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We have studied electromagnetic losses of a frequency-converter-fed cage-induction motor by using a numerical machine model that includes eddy-current and hysteresis phenomena in electrical steel sheets. We used the model to solve the two-dimensional (2-D) time-harmonic field and winding equations of a cage-induction machine, utilizing a finite-element method and phasor variables. We used complex reluctivity to couple the hysteresis and eddy currents in the sheets with the 2-D analysis. The model modifies the absolute value of the reluctivity according to a one-dimensional (l-D) eddy-current solution developed in the lamination thickness. To define the argument of the reluctivity, we applied both the 1-D field solution and measured hysteresis data. We compared computations of additional electromagnetic losses in a 37-kW test machine due to the higher harmonics of a frequency-converter supply with experimental results. The agreement is found to be reasonable. Index Terms--Eddy currents, finite-element methods, induction motors, losses.

The eddy currents induced in a thrust bearing result not only in a power loss of the system but also in phase lag and a decrease in magnitude of the magnetic force. Therefore, an accurate model including eddy-current effects would be highly beneficial at the design stage. We present a linear model including eddy-current effects for a typical active magnetic thrust bearing (AMTB). We define and calculate the dynamic current stiffness and displacement stiffness using both an analytical method and transient finite element (FE) analysis. We present a simple magnetic circuit model including eddy currents from which the two stiffnesses can be derived analytically. Furthermore, by using a transient analysis of the FE model, flux saturation, leakage, and fringing effects can all be taken into account. We show that the two stiffnesses have the same frequency response characteristics, both in the analytical model and from the results of the linear FE analysis, when the leakage and fringing effects are ignored. Even in the more practical case when the flux saturation, leakage and fringing effects are considered, the two stiffnesses can be considered to have the same frequency response characteristics within the 3 dB bandwidth without introducing much error. The open-loop frequency response of the two stiffnesses derived from the analytical model and the FE model agrees well with the experimental results. Index Terms--Active magnetic thrust bearing, dynamic stiffness, eddy-current effects, FEM model.

The eddy-current model is an approximation of the full Maxwell equations. We will give estimates for the modeling error and show how the constants in the estimates are influenced by the geometry of the problem. Additionally, we analyze the asymptotic behavior of the modeling error when the angular frequency tends to zero. The theoretical results are complemented by numerical examples using high order finite elements. These demonstrate that the estimates are sharp. Hence, this work delivers a mathematical basis for assessing the scope of the eddy-current model. Index Terms--Eddy current model, hp-FEM, magnetoquasistatic model, modeling error.

The interest in eddy current dampers is increasing especially in aeronautic and automotive industry. Such devices seem to be a valid alternative to conventional fluid film and viscoelastc dampers. Even if several papers have been published on this topic, an electromechanical model taking into account both the resistance and the inductance of the conductor is still lacking. The aim of the present paper is to model the electromagnetic interaction of an eddy current device operating as a damper or as a coupler and to validate it by means of experimental tests performed at steady state and vibrating about a fixed position. The study is based on the computation of the damping torque starting from the basic principles. The analytical models are developed using the bond graph formalism that allows to obtain purely mechanical analogs of the electromechanical system. The main results are the identification of eddy current damper dynamic model and the definition of a set of 'conversion rules' allowing to readily obtain the mechanical impedance from the torque to slip speed characteristic and vice versa. The experimental results confirm the band limited effect of the damping, which cannot be neglected for practical applications. The effect can be exploited in eddy current couplers to filter higher order disturbances. [DOI: 10.1115/1.2827990] Keywords: eddy current dampers, torsional dampers, electromagnetic couplers, induction machines, automotive systems, vibration control

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. Index Terms--Eddy current, magnetic field measurement, truncated region eigenfunction expansion.

In this paper, a 3-D eddy-current field model for calculating the eddy current losses in an air insulated bus duct system (AIBDS) is proposed. The temperature rises in the AIBDS, including its long linear section and connecting unit, are evaluated using a coupled fluid field and thermal field model. The contact resistance between the copper conductors and the temperature rise in those contact areas are measured and compared with the simulation results in order to validate the proposed methodology. Index Terms--Bus duct system (BDS), eddy current loss, temperature rise, thermal fluid.

