sizing surface-breaking cracks in ferrous metals. The new technique, which is called the * Corresponding author. Fax: 98-21-640-6469. E-mail address: [email protected] (S.H.H. Sadeghi).

surface magnetic ®eld measurement (SMFM), is based on the measurement of surface magnetic ®eld around cracks. In this technique, the surface magnetic ®eld is produced by a pair of U-shaped wires carrying an ac current of suf®ciently high frequency, and the perturbations due to a crack in the surface ®eld is measured by a properly oriented magnetic ®eld sensor placed at a close proximity of the metal surface. Fig. 1 shows a simulation of a typical SMFM signal when the tangential component of the magnetic ®eld at the surface of a ferrous metal with a semi-circular crack is measured. As can be seen in this ®gure, at the edges of the crack, a discontinuity in the sensor output is produced by which the crack can be detected. The magnitude of the discontinuity in the crack signal is exploited to measure the crack depth using an inversion curve based on the theoretical modeling of the surface magnetic ®eld. Also, the crack length can be determined using the strong in¯ections in the signal are indicative at the two ends of the crack. As alluded above, the crack depth in the SMFM technique can be determined using inversion charts based on a relevant mathematical model. The model assumes that the sensor output is exactly proportional to the surface magnetic ®eld. Deviations from this assumption could undermine the accuracy of the crack depth measurement. In this context, several magnetic ®eld sensors have been devised with various degree of sensitivity, resolution, ease of use,

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Fig. 1. A three-dimensional view of simulated SMFM crack signal. The signal corresponds to the output of an ideal magnetic ®eld sensor measuring the x-component, Bx, of the tangential magnetic ®eld at the surface of a conducting ferrous metal containing a semi circular fatigue crack. The magnetic ®eld in the metal is produced by a pair of U-shaped current carrying wires located above the metal.

cost, etc. Hall-effect probes [11], magnetoresistive sensors [12], magnetic±optic sensors [13], tape-head probes [6], Superconducting Quantum Interfering Device (SQUID) [14], and induction coils [15] are some of these sensors. An induction coil is a simple magnetic ®eld sensor whose output signal is related to the magnetic ®eld within volume according to the Faraday's law. The low cost, availability and simplicity of the induction coil makes it an appropriate candidate for magnetic ®eld measurement. However, only coils with in®nitesimally small size can perform a point measurement of magnetic ®eld. The ®nite size of coils in practice produces measurement errors due to its averaging effect on magnetic ®eld signals. Traditionally, pickup coils are made of several turns of wire, each with the same radius. One of the most common criterions is to determine the coil radius and number of turns that maximize the signal-tonoise ratio for detecting and sizing a crack. While this approach is useful, it does not consider the averaging effect of the coil. This is an important aspect of the SMFM technique, which infers the crack size from the signal discontinuity. In this paper, we aim to evaluate quantitatively the suitability of induction coils for crack detection and sizing by the SMFM technique. The paper is organized as follows. In Section 2, a theoretical model is presented to simulate the output signal of the SMFM probe when an induction coil with ®nite size and arbitrary shape is used as the magnetic ®eld sensor. Section 3 describes a novel inversion algorithm for reducing the averaging effect of the induction coil on the crack signal in the presence of noise. In Section 4, we

discuss the suitability of several induction coils with various sizes and shapes for crack depth measurement. 2. Theoretical model The schematic diagram in Fig. 2 illustrates the theoretical model used for the problem. In this model, a semi-in®nite ferrous metal contains a circular-arc crack of length 2p along the x-axis. The maximum depth of the crack, d0, is along the z-axis. The surface ®eld is produced by a pair of U-shaped wires (exciter) carrying an ac current of angular frequency v . The SMFM probe is assumed to operate in the stationary mode in which the exciter stands at a ®xed position with respect to the crack, and the magnetic ®eld sensor scans the crack normally. The magnetic sensor is an induction coil in the vicinity of the metal surface. The coil axis is chosen to be parallel to the crack edge in order to measure the x-component of the magnetic ®eld, Bx, at the metal surface plane. It is assumed that the coil (Fig. 2) consists of one layer of N tightly wound turns with cross-section area A lying in the y±z plane, forming a solenoid of length l, width l1 and height l2 along the x-axis. In general, the coil cross-section can have various shapes, including circular, rectangular and triangular shapes, Fig. 2. In addition, it is assumed that the coil-to-metal spacing and the coil height are so small that the variation of Bx in the z-direction can be ignored. This is a common practice in the SMFM technique to ensure that the surface magnetic ®eld is measured. It is not the magnetic ®eld at a point but instead the ®eld

