Eddy Current Model Research Articles

Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors

Realistic applications in metal detection involve multiple inhomogeneous‐conducting permeable objects, and the aim of this paper is to characterise such objects by polarizability tensors. We show that, for the eddy current model, the leading order terms for the perturbation in the magnetic field, due to the presence of N small conducting permeable homogeneous inclusions, comprises of a sum of N terms with each containing a complex symmetric rank 2 polarizability tensor. Each tensor contains information about the shape and material properties of one of the objects and is independent of its position. The asymptotic expansion we obtain extends a previously known result for a single isolated object and applies in situations where the object sizes are small and the objects are sufficiently well separated. We also obtain a second expansion that describes the perturbed magnetic field for inhomogeneous and closely spaced objects, which again characterises the objects by a complex symmetric rank 2 tensor. The tensor's coefficients can be computed by solving a vector valued transmission problem, and we include numerical examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical prediction. We also include algorithms for the localisation and identification of multiple inhomogeneous objects.

A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.

Multi-objective Optimization of Permanent Magnet Adjustable Speed Driver Base on RSM Model and NSGA-II

To optimise the structure of a permanent magnet adjustable speed driver (PMASD), a multi-objective optimization design method for increasing the output torque and reducing the eddy current loss was proposed. Firstly, the three-dimensional finite element method model of a PMASD was established, the influence of the main configuration parameters in the PMASD on the output torque and the eddy current loss was analyzed, and the reasonable range of the parameters was determined. When taking the minimal eddy current loss and maximum output torque as the optimal objectives, the secondary response surface numerical model equation was built using the Central Composite Design experimental method and the Response Surface Methodology. Then, while ensuring that the output torque of the PMASD is not less than the rated torque, the NSGA-II was used to perform multi-objective optimization based on the response surface model and the Pareto optimal solution sets for two objectives was obtained. Finally, the minimal eddy current loss model and maximum output torque model were selected to compare with the initial model that was not optimized. The simulation results proved that the output torque of the minimal eddy current loss model increased by 6.54% based on the reduction of eddy current loss by 0.81% and the output torque of the maximum output torque model increased by 24.41% based on an increase in eddy current loss of 10.51%: the performance of both optimization models had been improved significantly. The optimization results show that this method improves the transmission performance of the PMASD.

3D harmonic modeling of eddy currents in segmented conducting structures

PurposeThe purpose of this paper is to describe a semi-analytical modeling technique to predict eddy currents in three-dimensional (3D) conducting structures with finite dimensions. Using the developed method, power losses and parasitic forces that result from eddy current distributions can be computed.Design/methodology/approachIn conducting regions, the Fourier-based solutions are developed to include a spatially dependent conductivity in the expressions of electromagnetic quantities. To validate the method, it is applied to an electromagnetic configuration and the results are compared to finite element results.FindingsThe method shows good agreement with the finite element method for a large range of frequencies. The convergence of the presented model is analyzed.Research limitations/implicationsBecause of the Fourier series basis of the solution, the results depend on the considered number of harmonics. When conducting structures are small with respect to the spatial period, the number of harmonics has to be relatively large.Practical implicationsBecause of the general form of the solutions, the technique can be applied to a wide range of electromagnetic configurations to predict, e.g. eddy current losses in magnets or wireless energy transfer systems. By adaptation of the conductivity function in conducting regions, eddy current distributions in structures containing holes or slit patterns can be obtained.Originality/valueWith the presented technique, eddy currents in conducting structures of finite dimensions can be modeled. The semi-analytical model is for a relatively low number of harmonics computationally faster than 3D finite element methods. The method has been validated and shown to be computationally accurate.

A note on the block alternating splitting implicit iteration method for complex saddle‐point problems

SummaryRecently, Bai has proposed a class of block alternating splitting implicit (BASI) iteration methods [Numer. Linear Algebra Appl. 19(6):914‐936] for the complex generalized saddle‐point problems. The author demonstrates the unconditional convergence of the methods when the (1,1) block of the saddle‐point coefficient matrix is positive definite. In this paper, we display that the BASI iteration method is convergent for wider problems, for instance, the (1,1) block is positive semidefinite, provided the splittings meet certain requirements. The theoretical result encourages us to explore more efficient iteration schemes for more general problems. We present new BASI iteration schemes for the algebraic linear systems from time‐harmonic eddy current models. The corresponding matrix splitting preconditioners are presented for the generalized minimal residual method. Experimental results show the feasibility and effectiveness of our new iteration methods and the corresponding preconditioned generalized minimal residual methods.

Validation of a Harmonic Model for Eddy Currents in Slitted Conducting Plates

The paper describes a 3D semi-analytical harmonic modeling technique that is capable of modeling eddy current distributions in segmented conducting structures, such as slitted conducting plates, and the associated magnetic fields. The spatially varying conductivity of a conducting region is incorporated into the solutions of magnetic-field quantities and the induced current density. The harmonic model is compared to results obtained with finite element analysis. An experimental setup is used to measure the field distribution above differently slitted conducting plates, in which eddy currents are induced by a coil. The measurement results are compared to simulation results, and the perturbations are analyzed.

