Eddy-Current Interaction Between a Probe Coil and a Conducting Plate with a Cylindrical Borehole

A probe coil with a T-core above a layered conductor with surface hole is investigated for magnetic flux transfer along the ferrite core and enhancement of eddy currents in conductor. The cylindrical coordinate system is adopted and an artificial boundary is added to the solution domain with radius b, and the general formula for calculating the impedance of the T-core coil is derived using the truncated region eigenfunction expansion (TREE) method. For four special cases with different probe configurations, coil impedance changes due to the layered conductor and defect are calculated with Mathematica software over a frequency change ranging from 100 Hz to 20 kHz. The analytical results are in good agreement with those obtained by the finite element method and experimental measurements. The results show that under the same lift-off height and excitation frequency, the impedance change caused by the conductor or defect in the coil of long core column is greater than that of the short core column coil. It indicates that the probe coil with a long core column can transfer magnetic flux to the conductor, thereby enhancing eddy currents in the conductor.

The impedance change in an induction coil surrounding a metal tube adapter is investigated using the truncated region eigenfunction expansion (TREE) method. The conventional TREE method is inapplicable to this problem as a consequence of the numerical overflow of the eigenfunctions of the air-metal multi-subdomain regions. The difficulty is surmounted by a normalization procedure for the numerical eigenfunctions obtained from the 1D finite element method (FEM). An efficient algorithm is devised by the Clenshaw-Curtis quadrature rule for integrals involving the numerical eigenfunctions. The numerical results of the TREE and FEM simulation coincide very well in all cases, and the efficiency of the proposed method is also confirmed.

Currently, the finite element method (FEM) and analytical calculation are widely employed for the modeling of electromagnetic acoustic transducers (EMATs). However, it takes long time for finite element calculation. Previous analytical models for bulk wave EMATs are generally considered separately and incompletely, and expressions of radiated wave fields contain infinite integrations and multiple singular points, which result in complex numerical computation. A complete model containing the Lorentz force and radiated wave field calculation for the EMAT with a spiral coil and a NdFeB permanent magnet is established. By introducing a current loop instead of the permanent magnet and adopting the truncated region eigenfunction expansion (TREE) method, the distributions of static and dynamic magnetic fields and their generated Lorentz forces are calculated. A series expansion method is proposed for the computation of radiated wave fields, which replaces the integration by series operation and avoids the solutions of singular points effectively. The Lorentz forces and radiated wave fields of a typical transducer are computed. The validity of the model is verified by FEM and experiments. Their good agreements verify the accuracy and validity of the model.

The field and impedance change of a reflection probe in the vicinity of the edge of a magnetic metal plate is investigated by the truncated region eigenfunction expansion (TREE) method. The analysis is implemented with the second-order vector potential (SOVP), and the double Fourier series are formulated compactly with the matrices. To solve the eigenvalue problem for the magnetic metal plate, an adaptive approach based on the 1-D finite-element method (FEM) is proposed, and the complex eigenvalues can be calculated automatically. The proposed method is compared numerically with the 3-D FEM, and good agreement is achieved in all cases.

Multi-layer conductive structures, especially those with features like bolt holes, are vulnerable to hidden corrosion and cracking, posing a serious threat to equipment integrity. Early defect detection is vital for implementing effective maintenance strategies. However, the subtle signals produced by these defects necessitate highly sensitive non-destructive testing (NDT) techniques. Analytical modeling plays a critical role in both enhancing defect-detection capabilities and guiding the design of highly sensitive sensors for these complex structures. Compared to the finite element method (FEM), analytical approaches offer advantages, such as faster computation and high accuracy, enabling a comprehensive analysis of how sensor and material parameters influence defect detection outcomes. This paper introduces a novel T-core eddy current sensor featuring a central air gap. Utilizing the vector magnetic potential method and a truncated region eigenfunction expansion (TREE) method, an analytical model was developed to investigate the sensor’s interaction with multi-layer conductive materials containing a hidden hole. The model yielded closed-form expressions for the induced eddy current density and coil impedance. A comparative study, implemented in Matlab, analyzed the eddy current distribution generated by T-core, E-core, I-core, and air core sensors under identical conditions. Furthermore, the study examined how the impedance of the T-core sensor changed at different excitation frequencies between 100 Hz and 10 kHz when positioned over a multi-layer conductor with a hidden air hole. These findings were then compared to those obtained from E-core, I-core, and air-core sensors. The analytical results were validated through finite element simulations and experimental measurements, exhibiting excellent agreement. The study further explored the influence of T-core design parameters, including the air gap radius, dome radius, core column height, and relative permeability of the T-core material, on the inspection sensitivity. Finally, the proposed T-core sensor was used to evaluate crack and hole defects in conductors, demonstrating its superior sensitivity compared to I-core and air core sensors. Although slightly less sensitive than the E-core sensor, the T-core sensor offers advantages, including a more compact design and reduced material requirements, making it well-suited for inspecting intricate and confined surfaces of the target object. This analytical model provides a valuable tool for designing advanced eddy current sensors, particularly for applications like detecting bolt hole defects or measuring the thickness of non-conductive coatings in multi-layer conductor structures.