An analytical development of the magnetic vector potential is used to investigate the losses in round-wire planar windings. The current distribution in wires is affected by the skin effect and the field created by adjacent wires (proximity effect). This field depends on the current distribution in conductors, resulting in a closed-form problem. In this paper, we obtain the vector potential outside a conductor to estimate the effect of induced currents in the field shape over the neighboring conductors. We use the results to calculate the losses in planar windings such as those in domestic induction heaters. We obtain an equivalent resistance representing the losses in windings and compare it with measurements. This solution provides an accurate analytical approach to modeling the losses in close-packed windings. Index Terms--Eddy-current losses, electromagnetic analysis, electromagnetic induction, equivalent circuits, impedance measurements, induction heating.

The numerical and experimental studies on induction heating of continuously moving strips in transverse field are presented in this paper. The induced eddy current and its coupled thermal field in moving media is computed with FEM. The adopted mathematical model consists of a Fouriers thermal conduction equation and a set of differential equations, which describes the steady-state eddy current problem in a configuration comprising a magnetic vector potential and an electrical scalar potential. The calculated results are in good agreement with the measurement. Index Terms - Transverse flux, FEM computation, Induction heating, Eddy current, temperature field, 3D, Moving object

In order to consider the motion of object deforming, the analytical and finite element coupling technique is used to study the magnetoforming process. The analytical solution given through the Macro-Element (ME) technique is done and applied to the case of an axisymmetrical electromagnetic forming system study. The simulation of deforming process is obtained by the modification of the ME size through the modification of its matrix while retaining the same mesh topology. This modification is the consequence of the viscoplastic problem solution. The modeling results are validated from a comparison with experimental data. Index terms - Electromagnetic forming, finite element, Macro-Element, eddy currents, coupling model.

This paper deals with the numerical analysis of electromechanical devices. The dynamic characteristics are described by Maxwell's equations, by the constitutive equations, and by the equations of motion. For the solution of the electromagnetic problem a comparison between the BEM-FEM coupling method and a pure FEM approach is carried out. Index terms - Eddy currents, finite element methods, boundary element methods, moving bodies.

The remote field eddy current (RFEC) technique, conventionaly used for inspecting ferromagnetic pipes has been adapted for the detection of defects that are located deep in thick aluminum plates. This paper presents three dimensional (3D) numerical modeling of the RFEC flaw detection in metallic plates. In the RFEC method the pickup coils are located far apart which leads to a large volume to be modeled. A method involving partitioning of the volume in the 3D finite element model for the RFEC detection of defects, is demonstrated. Index terms - Remote field eddy current, multilayer geometry, nondestructive testing

In this paper a new technique for determining surface impedance is developed. The technique is derived from an eddy current modelling technique, the impedance method. Considerable extension and development of the impedance method has been undertaken, enabling the prediction of surface impedance measurements for horizontally stratified media and in the vicinity of vertical dislocations. The method is compared with surface impedance theory for horizontally stratified media and with field based results near laterally discontinuous structures. Index terms-eddy currents, surface impedance, impedance method

This paper describes an integral closure approach for 2-D analysis of electromagnetically driven flow in inductively coupled molten metal using the finite-element method. This methodology, which restricts the solution domain to the conducting regions(s), was demonstrated for flows generated by complex electromagnetic fields, and computed results were found to be in good agreement with measurements. Index Terms - Eddy currents, fluid flow, modeling, numerical methods.

In the paper the problem of mathematical modelling of the human brain subjected to AC magnetic field is presented. This is to model a novel therapeutical treatment in psychiatry, namely healing of deep depression. The human brain may be represented by a multilayer structure, each layer of which is described by electric conductivity. Since the conductivities are not of unique value, a special approach which averages these values has been used. The calculations serve to design a coil, its dimensions and supplying parameters, according to medical expectations. Index terms - Bioelectromagnetics, Eddy currents, Modelling, Finite Element Method

The present paper outlines a new application of the current iteration method used previously to handle open boundaries. Described is how to use this approach to couple disconnected regions in the finite element (FE) model of electromagnetic field. The basics of the procedure are demonstrated by using the example of the 2D eddy currents (skin effect) problem with a voltage excitation. The current iteration method is based on the analytical solution of the governing equation in a free space. This analytical approximation is coupled with the FE approximation in an iterative algorithm. It may be easily incorporated into an existing FE code. Numerical examples which are presented in the article show the particular usefulness of the method for coupling FE approximations in disconnected regions. Index terms - Eddy currents, disconnected regions, finite element method, infinite regions.

