Abstract
An eddy current technique for the simultaneous determination of the layer thickness and conductivity of a uniform conductive layer on a known conductive base material is presented. Layer thickness and conductivity are determined by fitting apparent conductivity measurements taken on the surface of the part at a discrete set of frequencies to a calibration volume. The calibration volume is constructed from apparent conductivity measurements taken on a set of known layer materials, of various known thicknesses, as a function of frequency. Experimental results that support the technique are presented.
Introduction
The addition of coatings, platings and surface layers, or the modification of near-surface material properties plays an important role in the industrial application of metals. The purposes of these modifications include improved corrosion resistance, wear resistance, electrical conduction, thermal isolation, and appearance. These modifications are prepared by various methods including heat treatment, shot peening, electrodeposition, hot dipping, cladding, and spraying. In many cases, measuring and controlling the layer properties during application, as well as monitoring the layer properties of in-service components, are essential to their successful use. Depending on the details of the coating and its application, mechanical, optical, magnetic, X-ray, ultrasonic, and electromagnetic techniques are available for their characterization. 1
Eddy current techniques 2 are particularly attractive for the characterization of surface layers and treatments because they are nondestructive, potentially noncontact, and, importantly, have frequency-dependent penetration depth. These related inspection methods rely on the principle of magnetic induction, in which currents are generated in a conductor when in the presence of a varying magnetic field. In many applications, including the current one, the varying magnetic field is produced by an eddy current transducer composed of a solenoid-like coil, created from wound wire, that is held with its primary axis perpendicular to the surface of the test article. The magnetic field generated by the coil penetrates into the test article setting up a looping field of currents, termed eddy currents. These current loops vary with depth and are confined to circulate in planes perpendicular to the axis of the transducer and parallel to the surface of the test article. Consequently, the transducer and test article form a coupled electro-magnetic system whose properties have been exploited for inspection and material characterization. Examples include the sensitivity of the system to the distance of the transducer from the surface and to the presence of eddy-current-disrupting discontinuities within the bulk of the material.
Simple eddy current models—assuming a plane electro-magnetic field varying with test frequency f interacting with a semi-infinite, isotropic, conductive medium—predict that eddy currents decay exponentially into the material being tested. The decay rate, commonly called the standard depth of penetration, δ, can easily be calculated using basic electro-magnetic material properties and the test frequency
where δ is the standard depth of penetration (m), f is the test frequency (Hz), µ is the magnetic permeability (H/m), and σ is the conductivity (S/m).
In this study, all of the materials are nonmagnetic so that their magnetic permeabilities are approximately the same as free space: µ = 4π(10−7). With µ fixed, equation (1) shows that the depth of penetration varies inversely with the square-root of the test frequency and the material’s conductivity.
As an example application, when the layer placed on the base metal is nonconductive, eddy current inspection is frequently used to determine the layer thickness as discussed in ASTM E376-11. In these applications, eddy current inspection techniques commonly measure the complex impedance of an upright, cylindrical coil or a flat, pancake-shaped coil placed on or near the surface of a test item. 3 The nonconductive coating behaves, in effect, like an air-gap as it is electro-magnetically neutral. In order to estimate the nonconductive coating’s thickness, measurements of the coil’s impedance are made at a set of known distances from the surface of the metal using a set of nonconductive shims of known thicknesses. Together, these impedance measurements form a calibration curve in the complex impedance plane. Impedance measurements of unknown coating thickness are then compared to the calibration curve, and the thickness of the coating is estimated using interpolation.
A calibration methodology is also used in the determination of material conductivities using eddy currents. In this application, impedance plane conductivity curves are determined experimentally both with and without a spacer of known thickness. A coil impedance from an unknown sample is then measured and compared with the conductivity calibration curves. In general, the measured coil impedance does not fall exactly on either of the conductivity calibration curves, but rather between them. The unknown conductivity is, again, estimated through interpolation.
