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The paper addresses the first eddy current benchmark problem proposed by the World Federation of Nondestructive Evaluation Centers (WFNDEC). The problem simulates the eddy current response to the presence of an axisymmetric circumferential defect in an Inconel-600 tube. All simulations employ the axisymmetric code of the electromagnetic field simulator Finite Element Method Magnetics. For three different frequencies of excitation, it is explained how the displacement of the detecting coil inside the tube leads to a variation in the impedance of the eddy current coil. Variations of the resistive and inductive components of the impedance with distance from the defect region are used to build the impedance trajectory for each frequency of analysis.

The first eddy current benchmark problem proposed by the World Federation of Nondestructive Evaluation Centers (WFNDEC) simulates the eddy current response to the presence of an axisymmetric circumferential defect in an Inconel-600 tube [

Parameter | Length (mm) |
---|---|

h_{1} | 0.2 |

h_{2} | 0.3 |

h_{3} | 0, 0.1, 0.5, 1.0 |

h_{4} | 2.0 |

d | 20.0 |

D_{1} | 19.7 |

D_{2} | 22.24 |

D_{3} | 9.0 |

D_{4} | 19.0 |

D_{5} | 22.24, 23.24, 24.24 |

D_{6} | 100.0 |

simulations employ the linear time-harmonic technique, and the finite element code employs the A-V formulation with the electric scalar potential V set to zero [

Eddy current testing (ECT) is a computational method with many industrial applications including crack detection to the rapid sorting of small components for flaws, size variations as well as material variations. The impedance of the detecting coil is distorted and altered by the presence of flaws or material variations [

As it can be observed in _{3}) and/or the gap between the tube and support plate (parameter D_{5}). In the configuration selected for the simulated work, the gap between the two coils is h_{3} = 0.5 mm, and the gap between the tube and support plate is also 0.5 mm. These choices appear highlighted in ^{6} S/m; and 2) the magnetic relative permeability, µ_{r}, of the support plate is 100. In all simulated problems, two identical air-cored coils with 1000 turns each are connected differentially and used as eddy current excitation and detection system. For short, the pair of air-cored coils will be referred to as eddy current coil. In the sequence of simulations, the eddy current coil moves downwards inside the Inconel-600 tube in the axial direction. The analysis includes three frequencies of excitation: 1, 10, and 100 kilohertz.

Results of the WFNDEC’s first benchmark problem have been published by different research groups. Initially, came the work of Sikora and Palka in 2001 [

In the WFNDEC’s first benchmark problem, neither the material properties nor the physical arrangement of the wires that form the pair of coils is described. It is worth noting that the wires made of, say copper or aluminum and accommodated in a given coil region can be specified as 1) stranded wire; 2) not stranded wire; 3) magnet wire; 4) plain stranded wire; 5) litz wire; or 6) square wire. According to the authors’ choice, each rectangular region that forms the pair of coils accommodates 1000-not stranded copper wires with unity relative permeability µ_{r} and electrical conductivity σ = 58 × 10^{6} S/m. Also, the value of 10 mA mentioned in the problem description has been assumed to mean the root-mean-square (rms) value of the exciting current. Once the field simulator FEMM always works with “peak” values, in all simulated problems the terminal current of the eddy current coil is specified as I_{p} = 14.142 mA.

An outline of the axisymmetric model is shown in

The grain of the finite element mesh in different regions of the model is defined by the parameter “edge size” [

values for the edge size parameter have been specified in regions such as the leftmost portion of the support plate near the air gap. As a result of these choices, economical meshes with an average of 65,000 nodes and 130,000 elements have been employed to save time and computer effort.