A transient eddy current package is being extended to model rotating systems. By allowing the system to rotate, induced eddy currents must be modelled. The system being developed uses the 'lock-step' approach to modal the rotation. As the rotation progresses the induced voltages in the coil windings change, and depending on the driving circuit configuration, the source currents will alter. Coupling to an external system model to determine the coil currents is therefore necessary. The extensions to the eddy current model are discussed, in order to compute induced voltages, and couple to an external power system model. The mechanism whereby the coupling will be achieved is also discussed. Index Terms - Electromagnetic fields, electromagnetic analysis, finite element methods, eddy currents

Two finite element implementations for modelling eddy current problems with velocity are examined here. The formulations are based on the A - [Psi] representation. In one finite element implementation, the moving current density is represented as a volume integral term. In the other, volume and surface integral terms represent the moving current density. These formulations are applied to a series of rotational velocity test models differing in conductor topology and the results are compared. Index Terms - Eddy current, electromagnetic fields, finite element methods, velocity formulations

Previously, thin plate assumptions have been used to obtain a one-dimensional eddy current model for predicting the stationary, transformer-type losses in magnetic bearings built out of laminated material. Using similar assumptions, as in the 1-D eddy current model, rotating losses can be predicted for a laminated heteropolar radial magnetic bearing. The thin plate model of rotating losses yields a hybrid analytical-boundary element model that is computationally inexpensive to implement. Predictions from this model compare favorably to losses experimentally measured in rotor run-down tests.

Two simple models are proposed for the analytical calculation of eddy currents induced by a time-varying magnetic field in thin conducting plates of various shapes. In the first model, it is assumed that the current paths are determined exclusively by the shape of the plate. This assumption makes evident a shape factor that characterizes the shape of the plate under investigation and is independent of the dimensions of the plate. Thus, several important parameters, such as the equivalent resistance of the plate, the total circulating current, etc., are expressed analytically as a function of this shape factor. The application of the model makes possible the estimation of the total eddy current, circulating in plates of any shape, but the accuracy of this estimate is strongly affected by the shape of the plate. This model is then improved for plates having symmetrical shapes by taking into consideration the principle of minimum energy dissipation. The results, obtained by the improved model for the total circulating current, arc in good agreement with the results obtained numerically as well as with analytical results obtained by the use of variational methods. Index Terms - Eddy currents, modeling, plates, resistance, shape.

A Cauer circuit model of a transformer laminate is presented. It considers saturation, eddy currents and hysteresis. The simulation results agree to experiments with an Epstein frame in the 10-200 Hz range. The paper includes a description of a computationally fast hysteresis model with few adjustable parameters and a physical approach to derive the Cauer circuit. The model can be used under various time-transient conditions and can easily be implemented into a larger system in a circuit simulation package, such as Saber, to study a transformers interaction with switching overvoltages, for example.

This paper compares finite element formulations for modelling eddy current problems with jumps in conductivity. We study schemes which represent the eddy current regions using nodal or edge based magnetic vector potentials. The use of an additional electric scalar potential, V, in the eddy current regions is discussed. The nodal A scheme is shown to produce misleading results at low frequencies whilst the AV and edge A schemes are acceptable.

This paper presents a simple dynamic model of eddy currents in a magnetic actuator. The model is based on the application of Maxwell's equations to a homogeneous ferromagnetic conductive material. The resulting diffusion equation is solved in two dimensions for a cross-sectional cut through a rectangular bar; boundary conditions are imposed by a sinusoidally varying actuator coil current. The utility of the new modeling approach is illustrated by predicting the dynamic performance of a magnetic bearing actuator. The predictions are found to be in good agreement with measured values. The model provides a new and convenient method of modeling the relationships among voltage, current, force, and flux in magnetic circuits containing eddy currents.

