Eddy Current Model Research Articles

Characterization of Heat-treated Bearing Rings via Measurement of Electromagnetic Properties for Pulsed Eddy Current Evaluation

Nondestructive evaluation of heat-treated bearing rings is a critical technique for quality control in industries. However, there are few reports addressing the complex mapping between hardness and electromagnetic properties due to the intricate changes in the microstructure. This paper proposes a dual-method approach, combining magnetic saturation eddy current techniques and Barkhausen noise reconstruction hysteresis loop techniques, to independently establish the relationship between hardness and electromagnetic properties. The results show that the electrical properties of unqualified specimens are significantly higher than those of other specimens, with qualified specimens have slightly higher properties than untreated ones. Additionally, an inverse relationship between hardness and magnetic properties is observed. Based on the obtained electromagnetic parameters, a pulsed eddy current hardness detection simulation model is established, which has the potential to improve purely data-driven methods for hardness detection in deep learning.

Unconventional coil shape design of eddy current sensors and the effect of shape parameters on the micro/nano-scale metal film thickness measurement performance

Abstract The thickness of metal film is a critical parameter, especially in micro/nano-manufacturing, where high-precision measurement is essential. The eddy current method, a non-destructive testing technique, is well-suited for in-situ measurement of micro/nano-scale metal film thickness due to its superior performance. However, enhancing the measurement capabilities of eddy current sensors remains a significant challenge. In practical applications, thickness sensitivity and spatial resolution are two key performance indicators of eddy current sensors, and improving both simultaneously is difficult. While the sensitive element (coil) of an eddy current sensor has a substantial impact on thickness sensitivity, its effect on spatial resolution has received less attention.
This study establishes an eddy current coil model based on electromagnetic field theory, defining both thickness sensitivity and spatial resolution in the context of micro/nano-scale metal film thickness measurement. Two unconventional coil shapes are introduced, contrasting with the traditional cylindrical design, to investigate the influence of coil shape parameters, specifically the spatial distribution of the coil turns, on the key performance indicators. Simulation results are corroborated through experimental validation. Based on a series of calculations and analyses, an optimization method for coil shape parameters is proposed using a defined comparison factor that balances both thickness sensitivity and spatial resolution, which offers a promising approach for improving coil shape design.

A differential low frequency eddy current test method for detecting moisture content of high voltage cable water-blocking tape

ABSTRACT The excessive moisture content in the water-blocking tape has been one of the main causes of high voltage cable erosion failures in recent years. However, the 2–4 mm thick aluminium sheath wrapped around the water-blocking tape makes it challenging to detect the moisture content. This study introduces a method for detecting the moisture content of high voltage cable water-blocking tapes based on low-frequency eddy currents. Initially, longitudinal and transverse impedance measurement experiments of the water-blocking tape were conducted to establish the relationship between impedance and moisture content. Since the impedance of the water-blocking tape is significantly higher compared to the aluminium sheath, the eddy current signal was obtained using a differential detection method. An excitation frequency of 10 Hz was chosen to achieve a deeper skin depth. Additionally, an eddy current finite element analysis model was developed, showing that moisture content has an almost linear relationship with the amplitude and phase of the eddy current signal. Finally, an experiment was conducted on an equivalent high voltage cable structure, and the experimental results aligned well with the simulation results.

Iterative Numerical Approximation Technique for 3D Eddy Current Models in Harmonic Regime Based on the Electromagnetic T-Formulation and the Finite Element Method

In this paper, an iterative numerical approximation technique is used for analyzing the distribution of eddy currents density in conductive materials plates by using the electromagnetic T-formulation and the Biot-Savart law, solved by the finite element method. The proposed approach allows for the meshing of the different parts of the studied system separately, including the sensor and the conductive plate, without the need for an air region. Firstly, this approach reduces the number of unknown variables by avoiding the air region of the system's mesh. Secondly, it simplifies the consideration of the sensor's motion without the need to remesh the system. For this purpose, a calculation code has been developed for solving an electromagnetic three-dimensional non-destructive testing model. This latter permits the resolution of JSEAM # 6 Benchmark problem to validate the proposed method. The impedance variation due to the presence of a defect are evaluated. The obtained results are compared with the experimental ones found in the literature. These results reveal a good agreement, which proves the validity of the proposed method.

