Steady-state Eddy Current Problem Research Articles

Topological Sensitivity Analysis for Steady State Eddy Current Problems With an Application to Nondestructive Testing

This paper proposes a novel solution to the inverse problem of eddy current nondestructive testing (NDT) based on topological shape optimization. The topological gradient (TG) is derived for a steady state eddy current problem using a topological asymptotic expansion for the Maxwell equation of a time harmonic problem. TG provides information on where the objective function is most sensitive to topology changes and can be used as a fast identification of the locations of the defects in the test specimen. The proposed method has been applied to typical eddy current testing (ECT) problems such as buried crack reconstruction and the detection of multiple cracks. The reconstructed shape of the crack shows good agreement with the experimental data from TEAM workshop problem 15. A comparison of different ECT inverse analyses is also discussed.

Eddy current and temperature field computation in transverse flux induction heating equipment for galvanizing line

This paper describes the 3-D eddy current FEM computation of transverse flux inductors used for heating strip in galvanized steel production. The adopted mathematical model consists of a differential equation system for the steady-state eddy current problem and a Fourier thermal conduction equation for moved media. The finite element method is applied in conjunction with the Galerkin method. The simplifications and boundary conditions required for an efficient solution are discussed.

3D multifields FEM computation of transverse flux induction heating for moving-strips

The numerical and experimental studies on induction heating of continuously moving strips in a 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 Fourier 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.

Design sensitivity analysis for steady state eddy current problems by continuum approach

In two dimensional steady state eddy current problems, the exact sensitivity formula to the interface shape variation is derived using the material derivative concept of continuum mechanics and an adjoint variable method. The obtained sensitivity formula is expressed in a line integral along the interface and it is calculated using the existing finite element analysis codes. For the rotor slot shape design of the induction motor driven by balanced 3-phase voltage source, the sensitivity for voltage source problems is calculated with the sensitivity for current source problems and the circuit equation. As an optimization algorithm, the Gradient Projection method is used to control easily the torques for various speeds of rotor.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

3D eddy current computation in the transverse flux induction heating equipment

This paper describes a 3D computation of transverse flux inductors used for heating strip and thin slabs. The adopted mathematical model consists of a differential equation system for the steady-state eddy current problem in a configuration comprising a magnetic vector potential and a scalar potential and a Fourier's thermal conduction equation for moved media. The finite element method is applied in conjunction with the Galerkin method. The simplifications and boundary conditions required for an efficient solution are discussed. The discretization of the numerical model is set up with the aid of macroelements and includes upwards of 100000 nodes for simple builds. The suitability of the numerical method developed for optimum design of transverse flux inductors is demonstrated by some results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Complementary formulations in steady-state eddy-current theory

The symmetry and the duality inherent in the Maxwell equations result in the existence of complementary formulations. This can be used to derive bilateral estimates for quantities like the resistance or the reluctance, in static situations. Whether similar bounds can be obtained for both the real and imaginary parts of the impedance in steady-state eddy-current problems is a controversial and still open question. The author reviews the basics of approximation theory, as applied to eddy-current equations and looks for proof methods that could be used to assert the existence of bilateral bounds, but finds them unable to do this.

An eddy current constraint formulation for 3D electromagnetic field calculations

The use of Coulomb's gauge is sufficient for a unique solution in 3-D magnetostatic field calculations in terms of the magnetic vector potential. For the sinusoidal steady-state eddy current problem Coulomb's gauge is not sufficient. The combination of Coulomb's gauge and an eddy current constraint provides a simple way of guaranteeing uniqueness. A formulation for 3-D electromagnetic fields utilizing this constraint is proposed. Results comparing this formulation with other finite-element formulations and with experimental data are given. >

Prediction of transient eddy current fields using surface impedance methods

Surface impedance methods are used primarily in steady-state eddy current problems where the thickness of the conductor is much larger than the skin depth. This application is extended to transient problems involving shell structures. The shell structure examined is a three-dimensional spherical shell. At time t=0, a uniform z-directed field of 1 T is turned on. Attention is primarily focused on extracting eigenvalues using the surface impedance method and more specifically on how to build the temporal responses from these eigenvalues. Because the source chosen is a unit step, it is necessary to incorporate a host of eigenvalues to get a good initial transient prediction. A less costly alternative is to relax the requirement of how much of the field is excluded from the conductor and to use fewer eigenvalues.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An approach to the solution of coupled transient non-linear problems solving the set of equations as a single system (electromagnetic fields)

A procedure designed to solve as a single system the set of differential equations describing a coupled problem, including transient nonlinear cases, is presented. The procedure is based on the discretization of the equations to produce a single algebraic system. The implementation of the method is explained, and applications to a nonlinear thermomagnetic problem and a sinusoidal steady-state eddy current problem are described. The potential advantages and drawbacks, as compared to those of other procedures, are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A boundary matching method in solving three-dimensional steady state eddy-current problems

A boundary method is developed for the computation of three-dimensional eddy-current distributions. The electromagnetic field is described by the magnetic field intensity in conductive regions and by the magnetic scalar potential in nonconductive regions. The fundamental solutions of the relevant differential equations are used as trial functions to approximate the solution of the eddy-current problem. A method is obtained for finding the near-best positions of the singularities of the fundamental solutions using a nonlinear least-squares minimization technique. This method requires relatively few trial functions to obtain satisfactory accuracy. >

One step finite element formulation of skin effect problems in multiconductor systems with rotational symmetry

This paper presents a new formulation for the classical steady state eddy current problem in multiconductor systems with rotational symmetry. The formulation given in this paper consists of obtaining the magnetic vector potential and the current distribution from the projective solution of the diffusion equation and Ampere's law given the total measurable current flowing in the conductors. This procedure provides an efficient, one step solution technique for the multipath eddy current problem in axisymmetric geometries.

A contribution to eddy current calculations in plane and axisymmetric multiconductor systems

For steady-state eddy current problems in Plane and axisymmetric multiconductor systems calculations by means of three different methods are described. The first method is a finite element approach using the superposition principle resulting in the admittance matrix of the total multiconductor system. The second method described is the integrodifferential approach providing a direct implementation of the total current condition. In case of multiconductor systems consisting of sub-systems with series connected conductors and given voltages the integrodifferential method has to be used combined with the superposition principle. The third method is based on the numerical solution of the underlying Fredholm integral equation of second kind. Some examples are given to show the particular advantages and disadvantages of the various methods with respect to computation time and accuracy of solution. Furthermore, the application of different higher-order isoparametric elements for each of the methods is investigated.

Solution of linear steady-state eddy-current problems by complex successive overrelaxation

The paper considers two topics which arise in the application of the finite-difference successive-overrelaxation method to problems involving eddy currents induced in nonmagnetic conductors by magnetic fields varying sinusoidally with time. First, a simple 1-dimensional problem is used to determine a criterion governing the mesh length between adjacent nodes in the grid covering a conducting region. It is shown that this length should not exceed 0.2 times the skin depth of the conductor. Secondly, the important question of the convergence of the solution as a function of the accelerating or relaxation factor is considered. A method of determining the optimum accelerating factor during the course of the iterative procedure is presented.