In this paper, we propose an original method for the physical analysis of electromagnetic interactions between an inductive sensor and any homogeneous conducting plane target. In the first part of this paper, the basic principles of inductive sensors are recalled. Then, we solve Maxwell's equations in order to obtain an analytical model for the relationship between the target properties and the electrical signals for a U-shaped sensor. Next, we emphasize the fact that our analytical relations can be interpreted as an extension of the electrical image method, even in the most difficult case of conducting and magnetic targets. In the last part, we present some measurements illustrating this new electrical image concept.

A method for the adaptive computation of 3-D eddy-current problems is presented. The 3-D eddy-current problem is formulated in terms of the electric intensity (E formulation). Edge elements that impose tangential continuity of the approximation function are employed for the discretization of the problem with the finite element method. An a-posteriori error estimation technique is proposed with the introduction of two error criteria: a) the tangential discontinuity of the magnetic intensity H, and b) the normal discontinuity of the eddy current density [J.sub.e]. The proposed error estimation technique is employed in a 3-D self-adaptive refinement procedure. Sufficient approximation of the skin effect and calculation of the eddy current distribution is obtained with the proposed method. The implementation of the proposed technique in a problem of 3-D eddy-current computation in a multiply connected conducting body is discussed.

A three-dimensional finite-element analysis is employed to investigate the losses of hydrogenerator stator end regions by using surface-impedance boundary conditions. The three-dimensional complexity of the end-winding geometry is fully taken into account by the model. The purpose of this work is also to evaluate the eddy-current paths allowing for slits in the stator teeth and their effectiveness in reducing stator core-end losses.

A tomographic approach is proposed for the inhomogeneities imaging in conductive media. We propose a linear discrete model for the resolution of the Forward problem and a Bayesian probabilistic approach for the resolution of the inverse problem. Some results (simulated and real data) are presented to show the efficiency of the proposed method in spite of the modelization errors due to the Born approximation for linearizing the problem. Our specific application concerns the Eddy current non-destructive evaluation of pipes.

This paper discusses a method based on the direct differentiation of the finite element matrices for shape optimisation of electromagnetic devices including eddy-currents. The sensitivity equation of the magnitude of the magnetic flux density is deduced in terms of a generic parameter p that can be either geometric or physical. The augmented Lagrange multiplier method is used in the optimization process. The result of an idealized problem is presented.

The finite element model for the time dependent eddy current problem is coupled with mesh (loop) equations for arbitrary external circuits. Stability of the resultant system and the time-stepping scheme is analyzed. The backward difference scheme is shown to be stable, while the Crank-Nicholson method exhibits peculiar instability. Theoretical analysis is supported by numerical results for test and practical problems. Keywords - Eddy currents, finite elements, circuits, time-stepping, stability.

The paper deals with the analysis of eddy-currents and histeresis losses in the core of 3 phase transformer supplying a static converter bridge during failure of valve in this bridge. The applied simulation methods are described as well as the results of analysis in the mostly important case: the break-down of one valve of the bridge produces an internal short circuit disconnected by the fuse. Then the operation continues without one diode, changing greatly the values of all currents and core losses. The computation of all quantities needed the application of coupled circuit and field methods.

We present a Boundary Integral Method (BIM) for the computation of the current distribution in an antenna with negligible skin depth, the unknown quantities being the overall current intensity and the scalar magnetic potential on the boundary. After computing these quantities, the current density distribution can be deduced and the resistance as well as the inductance of the antenna evaluated. The results are compared with experimental values.

This paper provides new regularized formulas for the boundary element analysis of the modified Helmholtz equation describing eddy current fields. The present formulas are shown to regularize near singularity in the kernels and effectively to improve the accuracy in the computation of fields near boundaries.

This paper describes the hybrid finite element (FE) and boundary element (BE) method for the eddy current problem including moving conductors. Electromagnetic phenomena due to the three-dimensional movement of conductors can be computed easily and naturally by the presented formulation. Its application to a simple model which has only one conductor is also given.

To model the idealized exciting current sources, such as the ideal line current and ideal current sheet, in the FEM, a general approach is proposed in this paper. The idealized exciting current sources are represented by means of the Dirac Delta function. Some numerical examples will show the good accuracy of the developed method.

