Eddy Current Model Research Articles

Numerical Solution of an Axisymmetric Eddy Current Model with Current and Voltage Excitations

Numerical Solution of an Axisymmetric Eddy Current Model with Current and Voltage Excitations

Eddy Current Loss Model for Ferrite Ring Cores Based on a Meta-Material Model of the Core Properties

A model for estimating the total eddy current losses in ring ferrite cores is proposed, which is based on a microstructure model of the electrical parameters of the material (conductivity and permittivity) and on the solution of the wave equations in the core. The presented model, combined with state-of-the-art estimation of the hysteresis losses, gives good agreement with the core loss measurements, over a wide range of frequencies (10–700 kHz).

Neural network parametric identification of an eddy current sensor model using a reservoir computing system

The development of modern automated systems for monitoring, control and management of technological processes is associated with the improvement of methods for identifying and studying the primary means of collecting and processing information. An algorithm for neural network parametric identification using reservoir calculations is proposed. An algorithm for parametric identification orient to conducting a computational experiment on a given nonlinear model, forming training samples based on the results of the experiment, training dynamic and static neural networks and calculating estimates of parameters of a nonlinear model using trained networks according to experimental data. Keywords eddy current sensor, identification, dynamic model, equivalent circuit, static neural network, dynamic neural network, reservoir computing, mathematical model

Parallel-motion-type eddy current damper model of rectangular magnet and conductor

Eddy currents in a conductor moving in a non-uniform magnetic field in a static coordinate system are expressed as the superposition of the term by the partial derivative of the magnetic vector potential with respect to time and by the gradient of scalar potential in a stationary-conductor coordinate system. In this study, we proposed the general equation of “gradient of scalar potential is zero” condition (GSPZ condition) throughout the conductor. Additionally, under satisfying the GSPZ condition, we propose the method of obtaining the magnetic damping force from both the magnetic flux densities and the eddy currents calculated using the Biot-Savart law and Fleming's left-hand rule for the parallel-motion-type eddy current damper (GSPZ-A method). The precision of the GSPZ-A method is similar to that of the three-dimensional finite element method (3D-FEM); however, the effect of the secondary magnetic field was not considered. In this study, the GSPZ condition for the parallel-motion-type eddy current damper of a rectangular magnet and conductor of arbitrary dimensions was established. Furthermore, the GSPZ condition was applied to two types of eddy current dampers—one composed of the single square magnet and the other of the combined square magnet with oppositely aligned magnetic poles. The magnetic damping forces calculated using the GSPZ-A method were compared with those obtained from the 3D-FEM and experiments. As a result, the errors from the GSPZ-A method to 3D-FEM for the single and combined magnets were 10 and 0.4 %, respectively.

Fast Solution of 3-D Eddy-Current Problems in Multiply Connected Domains by a, v-φ and t-φ Formulations With Multigrid-Based Algorithm for Cohomology Generation

The fast solution of three-dimensional eddy current problems is still an open problem, especially when real-size finite element models with millions of degrees of freedom are considered. In order to lower the number of degrees of freedom a magnetic scalar potential can be used in the insulating parts of the model. This may become difficult when the model geometry presents some conductive parts which are multiply connected. In this work a multigrid-based algoritm is proposed that allows for a calculation in linear-time of cohomology, which is needed to introduce the scalar potential without cuts. This algorithm relies on an algebraic multigrid solver for curl-curl field problems, which ensures optimal computational complexity. Numerical results show that the novel algorithm outperforms state-of-the-art methods for cohomology generation based on homological algebra. In addition, based on this algoritm, novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a,v$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> formulations to analyze three-dimensional eddy current problems in multiply connected domains are proposed. Both formulations, after discretization by the cell method, lead to a complex symmetric system of linear equations amenable to fast iterative solution by Krylov-subspace solvers. These formulations are able to provide very accurate numerical results, with a minimum amount of degrees of freedoms to represent the eddy current model. In this way the computational performance is improved compared to the classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A,V$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> formulation typically implemented in finite element software for electromagnetic design.

Eddy current loss modelling for torque‐motors

A torque motor is a key component that directly influences the dynamic performance of two-dimensional (2D) cartridge servo valves. Considering the effects of eddy current, a magnetic circuit model of a torque motor is proposed, and the eddy current resistance value considering the skin effect is calculated. When the eddy current effect is ignored, the natural frequency and critical damping of a torque motor are less than the calculated values under the eddy current effect, so the eddy current influence can be assumed to increase torque motor damping, as the iron loss can be equivalent to the damping energy consumption. Test rigs were set up and experiments were conducted based on a torque motor prototype. Through a finite element simulation analysis, the eddy current effect of the armature was found to be greater than that of the yokes. The magnetic leakage coefficients of the permanent magnets were calculated, and the obtained values were compared through experiments and simulations. Both the numerical simulations and experimental results showed that the eddy current model is closer to actual working conditions than the model for which the eddy current loss was ignored.

