2022 Volume 63 Issue 4 Pages 529-537
This study investigated whether it is really reasonable to insist that being “rich in frequency” represents an advantage of pulsed eddy current testing when compared with conventional eddy current testing. More specifically, this study compared the capabilities of pulsed eddy current testing and conventional eddy current testing in evaluating the thickness of non-ferromagnetic plates with thicknesses of 1–20 mm. To avoid any instrumentation effect, the investigations were performed based on signals obtained by finite element simulations, and the correlations between the thickness and a scalar feature value extracted from measured signals were discussed. This study considered two typical excitation waveforms, a Gaussian pulse and a step function simulating the sudden termination of the excitation currents, and four simple and conventional scaler feature values: peak amplitude, time to peak, time to attenuate the signal into a certain level, and logarithmic slope of the signal attenuation. Narrowing the Gaussian pulse led to difficulty in terms of the thickness evaluation, although it should make the pulse richer in frequency. Subsequent analyses compared the capabilities of the pulsed and conventional eddy current testing to evaluate plate thickness under the assumption that they have the same signal-to-noise ratio. The results revealed that the error in the thickness evaluation using the pulsed eddy current testing was somewhat larger than the error using the conventional eddy current testing with three frequencies. Whereas the target of this study was limited to the thickness evaluation of non-ferromagnetic plates, the results of this study point out that what is important is using frequencies in a proper range as well known in the conventional eddy current testing. They also indicate that it is not reasonable to postulate that “richness in frequency” always leads to a better nondestructive evaluation when using the pulsed eddy current testing.
Pulsed eddy current testing is an electromagnetic non-destructive testing method that utilizes eddy currents induced inside a conductive material. The major difference between pulsed eddy current testing and conventional eddy current testing using time-harmonic electromagnetic fields is that the former method induces eddy currents using magnetic fields with an abrupt change, for example, pulse and square waveforms. The pulsed eddy current testing has a history of over half a century.1–4) Many theoretical and numerical works were performed in the 1990s,5–11) and they significantly contributed to clarify the physical background and characteristics of pulsed eddy current testing. Experimental studies in this regard were initially rather limited due to the difficulty associated with measuring small signals with a high-speed sampling and high resolution; significant advances in electronic instrumentation led to recent active studies on pulsed eddy current testing.11) However, the advantage of pulsed eddy current testing over the conventional eddy current testing using time-harmonic single or multiple frequencies remains somewhat unclear.
One known advantages of pulsed eddy current testing concerns the fact that its power consumption per cycle, which affects the stability of the system, is generally low, even when the maximum power output is high. This is practically useful, but its contribution is the same as that of an amplifier. Sometimes pulsed eddy current testing is considered to enable a rapid inspection because theoretically a single ideal pulse contains all the frequency components. In reality, however, averaging the responses to multiple repeated pulses is generally necessary to obtain clear signals, which limits the speed of inspection as well as the fundamental frequency contained within the signal. This also means that pulsed eddy current testing actually has discrete frequency components. A typical approach to analytically or numerically obtain signals of pulsed eddy current testing is to transform superposed time-harmonic signals obtained in the frequency domain using the inverse Fourier transform.7–10,12–18) This approach is frequently utilized even when magnetic materials are targeted,17,18) as the magnetic fields used by the pulsed and conventional eddy current testing methods are sufficiently small to approximate the magnetic properties as linear ones. This indicate that a signal of pulsed eddy current testing is essentially identical to the one by the conventional eddy current testing with massive frequencies19) because Fourier transform is a linear and invertible transform. Thus, it is plausible that the major advantage of pulsed eddy current testing is that the signals obtained are “rich in frequency”, as commonly insisted by most or possibly almost all of recent papers.