Nondestructive evaluation of conductive coatings is significantly more complicated because the layer is no longer electro-magnetically neutral and, consequently, modifies the simplified exponential model described above. As a result, coil measurements result in “apparent impedances” and “apparent conductivities” dependent on the properties of the mixed layer–base system. Despite this complication, a variety of techniques for determining conductive layer properties using eddy current impedance measurements have been developed. Most have used either theoretical or numerical eddy current inversion. 4 In these model-based approaches, either numerical or analytical models that predict the detailed interaction of the eddy current system with the test part are used. Such approaches are grouped together as inverse eddy current problems because they determine layer properties by iteratively varying the layer properties of the model until its predictions match the measured values within some predetermined tolerance. Under certain circumstances, these methods have proved successful at characterizing material property profiles for both uniform and depth-dependent coatings and surface treatments.
Initially, Moulder et al. 5 considered single conductive uniform layers. Their work relied on the exact theoretical models presented by Dodds and Deeds 6 for the impedance of upright cylindrical coils placed in proximity to metal surfaces coated with a single, uniform conductive layer. The model was used by Moulder et al., in conjunction with a least-squares algorithm, to determine the thickness and conductivity of a single conductive layer on a conductive base. This eddy current inversion was accomplished by varying the theoretical model parameters until the model prediction matched observed impedance measurements.
Measurements are further complicated if the properties of the layers vary with depth. Under these conditions, the impedance characteristics of eddy current probes placed on or near such material systems are complex and difficult to predict. As a result, determining the probe’s characteristics using an inverse eddy current technique can involve complicated modeling, analysis, or numerics. To avoid exactly modeling a continuous conductivity profile, Uzal and Rose 7 relied on the analytical solution provided by Cheng et al. 8 for the impedance of an eddy current probe above a plate composed of an arbitrary number of layers, each with constant conductivity. In their approach, approximately 50 layers were used to represent the continuous profiles considered, requiring the predicted impedance to be computed numerically. This numerical solution was then used as the exact solution in the eddy current inversion scheme. Inversion techniques have also been applied to depth-dependent conductivity problems using exact analytical models. In these cases, analytical models of coil impedance are calculated using an assumed conductivity profile shape. As examples, Uzal et al. 9 considered conductivities that varied as a hyperbolic tangent with depth into the part, while Theodoulidis et al. 10 considered profiles that varied linearly, exponentially, or quadratically.
Like the previous techniques described, this work also leverages the frequency-dependent penetration depth of eddy currents to measure the conductivity and thickness of conductive layers. It differs, however, in that rather than using complex impedances the method relies on conductivity measurements made at several frequencies using a commercially available off-the-shelf eddy current meter. In addition, importantly, the need for eddy current inversion is also avoided using a calibration-based approach. The calibration information is obtained by measuring the apparent conductivity of layered conductors as a function of layer thickness, frequency, and layer conductivity. In the experiment presented, a single calibration volume was constructed using five layer thicknesses, four frequencies, and three layer materials. A common base material was used for all calibrations and tests. The apparent conductivities, dependent on the electromagnetic properties of both the layer and base material, are measured on the surface of the layered sample using a commercially available conductivity meter and flat pancake conductivity probe. Unknown layer conductivities and thicknesses are estimated by comparing measured apparent conductivities, taken on an unknown sample at the four calibration frequencies, to the calibration volume. A least-squares algorithm is then used to determine the most likely value for the layer thickness and conductivity.
As with other calibration techniques, such as the determination of nonconductive coating thicknesses, eddy current coil measurements are compared with a set of known measurements. The current method is similar in that a calibration technique is used, but differs in two important ways. First, apparent conductivities are used rather than complex impedance measurements. Provided the eddy current penetration depth meaningfully exceeds the thickness of the layer, the measured conductivities do not represent the actual conductivity of the layer or the base material, but rather on the properties of both. Second, a three-dimensional calibration volume is used rather than the two-dimensional complex impedance plane. The calibration volume is constructed by measuring the apparent conductivity of a layer–base material combination while varying eddy current test frequencies, layer thicknesses, and layer conductivities.