The impedance trajectory is plotted in the complex R-X plane, and represents the variation of the eddy current coil impedance that occurs when the material medium of the region representing the defect (flaw) changes. To obtain the impedance trajectory associated to a given frequency of excitation, it is necessary to obtain two different sequences of magnetic vector potential solutions. The first sequence of vector potentials {A_{1}} provides the values of the coil impedance when the defect is present, i.e., the rectangular region of area h_{1} × h_{2} shown in _{2}} provides the values of the coil impedance when the defect is not present, i.e., the rectangular region of area h_{1} × h_{2} is filled by the same material of the tube: Inconel-600. In both sequences, the displacement of the detecting coil, d, varies in the range [0; d_{max}]. According to the reference frame, when the pair of coils is symmetrically positioned with respect to the origin and defect region, at (r = 0; z = 0), d = 0. When the detecting coil is sufficiently far away from the defect region, the difference ΔZ in the coil impedances given by the two sequences of vector potentials vanish. All additional generated points of the trajectory will be situated at the origin of the R-X plane. The final shape of the correct or idealized impedance trajectory is a smooth, convex and closed loop. The distance d = d_{max} where ΔZ vanishes is problem-dependent, and varies with the frequency of excitation: for 10 kHz, d_{max} = −15 mm, whilst for 100 kHz d_{max} = −11.75 mm.

The coil movement is simulated employing steps of 0.25 mm. The movement is the result of a sequence of vertical translations of the eddy current coil in the −z direction. Each vertical translation requires a new mesh for the problem. This approach has been chosen because the artifice of material re-identification to change materials in problems representing consecutive positions of the pair of coils would require a very complicated drawing, with many distinct regions in the air layer that accommodates the eddy current coil.

At this low level of operating frequency, the ECT approach is not reliable for this class of problem, and its use is quite rare. In the graph of

case, for example, of the computations in the ranges −12.75 ≤ d ≤ −10.50 and −9.50 ≤ d ≤ −8.75. At this low level of operating frequency, the ECT approach is clearly questionable, perhaps even downright worthless for this class of problem.

The calculation of variations in the coil impedance using the two sets of vector potential solutions {A_{1}} and {A_{2}} and the plotting of the impedance trajectory at the frequency of 10 kilohertz are discussed in the following. The graph presented in

The relationship between the positional displacement d of the eddy current coil and the variations (Δr; Δx) that form the impedance trajectory is not clear at a first glance. In the attempt to clarify the subject, the 10 kHz impedance trajectory is illustrated separately in Appendix B.

To introduce the basic operations that lead to the 10 kHz impedance trajectory, let us initially consider the data presented in _{1} are related to the sequence of vector potentials {A_{1}}, whereas impedances Z_{2} are related to the sequence of vector potentials {A_{2}}; and 2) in the range −15.00 ≤ d ≤ −14.00 wherein the variations Δr and Δx tend to vanish. This can be observed on the last line of

Displacement (mm) | Re(Z_{1}) (Ω) | Im(Z_{1}) (Ω) | Re(Z_{2}) (Ω) | Im(Z_{2}) (Ω) | ΔR (Ω) | ΔX (Ω) | ΔZ (Ω) | Δr (pu) | Δx (pu) |
---|---|---|---|---|---|---|---|---|---|

−3.00 | 168.404 | 674.845 | 168.455 | 674.820 | −0.0510 | 0.025 | 0.0568 | −0.8909 | 0.4367 |

−3.25 | 168.371 | 674.798 | 168.422 | 674.772 | −0.0510 | 0.026 | 0.0572 | −0.8909 | 0.4542 |

−3.50 | 168.336 | 674.728 | 168.385 | 674.701 | −0.0490 | 0.027 | 0.0559 | −0.8559 | 0.4717 |

−3.75 | 168.300 | 674.654 | 168.347 | 674.627 | −0.0470 | 0.027 | 0.0542 | −0.8210 | 0.4717 |

−4.00 | 168.263 | 674.566 | 168.307 | 674.540 | −0.0440 | 0.026 | 0.0511 | −0.7686 | 0.4542 |

… | … | … | … | … | … | … | … | … | … |

−14.00 | 162.872 | 663.204 | 162.872 | 663.203 | 0.000 | 0.001 | 0.001 | 0.000 | 0.0175 |

−14.25 | 162.760 | 663.054 | 162.760 | 663.054 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 |

−14.50 | 162.658 | 662.898 | 162.658 | 662.897 | 0.000 | 0.001 | 0.001 | 0.000 | 0.0175 |