However, what is truly important from the viewpoint of non-destructive evaluation is the information contained within the signal. As “rich in frequency” does not always mean “rich in information”, it is not proper to discuss the advantages of pulsed eddy current testing based solely on the number of frequency components contained within the signals. For example, it is not too much to say that the difference between signals measured at 100 Hz and those measured at 100.1 Hz in the conventional eddy current testing is negligibly small from a practical viewpoint. One can measure, however, signals at a huge number of frequencies between 100 and 100.1 Hz by setting a small frequency interval. In addition, signals measured at too low or too high excitation frequencies cannot provide the desired information concerning the target. Several recent studies have transformed the signals of pulsed eddy current testing into the frequency domain using Fourier transform and evaluated a few parameters, some of which are identical to the phase of the conventional eddy current testing at a low frequency obtained in the frequency domain.20–22) This implies that insisting “rich in frequency” as an advantage of pulsed eddy current testing is questionable. This is supported by the findings of an earlier study, which demonstrated that if the frequencies are properly chosen and the signals are properly mixed, multiple frequency eddy current testing can be comparable with pulsed eddy current testing in terms of the thickness evaluation, even when the phase information of the signals is disregarded.23) It should be obvious that using multi-frequencies would be better, namely more informative, than using a single frequency in the conventional eddy current testing. However, this does not mean “the more, the better”. In other words, the superiority of the multi-frequency eddy current testing, which usually uses several frequencies, over the single-frequency eddy current testing does not necessarily support that the pulsed eddy current testing is more advantageous than the conventional, including multi-frequency, eddy current testing because its signal contains many, namely much more than several, frequency components. If several frequencies are “rich enough”, multi-frequency eddy current testing would be practically more advantageous than the pulsed eddy current testing as coherent detection, which is used to measure signals in frequency domain, enables to detect signals buried in large noise.
Based on the above, this study investigated whether it is really reasonable just to insist “richness in frequency” as an advantage in pulsed eddy current testing over conventional eddy current testing. As a specific target, this study considered the evaluation of the thickness of non-ferromagnetic metallic plates, which represented a typical target of pulsed eddy current testing. Several simple and commonly used scaler features of pulsed eddy current testing signals, such as time-to-peak, were extracted from its time-domain signals in order to estimate the thickness of the plate, and their effectiveness was compared with the signals of eddy current testing with time-harmonic excitation that is referred to as conventional eddy current testing hereafter. As the electric circuits for pulsed eddy current testing differ from those for conventional eddy current testing, the investigations were performed numerically using finite element simulations to obtain the ideal signals and exclude the effects of the capabilities of the instruments.
Figure 1 illustrates the problem considered in the present study. A circular coil carrying a time-varying current is situated above an aluminum plate. The thickness of the plate is evaluated based on signals measured by a magnetic sensor attached to the bottom surface of the coil as indicated by the black circle. The magnetic sensor is assumed to have zero volume and sense the vertical component of the magnetic flux density. The thickness of the aluminum plate, indicated as T, varies from 0 (no plate) to 20 mm with a 1 mm pitch. The radius of the plate is sufficiently large for the discussion in this study. Whereas the plate is nonmagnetic, it is likely that the magnetic property of the target does not affect the conclusions of this study so long as the magnetic fields used to induce the eddy currents are not so large24) and the magnetic property of the target can be well-approximated linearly in numerical simulations.
Geometry of the problem considered in this study (unit: mm).
As the excitations of pulsed eddy current testing, the two waveforms commonly used in studies on pulsed eddy current testing, namely a Gaussian pulse and a step function, were considered. The Gaussian pulse was represented as
| \begin{equation} I = I_{\text{G}}\exp \left\{-\frac{1}{2}\left(\frac{t - t_{\text{G}}}{\sigma_{t}}\right)^{2}\right\}, \end{equation} | (1) |
| \begin{equation} I = \begin{cases} I_{\text{S}} & \text{$t < t_{\text{S}}$}\\ 0 & \text{$t \geq t_{\text{S}}$} \end{cases} , \end{equation} | (2) |
By contrast, several frequencies ranging from 10 Hz to 200 Hz were adopted to obtain the signals of the conventional eddy current testing. It should be noted that the amplitudes of the excitation currents are not essential because of the same reason mentioned above.
