3-D eddy current calculations using the magnetic vector potential

Two methods for using the magnetic vector potential for 3-D eddy current calculation are treated. One method uses the magnetic vector potential that is continuous over the entire region and generally accompanies the electric scalar potential. It has the advantage that no cutting is necessary for the multiply-connected-region problem. The other method uses the magnetic vector potential that is discontinuous across the interface surface between different media. This magnetic vector potential can be arranged so that the electric scalar potential does not appear in the equations when the conductivity is constant. It has the disadvantage that cutting is necessary for the multiply-connected-region problem. New boundary value problem formulations are given for both methods, precisely defining the interface and boundary conditions     

Performance of different vector potential formulations in solvingmultiply connected 3-D eddy current problems

The authors describe their numerical experiences in applying FEM (finite-element method) solution techniques to a 3-D (three-dimensional) eddy-current problem with a coil-driven multiply connected conductor, the benchmark problem No.7 of the International TEAM Workshops. Several formulations have been tried using a magnetic vector and electric scalar potential or an electric vector and a magnetic scalar in the conductor and a magnetic vector or scalar potential outside. The problem has been solved at two frequencies. The authors briefly describe the formulations used and compare the performance. Magnetic field and current density plots are also compared. The advantages and disadvantages of the various versions are pointed out. The use of a magnetic scalar potential H rather than a magnetic vector potential A outside the conductor and the hole substantially reduces the number of degrees of freedom and thus the computational effort. The versions using it in the conductor yield relatively ill-conditioned systems. Also, at the higher frequency, the conditioning deteriorates considerably     

Direct error in constitutive equation formulation for inverse heat conduction problem

We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.     

Three-dimensional eddy current analysis by the boundary elementmethod using vector potential

A boundary-element method using a magnetic vector potential for eddy-current analysis is described. For three-dimensional (3-D) problems, the tangential and normal components of the vector potential, tangential components of the magnetic flux density, and an electric scalar potential on conductor surfaces are chosen as unknown variables. When the approximation is introduced so that the conductivity of the conductor is very large in comparison with the conductivity of air, the number of unknowns can be reduced; also, for axisymmetric models the scalar potential can be eliminated from the unknown variables. The formulation of the boundary-element method using the vector potential, and computation results by the proposed method, are presented     

EDDY CURRENT AND TEMPERATURE SIMULATION IN THIN MOVING METAL STRIPS

The paper presents a special finite element formulation for the computer simulation of an inductive heating device for thin moving metal strips. The calculation includes both the electromagnetic field and the temperature distribution resulting from the losses of the induced currents. The electromagnetic field is described by an electric vector potential and a magnetic scalar potential, the induced eddy currents are represented by a special boundary condition for the magnetic scalar potential along the surface of the current-carrying strip. This avoids the necessity to discretize the thin secondary region. The thermal model includes the movement of the strip as well as convection and radiation as its surfaces. The field equations are discretized using the Galerkin variant of the Method of Weighted Residuals. The mutually coupled electromagnetic and temperature fields are then calculated using an iterative, staggered solution scheme. Comparisons between calculated and measured temperature profiles show the validity of the presented approach.     

Finite element scheme for transient 3D eddy currents

A transient 3-D finite-element model is presented. The method is based on the solution of the magnetic scalar potential in nonconducting regions and the magnetic vector potential and an electric scalar potential in eddy-current regions. Multiply connected regions of magnetic scalar can be avoided by extending the region modeled by the magnetic vector potential to fill any holes in the conducting regions. The model was used to simulate the FELIX brick experiment     

Effect of displacement current in magneto–electro-elastic plates subjected to dynamic loading

Dynamic response of moderately thick magneto?Celectro-elastic plate using magnetic vector potential in finite element formulation is presented in this paper. Dynamic loading generate time varying electric and magnetic fields in magneto?Celectro-elastic continuum. Displacement current is associated with the generation of magnetic field due to time varying electric field. The non-conservative electric field is represented using electric scalar potential and magnetic vector potentials. Studies are carried out for CCCC, CCFC, CFFC and FCFC boundary conditions of the plate excited with time-harmonic mechanical excitation, the frequency range being chosen based on the critical frequency of the plate analyzed. The magnetic flux density in longitudinal x-direction is not affected by the electric displacement current for all the boundary conditions. The longitudinal y-direction and transverse direction components of magnetic flux density are showing variations for FCFC boundary condition when displacement current is accounted. The effect of displacement current is significant when two opposite edges of the plate are clamped.     

A new finite‐element formulation for electromechanical boundary value problems

In this paper, a new finite‐element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that uses scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite‐element stiffness matrix from the vector potential formulation is also positive definite. If the material is non‐linear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non‐linear ferroelectric material, and results are compared to an analytical solution. Copyright © 2002 John Wiley & Sons, Ltd.     

Eddy-Currents Computation With T-$Omega$ Discrete Geometric Formulation for an NDE Problem

We propose a discrete geometric formulation based on a magnetic scalar potential and on the circulation of an electric vector potential to solve eddy-current problems for nondestructive evaluation of steel bars with longitudinal defects.     

Vector diffraction theory for electromagnetic waves

The scalar Huygens-Fresnel principle is reformulated to take into account the vector nature of light and its associated directed electric and magnetic fields. Based on Maxwell's equations, a vector Huygens secondary source is developed in terms of the fundamental radiating units of electromagnetism: the electric and magnetic dipoles. The formulation is in terms of the vector potential from which the fields are derived uniquely. Vector wave propagation and diffraction formulated in this way are entirely consistent with Huygens's principle. The theory is applicable to apertures larger than a wavelength situated in dark, perfectly absorbing screens and for points of observation in the right half-space at distances greater than a wavelength beyond the aperture. Alternatively, a formulation in terms of the fields is also developed; it is referred to as a vector Huygens-Fresnel theory. The proposed method permits the determination of the diffracted electromagnetic fields along with the detected irradiance.     