Eddy-current inspection for nondestructive evaluation has traditionally been investigated in terms of coil impedance signals via theoretical and experimental methods. However, advanced eddy-current techniques use solid-state sensors such as Hall devices, giant magnetoresistive sensors, anisotropic magnetoresistive sensors, and superconducting quantum interference devices for magnetic field measurement to achieve better sensitivity and high temporal and spatial resolution in material evaluation and characterization. Here, we review the Dodd and Deeds integral model and use the truncated region eigenfunction expansion (TREE) method for computation of the magnetic field. This results in series expressions instead of integral ones. Thus, the computation is both simplified and speeded up so that it becomes convenient for solving one-dimensional eddy-current inverse problems. We compare the theoretical results from the analytical model with the results from a numerical simulation based on the finite-element method in terms of accuracy and computation time.

Computationally fast electromagnetic models of eddy current sensors are required in model-based measurements, machine interpretation approaches or in the sensor design phase. If a sensor geometry allows it, the analytical approach to the modeling has significant advantages in comparison to numerical methods, most notably less demanding implementation and faster computation. In this paper, we studied an eddy current sensor consisting of a transmitter coil with a finitely long I ferrite core, which was screened with a finitely thick magnetic shield. The sensor was placed above a conductive and magnetic half-layer. We used vector magnetic potential formulation of the problem with a truncated region eigenfunction expansion, and obtained expressions for the transmitter coil impedance and magnetic potential in all subdomains. The modeling results are in excellent agreement with the results using the finite element method. The model was also compared with the impedance measurement in the frequency range from 5 kHz to 100 kHz and the agreement is within for the resistance change due to the presence of the half-layer and for the inductance change. The presented model can be used for measurement of properties of metallic objects, sensor lift-off or nonconductive coating thickness.

In eddy current testing, probe-coils with the E-type pot core are commonly used for detecting defects in test objects. In this paper, an analytical mathematical model of such a probe placed above a two-layered conductive material with a surface hole has been presented. The final formulas in the closed form that make it possible to calculate the coil impedance were worked out using the truncated region eigenfunction expansion (TREE) method and implemented in Matlab. Changes in resistance and reactance were determined for both material without a hole and for space containing no conductor. The results were compared with those obtained for the air-cored coil and the I-cored coil. The correctness of the calculations was confirmed through experimental measurements and with the finite element method (FEM) in the COMSOL Multiphysics package

PurposeIn eddy current nondestructive testing, ferrite-cored probes are usually used to detect and locate defects such as cracks and corrosion in conductive materials. However, the generic analytical model for evaluating corrosion in layered conductor using ferrite-cored probe has not yet been developed. The purpose of this paper is to propose and verify the analytical model of an E-cored probe for evaluating corrosion in layered conductive materials.Design/methodology/approachA cylindrical coordinate system is adopted and the solution domain is truncated in the radial direction. The magnetic vector potential of each region excited by a filamentary coil is derived first, and then the expansion coefficients of the solution are obtained by matching the boundary and interface conditions between the regions and the subregions. Finally the closed-form expression of the impedance of the multi-turn coil is derived by using the truncated region eigenfunction expansion (TREE) method, and the impedance calculation is carried out in Mathematica. In the frequency range of 100 Hz to 10 kHz, the impedance changes of the E-cored coil and air-cored coil due to the layered conductor containing corrosion are calculated, respectively, and the influences of corrosion on the coil impedance change are investigated.FindingsAn analytical model for the detection and evaluating of corrosion in layered conductive materials using E-cored probe is proposed. The model can quickly and accurately calculate the impedance change of E-cored coil due to corrosion in layered conductor. The correctness of the analytical model is verified by finite element method and experiments.Originality/valueAn accurate theoretical model of E-cored probe for evaluating corrosion of multilayer conductor is presented. The analytical model can be used to detect the inhomogeneity of layered conductor, design ferrite-cored probe or directly evaluate the corrosion defects of layered conductors.