The numerical simulations to evaluate the signals were performed using commercial finite element software COMSOL Multiphysics v5.4 + AC/DC module. The governing equation was
| \begin{equation} \nabla^{2}\boldsymbol{A} = \mu\left(\boldsymbol{i} - \sigma \frac{\partial\boldsymbol{A}}{\partial t} - \varepsilon \frac{\partial^{2}\boldsymbol{A}}{\partial t^{2}}\right), \end{equation} | (3) |
To evaluate the plate thickness based on the signals, this study avoided using sophisticated numerical inversions11,26,27) and instead assumed that the thickness could be directly evaluated from a scalar feature value of the measured signals, x, as f(x), where f is an empirically obtained simple function by linearly interpolating the feature values obtained by the numerical simulations. Moreover, the signals due to T = 0 and 20 mm were assumed to be known in advance. This assumption corresponds to the general calibration procedure followed in actual inspections to exclude the effect of the impedance of the electric circuit and does not lose the generality of the discussion in the present study.
Because the signals of the pulsed eddy current testing are measured in the time domain, there would be a variety of features available to non-destructively evaluate the target. Each of them should have advantages and disadvantages depending on its application; it is not reasonable to attempt to evaluate all possible features. Therefore, this study considered the following four features, which are simple and commonly used to analyze pulsed eddy current signals, to evaluate plate thickness from pulsed eddy current testing signals:
Definitions of the features of pulsed eddy current testing considered in this study.
To evaluate the plate thickness based on the signals of the conventional eddy current testing, the amplitudes and the phases of the signals were evaluated after the signals were normalized so that those of T = 0 and T = 20 mm were (0, 0) and (1, 0), respectively, in the 2D impedance plane.
2.3 Results and discussion 2.3.1 Pulsed eddy current testing with pulse excitationFigure 3 shows the signals obtained when the coil carries the Gaussian pulses with the largest and smallest FWHM, that is, σt = 0.025 ms and σt = 2.5 ms, respectively. The figure confirms that larger σt is preferable for the quantitative evaluation of the thickness of thick plates. If σt = 0.025 ms, there is a large difference between signals due to T = 0 mm and T = 1 mm, but almost no difference can be confirmed between those due to T ≧ 5 mm. By contrast, the signals with σt = 2.5 ms show some differences between signals due to T = 10 and 20 mm, indicating the superior capability to quantify the plate thickness.
Signals with the pulse excitation.
To confirm the effect of the σt, Fig. 4 shows how the relationship between the plate thickness and the feature values changes with σt. As the tBth and ΔBth depend on the Bth, the figure shows them at two Bth, 0.01 and 0.0001, to demonstrate the effect of the Bth. Using a small σt makes the change in the feature values smaller, especially when the plate is thick, thereby rendering the quantitative evaluation of the plate thickness difficult. Figures 4(e) and (f) indicate that the ΔBth becomes almost irrelevant with regard to the σt after a certain timepoint because of the rapid attenuation of the high-frequency components. This implies that most information on the plate thickness is contained in a certain low-frequency band and that higher frequencies do not increase but rather mask the information. In other words, it is important to use a proper frequency range, and widening the frequency range can be harmful. It should be noted that this also implies that an ideal pulse, which is “richest in frequency”, is one of the worst waveforms to evaluate plate thickness when using the single scalar feature easily extracted from the time-domain signals.
Relationship between plate thickness and feature values obtained by the pulse excitation.
Figure 5 presents the signals, both original and differential signals obtained when the coil carries the step excitation current with the largest and smallest fall times, respectively. The signals when T = 20 mm are excluded because the differential signals are obtained by evaluating the difference based on the signals of T = 20 mm. As the small difference between the signals due to T = 15 mm and T = 19 mm implies, the differential signal due to T = 19 mm is quite small.
Signals with the step excitation that stops the exciting currents at 50 ms.
Figure 6 summarizes the relation between the feature values and thickness. Time to peak, tmax (Fig. 6(b)) is almost proportional to the thickness, indicating that it is suitable to evaluate the thickness as proposed in earlier studies. However, note that quantitatively evaluating tmax requires a clear confirmation of the signal peak. Specifically, the limitation and effectiveness of tmax to evaluate the thickness strongly depend on the signal-to-noise ratio. Figure 6(c) shows that tBth is also effective to evaluate the thickness as it changes almost linearly with the thickness, and a smaller Bth is preferable to evaluate the thickness from tBth because the slope of tBth becomes larger. However, the signals should be clearly confirmed as small as Bth to quantitatively evaluate tBth, which indicates that the effectiveness of using tBth also depends on the signal-to-noise ratio. The relationship between ΔBth and thickness is almost independent from Bth if Bth is smaller than 0.01. This is consistent with the results shown in Figs. 4(e) and (f), and it also indicates that a certain low-frequency band contributes to the thickness evaluation.