S.H.H. Sadeghi et al. / NDT&E International 34 (2001) 493±504

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A

To current source A

z

Inducer

I Scan Direction

c I

l2

Induction coil sensors with various cross sections

A l S l1

-p ys o

h

Metal Surface

a

o b

b

xs

a

Crack

y

d0 p

Projection area of the coil on the x-y plane

D x

Fig. 2. Schematic diagram of a SMFM probe in the stationary mode, comprising a pair of U-shaped current-carrying inducer and an induction coil magnetic ®eld sensor.

averaged over the coil area that determines the signal measured by the sensor. In addition, the coil, in general, can have an arbitrary shape. The coil output voltage, v, is obtained according to Faraday's law Z w dx dc vˆ2 …1† ˆ2 dt l dt where c is the total ¯ux linkages in the coil and w denotes the ¯ux passing through each turn. Considering that c is a sinusoidal function of time, Eq. (1) may be rewritten in phasor format, i.e. Z V ˆ 2q C ˆ q F dx …2† l

where V, C , and F are, respectively, the phasor representations of v, c and w , and q ˆ 2jv: To obtain V, one should expand the right-hand-side of Eq. (2) in terms of Bx. For example, when the coil is centered at origin along the scanning line (Fig. 1), the coil output voltage, V(0,0), is obtained as below: Z ZZ V…0; 0† ˆ q dx B x …x; y; z† dy dz …3† l

A

Since the variation of Bx in the z-direction is ignored,

Eq. (3) is reduced to: ZZ z…x; y†Bx …x; y† dx dy V…0; 0† ˆ q S

…4†

where S is the projection area of the coil on the x±y plane and z denotes the z-coordinate of an arbitrary point on the coil. To compute the right-hand-side of Eq. (4), we de®ne a shape function in the sense of the theory of distributions [16] for any point on the coil, G(x,y), which describes the coil shape and size. For example, the shape function for a rectangular coil located at the origin (point o in Fig. 2) is 8 l l1 > > < l2 ; uxu , 2 and uyu , 2 G…x; y† ˆ …5† > > : 0; uxu . l or uyu . l1 2 2 and for a triangular coil is 8 l2 l l1 > > < l2 2 2 l uyu; uxu , 2 and uyu , 2 1 G…x; y† ˆ > l l > :0 uxu . or uyu . 1 2 2

…6†

In the case of a circular coil, since the variation of Bx in the z-direction is ignored, one can assume that the circular coil

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Fig. 3. Shape functions, G…x; y†; and their corresponding normalized ®lter functions, G…kx ; ky †; for three induction coil sensors with rectangular cross-section (a,b), half-circular cross-section (c,d), and triangular cross-section (e,f).

is symmetrical with respect to plane z ˆ l2 =2: Thus, the coil can be divided into two identical parts and its total magnetic ¯ux linkage is easily obtained by doubling up the magnetic ¯ux linkage induced within each half. In other words, the shape function for a circular coil is 8 q l l1 > 2 2 > < l1 2 4y ; uxu , 2 and uyu , 2 G…x; y† ˆ …7† > l l1 > : 0; uxu . or uyu . 2 2 Using the concept of coil shape function, Eq. (4) can be rewritten as below: Z1 1 Z1 1 G…x; y†Bx …x; y† dx dy …8† V…0; 0† ˆ q 21

21

When the coil is moved to point (xs, ys), V is obtained as below: Z1 1 Z1 1 V…xs ; ys † ˆ q G…xs 2 x; ys 2 y†Bx …x; y† dx dy …9† 21