−14.75 | 162.572 | 662.808 | 162.572 | 662.807 | 0.000 | 0.001 | 0.001 | 0.000 | 0.0175 |

−15.00 | 162.480 | 662.684 | 162.480 | 662.684 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 |

For a coil displacement d = −3.25 mm, the variation in magnitude of the coil impedance is maximum, ΔZ_{max} = 0.05724509 Ω, and this can be observed in line 2 and column 8 of

Δ R = Re ( Z 1 ) − Re ( Z 2 ) = − 0.0510 Ω . (1)

The variation of the inductive component ΔX is

Δ X = Im ( Z 1 ) − Im ( Z 2 ) = − 0.0260 Ω . (2)

To compute the per unit variations of the components Δr and Δx, their ohmic values should be normalized with respect to the maximum magnitude in variation of the coil impedance ΔZ_{max} along the total excursion. For a coil displacement d = −3.25 mm, one has

Δ r = Δ R / Δ Z max = − 0.8909 p .u . (3)

Also, at d = −3.25 mm, the per unit variation of the inductive component Δx is

Δ x = Δ X / Δ Z max = 0.4542 p .u . (4)

Values of the variables used in the calculations expressed by (1)-(4) appear highlighted on the 2^{nd} line of

For the operating frequency of 100 kilohertz, the graph presented in

It is worth noting that the characteristics shown in

The field simulator FEMM calculates the eddy current coil impedance, Z, as part of the “circuit properties” by using

Z = V p / I p , (5)

where V_{p} denotes the peak value of the terminal voltage, and I_{p} denotes the peak value of the terminal current. To ensure that values of the coil impedance at different positions have been correctly computed, impedance values can also be computed using the alternative method explained in Appendix A.

To illustrate the use of the alternative method of impedance computation, let us consider the data presented in _{1}(Z) and Im_{1}(Z) in the range −3.00 ≤ d ≤ −2.00. The maximum variation in the coil impedance is ΔZ_{max} = 1.61848 Ω and occurs at d = −2.50 mm. The data also include the coil dissipated energy P and the magnetic stored energy W in the domain of analysis. Now, let us consider the vector potential solution A_{1} that represents the displacement d = −2.50 mm. For a “rms” current I = 10 mA and an angular frequency ω = 2π × 1 × 10^{5} rad/s, the resistive component R is given by

R = P / I 2 = 530.076 Ω , (6)

and the inductive reactance X is given by

X = ( ω 2 W ) / I 2 = 6203.53 Ω . (7)

Values calculated in (6) and (7) ought to be compared to the values indicated in columns 2 and 3 of the 3^{rd} line of

The composition, in the complex R-X plane, of the two characteristics presented in Figure5 produce the 100 kHz impedance trajectory shown if Figure6. This trajectory ought to be compared to the 10 kHz impedance trajectory shown in FigureB3. It is worth noting that, as the distance between the detecting coil and the defect region increases, the variations Δr and Δx become smaller and more difficult to be computed accurately. The 10 kHz impedance trajectory lacks smoothness, especially along its final portion that represents the coil scan below d = −11.75 mm. The reentrant corners present in the plot reflect the failure of the computation technique in detecting those very small variations in the resistive and inductive components of the eddy current coil. The 100 kHz impedance trajectory, on the other hand, possesses the shape of a smooth convex characteristic, and the distribution of its discrete points is clearly in accordance with the physical understanding of the problem.

Displacement (mm) | Re_{1}(Z) (Ω) | Im_{1}(Z) (Ω) | ΔZ (Ω) | P (W) | W (J) |
---|---|---|---|---|---|

−2.00 | 529.830 | 6203.81 | 1.5289 | 0.0529830 | 4.93666E−007 |

−2.25 | 529.930 | 6203.84 | 1.6101 | 0.0529930 | 4.93667E−007 |

−2.50 | 530.076 | 6203.75 | 1.6185 | 0.0530076 | 4.93661E−007 |

−2.75 | 530.156 | 6203.80 | 1.5810 | 0.0530156 | 4.93665E−007 |

−3.00 | 530.195 | 6203.72 | 1.4952 | 0.0530195 | 4.93658E−007 |

Eddy current testing (ECT) relies on the change in impedance of a detecting coil caused by the presence of electrical currents induced on a test specimen subjected to a time-varying magnetic field. The technique is used for the detection of cracks and other defects that interrupt the flow of the induced currents in the test specimen. The results are usually presented in the form of impedance trajectories in the complex R-X plane. The technique does not depend on the magnetic properties of the material where the defect is located, and can be applied to any conducting material.