Relationship between plate thickness and feature values obtained by the step excitation.
Figure 7 shows how the amplitudes and the phases of the signals of the conventional eddy current testing change with the plate thickness, which is consistent with the general characteristics of the conventional eddy current testing. The relationship between the plate thickness and the phase of signals, shown in Fig. 7(b), excludes plates thinner than 1 mm as the phase of the signal of T = 0 cannot be defined and the thicknesses considered in the numerical simulations have a 1 mm pitch. Precisely measuring the signals enables the simple evaluation of the plate thickness from the phase, unless the frequencies are not too large. This is a well-known fact. In other words, the capability of the conventional eddy current testing to evaluate plate thickness strongly depends on the signal-to-noise ratio similar to pulsed eddy current testing.
Relationship between the signal feature of conventional ECT signals and plate thickness.
As the signals utilized in this study are numerical, any small change in the plate thickness will alter the signals. However, it is not reasonable to discuss the capabilities based on a small change in the signals that will, in reality, be buried in noise. Thus, this study discussed the capabilities based on the following two criteria: maximum evaluable thickness and robustness of the evaluation.
3.1.1 Maximum evaluable thickness, TmThe diffusive nature of electromagnetic fields indicates that the feature value should asymptotically approach a certain value with plate thickness. This implies that quantitatively evaluating the plate thickness becomes increasingly difficult as the thickness increases. Thus, the present study assumed that a plate thicker than Tm could not be quantitatively evaluated if the amplitude of the signal used to extract the feature was below a certain threshold, N, as
| \begin{equation} \frac{|B_{T_{m}} - B_{20}|}{\max|B_{T} - B_{20}|} \leq N, \end{equation} | (4) |
Practically speaking, the evaluation should be robust against noise, that is, a small change in the signals should not lead to a large error in the thickness evaluation. To evaluate the robustness, this study considered a 1% noise pollution. More specifically, the maximum error caused by changing the amplitude of the signals by 0.01 was evaluated. It should be noted that 0.01 corresponds to 1% of the maximum of the signals considered in the evaluations, and the one that changes by 0.01 is not the feature value but the amplitude of the signals.
3.2 Results and discussionFigure 8 compares the relationships between the threshold, N, and the maximum evaluable thickness, Tm. Based on both the results and the discussion in the previous section, the figure presents the pulse excitation signals when σt = 2.5 ms. It is reasonable that a larger N leads to a smaller Tm as the change in a feature value caused by the change in the thickness becomes smaller as the thickness decreases. The signals of conventional eddy current signals obtained using an excitation frequency as low as 50 Hz provide larger (i.e., better) Tm values than those of the pulsed eddy current testing. Note that the fundamental frequency of both the pulse and step excitations is 10 Hz.
The effect of threshold, N, on the maximum evaluable thickness, Tm.
Figure 9 summarizes the maximum possible errors of pulsed eddy current testing caused by the 1% noise pollution. As it is not reasonable to evaluate the signals buried by noise, the Bth was set to 0.01 in order to evaluate the tBth and ΔBth. Figures 4 and 6 indicate that a smaller Bth is preferable when it comes to evaluating to the plate thickness, which indicates that, in reality, that the Bth should be chosen based on the signal-to-noise ratio, as mentioned above. The maximum error in evaluating the thickness using the tBth and ΔBth is denoted by a single line in the figures. This is because the errors in evaluating the plate thickness using the tBth and ΔBth were evaluated by using the feature values when Bth = 0.02. Figure 9 demonstrates that the evaluation by ΔBth of step excitation leads to the smallest error in thickness evaluation; a 15 mm plate is evaluated as 13.9 mm. It should be noted again that the 1% noise was added not to feature values but to signal amplitude. Thus, for example, the linearity of a relationship between a feature value and plate thickness, shown in Fig. 6, does not assure that the error increases linearly with the plate thickness.
