Analytical solution for magnetic field of cylindrical defect in eddy current nondestructive testing

Abstract

In metal components, a defect frequently causes the failure of the components and results in serious accidents consequently. Eddy current nondestructive testing method is one of the most effective testing approaches to quantitatively monitor the defects both in non-ferrite metal and in ferrite metal. Therefore, knowledge of the magnetic field is necessary to deeply understand the mechanism of eddy current nondestructive technique in defect characterization. In this paper, an analytical expression for the magnetic field for a cylindrical defect in metal in eddy current testing system is proposed by solving the partial differential equations. Firstly, a conductive metal plate with a cylindrical defect is modeled for the cylindrical coil with rectangular cross-section, and the analytical expression to the magnetic potential vector is obtained. Then, the series form expressions are extended to calculate the axial component of magnetic induction intensity Bz and is computed on the MATLAB platform. Finally, a finite element model is setup to validate the proposed analytical expression for the magnetic field. The results show that the magnetic field calculated by the analytical method is in good agreement with that calculated by finite element method, and the calculation efficiency of the analytical method is higher than that of FEM. The proposed expressions not only can be readily used to investigate the quantitative relationship of the cylindrical defects in a conductive metal, but also it is expected to be extended to seek the theoretical expression for the elliptic defect and also for the natural defect in key structures (e.g. oil pipe, rail track, iron bridge, oil tank, ocean ship) by using (pulsed) eddy current nondestructive technique.

The semi-analytic solution to the magnetic field of frequency-band-selecting eddy current testing for local corrosion flaw in large diameter pipeline

“…Values in the "Verification by numerical simulation" section External radius of the excitation coil (mm) magnetic field in regions 1 and 2 and obtain equations ( 8) to (10). You can find the detailed description for equations ( 5) to ( 7) by Yu et al 23 C 11 = C 22 + e 2λd C 21 (5)…”

“…By substituting equations ( 8) to (10) into equations ( 11) and (12), which are the magnetic vector potentials in regions 1 and 2 investigated by Yu et al, 23 we can obtain the series expression for the magnetic vector potential in regions 1 and 2 for a single coil as equation ( 13) 11)…”

“…Therefore, for the single coil, to calculate the magnetic field at the coil, we can replace z with z 1 in equation (5) for the magnetic field in region 1 and replace z with z 2 in equations (6) and (7) for the magnetic field in regions 1 and 2 and obtain equations (8) to (10). You can find the detailed description for equations (5) to (7) by Yu et al 23 By substituting equations (8) to (10) into equations (11) and (12), which are the magnetic vector potentials in regions 1 and 2 investigated by Yu et al, 23 we can obtain the series expression for the magnetic vector potential in regions 1 and 2 for a single coil as equation (13) where G k and G a are n × n matrixes. They can be expressed as Therefore, according to equation (13) and the superposition principle, the magnetic vector potential in series expansion expression at any point z in regions 1 and 2 can be deduced as In equations (5) to (16), I false~ is the excitation current in the frequency domain, and λ is the eigenvalue of J 1 ( λb ) = 0 , E , U , V , G , M , and N are all n × n matrixes, and E and G are diagonal matrixes.…”

“…In terms of the analytic modeling for ECT, based on the great contributions of Dodd and Deeds 7 as well as Theodoulidis et al 19 studied the eddy current induction problem in a half-space conductor with a vertical borehole and presented a solution based on modal techniques; Zhang et al 20 and Hao et al 21 presented the approximate solution of PECT for a steel plate with a flat-bottom hole by using the TREE method; Tytko et al 22 proposed an analytical model of a probe placed above a two-layered conductive material with a surface hole. Our group derived the analytic solutions to the magnetic field for ECT with a non-through hole 23 and the semi-analytic solutions to the magnetic field for ECT with a slit defect. 24 However, for FBS-ECT, there is no analytical expression to reveal the features of FBS-ECT in different application cases.…”

Design of a Remote, Multi-Range Conductivity Sensor

“…A wide range of antenna and metamaterial arrays [ 3 , 4 , 5 , 6 ] has been developed; these arrays are involved in eddy current sensing techniques. Most publications in the field focus on non-destructive testing (NDT) applications, which include thickness measurements [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ], detection of cracks and defects [ 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ], as well as measurements of conductivity [ 10 , 11 , 12 , 13 , 14 , 29 , 30 , 31 , 32 ] and permeability [ 33 , 34 , 35 ] in highly conductive samples. These applications cover only a fraction of the conductivity spectrum, although detecting mid-range and low can lead to exciting applications.…”