A pulsed eddy current testing method for relative thickness of ferromagnetic components considering local hysteresis characteristics

A pulsed eddy current testing method is proposed for the assessment of the relative thickness of ferromagnetic components with uneven external insulation layer and remanence. By establishing a least squares problem of theoretical and measured value of induced voltage based on the time-domain analytical expression to pulsed eddy current testing model of a ferromagnetic plate, unknown parameters including the thickness, conductivity and relative permeability are determined by inversion algorithm. Then, the coupling relationship among parameters in the inverse problem is verified by experiments on a steel plate, and the detection method for relative thickness is proposed. Furthermore, the function relationship between the inversion results of the thickness and the relative permeability under different amplitudes of the pulsed current is obtained through the linear fitting. Finally, the suppression method is discussed for the influence of the variation of permeability. It is suggested that the reference value of the thickness should be dynamically corrected by this function, as to improve the accuracy and reliability. This paper provides a reliable non-destructive testing method for wall-thinning corrosion detection of coated ferromagnetic tubes or tanks in industry.

Locating Defects in Conductive Materials Using the Eddy Current Model of the Filamentary Coil

This article presents an innovative technique for detecting flaws in conductive materials by using an ideal filamentary coil. To characterize such a coil accurately and explicitly, it is sufficient to be in possession of merely two parameters: the radius of the circle within which all the turns are located and the distance of the coil from the tested surface. The mathematical model derived using the truncated region eigenfunction expansion method enables the calculation of the changes in the components of the filamentary coil impedance that are the result of positioning the coil close to the conductive material with a hole. Because of this, the air-cored coil model can be replaced with a much simpler filamentary coil model. This solution makes it is possible to detect various types of holes (internal, surface, subsurface or through) occurring in both multilayer magnetic and non-magnetic materials. The derived results were verified by means of measurements and numerical calculations based on the finite element method. Very good agreement was observed in both cases. The paper contains the source code implemented in Matlab, which is used to for calculations.

An Improved Core Loss Model of Ferromagnetic Materials Considering High-Frequency and Nonsinusoidal Supply

With the emerging power electronic technologies, numerous electrical devices use high-frequency and nonsinusoidal supplies. Such sources generate complex harmonic fields that overburden the accurate prediction of core losses using the conventional finite element method (FEM). Thereby, to predict core losses under those high-frequency and nonsinusoidal supplies accurately, this study first built a test system for core losses measurement of ferromagnetic materials (e.g., soft ferrite, laminated steels, and nanocrystalline alloy) fed by the square and pulsewidth modulated (PWM) supplies. Then, the loss prediction problems of the classical core loss model under high-frequency and nonsinusoidal supplies are revealed. Afterward, an improved FEM-based core loss model is proposed, which includes a piecewise variable parameters based hysteresis loss model, an eddy current loss model considering different cross sections of magnetic circuit and supply waveforms, and improved excess loss coefficients. Experimental validations on different materials, the nanocrystalline alloy in an 11-kVA intermediate frequency transformer, the ultrathin silicon sheets in a 10-kW high-speed permanent magnet motor, and the nanocrystalline alloy in a 3-kW wireless power transfer system, are conducted systematically. The effectiveness and generality of the improved core loss model were verified.

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We consider sufficient conditions to prove that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. Furthermore, we show that mixed finite element formulations arising from dynamics fluids (time-dependent Stokes problem) and from electromagnetic applications (eddy current models), can be analyzed as applications of the developed theory. Finally, we include numerical tests to illustrate the performance of the method and confirm the theoretical results.

Parameter identification of pulsed eddy current testing equivalent circuit model and its application to defect evaluation

The equivalent circuit model of pulsed eddy current testing (PECT) has simple mathematics but its applications are limited to qualitative analysis such as principle illustration and signal interpretation. In this paper, the parameters of equivalent circuit model are estimated using system identification method and quantitative relationships are found between some of the parameters and the size of the defect. The equivalent circuit equations were solved from the perspective of system analysis to yield the system transfer function. An m-sequence of 10 order was selected to excite the system and the probe current was used as the output. A set of experiment input-output data for system identification were obtained by performing experiments on two aluminium alloy 6061 slabs, one of which was machined with five slots of different widths, and the other one was machined with five depth-varied slots. The equivalent circuit parameters were finally estimated based on the identified transfer function parameters. It is found that the values of resistance and self-inductance of the secondary windings decrease greatly and monotonically with the increase of the slot width or depth. The self-inductance is more sensitive to the slot size variation than the resistance. Both of them have the potential to serve as the signal feature for defect evaluation in PECT applications.