Error in evaluating plate thickness caused by the 1% noise pollution.
Figure 10 presents the effect of the Bth on the error caused by the 1% noise pollution. Based on the results presented in Fig. 9, Fig. 10 presents only the results of the thickness evaluation using ΔBth of step excitation. The three vertical dotted lines terminating the curves that show the errors correspond to the Tm, as shown in Fig. 8, under the assumption that Bth = N. A larger Bth leads not only to robustness against noise pollution but also to a smaller maximum evaluable thickness, as shown in the figure. Thus, an appropriate Bth should be chosen based on the thickness of the target. This also indicates that the capability strongly depends on the signal-to-noise ratio.
Effect of Bth on the error in evaluating plate thickness using ΔBth.
Figure 11 summarizes the error in evaluating the plate thickness based on the phase of the signals of conventional eddy current testing. A comparison between Fig. 11 and Fig. 10 indicates that the use of the three frequencies of 10 Hz, 50 Hz, and 200 Hz can evaluate plate thicknesses of 1–20 mm more accurately than pulsed eddy current testing. For example, the error in evaluating a 15-mm thick plate using a 10 Hz signal is about 0.6 mm, while the error when using ΔBth is larger than 1.0 mm. For a 5-mm thick plate, conventional eddy current signals at 200 Hz provide a thickness evaluation as accurate as when the ΔBth is used with the Bth set to 0.1. A single-frequency signal does not cover a large thickness range. However, the figures point out that using several, or probably a few, frequencies enables is sufficient to evaluate the plate thickness as accurately as pulsed eddy current testing. By contrast, the use of several frequencies enables other feature values to be obtained, including the lift-off point of intersection.29) Although this implies that using several frequencies would enhance the practical robustness against the lift-off variation, the results presented in the figures indicate that it is not reasonable to attribute the key advantage of pulsed eddy current testing to the “richness in frequency” so long as the simple scalar features extracted from the time-domain signals are utilized.
Error in evaluating plate thickness using the phase of the conventional eddy current signals.
This study investigated whether using an excitation waveform containing massive frequency components leads to a more quantitative non-destructive evaluation in electromagnetic non-destructive evaluation. Specifically, this study evaluated whether “rich in frequency”, which is commonly insisted as the advantage of pulsed eddy current testing over the conventional eddy current testing using sinusoidal excitation, contributes to the accurate evaluation of plate thickness. To exclude the effect of the instrumentation needed for the measurements, the investigations in this study were performed based on signals obtained using numerical simulations.
The results revealed that richness in frequency does not always contribute to the quantitative thickness evaluation of the target so long as the four simple and conventional scaler feature values are used. The reason is that too high, and probably too low, signal frequencies do not provide information about the thickness and contribute to burying needed information. Further numerical investigation on the ill-posedness of the evaluation revealed that using a few frequencies enables a plate thickness evaluation as accurately as, or possibly more accurately than, pulsed eddy current testing. In short, the results of this study indicate that it is not reasonable to always insist “rich in frequency” as the advantage of pulsed eddy current testing and that what is truly essential is, simply, the signal-to-noise ratio.
This study focused on the thickness evaluation of thick non-ferromagnetic plates. It is plausible that targeting thinner plates leads to the same conclusions while the proper frequencies of the conventional eddy current testing would be higher; it would be also plausible that the conclusion remains the same even when ferromagnetic plates are considered30) because their electromagnetic properties are usually assumed to be linear. It should be noted that this study evaluated the correlations between the thickness and a scalar feature value that is commonly used and easily extracted from time-domain pulsed eddy current signals. That is, the conclusions drawn in this study do not exclude the possibility that the combinational use of multiple features, including both time- and frequency-domain features, would lead to a more robust and accurate thickness evaluation. It should also be also noted that the discussions above would not be applicable to other purposes, such as crack sizing. However, it is not reasonable to postulate that using more frequencies leads to a better non-destructive evaluation in pulsed eddy current testing31) so long as the four simple scalar feature easily extracted from its time-domain signals is used.
This work was partially supported by JSPS KAKENHI Grant Number 19F19045, National Natural Science Foundation of China under grant 52077214, and Priority Academic Program Development of Jiangsu Higher Education Institutions.