21

In other words, V is linearly related to the convolution of G and Bx. Using the convolution theorem of Fourier integral transform [17] and Eq. (9), we have V…kx ; ky † ˆ q´G…kx ; ky †´Bx …kx ; ky †

…10†

whereV…kx ; ky †; G…kx ; ky † and Bx …kx ; ky †, are, respectively, the two-dimensional Fourier transforms of V(x,y), G(x,y),

and Bx(x,y), and the variables kx and ky are the components of the spatial frequency k. Eq. (10) indicates that the coil output voltage does not necessarily give the true measurement of the magnetic ®eld. In fact, the coil acts as a spatial ®lter whose behavior is characterized by the coil ®lter function, G…k†: Fig. 3 shows plots of shape functions and their corresponding ®lter functions for three coils with rectangular, half-circular and triangular cross-sections. 3. Recovering the crack signal As seen in Eq. (10), the spatial frequency spectrum of the magnetic ®eld distribution, Bx …k†; could be recovered by a division operation. The division process is only permitted when G…k† never becomes zero. However, as illustrated in Fig. 3(b), (d) and (f), G…k† is a low-pass ®lter with in®nitely many zeros. The zeros of G…k† can therefore cause an illposed problem in the determination of Bx …k†: To overcome this problem, one may low-pass ®lter the measured data to eliminate the high-spectrum-frequency components before computing the division process [18]. The largest frequency component in this process must be less than the ®rst zero of G…k†: While this generalized deconvolution process may be appropriate for recovering the measured ®eld with slowly varying distribution, it is not applicable in our problem

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where the discontinuity feature of the ®eld with large spatial frequency components is of great interest.

the dif®culties associated with noise or the zeros of the coil ®lter function.

3.1. Experimental considerations

3.3. Numerical implementation of the algorithm

In a practical measurement system, the presence of noise is inevitable. Thus, a further dif®culty arises in determining Bx …k† when the noise effect is included in the recovery process. In fact, for large values of k and in the vicinity of the zeros of G…k† where G…k† is small, the division operation ampli®es the measurement noise. Again, the low-pass ®ltering process can solve the problem provided that the ®eld distribution is not fast varying in space.

To implement the proposed algorithm on a digital computer, one should consider ®elds sampled at a ®nite number of discrete points instead of continuous functions de®ned over the entire x±y or kx ±ky. In other words, restricting the sampling period to a ®nite interval is equivalent to multiplying the original function in space domain by a rectangular pulse of the same length. Since discrete frequency spectra arise when considering periodic functions, it is assumed that the surface magnetic ®eld is periodic with periods X and Y in x and y directions. Although in experiments we do not study periodic functions, we can select X and Y suf®ciently large that our assumption of periodicity does not affect signi®cantly our results. In fact, X and Y must be selected much larger than both the spatial dimensions of the inducer so that the magnitude of the ®eld function tends to zero. This results in negligible discontinuities in the sampled function due to rectangular windowing of the original function which can otherwise lead to additional errors known as spectral leakage [17]. We sample the magnetic ®eld at discrete points, where Nx and Ny are the number of points along each axis of a rectangular grid sides of length X and Y, and Dx and Dy are the distances between adjacent points in the grid …Dx ˆ X=Nx and Dy ˆ Y=Ny †: The distances between points in the kx ±ky plane are Dkx and Dky, where Dkx ˆ 2p…Nx Dx† and Dky ˆ 2p…Ny Dy†: To avoid aliasing errors [17], the maximum

3.2. Proposed algorithm In view of the problems mentioned above and considering the fact that a wide spectrum of Bx …k† must be determined for accurate recovery of the discontinuity in the magnetic ®eld, we have adopted a different technique. In this technique, a threshold level, T, is ®rst chosen to remove sections of G…k† where noise is dominant (Fig. 4). Then, Bx …k† is computed as below: 8 > < V…k† for uG…k†u . T Bx …k† ˆ q´G…k† …11† > : 0 for uG…k†u , T Finally, Bx(x,y) is determined using an inverse Fourier transform algorithm. With the proposed technique, a relatively wide spectrum of Bx …k† will be recovered without having

Fig. 4. Selected sections of the magnitude of the ®lter function, uG…kx ; ky †u; used in the deconvolution process. In this plot, uG…kx ; ky †u corresponds to the rectangular coil ®lter function shown in Fig. 3 (b) and T ˆ 6:7% of the peak value of the coil ®lter function.