Experimental setup
All conductivity measurements were made using a commercially available multi-function eddy current instrument operating in conductivity mode. A commercial 9.5-mm pancake conductivity probe was used for sensing. The device was calibrated for each test frequency over the range of conductivities tested and re-zeroed before each test at the conductivity closest to the apparent conductivities to be measured. The test frequencies used were dictated by the conductivity meter and were restricted to 60, 120, 240 and 480 kHz.
The conductive layers used in the study were composed of Alloy 110 Annealed copper, 1100 grade aluminum, Alloy 510 Tempered Grade A bronze, and Alloy 260 Half Hard Temper brass. All of the layers were placed onto a 76-mm diameter, 10.4-mm thick, 2024 T3 aluminum base. The layers and base material had conductivities of 597, 376, 255, 170, and 188 × 105 S/m, respectively, determined in accordance with ASTM B193-02(2008). These layer materials were used because they span the range of common conductivities. Aluminum was chosen as the base material because of its common application.
The layer thicknesses were chosen so that the eddy current penetration depth into the materials under investigation, at the test frequencies available, would be affected by the properties of both the layer and base material. Referencing equation (1), an approximate minimum layer thickness, δmin = 0.0983 mm, was established by assuming the extreme case in which the material with the highest conductivity (copper) was tested at the highest frequency (480 kHz). Likewise, a maximum layer thickness, δmax = 2.61 mm, was determined using the other extreme case in which the material with the lowest conductivity (brass) was tested at the lowest frequency (60 kHz). Using these bounds, layer material was obtained from widely available shim stock cut into 76 mm squares. The copper, bronze, and brass layers were purchased off the shelf with the desired layer thicknesses of 0.13, 0.25, 0.38, 0.51, and 0.64 mm. The aluminum used for the calibration was only available in a single thickness of 0.13 mm, however. Consequently, the individual aluminum shims were stacked to produce the required layer thickness.
In testing the method, a fifth material was necessary, one with conductivity different from the calibration materials, but within the range calibrated. The material chosen was a copper-tungsten metal matrix composite (MMC) composed of approximately 85% copper by volume with a measured conductivity of 507 × 105 S/m. This material was not purchased, but was fabricated for use in another related program in which its conductivity was measured independently. For the test, layers of thickness 0.13, 0.25, 0.33, 0.38, 0.46, 0.51, 0.58, and 0.64 mm were fabricated from a billet of the material using electro-discharge machining. Importantly, three additional layer thicknesses which do not coincide with the thicknesses in the calibration data are considered: 0.33, 0.46, and 0.58 mm. The resulting samples were irregular in shape, but all were at least 25 mm by 55 mm.
Using a commercial-off-the-shelf eddy current inspection instrument a total of five tests were conducted, each on a specific layer material. Four tests using the copper, aluminum, bronze, and brass layers were conducted to create the calibration volume with a fifth conducted on the MMC test case. Each of the four calibration tests consisted of 100 apparent conductivity measurements, five for each layer thickness–frequency combination (Table 1). The resulting five-measurement set for each layer frequency–thickness pair was averaged, and statistically analyzed, to produce the final results.
Test matrix for a single calibration layer material (copper, aluminum, bronze, and brass) with five measurements taken at each layer thickness–frequency combination.
The fifth test on the MMC layer consisted of 160 apparent conductivity measurements, again, five for each layer thickness–frequency combination (Table 2). As for the calibration volume tests, each five-measurement set was averaged, and statistically analyzed, to produce the final results.
Test matrix for the MMC layer material with five measurements taken at each layer thickness–frequency combination.
MMC: metal-matrix composite.
In all instances, the conductivity probe was firmly placed on the layer–base sample in order to minimize any air gaps. Particular attention was given to the testing of the thicker aluminum layers as they were constructed by stacking several thinner layers. Any remaining air gaps would have a minimal effect on the conductivity measurement since the gaps lay in a plane parallel to the eddy currents in the test specimen and would not disrupt their flow.