Eddy current modelling using multi-layer perceptron neural networks for detecting surface cracks

A new method for computing fracture mechanics parameters using computational Eddy Current Modelling by Multi-layer Perceptron Neural Networks for detecting surface cracks. The method is based upon an inverse problem using an Artificial Neural Network (ANN) that simulates mapping between Eddy current signals and crack profiles. Simultaneous use of ANN by MLP can be very helpful for the localization and the shape classification of defects. On the other side, it can be described as the task of reconstructing the cracks and damage in the plate profile of an inspected specimen in order to estimate its material properties. This is accomplished by inverting eddy current probe impedance measurements that are recorded as a function of probe position, excitation frequency or both. In eddy current nondestructive evaluation, this is widely recognized as a complex theoretical problem whose solution is likely to have a significant impact on the detection of cracks in materials

Distributed Current Source Method for Modeling Magnetic and Eddy-Current Fields Induced in Nonferrous Metallic Objects

This paper presents a new modeling method to determine the harmonic eddy current (EC) field induced in a nonferrous metal and its corresponding magnetic flux density (MFD) by an EC-based sensing system for geometrical measurements, which accounts for the boundary effects of the object. Modeled using a distributed current source (DCS) method in state-space representation, the EC field is formulated as a two-step constrained least-square problem to solve for its real and imaginary parts. Two practical techniques to improve the efficiency and accuracy of the EC solutions are illustrated. The first refines the DCS distribution based on the skin-depth effects, and the second takes advantages of commercial mesh-generation software to facilitate the modeling of EC induced in complex-shaped objects. The DCS-based EC models are verified numerically by comparing computed results with two-dimensional (2-D) analytical axisymmetric solutions and commercial finite-element analysis, and evaluated experimentally with an EC sensor that measures the MFD generated by the induced EC in different materials and geometrical configurations.

Influence of k-space trajectory corrections on proton density mapping with ultrashort echo time imaging: Application for imaging of short T2 components in white matter

PurposeTo evaluate the impact of MR gradient system imperfections and limitations for the quantitative mapping of short T2* signals performed by ultrashort echo time (UTE) acquisition approach. Materials and methodsThe measurement of short T2* signals from a phantom and a healthy volunteer study (8 subjects of average age 28 ± 4 years) were performed on a 3T scanner. The characteristics of the gradient system were obtained using calibration method performed directly on the measured subject or phantom. This information was used to calculate the actual sampling trajectory with the help of a parametric eddy current model. The actual sample positions were used to reconstruct corrected images and compared with uncorrected data. ResultsComparison of both approaches, i.e., without and with correction of k-space sampling trajectories revealed substantial improvement when correction was applied. The phantom experiments demonstrate substantial in-plane signal intensity variations for uncorrected sampling trajectories. In the case of the volunteer study, this led to significant differences in relative proton density (RPD) estimation between the uncorrected and corrected data (P = 0.0117 by Wilcoxon matched-pairs test) and provides for about ~15% higher values for short T2* components of white matter (WM) in the case of uncorrected images. ConclusionThe imperfection of the applied gradients could induce errors in k-space data sampling which further propagates into the fidelity of the UTE images and jeopardizes precision of quantification. However, the study proved that measurement of gradient errors together with correction of sample positions can contribute to increased accuracy and unbiased characterization of short T2* signals.

Analysis and numerical modelling of eddy current damper for vibration problems

This work discusses a contactless eddy current damper, which is used to attenuate structural vibration. Eddy currents can remove energy from dynamic systems without any contact and, thus, without adding mass or modifying the rigidity of the structure. An experimental modal analysis of a cantilever beam in the absence of and under a partial magnetic field is conducted in the bandwidth of 01 kHz. The results show that the eddy current phenomenon can attenuate the vibration of the entire structure without modifying the natural frequencies or the mode shapes of the structure itself. In this study, a new inverse method to numerically determine the dynamic properties of the contactless eddy current damper is proposed. The proposed inverse method and the eddy current model based on a lineal viscous force are validated by a practical application. The numerically obtained transfer function correlates with the experimental one, thus showing good agreement in the entire bandwidth of 01 kHz. The proposed method provides an easy and quick tool to model and predict the dynamic behaviour of the contactless eddy current damper, thereby avoiding the use of complex analytical models.

Eddy current modeling in linear and nonlinear multifilamentary composite materials

Abstract In this work, a numerical model is developed for a rapid computation of eddy currents in composite materials, adaptable for both carbon fiber reinforced polymers (CFRPs) for NDT applications and multifilamentary high temperature superconductive (HTS) tapes for AC loss evaluation. The proposed model is based on an integro-differential formulation in terms of the electric vector potential in the frequency domain. The high anisotropy and the nonlinearity of the considered materials are easily handled in the frequency domain.