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spatial frequencies contributing to the surface magnetic ®eld, Kx and Ky, must be less than p/Dx and p/Dy, respectively. Although the frequency spectrum of the ®eld is not strictly band-limited, its spatial frequency content is greatly attenuated at large k, minimizing aliasing errors. 4. Results Results of various simulations are presented here to examine the effects of coil size and shape on the measured magnetic ®eld associated with a speci®ed SMFM crack signal. The crack is assumed to be semi-circular (Fig. 1) with d0 ˆ 5 mm and p ˆ 18 mm, and the SMFM probe geometry are a ˆ 4 mm; b ˆ 10 mm; c ˆ 80 mm; h ˆ 11 mm; and D ˆ 10:5 mm: The crack signal has been obtained theoretically using the technique described in Ref. [8]. To simu-

late the output signals of various coils, a similar technique to that proposed in Ref. [18] is used. The technique is summarized in the block diagram shown in Fig. 5. As can be seen in this ®gure, the actual magnetic ®eld (crack signal) is ®rst convolved with the coil shape function. The convolved signal is then added by white noise in order to simulate the coil output signal in a practical measurement system. Finally, the crack signal is recovered using the algorithm described in this paper. In the implementation of the algorithm, Nx ˆ Ny ˆ 512 and DX ˆ DY ˆ 0:4 mm for which the errors due to aliasing and spectral leakage can be ignored. 4.1. Effect of coil shape In the ®rst set of simulations, the effect of coil shape on the crack signal is examined. In this connection, three coils

Fig. 5. Schematic diagram summarizing the forward and inverse problems. Starting with a known SMFM crack signal, Bx(x,y), we calculate the sensor output signal, V(x,y), then add noise, and calculate the recovered crack signal from noisy sensor output signal.

S.H.H. Sadeghi et al. / NDT&E International 34 (2001) 493±504

with rectangular, circular and triangular cross-sections have been considered. For the purpose of comparison, the effect of noise is ignored and each coil is assumed to have the same volume, vol ˆ 45p mm 3. It is also assumed that the projection of each coil on the x±y plane, S ˆ 30 mm2 : The geometry of the selected coils and their corresponding shape and ®lter functions are shown in Fig. 3. The normalized output signal of each coil when scanning the crack along the x-axis is plotted in Fig. 6(a), (c) and (e). The normalizing factor in these plots is the coil output voltage when it is located under the inducer (point o 0 in Fig. 2) in the absence of the crack. Examination of plots in Fig. 6 clearly reveals the averaging effect of each coil. In particular, it is seen that the discontinuity feature of the signal is smeared out. In order to recover the actual crack signal, the simulated coil output signals have been further processed using the deconvolution technique described earlier. In the recovering process, three threshold levels have been selected so that the performance of each coil in various test conditions is examined. The results are plotted in Figs. 6±8 for which T ˆ 1:36; 5.44 and 10.88 percentage of the peak value of the coil ®lter function in each case, respectively. Notice that for a given coil shape function, the recovered discontinuity in the magnetic ®eld is a function of the threshold

499

level in the deconvolution process which, in general, depends on the measurement noise. We would like to have some quantitative estimate of the quality of each coil in the proposed recovery technique. One ®gure of merit is the value of normalized magnitude of the discontinuity, R, i.e. Rˆ

Bx;2 2 Bx;1 Bx;1

…12†

where Bx,1 and Bx,2 are the magnitudes of the crack signal at the lower and upper points of the discontinuity in the crack signal, Fig. 6(b), (d) and (f). In particular, the computed value of R at the deepest point of the crack, R0, can be used to measure the depth of the crack, d0 [8]. In addition to the local value of R, one can compute the normalized mean-square deviation (MSD) between the original discontinuity curve, R(x) and its reconstructed counterpart, R c(x), i.e. Z1 1 MSD ˆ