Results
The data used to construct the calibration volume (Figure 1) are provided in Tables 3–6. One table of average apparent conductivities, plus/minus their standard errors, is provided for each calibration layer material as a function of both layer thickness and inspection frequency (see the “Experimental setup” section for testing details). Table 3 provides the results for the known copper calibration layers on the aluminum base; Table 4 provides the results for the known aluminum calibration layers on the aluminum base; Table 5 provides the results for the known brass calibration layers on the aluminum base; and Table 6 provides the results for the known bronze calibration layers on the aluminum base. The values in each table cell are the average of five apparent conductivity measurements taken on a single surface with a calibration layer of known thickness at one test frequency. The measurement is then repeated for each calibration layer thickness, test frequency, and, ultimately, calibration material. Measurement repeatability for the calibration data is quite good with typical values of the standard error well below 1%.
Calibration volume. Apparent conductivity (×105 S/m) value is represented by the size and darkness of the circle at each location. Larger circle/red fill implies greater apparent conductivity.
Average apparent conductivities (×105 S/m) and their standard errors for the copper calibration layer.
Average apparent conductivities (×105 S/m) and their standard errors for the aluminum calibration layer.
Average apparent conductivities (×105 S/m) and their standard errors for brass calibration layer.
Average apparent conductivities (×105 S/m) and their standard errors for bronze calibration layer.
The calibration volume (Figure 1) shows the apparent conductivities measured from the calibration materials as a function of test frequency, layer conductivity, and layer thickness. Each circular data point represents the apparent conductivity associated with the testing of a single layer material of fixed thickness at one of the four test frequencies. The magnitude of the apparent conductivity is represented by the size and color of the circle. As presented, the data naturally divide into four planes parallel to the test frequency–layer thickness plane with each plane corresponding to a single layer material of constant conductivity. The apparent conductivities on each of these planes are similar in magnitude because they represent the apparent conductivities for a single layer material. In contrast, the apparent conductivities between neighboring planes are much more pronounced because of the large difference in conductivities between different layer materials. Most clear is the trend on the copper plane (topmost) toward lower apparent conductivities as layer thicknesses and test frequencies decrease. Decreasing layer thickness lowers the apparent conductivity because as the copper layer decreases, the influence of the lower conductivity aluminum base on the measurement increases. A similar effect occurs when test frequencies are decreased. In this case, however, it is the penetration depth of the eddy current which increases again increasing the aluminum bases effect on the apparent conductivity measurement.
A second interpretation of the calibration data can be obtained by considering plots of apparent conductivity versus test frequency and layer thickness for fixed layer materials. These plots are shown in Figure 2 with each surface corresponding to an individual layer material. More precisely, each surface represents the apparent conductivity of a layer–base material pair, each of fixed conductivity, as a function thickness and inspection frequency. In this presentation, the size and darkness of the circles in Figure 1 are equal to the height of each calibration surface above the thickness–test frequency plane. Upon inspection, the surfaces display some expected common behavior. First, because increasing the frequency has the same effect as increasing the thickness of the layer, there is a suggestion of symmetry about the diagonal on the layer thickness–test frequency plane. In addition, as either the test frequency or layer thickness becomes large, the apparent conductivity of the layer–base system approaches that of the layer’s conductivity. This is most evident in the uppermost surface for copper whose apparent conductivity approaches the 597 × 105 S/m conductivity of the copper layer at either boundaries. Most interesting, however, are the shapes of the calibration curves for lower frequencies and thinner layers. In this regime, the penetration depth of the eddy currents is on the order of the layer thickness, and consequently, the apparent conductivity measure represents the properties of both the layer and base material. For layer materials with conductivities greater than the aluminum base—copper, aluminum, and bronze—the apparent conductivity curves dips down as the lower conductivity of the base material begins to affect the overall measurement. The reverse is true for brass, the bottom-most calibration surface, whose apparent conductivity rises as the aluminum base begins to affect the conductivity measurement. And, as expected, all of the curves approach the same apparent conductivity, that being the conductivity of the base material, as the thickness and frequency approach zero.