Eddy Current Mechanism Model for Dynamic Magnetic Field in Ferromagnetic Metal Structures

The degaussing process is crucial for ensuring magnetic protection in ships. It involves the application of oscillating and attenuating magnetic fields to eliminate residual magnetism in the ship’s structure. However, this process can lead to the generation of distorted magnetic fields within the ship’s cabin, posing a potential threat to electronic equipment performance. Therefore, it is essential to have a comprehensive understanding of the dynamic magnetic field response in ship structures to develop effective degaussing systems. To address this need, this paper proposes an eddy current model for analyzing the dynamic magnetic field response in ferromagnetic metal structures. This model focuses on the role of eddy currents in shaping the magnetic field response and provides valuable insights into the underlying mechanisms. Using the proposed eddy current model, the effects of key system parameters such as thickness, conductivity, and the length-scale of the ship structure can be analytically investigated. This analysis helps in understanding how these parameters influence the dynamic magnetic field response and aids in the design and optimization of degaussing systems. The effectiveness and applicability of the proposed eddy current model are demonstrated through comprehensive investigations involving two simulation cases of varying complexity. The model accurately predicts the changing trends of the dynamic magnetic field response, as confirmed through finite element simulations. This validation highlights the model’s ability to reproduce simulation results accurately and its potential as a powerful tool for analyzing and optimizing dynamic magnetic field responses. In summary, the proposed eddy current model represents a significant advancement in the field. It provides a valuable theoretical framework for understanding and analyzing the dynamic magnetic field response in ferromagnetic metal structures. By offering insights into the underlying mechanisms and the influence of key parameters, this research contributes to the development of improved degaussing systems and enhances the overall magnetic protection capabilities of ships.

Adaptation of an Eddy Current Model for Characterizing Subsurface Defects in CFRP Plates Using FEM Analysis Based on Energy Functional

In this work, a known Eddy Current (EC) model is adapted to characterize subsurface defects in carbon fiber-reinforced polymer (CFRP) plates intended for the civil aerospace industry. The considered defects include delaminations, microcracks, porosity, fiber breakage, and the simultaneous presence of these defects. Each defect is modeled as an additive variation in the material’s electrical conductivity tensor, allowing for a detailed mathematical representation of the defect’s influence on the CFRP’s electromagnetic behavior. The additivity of the variations in the conductivity tensor is justified by the assumption that the defects are not visible to the naked eye, implying that the material does not require non-destructive testing. The adapted EC model admits a unique and stable solution by verifying that all analytical steps are satisfied. To reconstruct 2D maps of the magnetic flux density amplitude, a FEM formulation is adopted, based on the energy functional because it ensures a stable and consistent numerical formulation given its coercivity. Moreover, the numerical approach allows precise and reliable numerical solutions, enhancing the capability to detect and quantify defects. The numerical results show that the obtained 2D maps are entirely superimposable on those highlighting the distribution of mechanical stress states known in the literature, offering a clear advantage in terms of detection costs. This approach provides an effective and economical solution for the non-destructive inspection of CFRP, ensuring accurate and timely defect diagnosis for maintaining structural integrity.