21

2

uR…x† 2 Rc …x†u dx

Z1 1 21

2

…13†

uR…x†u dx

The smaller the MSD, the better the recovering process; if

Fig. 6. Normalized crack signal (solid line) and its recovered counterpart from output signal (dot line) of a rectangular coil (a,b), a circular coil (c,d), and a triangular coil (e,f) when scanning the crack along the y-axis (a,c,e) and along the x-axis (b,d,f). In (a), (c) and (e) the coil output signal in each case (dashed line) is also plotted. The noise in the measurement system is ignored and T ˆ 1:36% of the peak value of the coil ®lter function in each case.

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Fig. 7. Normalized crack signal (solid line) and its recovered counterpart from output signal (dot line) of a rectangular coil (a,b), a circular coil (c,d), and a triangular coil (e,f) when scanning the crack along the y-axis (a,c,e) and along the x-axis (b,d,f). In (a), (c) and (e) the coil output signal in each case (dashed lines) is also plotted. The noise in the measurement system is ignored and T ˆ 5:44% of the peak value of the coil ®lter function in each case.

the MSD is zero, the crack signal and its recovered counterpart are identical. The MSD is not the only way to compare the recovery quality. However, this parameter does allow us to compare the recovered signals and say something about their global relative quality. The values of R0 and MSD for each coil are shown in Table 1. It should be, however, noted that, due to the Gibbs phenomenon [17], the reconstructed value of discontinuity, Rc0 , is obtained by averaging the signal oscillations in the vicinity of the discontinuity. Results shown in Table 1 indiTable 1 Quantitative evaluation of the recovery algorithm for various coil shapes Coil shape

T (%)

R0

Rc0 2 R0 £ 100 R0

MSD

Rectangular

1.36 5.44 10.88

0.5659 0.5319 0.5056

23.42 29.39 213.87

0.0148 0.0080 0.0419

Circular

1.36 5.44 10.88

0.5616 0.5316 0.5098

24.33 29.44 213.15

0.0148 0.0181 0.0060

Triangular

1.36 5.44 10.88

0.5579 0.5665 0.5382

24.96 23.49 28.31

0.0113 0.0389 0.0661

cate that the rectangular coil yields the best performance in all cases. The superiority of the rectangular coil over the circular and triangular coils should be traced to its shape and ®lter functions in the recovery algorithm. 4.2. Effect of noise In the next set of simulations, the effect of noise on the recovery process is examined. As mentioned earlier, the presence of noise in a practical measurement system is inevitable. In this connection, the recovery algorithm for the output signal of the rectangular coil described in Section 4.1 in the presence of three noise levels has been investigated. The peak-to-peak value of noise in the measurement system, n, is assumed to be 0.59, 1.47 and 2.35% of the peak value of the coil output signal. The selected noise levels were then added to the noiseless signal according to Fig. 5. The C-scan plots of the metal prior and after implementing the recovery algorithm are shown in Fig. 9. As seen in this ®gure, the recovery algorithm considerably enhances the crack signal (the discontinuity) while imposing unimportant noise on the rest of the signal. To quantitatively examine the suitability of the algorithm in the recovery of the crack signal, the normalized output signal of the coil

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Fig. 8. Normalized crack signal (solid line) and its recovered counterpart from output signal (dot line) of a rectangular coil (a,b), a circular coil (c,d), and a triangular coil (e,f) when scanning the crack along the y-axis (a,c,e) and along the x-axis (b,d,f). In (a), (c) and (e) the coil output signal in each case (dashed line) is also plotted. The noise in the measurement system is ignored and T ˆ 10:88% of the peak value of the coil ®lter function in each case.

along the x- and y-axes are plotted in Fig. 10. Examination of plots in Fig. 10 clearly reveals the suitability of the proposed recovery algorithm. The values of R0 and MSD for each noise level are shown in Table 2. The selected threshold level in each case has been selected such that the magnitude of the discontinuity in the crack signal becomes as close as possible to its actual value. Results shown in Table 2 demonstrate the suitability of the performance of the proposed recovery algorithm in a practical test condition and in the presence of noise. 4.3. Effect of coil size The effect of coil size on the measured crack signal is examined in this section. It should be noted that the coil output signal in general approaches the actual crack signal Table 2 Quantitative evaluation of the recovery algorithm for various noise levels Coil shape

n(%)