Eddy Current Non-Destructive Characterization of Carbon Fiber Reinforcement Composites Considering Capacitive Effect

This paper presents a modelling procedure to take into account the capacitive effect at high frequencies, in Eddy Current Non-Destructive Characterization (EC-NDC) of Unidirectional Carbon Fiber Reinforcement Composite (UD-CFRC) rods. To simulate the complete EC-NDC systems, first, the multilayer circular air coil is physically modeled by a finite element (FE) axisymmetric eddy current model coupled to equivalent RL circuit. Each layer of the coil is represented by an equivalent resistance (R) in series with the equivalent inductance (L). Secondly, R and L of the coil layers are computed for several frequencies up to 5Mhz, and then introduced into the equivalent RLC circuit with considering inter-turn and interlayer capacitances. Then the inversion problem is solved in order to identify all inner capacitances of the coil. Finally, the UD-CFRC rod is introduced into the FE eddy current axisymmetric model coupled to an equivalent RLC circuit, as a homogenized conductive material with an equivalent transverse conductivity. The coil with the presence of the homogenized UD-CFRC rod is then modeled as a transformer with a secondary connected to a capacitor in parallel with a resistance in order to evaluate the inner capacitor of the UD-CFRC. All evaluated parameters are then introduced in the last model. The comparison between the computed impedance parts and the measured ones shows a mean error less than 2% and a maximum one of 5% according to the frequency.

A time discrete scheme for an electromagnetic contact problem with moving conductor

In the context of multi-component electromagnetic problems, we investigate an eddy current model with moving conductor. The time derivative acting on the moving subdomain and the jumps of the material coefficients on the interfaces raise the challenge to handle the boundary term which is gained by the Reynolds transport theorem. A time-discrete scheme based on backward Euler’s method is proposed to solve this problem. Using Rothe’s method, we show the stability as well as the convergence to a unique weak solution of the variational system.

The eddy current loss segmentation model of permanent magnet for temperature analysis in high‐speed permanent magnet motor

The eddy current loss segmentation model of permanent magnet for temperature analysis in high‐speed permanent magnet motor

Direct Current Hysteresis Loops of a MuMETAL Ring Measured by Continuous Recording and Automatic Ballistic Methods

We measured saturated hysteresis loops of a MuMETAL ring using a computerized, high-resolution apparatus by both extrapolated continuous recording and automatic ballistic methods. The recording time per loop varied from 17.7 to 141.6 s. The loops obtained for short recording times were significantly wider than the loop obtained by the ballistic method. By extrapolating the magnetic field of different loops at the same flux density values linearly to zero flux density rate, the resultant dc loop had coercivity in agreement with its ballistic value. The results are quantitatively analyzed and discussed on the basis of eddy current models. The intolerably large variation and bias in the measured coercivity stated in a current ASTM standard may be resolved by this technique.

On a class of degenerate abstract parabolic problems and applications to some eddy current models

We present an abstract framework for parabolic type equations which possibly degenerate on certain spatial regions. The degeneracies are such that the equations under investigation may admit a type change ranging from parabolic to elliptic type problems. The approach is an adaptation of the concept of so-called evolutionary equations in Hilbert spaces and is eventually applied to a degenerate eddy current type model. The functional analytic setting requires quite minimal assumptions on the boundary and interface regularity. The degenerate eddy current model is justified as a limit model of non-degenerate hyperbolic models of Maxwell's equations.

An eddy current model of pot-cored coil for testing multilayer conductors with a hole

An eddy current model of pot-cored coil for testing multilayer conductors with a hole

Eddy current damper capable of collecting electric energy

This paper presents an eddy current damper model that can store electrical energy. The damper is mainly used under strong impact load. On the basis of generating sufficient electromagnetic damping force to resist the impact load, the generated eddy current is guided out of the device through the winding and connected to the energy storage components of the external circuit. In this way, it is realized that the eddy current damper has the function of resisting the impact load. And at the same time, the eddy current damper can convert a part of the impact energy into electric energy and store it in the energy storage element. After research, the theoretical scheme is completely feasible, and the single impact energy conversion efficiency can reach 2.33 %.

3-D adaptive FEA with weighted node density technique

PurposeThis paper aims to propose a method to create 3-D finite element meshes automatically using the Delaunay tetrahedralization with the weighted node density technique. Using this method, the adaptive finite element analysis (FEA) was carried out for the calculation of the magnetic field of an eddy current verification model to clarify the usefulness of the method. Moreover, the error evaluation function for the adaptive FEA was also discussed.Design/methodology/approachThe method to create the 3-D finite element meshes using the Delaunay tetrahedralization is realized by the weighted node density technique, and Zienkiewicz-Zhu’s error estimator is used as the error evaluation function of the adaptive FEA.FindingsThe magnetic flux density vectors on the node in the error evaluation function for the adaptive FEA should be calculated with the weighted average by the reciprocal of the volume of elements.Originality/valueThis paper describes the method to create 3-D finite element meshes and the comparison among calculation methods of the magnetic flux density vectors on the node for the error estimator.

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models