Autonomous Underwater Vehicle Path Planning Based on Improved Salp Swarm Algorithm

Aiming at the problem of path planning for autonomous underwater vehicle (AUV) to cope with the influence of obstacles and eddies in complex marine environments, a path planning method based on an improved salp swarm algorithm (ISSA) is proposed. Firstly, the motion model of the AUV and eddy current model are constructed, including the relationship between position, velocity, attitude, and control inputs. Secondly, the improved SSA is proposed, which introduces the Levy flight strategy to enhance the algorithm’s optimization seeking ability and adds a nonlinear convergence factor to enhance the convergence ability of the algorithm. The stability and robustness of the improved algorithm are verified by test functions. Finally, the ISSA is applied to AUV path planning, which optimizes the AUV travel distance, improves the search efficiency and accuracy, and avoids the local optimum of the algorithm. The ISSA enhances the adaptive ability and robustness of the algorithm by introducing a dynamic adjustment strategy and feedback mechanism. Experimental verification is carried out using a simulated marine environment. The results show that the ISSA is better than the traditional algorithm in terms of path length as well as algorithm stability, and can effectively improve the navigation performance of AUV.

Free vibration of ferromagnetic functionally graded cylindrical shells subjected to magnetic and temperature fields

The free vibration characteristics of functionally graded cylindrical shells under magnetic and temperature fields are investigated. According to Donnell’s theory, the kinetic energy, deformation energy, and their variational expressions are obtained. Based on the electromagnetic elasticity theory, the eddy current electromagnetic and magnetization force models are established. Using Hamilton’s principle, the magneto-thermo-elastic free vibration partial differential equations are derived. The modal frequency differential matrix equation with diverse boundary conditions is derived from the Galerkin integral technique. Through numerical examples, the curves and three-dimensional surface diagrams depicting the variations for the natural frequency with diverse parameters are obtained. The feasibility of the proposed solution method in this paper is supported by literature examples. The influence of magnetic induction intensity, temperature, power-law index, wave number, and shell dimension on the natural frequencies are determined.

Parallel-motion-type eddy current damper model of ring magnet and conducting disk

Parallel-motion-type eddy current damper model of ring magnet and conducting disk

Design of passive and structural conductors for tokamaks using thin-wall eddy current modeling

A new three-dimensional electromagnetic modeling tool (ThinCurr) has been developed using the existing PSI-Tet finite-element code in support of conducting structure design work for both the SPARC and DIII-D tokamaks. Within this framework a 3D conducting structure model was created for both the SPARC and DIII-D tokamaks in the thin-wall limit. This model includes accurate details of the vacuum vessel and other conducting structural elements with realistic material resistivities. This model was leveraged to support the design of a passive runaway electron mitigation coil (REMC), studying the effect of various design parameters, including coil resistivity, current quench duration, and plasma vertical position, on the effectiveness of the coil. The REMC is a non-axisymmetric coil designed to passively drive large non-axisymmetric fields during the plasma disruption thereby destroying flux surfaces and deconfining RE seed populations. These studies indicate that current designs should apply substantial 3D fields at the plasma surface during future plasma current disruptions as well as highlight the importance of having the REMC conductors away from the machine midplane in order to ensure they are robust to off-normal disruption scenarios.

An Examination of the Stiffness Terms Needed to Model the Dynamics of an Eddy Current Based Maglev Vehicle

This paper re-examines the basis for each eddy current stiffness term computed from prior published steady-state eddy current models. The paper corrects prior analysis work by confirming, through the use of 2-D and 3-D dynamic finite element analysis modelling, that when a magnetic source is moving over an infinite-wide and infinite-long conductive sheet guideway the steady-state lateral and translational stiffness terms will be zero and only the vertical coupled stiffness terms need to be modelled. Using these observations, a much simplified 6 degrees-of-freedom (DoF) linearized eddy current dynamic force model can be used to compute the steady-state force changes in eddy current based maglev vehicles when operating over a wide uniform conductive track.

Distributed point source method for eddy current modelling

ABSTRACT Distributed point source method (DPSM) has been gradually applied to the field of nondestructive testing (NDT). As a semi-analytical modelling technique, DPSM is extremely powerful and straightforward for solving various engineering problems, such as ultrasonic fields and electromagnetic field. In this paper, the technique is extended to model the eddy current field including sweep frequency measurement and scanning defect. The configuration of eddy current sensors consists of excitation coil and solid-state magnetic field measurement sensors such as Hall device and giant magnetoresistive sensor. Besides that, the wave propagation has been described and magnetic field created by eddy current has been calculated. The results obtained by DPSM is compared with that calculated by finite element method (FEM) in terms of accuracy and computation time, which indicates DPSM can improve calculation speed while ensuring calculation accuracy.