R0

Rc0 2 R0 £ 100 R0

MSD

Rectangular

0.59 1.47 2.35

0.5760 0.5768 0.5248

20.35 21.73 210.59

0.0402 0.0417 0.0488

as the coil size is reduced. However, the coil size may not be smaller than a certain limit due to practical reasons. In this connection, three rectangular coils with various sizes have been investigated. Assuming that the coils have the same length …l ˆ 5 mm†; their widths were taken to be l1 ˆ 4; 6 and 8 mm. In order to have the same signal-to-noise ratio for a given noise level, the volume of each coils was assumed to be identical (vol ˆ 45 p mm 3) from which the coil height (l2) in each case was chosen accordingly. The normalized output signal of each coil when scanning the crack along the x- and y-axes are plotted in Fig. 11. The normalizing factor used in this ®gure is the same as the previous cases while the peak-to-peak value of noise in the measurement system is assumed to be 1.18% of the peak value of the coil output signal. Examination of plots in Fig. 11 con®rms that the crack signal departs from its actual shape as the coil size increases. For the purpose of comparison, the results of the recovery algorithm are also plotted in Fig. 11. The selected threshold level in each case has been selected such that the magnitude of the discontinuity in the crack signal becomes as close as possible to its actual value. To evaluate the effectiveness of the recovery algorithm, the values of Rc at the center of crack are shown in Table 3. From Table 3, it can be observed that the

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Fig. 9. The C-scan plots of the metal surface: (a) prior implementation of the recovery algorithm; (b) after implementation of the recovery algorithm.

Table 3 Quantitative evaluation of the recovery algorithm for various coil sizes Coil shape

l1 (mm)

R0

Rc0 2 R0 £ 100 R0

MSD

Rectangular

4 6 8

0.5798 0.5129 0.5227

21.22 212.62 210.95

0.0285 0.0411 0.0375

use of the proposed algorithm can substantially recover the discontinuity in the crack signal even in the case of large coils. 5. Conclusions The performance of an induction-coil magnetic ®eld

Fig. 10. Normalized crack signal (solid lines) and its recovered counterpart from output signal (dot lines) of a rectangular coil for various noise levels in the measurement system when scanning the crack along the y-axis (a,c,e) and along the x-axis (b,d,f); (a,b) n ˆ 0:59%; (c,d) n ˆ 1:47%; and (e,f) n ˆ 2:35% of the peak value of the coil output signal. In (a), (c) and (e) the coil output signal (dashed lines) is also plotted.

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Fig. 11. Normalized crack signal (solid line) and its recovered counterpart from output signal (dot line) of a rectangular coil with various sizes when scanning the crack along the y-axis (a,c,e) and along the x-axis (b,d,f); (a,b) l1 ˆ 4 mm; (c,d) l1 ˆ 6 mm; and (e,f) l1 ˆ 8 mm: In (a), (c) and (e) the coil output signal (dashed line) is also plotted.

sensor used in the SMFM technique has been investigated. It has been shown that the coil behaves as a lowpass spatial ®lter, smearing out the discontinuity in the measured magnetic ®eld. The discontinuity, however, can be recovered by deconvolving the coil output signal with its spatial ®lter function. In the deconvolution process, the zeros of the ®lter function cause dif®culties, particularly in the presence of noise. To overcome this problem, a threshold level is chosen to remove sections of the ®lter function where noise is dominant. This approach enables one to recover a relatively wide spectrum of the crack signal without having the dif®culties associated with noise or the zeros of the coil ®lter function. The performance of the deconvolution technique depends on the coil size and shape, as well as the measurement system noise. In particular, for a given number of turns and surface area, rectangular coils have been found to be the most appropriate coils for use in the SMFM technique. Acknowledgements This work is supported by Amirkabir University research grant 15/641.

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