In pipeline inspection, defects cause a non-uniform magnetic field distribution, resulting in permeability distortion along the surface of the ferromagnetic material. This paper studies a 2-D axisymmetric problem of the sinusoidal eddy field for the ferromagnetic material with longitudinally partitioned regions of different magnetic permeability, which has not been reported before. Applying the truncated region eigenfunction expansion (TREE) method, a 2-D analytical model of the sinusoidal eddy field based on permeability distortion is derived, and the analytical expressions of the vector magnetic potential and the coil impedance are obtained. The coil impedance variation is analyzed and validated by the finite element method (FEM) and the experiment. In Model 1, the maximum errors of the changes in the resistance and the reactance are 0.52% and 0.29%, respectively. In Model 2, the maximum errors of the changes in the resistance and the reactance are 0.88% and 2.05%, respectively. The effect of the number of sub-regions is examined, and the results show that increasing the number of sub-regions can improve the accuracy.

The paper examines the problem of an axially symmetric I-cored coil located above a three-layered plate with a hole in the middle layer. A cylindrical coordinate system was applied, wherein the solution domain was truncated in the radial direction. The employment of the truncated region eigenfunction expansion (TREE) method resulted in deriving the final formulas for the change of the coil impedance with regard to the air space, and also pertaining to the test object without a flaw. Formulas for various configurations of the test object, among others for a surface hole, a subsurface hole and a through hole, have been presented. For the purpose of defectoscopy, the influence of the hole in the plate on the impedance components was investigated. The calculations were made in Matlab for frequencies from 100 Hz to 50 kHz. The obtained results were verified using the finite element method (FEM) in Comsol Multiphysics package. A very good agreement was observed in the case of both the resistance and reactance.

Conductors consisting of thin layers are commonly used in many industries as protective, insulating or thermal barrier coatings (TBC). Nondestructive testing of these types of structures allows one to determine their dimensions and technical condition, while also detecting defects, which significantly reduces the risk of failures and accidents. This work presents an eddy current system for testing thin layers and coatings, which has never been presented before. It consists of an analytical model and a pot-core sensor. The analytical model was derived through the employment of the truncated region eigenfunction expansion (TREE) method. The final formulas for the sensor impedance have been presented in a closed form and implemented in Matlab. The results of the calculations of the pot-core sensor impedance for thin layers with a thickness above 0.1 mm were compared with the measurement results. The calculations made for the TBC were verified with a numerical model created using the finite element method (FEM) in Comsol Multiphysics. In all the cases, the error in determining changes in the components of the pot-core sensor impedance was less than 4%. At the same time, it was shown that the sensitivity of the applied pot-core sensor in the case of thin-layer testing is much higher than the sensitivity of the air-core sensor and the I-core sensor.

An analytical model for eddy current testing of an I-core coil located above a two-layer conductive material is presented. The upper layer is an infinite plane conductor, and the bottom layer is a conductive cylinder. The method of truncated region eigenfunction expansion (TREE) is used to solve this axisymmetric problem. First the magnetic vector potential of a filamentary coil coaxial with the I-core over the two-layer conductor is considered. Then the closed form expression for the impedance of the multi-turn coil with rectangular cross section is derived by using the principle of superposition from the filamentary coil field. For frequencies ranging from 0.1 kHz to 10 kHz, both the impedance changes of the I-core coil located above the infinite plane conductor without the conducting cylinder, and in the absence of the two-layer conductor are calculated using Mathematica, respectively. The influence of the conducting cylinder below the infinite plane conductor on the impedance change is analyzed. The analytical calculation results are verified by the finite element method and experiment, the results agree very well, which verifies the correctness of the analytical model.

An analytical model is developed for coaxial coils with a hollow iron core and a magnetic shield. The problem is solved by an enhanced version of truncated region eigenfunction expansion (TREE). The eigenvalues and eigenfunctions of the complicated mixed regions are solved by the one-dimensional finite element method (FEM). The results are validated by comparisons with measurements and FEM simulation. Additional results reveal that this configuration enables a significant portion of iron core material to be saved while maintaining a high level of magnetic coupling between the coils.

This paper proposes an analytical approach that can describe the Pulsed Eddy Current Testing (PECT) of stratified conductive structures by a single air-coil used for detecting and sizing of metal loss due to interlayer corrosion. Specifically for a single air-cored coil above a multilayered conductive structure, the coil impendance response of the cylindrical eddy-current probe is calculated based on the truncated region eigenfunction expansion (TREE) method using the generalized matrix coefficients in series form. Analytical modeling of pulsed eddy current is developed using the Fourier series for multilayered conductive structures. A simulation study is carried out for demonstrating the capability of the proposed approach for sizing the metal loss in a two-layer specimen. Furthermore, the experiment is set up to confirm the simulating results of the analytical model for multilayered steels. Good agreement between the theoretical results and measured signals is observed.