Simulation Study of Transient Eddy Current Effect in 3T Animal Gradient Coil

In the process of high-field magnetic resonance imaging (MRI), the eddy current modeling and simulation of the cold screen and metal parts are not highly accurate. To solve this problem, this paper uses the target field method to design a 3T animal gradient coil model. To get more realistic simulation results, a complex model including a gradient system, cryogenic vessel, cold screen, epoxy resin, and temperature holes was constructed. The model was refined using the finite element method, and the effects of eddy currents induced by the gradient coil on the cylindrical cryogenic container and cold screen in the superconducting magnetic resonance instrument under the state of multi-physical field coupling were simulated. The model effectively reflects the transient eddy current distributions on the cold screen and cryogenic vessel in the MRI system and their time-dependent temperature rise effects.

Electro dynamic model of eddy currents in EU DEMO TF coil casing during major plasma disruption

The conceptual design of the EU DEMO reactor is currently ongoing within the EUROfusion consortium. Many different fault transients must be considered and carefully analyzed in the design phase; one of the most severe is the major plasma disruption (MPD), which causes several drawbacks on the magnet system. During a disruption, the plasma current decreases extremely fast, causing a fast variation of the magnetic field, which in turn induces an electric field. In presence of conductive materials, e.g., coil casing and vacuum vessel (VV), the electric field induces large eddy currents which deposit power by Joule effect. The conductive regions are tightly coupled on different timescales through the magnetic field induced by the eddy currents: the eddy currents in the VV influence the magnetic field evolution in the TF coil casing, thus affecting the power deposition in the latter. The aim of this work is the evaluation of the power deposited within the TF coil casing during a major plasma disruption due to eddy currents. The power deposition has been evaluated by means of the 3D-FOX, a finite element (FE) tool developed at Politecnico di Torino. The computed power deposition is used as input to the thermal-hydraulic (TH) simulation, performed with the 4C code, with the aim of assessing the erosion of the temperature margin given by MPD.

Characterising small objects in the regime between the eddy current model and wave propagation

AbstractBeing able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects.

Modeling Eddy Current Losses in HTS Tapes Using Multiharmonic Method

Due to the highly nonlinear electrical resistivity of high temperature superconducting (HTS) materials, computing the steady-state eddy current losses in HTS tapes, under time-periodic alternating current excitation, can be time consuming when using a time-transient method (TTM). The computation can require several periods to be solved with a small time-step. One alternative to the TTM is the multiharmonic method (MHM) where the Fourier basis is used to approximate the Maxwell fields in time. The method allows obtaining the steady-state solution to the problem with one resolution of the nonlinear problem. In this work, using the finite element method with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H-\varphi$</tex-math></inline-formula> formulation, the capabilities of the MHM in the computational eddy current loss modeling of HTS tapes are scrutinized and compared against the TTM.

Transient eddy current analysis for a spiral gradient pulse

Eddy currents are induced in the metallic structures of MRI machines due to the rapid switching of gradient fields generated by gradient coils. Several undesirable effects are associated with the induced eddy currents such as heat, acoustic noise, and MR image distortions. Accurate transient eddy currents numerical computations are required to predict and ameliorate such effects. Spiral gradient waveforms are of importance, particularly for fast MRI acquisition applications. For mathematical convenience, previously published work is mostly concerned with transient eddy currents computations associated with trapezoidal gradient waveforms; where spiral gradient waveforms were not considered. We recently presented preliminary transient eddy currents computations induced by an amplitude-modulated sinusoidal pulse in the scanner’s cryostat. In this work, we present a full computational framework for transient eddy currents induced by a spiral gradient waveform. A mathematical model for transient eddy currents involving the spiral pulse was derived and presented in detail using the circuit equation. Computations were implemented using a tailored multilayer integral method (TMIM) and results were compared to Ansys eddy currents analysis for cross-validation. The transient response of resultant fields generated by both an unshielded transverse coil driven by a spiral waveform was computed showing high agreement between Ansys and TMIM; albeit with high computational efficiency concerning time and memory. For further validation, computations for a shielded transverse coil were performed showing how eddy currents effects are reduced.