Apparent conductivity (×105 S/m) calibration surfaces: copper, aluminum, bronze, and brass (top to bottom).
The average apparent conductivities for the MMC test layers, produced using the same procedure used to generate the calibration data (see the “Experimental setup” section for testing details) and presented using the same format, are provided in Table 7, but with eight layer thicknesses. Each MMC combination is assumed to have an unknown layer thickness and conductivity. In the current scenario, 20 measurements would be taken on each surface, 5 at each of the four inspection frequencies. Therefore, each column in Table 7 represents the result of a single experiment or, so to speak, a single inspection. Several of the MMC layer measurements were less repeatable than those for the calibration volume, such as the measurements for a MMC layer of thickness 0.25 mm (Table 7), but were still quite good, with the worst cases having standard errors that were in the neighborhood of only 1%.
Average apparent conductivities (105S/m) and their standard errors for the MMC layers.
MMC: metal-matrix composite.
As an example, consider the MMC inspection data associated with the layer of thickness 0.33 mm. If we plot these data as a function of frequency with the calibration surfaces using the MMC’s known layer thickness, then we can see the location of the layer–base system’s apparent conductivity with respect to the calibration surfaces (Figure 3). As is expected, the curve does not lie on any of the existing calibration curves, but rather at a location between the copper and aluminum calibration surfaces on what would be the MMCs calibration surface.
Apparent conductivity (×105 S/m) of 0.33-mm MMC layer.
In order to estimate the conductivity and thickness of this layer from the given data, it is necessary to search for the optimum location of the test data within the calibration volume. This is accomplished using a least-squares search in which the layer thickness and conductivity are varied until the error between the measured apparent conductivity of the MMC-aluminum pair and the apparent conductivity of the calibration volume is a minimum. While the measured apparent conductivities used for the calibration are known, the apparent conductivities within the calibration volume at arbitrary locations are not known. Consequently, a spline interpolation scheme was developed. This allowed the apparent conductivity associated with an arbitrary location in the calibration volume to be estimated. An unconstrained nonlinear optimization technique was then used to minimize the mean-square error between the measured apparent conductivities and the apparent conductivities of the calibration volume.
The minimization is accomplished by comparing the apparent conductivities measured on the unknown layer to the apparent conductivities predicted in the calibration volume at a guess location. The square of the difference between these two values is then minimized. The function minimized is
where R2 is the error norm to be minimized, and
An example of a calibration volume search is shown in Figure 4 for the 0.33-mm MMC layer. The figure shows the initial search data provided to the search algorithm, the final, optimum, location of the apparent conductivity test data provided by the search, as well as the true location of the MMC layer–aluminum base system (Table 7, column 3). Importantly, the optimum estimate of layer properties was found to be independent of the initial starting position. Test cases using multiple initial search locations were found to result in identical conductivity and thickness estimates.
Calibration volume with results showing initial guess (crosses, dashed line), estimated result (crosses, solid line), and true value (diamonds, solid line). Apparent conductivity (×105 S/m) value is represented by the size and color of the circle at each location. Larger circle/red fill implies larger apparent conductivity.
Thickness and conductivity estimates for the eight MMC layers, as well as their true characteristics, are presented in Table 8. Standard errors for the estimates are not provided because average apparent conductivity values were used when performing the minimization searches. Although unavailable, it is expected that the errors in these data would be small based upon the repeatability of the underlying apparent conductivity measurements. These data are presented in Figures 5 and 6. Estimated and actual conductivities for the MMC layer are shown in Figure 5. The expected value is shown as a straight line as all of the layers are MMC have the same conductivity. Overall, the match is quite good, within 4%, if the data for the thinnest layer is ignored. Conductivity estimates for layers thin in comparison to the eddy current depth of penetration are expected to be the least accurate as their effect on the apparent conductivity would be slight. In principle, this can be avoided using higher testing frequencies and therefore smaller depths of penetration. In contrast to thin layers, the thicker layers show an excellent match.