Several finite element formulations in terms of various potentials are reviewed for the calculation of transient eddy currents in nonlinear media. A fast and simple iterative technique is used to solve the nonlinear algebraic equations arising. The different formulations are compared by solving some large three dimensional problems by various methods.

Methods of representing segmented conductors in 2 dimensions are compared with dimensional calculations, showing good agreement for a simple problem. The calculation of power dissipation is improved for the 2 dimensional approximatio by adjustment of the material resistivity to compensate for three dimensional effects. The limitations of the 2 dimensional analysis are demonstrated using, as an example, an induction heating crucible.

A two-dimensional finite element model is developed to model the electromagneti behavior of permanent magnet type eddy current couplers under constant speed operation. The model accounts for the nonlinearity of the steel flux paths and is verified using test measurements from a prototype eddy current coupler. The proposed solution differs from conventional magnetostatic finite element models in that an unknown current-density distribution must be determined through an iterative process. The model is used to study the influence of certain design parameters on the torque-speed characteristics of such devices.

A method for finite element modelling of transient 3D eddy currents in magnetically nonlinear media is described. The magnetic vector potential A, wit or without the electric scalar potential V, is used inside eddy current regions coupled to magnetic scalar potentials elsewhere. Time transient torque, current and flux values are compared to measurements taken from a rotational test rig.

A complete, three phases transformer model for the calculation of electromagnetic transients is presented. The model consists of a set of state equations solved with the trapezoidal rule of integration in order to obtain an equivalent Norton circuit at the transformer terminals. Thus the transformer model can be easily interfaced with an electromagnetic transients program. Its main features are: (a) the basic elements for the winding model are turns, (b) the complete model includes the losses due to eddy currents in the windings and in the iron core, (c) the solution of the state equations is obtained in decoupled iterations. For validation, the frequency response of the model is compared with tests on several transformers. Applications to the calculations of transients are given for illustration. Keywords: Transformer modeling, Electromagnetic transients, Eddy currents, Laminations.

Frequency dependent representations of eddy currents in laminated cores of power transformers are developed. One representation is based on a continued fraction expansion for the frequency dependent magnetizing impedance while the other is based on a discretization of flux distribution within the lamination. Both models are low order and reproduce the frequency characteristics of the transformer magnetizing branch up to 200 kHz with less than 5% error. The importance of modeling the frequency dependence of eddy currents for the calculation of the transient recovery voltage across a low voltage transformer breaker interrupting a nearby fault is demonstrated.

Modelling non-conductors with the magnetic scalar potential is a very popular approach to 3D eddy current problems. However, cuts are required if the magnetic scalar region is multiply connected. The manual definition of some cuts can be difficult even for experienced users of FE codes. This paper describes a completely automatic scheme which has been implemented in the 3D eddy current package MEGA, cuts are generated allowing multiply connected eddy current problems to be solved.

A new iterative least-squares algorithm explaining the mechanics of eddy current inversion was presented. The algorithm minimized the mean-square difference between the predicted and actual impedance values by using the conductivity distribution. The gradient of the impedance showed how the impedance gradient can be updated such that a reduction in the mean-square difference is ensured.

Eddy current effects have been included in the model of the transformer for the study of electromagnetic transients. Existing analytical formulas for the calculation of losses in the windings have been evaluated. Various equivalent circuits have been fitted to represent in the time domain the damping produced by eddy currents in the windings. A frequency dependent model has been derived for the iron core, based on the physical distribution of losses and magnetization effects. The parameters of this model are obtained by optimal discretization of the laminations. Simulations of transients have been presented to show the effects of eddy currents in the damping of transients.