Nondestructive Measurement of Hall Coefficient for Materials Characterization

The Hall effect is widely exploited in NDE for measuring unknown weak magnetic fields using a small piece of conducting material of known high Hall coefficient. The Hall effect could be also exploited in NDE for measuring the unknown weak Hall coefficient of conducting materials using a strong applied magnetic field, but such measurements are fraught with difficulties because of the need to cut the specimen into a small piece similar to a Hall sensor, which of course is inherently destructive. This paper tries to answer the question how the need for destructive cutting in order to produce a measurable Hall voltage could be avoided. The underlying problem is that the Hall effect produces a Hall current that is normal to the conduction current but does not directly perturb the electric potential distribution unless the Hall current is intercepted by the boundaries of the specimen. This study investigated the feasibility of using alternating current potential drop techniques for nondestructive Hall coefficient measurement in plates. Specifically, the directional four-point square-electrode configuration is investigated with superimposed external magnetic field. Two methods are suggested to make Hall coefficient measurements in large plates without destructive machining. At low frequencies constraining the bias magnetic field can replace destructively constraining the dimensions of the specimen. At sufficiently high inspection frequencies the magnetic field of the Hall current induces a strong enough Hall electric field that produces measurable potential differences between points lying on the path followed by the Hall current even when it is not intercepted by either the edge of the specimen or the edge of the magnetic field. Both techniques are investigated first analytically to illuminate the underlying physics and then by numerical simulations to make useful quantitative predictions.     

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.     

An improved finite element analysis of eddy current problemsconnected to voltage source

In conventional methods, a 3-D scalar potential problem must be solved in order to obtain a current source producing magnetic fields. In the proposed method, electric power is supplied to a region under consideration by the Poynting vector of a transverse electromagnetic (TEM) wave through a semi-infinite twin-lead-type feeder. Therefore, it is sufficient to solve the 2-D Laplace equation of the TEM wave before the calculation of the 3-D eddy current problem. Numerical analyses of conducting electrodes are shown. Reasonable results with phase lag and skin effect are obtained     

An incomplete gauge formulation for 3D nodal finite-elementmagnetostatics

A method for imposing the gauge condition on the 3-D magnetic vector potential magnetostatic field computation using nodal finite elements is presented. In this method, the gauge A.w=0 is applied in the part of the problem that is not situated in the neighborhood of the materials interfaces that are tangential to w . This results in a formulation which maintains the discontinuous properties of the magnetic induction tangential components, reduces the number of unknowns, and improves the system matrix conditioning. The proposed formulation is compared with the Coulomb-gauge and ungauged formulations, showing that it results in better precision and worse conditioning than the Coulomb-gauge and has the same precision with a better conditioning than the ungauged formulation     

A three dimensional eddy current formulation using two potentials: The magnetic vector potential and total magnetic scalar potential

A formulation of the three dimensional eddy current problem is presented. The magnetic vector potential is used in regions with source currents and conducting material and the total magnetic scalar potential is employed elsewhere. The continuity of the normal component of flux density and tangential component of field intensity are used to couple the two potentials on the interface between regions. The formulation leads to a symmetric system amenable to traditional solution techniques. The formulation is also valid for static problems with modification that are easily implemented.     

On the use of the total scalar potential on the numerical solution of fields problems in electromagnetics

The paper summarizes the formulation of a set of computer algorithms for the solution of the three-dimensional non-linear Poisson field problem. Results are presented that were obtained by applying algorithms to the analysis of two-dimensional magnetostatic fields. Scalar and vector potentials were used, and it is shown that the convenient single valued scalar potential associated with the induced sources gives severe accuracy problems in permeable regions. The results become as good as those obtained using vector potential if the scalar potential associated with the total field is used for permeable regions. The combination of two scalar potentials has a significant advantage for three-dimensional problems.     

Vector potential formulations and finite element trial functions

In field systems containing a divergenceless vector, the problem may be posed in terms of a vector potential for convenience. For the solution of the magnetic vector potential in three dimensional problems with current sources, there exist three standard variational formulations in the literature. While all these are known to give verifiable physical solutions, there is some question as to which is to be preferred. Indeed, one of them is invalid for infinite dimensional fields in that, without the finite element trial functions, it will not give unique solutions since it does not explicitly impose the divergence of the vector potential. In this paper, we look at the formulations in the light of the restrictions imposed by the finite element trial functions for tetrahedral elements and arrive at the curious result that that formulation which is totally invalid when the vector potential is unrestricted by trial functions, is in fact valid in finite element analysis and, at the same time, is the best. We further show that this formulation yields naturally non-divergent vector potential solutions, strictly as a result of the trial functions.     

Computation of 3-D current driven skin effect problems using acurrent vector potential

A finite element formulation of current-driven eddy current problems in terms of a current vector potential and a magnetic scalar potential is developed. Since the traditional T-Ω method enforces zero net current in conductors, an impressed current vector potential T₀ is introduced in both conducting and nonconducting regions, describing an arbitrary current distribution with the prescribed net current in each conductor. The function T ₀ is represented by edge elements, while nodal elements are used to approximate the current vector potential and the magnetic scalar potential. The tangential component of T is set to zero on the conductor-nonconductor interfaces. The method is validated by computing the solution to an axisymmetric problem. Problems involving a coil with several turns wound around an iron core are solved