Least-square thickness and conductivity estimates for MMC layers.
MMC: metal-matrix composite.
Estimated conductivity of the eight MMC layers (solid line). True layer conductivity (dashed line).
Estimated thickness of the eight MMC layers (solid line). True layer conductivity (dashed line).
Thickness estimates are shown in Figure 6. The results trend with but do not, in general, match the known values and have errors of approximately 20%. The relative accuracy of the conductivity estimate and inaccuracy of the thickness measurement can be attributed to the characteristics of the gap in the calibration volume in the region in which the MMC data lie. In the case of conductivity, the gap (vertical) in the calibration data occurs between the conductivity of the copper and aluminum layers (Figure 1). This gap represents 221 × 105 S/m or 52% of the available information along that axis. In the thickness (horizontal) direction, however, no thickness data are available. Consequently, the conductivity interpolation in the calibration volume is more robust than the interpolation in the thickness direction.
Conclusion
A calibration-based technique for determining the conductivity and thickness of conductive layers placed on a conductive base material has been presented. The method relies on eddy current measurements of apparent conductivities taken on the surface of the layer–base system at several fixed frequencies using commercially available equipment. Apparent conductivities are measured because, for the penetration depths considered, the eddy current measurements are affected by both the conductivity of the layer and base material. Apparent conductivity measurements on layers of known thickness and conductivity, taken at fixed frequencies, are used to generate a calibration volume (Figure 1). The calibration volume provides the apparent conductivity of a layer–base calibration sample as a function of test frequency, layer thickness, and layer conductivity. The apparent conductivities of a layer of unknown conductivity and thickness are then measured at the same frequencies and compared to those documented in the calibration volume. A nonlinear, least-square optimization algorithm is then used to find the optimum conductivity and thickness of the measured apparent eddy current data within the calibration volume.
A test of the method was performed using four calibration materials (copper, aluminum, bronze, and brass), of five thicknesses (0.13, 0.25, 0.38, 0.51, and 0.64 mm), and four test frequencies (60, 120, 240, and 480 kHz). A fifth known material, a copper-tungsten metal matrix composite, sectioned into eight layer thicknesses (0.13, 0.25, 0.33, 0.38, 0.46, 0.51, 0.58, and 0.64 mm) was used as the unknown layer. Test results showed that layer conductivities were well predicted, within about 4% (Figure 5). Layer thickness predictions erred by approximately 20%; however, the trend in thickness was accurately captured (Figure 6). These conflicting results are explained by the fact that the density of the calibration data in the calibration volume was not distributed uniformly in the thickness and conductivity dimensions. While there was a large gap in calibration information in both dimensions, the gap in the thickness direction was more severe (Figure 1).
While successfully demonstrated, application of the technique could be improved and extended. First, the construction of the calibration volume could be more closely tailored to the layer–base system under investigation. In most inspection situations, knowledge of the material system being investigated is available, most importantly being the range of conductivities and thickness expected. With this information, a more precise calibration volume could be constructed using calibration samples that closely bracket the expected values. This would greatly improve the accuracy of the methods and would presumably improve thickness estimations. The difficulty of this approach is in the manufacture of the calibration samples whose thicknesses and conductivities must be closely controlled. Second, it is important to realize that the technique is not limited to the length and frequency scales explored. With the construction of proper calibration samples, thinner layers could easily be investigated by increasing the test frequencies used.
Footnotes
Acknowledgements
The author acknowledges Prof. P Joyce and Capt. L Brown of the Mechanical Engineering Department at the US Naval Academy for their helpful discussions as well as for providing the materials used in the investigation.
Handling Editor: James Baldwin
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author received no financial support for the research, authorship, and/or publication of this article.