A new alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new alternating positive semidefinite splitting (APSS) iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.

A Full-Wave Integral Equation Method Including Accurate Wide-Frequency-Band Wire Models for WPT Coils

A full-wave surface integral equation method is used for modeling coils in a wide frequency range. This is of high importance in the analysis of resonant wireless power transfer links, for example. This paper presents a nonlocal generalization of the impedance boundary condition (BC) concept by incorporating a 2-D model (using the finite element method here) of the quasi-static electromagnetic field within the wire. This yields a numerical approximation of the nonlocal impedance BC on the wire surface that holds true at any frequency. The 2-D eddy-current model has to be evaluated only twice at one frequency, which makes the scheme computationally efficient. The method accurately predicts the ohmic loss as shown by the validations against alternative methods.

Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems

Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.

Analysis and Experiment of Eddy Current Loss in Radial Magnetic Bearings with Solid Rotor

In order to reduce power losses for the aerospace applications, this paper analyzes the eddy current losses produced by high-speed rotating solid rotor core magnetic bearings of magnetically suspended control & sensitive gyroscope (MSCSG). An analytical model of the eddy current loss of a solid rotor radial magnetic bearing (RMB) is presented. Considering the difference between NNNN and NSNS magnetic circuits of RMB, the magnetic field expressions of stator magnetic poles are listed. The magnetic field of stator poles is replaced by Fourier series expansion. According to the magnetic field distribution around the magnetic pole surface and the boundary conditions of the rotor surface, the mathematical expression of eddy current loss is obtained. The measurement method of rotational power loss in radial magnetic bearing is proposed, and the results of the theoretical analysis are verified by experiments in the prototype MSCSG. The experimental results show the correctness of calculation results. Eddy current loss models and test methods provide theoretical support for analyzing eddy current losses in solid rotors and reducing power consumption.

Influence of magnet eddy current on magnetization characteristics of variable flux memory machine

In this paper, the magnet eddy current characteristics of a newly developed variable flux memory machine (VFMM) is investigated. Firstly, the machine structure, non-linear hysteresis characteristics and eddy current modeling of low coercive force magnet are described, respectively. Besides, the PM eddy current behaviors when applying the demagnetizing current pulses are unveiled and investigated. The mismatch of the required demagnetization currents between the cases with or without considering the magnet eddy current is identified. In addition, the influences of the magnet eddy current on the demagnetization effect of VFMM are analyzed. Finally, a prototype is manufactured and tested to verify the theoretical analyses.

2-D Semi-Analytical Modeling of Eddy Currents in Multiple Non-Connected Conducting Elements

This paper concerns the semi-analytical modeling of parasitic eddy current distributions and forces in electromagnetic devices. The harmonic, Fourier-based, model is extended to incorporate a position-dependent conductivity, and restrictions on the induced current density are added to model eddy currents in non-connected conducting elements. To model time-dependent behavior of eddy currents, multiple time harmonics can be incorporated in the solutions. The developed method is verified using the finite-element method for a wide range of frequencies. The model is applied to a coreless linear motor with electrically conducting cooling plates, and the parasitic force is analyzed.

Experimental Study of Iron Losses Generated by a Uniform Rotating Field

This paper introduces a mock-up with a solid iron cylinder rotating in a fixed excitation frame powered by dc current. A uniform field is created in the rotating cylinder, which is driven by an external motor. The braking torque is measured, allowing the study of iron losses generated by a uniform rotating field. The aim is to advance toward a vector iron loss model that takes into account different magnetization directions and their interdependency. The eddy current losses are the dominant loss source in the mock-up, rendering it easier to model. A standard eddy currents model is proposed. It models well the losses at low and medium magnetic flux density, but lower losses are measured near saturation levels. This could be due to the high thickness to radius ratio changing the eddy current paths near the edges of the rotor, which can be later analyzed by a full 3-D finite-element method analysis. A 2-D finite-element method simulation is performed, which estimates the magnetic flux heterogeneity in the rotor. Several lessons are drawn from this mock-up, which prepares a second version with a higher speed and a longer laminated stack rotor. A higher air gap will decrease the voltage fluctuation in the primary winding, a smaller angular opening of the excitation frame will improve the uniformity of the flux in the rotor.

A note on the justification of the eddy current model in electrodynamics

The issue of justifying the eddy current approximation of Maxwell's equations is reconsidered in the time‐dependent setting. Convergence of the solution operators is shown in the sense of strong operator limits.

Study on the eddy current testing model of “CFRP-bond-steel” multilayer and heterogeneous media

Study on the eddy current testing model of “CFRP-bond-steel” multilayer and heterogeneous media