Using the example of a monopole that is spontaneously generated above a thin conducting sheet, the simplicity and power of Maxwell's 1872 theory of eddy currents in thin conducting sheets is illustrated. This theory employs a receding image construction, with a characteristic recession velocity [v.sub.o] = 2/([mu.sub.o sigma d]), where the sheet has conductivity [sigma], and thickness d. A modern derivation of the theory, employing the magnetic scalar potential, is also presented, with explicit use of the uniqueness theorem. Also discussed are limitations on the theory of which Maxwell, living in a time before the discovery of the electron, could not have been aware. Previous derivations either have not appealed explicitly to the uniqueness theorem, or have employed the now unfamiliar current function, and are therefore either incomplete or inaccessible to the modern reader. After the derivation, two important examples considered by Maxwell are presented - a monopole moving above a thin conducting sheet, and a monopole above a rotating thin conducting sheet (Arago's disk) - and it is argued that the lift force thus obtained makes Maxwell the grandfather, if not the father, of eddy current MAGLEV transportation systems. An energy conservation argument is given to derive Davis's result that, for a magnet of arbitrary size and shape moving parallel to a thin conducting sheet at a characteristic height h, with velocity v, the ratio of drag force to lift force is equal to [v.sub.o/v, provided that d << [delta.sub.c], where [delta.sub.c] = [Mathematical Expression Omitted]. If d >> [delta.sub.c], the eddy currents are confined to a thickness [delta.sub.c], leading to an increase in the dissipation and the drag by a factor of d/[delta.sub.c], so that the ratio of drag to lift force becomes proportional to [Mathematical Expression Omitted]. The case of a monopole fixed in position, but oscillating in strength (such as can be simulated by one end of a long, narrow, ac solenoid), is also treated. This is employed to obtain the results for an oscillating magnetic dipole whose moment is normal to the sheet. A general discussion of electromagnetic induction and electrical conductors, both thick and thin, is given, emphasizing the difference between the high-frequency limit, where flux expulsion occurs and the self-inductance dominates, and the low-frequency limit, where the flux penetrates and the electrical resistance dominates. A discussion of Lenz's law, as a statement about motion, is given. It is argued that the most general form of such a statement of Lenz's law is that induced currents tend to accelerate a conductor in the direction that most effectively decreases the rate of Joule heating. A calculation, in the low-frequency limit, of the drag force on a magnetic dipole falling down a long conducting tube, is also given. This last case can be given a striking demonstration with the newly available neodymium-iron-boron magnets.

The properties of edge element methods are further investigated through two types of mathematical models. H - [Phi] formulation is proposed for 3D moving conductor eddy current problems. Second-order nodal-based finite elements are used in non-eddy current region for easily dealing with interface conditions. Asymmetrical equation is calculated by improved BICG solver. Index terms - Moving conductor problems, H - [Phi] model, Second-order FEM.

A technique for modelling thin skin depth eddy currents in 3D devices using the magnetic vector potential A is presented. An analytic solution for the magnetic field in the thin skin is used to develop a surface impedance which appears as a surface integral at faces of elements which interface the thin skin eddy current region. The method has been implemented using classical nodal variable finite elements. Results are compared with an analytic solution and measurements from a nonlinear test rig.

The paper presents a method for the estimation of the iron losses under PWM (Pulse Width Modulation) flux supply conditions. The computational results are validated versus measurements on Epstein's yoke. The proposed model is based on Finite Element method formulated in the harmonic domain and takes into account ferromagnetic hysteresis and eddy currents. A classical Preisach model is used for hysteresis modelling neglecting the effects of excess losses in field analysis. Different PWM supply conditions are measured and simulated with satisfactory agreement.

Magnetization response during high frequency and high amplitude write current excitations at the air-bearing surfaces (ABS) of thin film inductive and MR write heads were observed by using a Scanning Kerr Effect Microscope. Under certain excitation conditions, bubble-like dynamic domain walls that separate the regions of 180 [degrees] out-of-phase magnetization reversals were observed predominately at the ABS of trailing poles (P2). This magnetization reversal mechanism is believed to be introduced by the combined effects of eddy current damping and local magnetization saturation, and is consistent with the model proposed by Wood and Williams.

The impedance, efficiency, and current and field risetime for laminated heads have been calculated using our model developed earlier [1]. The calculated head impedance results are compared to the experimental data. The experiments have also shown that laminated heads performed better than the non-laminated heads in isolated pulse amplitude and overwrite at high frequencies under identical test conditions; however, the laminated heads did not show improved performance in PW50 and nonlinear transition shift (NLTS) under our test conditions.