Review of Hysteresis Models for Magnetic Materials

There are several models for magnetic hysteresis. Their key purposes are to model magnetization curves with a history dependence to achieve hysteresis cycles without a frequency dependence. There are different approaches to handling history dependence. The two main categories are Duhem-type models and Preisach-type models. Duhem models handle it via a simple directional dependence on the flux rate, without a proper memory. While the Preisach type model handles it via memory of the point where the direction of the flux rate is changed. The most common Duhem model is the phenomenological Jiles–Atherton model, with examples of other models including the Coleman–Hodgdon model and the Tellinen model. Examples of Preisach type models are the classical Preisach model and the Prandtl–Ishlinskii model, although there are also many other models with adoptions of a similar history dependence. Hysteresis is by definition rate-independent, and thereby not dependent on the speed of the alternating flux density. An additional rate dependence is still important and often included in many dynamic hysteresis models. The Chua model is common for modeling non-linear dynamic magnetization curves; however, it does not define classical hysteresis. Other similar adoptions also exist that combine hysteresis modeling with eddy current modeling, similar to how frequency dependence is included in core loss modeling. Most models are made for scalar values of alternating fields, but there are also several models with vector generalizations that also consider three-dimensional directions.

Time-domain analytical solutions to pulsed eddy current model of moving cylindrical conductor

The real-time or high-speed eddy current testing for conductor specimens are essential in industrial production and the analytical methods of eddy current field for moving conductor need to be further researched. In this paper, the time-domain analytical model of relative motion between the cylindrical conductor and probe is solved under pulsed current excitation. The basic equations and boundary conditions of moving conductor are presented and the boundary-value problem is established. Then the complex frequency-domain expression of voltage induced in the pick-up coil can be obtained by adopting the Laplace and Fourier transformation. Later, the induced voltage in time-domain is derived through the inverse Laplace transformation via residue theorem. Furthermore, the diffusion process of pulsed eddy current is studied by solving the eddy current density in conductor region and the effects of moving velocity on eddy current field are analyzed quantitatively. These calculations show that the larger the moving velocity, the closer the center area of eddy-current distribution is to the pick-up coil, as well as the larger the amplitude of induced voltage in pick-up coil, which can be equivalent to the decrease of axial distance between the pick-up coil and drive coil. The theoretical values obtained in this paper are in excellent agreement with the experimental results and can be applied to the pulsed eddy current testing for moving conductor or probe.

Validation of a plasma burn-through simulation with an ECH power absorption model in KSTAR

An electron cyclotron heating (ECH) power absorption model was integrated with a plasma burn-through simulator, DYON, and the new version, DYON-EC, was validated against KSTAR ECH-assist start-up experiments. The absorbed ECH power was calculated with an analytic formula which is a function of the electron density and electron temperature and ECH hardware settings, namely, the injected power, wave frequency, harmonic number, mode fraction, and beam injection angles. Wave parameter changes by wall reflection was also included to simulate multiple reflections. The absorbed ECH power was self-consistently included in the electron energy balance equation. The simulation settings of the plasma-wall interaction model and the electromagnetic scenario including the eddy current model were optimized to reproduce the plasma parameter evolution and line emission data in a pure ohmic start-up discharge. The study revealed that assumption of double-path EC beam absorption is required to reproduce a KSTAR ECH-assisted start-up discharge. Using the same optimized settings, DYON-EC modelling successfully reproduced multiple KSTAR EC-assisted discharges in a large range of operation parameters. The good statistical reproduction of measured plasma parameter evolution confirms the validity of the DYON burn-through modelling with ECH.