The WFNDEC’s first eddy current benchmark problem can be viewed as a set of several ECT benchmark problems because it allows changes in geometry, material properties and frequency of excitation. The problem configuration selected for the simulated work includes the external support plate (SP), a pair of coils that accommodate 1000-not stranded copper wires, and a defect region symmetrically positioned around the origin of the r-z plane. The rms value of the excitation current is 10 mA, and the analysis involves three different frequencies of excitation: 1, 10 and 100 kHz.

The discussion places emphasis on the relationship between the positional displacement of the eddy current coil and the variations in the coil impedance used to obtain the impedance trajectory. For each of the three operating frequencies, numerical results include ohmic and per unit variations of the impedance components with respect to positional displacement and the resulting impedance trajectory.

For the operating frequency of 1 kHz, the ECT approach is clearly questionable for this class of problem, and the simulated experiment has failed in identifying impedance distortions along the coil scan. In the attempt to facilitate the teaching of ECT techniques on introductory courses on NDE, an additional effort has been made to better illustrate the 10 kHz experiment. The most satisfactory results are associated to the frequency of 100 kHz.

Additional configurations of the WFNDEC’ first problem remain to be tested in future work. Important investigations include 1) the system’s response to higher frequencies of excitation; 2) further reduction in the area of the rectangular defect; and 3) changes in the geometry of the flaw considering both surface and under surface defects.

The authors acknowledge the financial support, in the form of scholarships, from the Brazilian Federal Agency for Postgraduate Studies (CAPES). The authors give thanks to David Meeker (dmeeker@ieee.org) for the use of the finite element CAD system.

The authors declare no conflicts of interest regarding the publication of this paper.

Nogueira, A.F.L., Weinert, R.L. and Maldonado, L.J.A.S. (2021) An Introductory Note on Finite Element Problems Based on the Eddy Current Testing Approach. Journal of Electromagnetic Analysis and Applications, 13, 145-159. https://doi.org/10.4236/jemaa.2021.1311011

Let J_{e} denote the external current density on a conducting medium. One may define the electric field intensity E by

E = − ∇ V − j ω A , (A.1)

the magnetic flux density B by

B = ∇ × A , (A.2)

and the magnetic field strength H by

H = B μ , (A.3)

where ω is the angular frequency of the excitation. V is the electric scalar potential, A is the magnetic vector potential and µ is the magnetic permeability of the material medium. The impedance of the eddy current coil can be obtained by calculating the dissipated energy P and the total magnetic stored energy W, using the following equations [

P = ∫ Γ J e E ∗ d Γ (A.4)

and

W = 1 2 ∫ Γ H B ∗ d Γ , (A.5)

where G refers to the whole solution domain, and (*) denotes the complex conjugate operator. The coil impedance is computed by

R + j ω L = 1 I 2 ( P + j ω 2 W ) , (A.6)

where R is the resistance, L is the inductance and I is the “rms” current.

According to (Ida, 1983) [

In the following, the 10 kHz impedance trajectory is presented as the union of two plots 1) the first plot represents the portion of the impedance trajectory obtained when the coil displacement d varies along the initial 1/3 of the total excursion, i.e., −5.0 ≤ d ≤ 0.0 mm; and 2) the second plot represents the portion of the impedance trajectory obtained when the coil displacement d varies along the remaining 2/3 of the total excursion, i.e., −15.00 ≤ d ≤ −5.00 mm. The first plot is presented in FigureB1, the second plot is presented in FigureB2, and the final plot, representing the complete impedance trajectory is presented in FigureB3.