Finite element analysis of 3-D eddy currents

The authors review formulations of three-dimensional (3-D) eddy current problems in terms of various magnetic and electric potentials. The differential equations and boundary conditions are formulated to include the necessary gauging conditions and thus to ensure the uniqueness of the potentials. Different sets of potentials can be used in distinct subregions, thus facilitating an economic treatment of various types of problems. A novel technique for interfacing conducting regions with an electric vector and a magnetic scalar potential to eddy-current-free regions with a magnetic vector potential is described. Finite-element solutions to several large eddy-current problems are presented     

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.     

Comparison of different finite element formulations for 3Dmagnetostatic problems

The results obtained with several three-dimensional software packages for magnetostatic field calculation using the finite-element method (FEM) are compared with regard to their accuracy and their computational time requirements. The packages are based on the vector potential (VPOT), the reduced scalar potential (RSP), and the total and reduced scalar potential (TSP+RSP), respectively. Results for an iron cylinder immersed in the field of a cylindrical coil are given. It is found that the finite-element formulation using a total and reduced scalar potential and the direct iteration method are useful for dealing with nonlinear magnetostatic field problems     

Comparison of full and reduced potential formulations forlow-frequency applications

The authors examine five alternative finite element formulations of magnetostatic problems: (1) full vector potential, (2) reduced vector potential, (3) reduced scalar potential with source integral over all volume, (4) reduced scalar potential with source integral over permeable volumes only, and (5) reduced scalar potential with surface source integral. They show that (3) is more prone to numerical errors than the other methods, and that (4) and (5) are competitive in accuracy with (1) and (2) in air regions but are easier to implement in three dimensions. However, all three scalar formulations suffer from cancellation errors in iron regions     

Transient 3D eddy currents using modified magnetic vectorpotentials and magnetic scalar potentials

A package named VECTOR for solving 3-D eddy current problems is presented. The package is a developmental version of the commercial package CARMEN and in the same way solves the vector diffusion equation, involving a modified vector potential, within conductors and the scalar Poisson equation, using a magnetic scalar potential, in non-eddy-current regions. It has been shown that this set of equations yields a unique solution for both the magnetic vector potential (and hence the currents) and the fields (which are derived from the magnetic potentials by differentiation). This package has recently been extended to solve transient problems, using simple time-stepping techniques. Some results using the package for problems with analytic solutions are given     

Numerical analysis of 3D magnetostatic fields

Formulations of three-dimensional magnetostatic fields are reviewed for their finite-element analysis. Partial differential equations and boundary conditions are set up for various kinds of potentials. Besides the method using two scalar potentials, several vector potential formulations are also discussed. Galerkin techniques combined with the finite element method are applied for the numerical solution of the boundary value problems. The effect of gauging the vector potential upon the numerical performance is investigated. Solutions by different formulations to a simple test problem and a benchmark problem involving relatively thin saturated iron plates are presented. The latter is compared to measured results.<>     

Calculation of equipotentials and flux lines in axially symmetrical permanent magnet assemblies by computer

Two equivalent theoretical models of permanent magnets are used to develop algorithms for numerically computing the magnetic scalar potential and the magnetic vector potential in the vicinity of an axially symmetric array of pole pieces and permanent magnets. A computer program based on these algorithms calculates equipotential surfaces and flux lines in and around the magnets and pole pieces. In deriving the algorithm for numerically calculating the vector potential a relationship between the magnetic scalar potential and the vector potential was found which enables the program to calculate the vector potential from the scalar potential distribution and thus generate equipotentials and flux lines with only one iterative calculation. An algorithm which calculates the scalar potential of a "floating" pole piece, that is, one on which the scalar potential has not been specified, is developed. The vector potential around the pole piece is determined from the scalar potential calculation, and this information is used to calculate the vector potential and the flux lines within the pole piece. The computer program calculates the coordinates of all points at which the equipotential lines and flux lines cross the Liebmann net. This information is fed to a cathode ray tube plotter which generates a field plot. To deal with systems in which macroscopic currents are present as well as permanent magnets, the iterative Liebmann net calculation of the vector potential is developed, and a method of applying Neumann boundary conditions to the vector potential at high-permeability surfaces is described.     

Vector near-field calculation of scanning near-field optical microscopy probes using Borgnis potentials as auxiliary functions

A new boundary integral equation method for solving the near field in three-dimensional vector form in scanning near-field optical microscopy (SNOM) using Borgnis potentials as auxiliary functions is presented. A boundary integral equation of the electromagnetic fields, expressed by Borgnis potentials, is derived based on Green's theorem. The harmonic expansion in rotationally symmetric SNOM probe--sample systems is studied, and the three-dimensional electromagnetic problem is partly simplified into a two-dimensional one. The boundary conditions of Borgnis potentials both on dielectric boundaries and on perfectly conducting boundaries are derived. Relevant algorithms were studied, and a computer program was written. As an example, a SNOM probe-sample system composed of a round metal-covered probe and a sample with a flat surface has been numerically studied, and the computational results are given. This new method can be used efficiently for other electromagnetic field problems with round subwavelength structures.     

Finite element computation of three-dimensional electrostatic and magnetostatic field problems

Three-dimensional field solutions have been of considerable interest in electric machine field problems for a number of years. However, only with the advent of large scale computers, numerical analysis methods, developments in grid generation and graphics display techniques computation of three-dimensional magnetic fields has become somewhat tractable. In this paper, the progress made by the authors' company in three-dimensional field computation for scalar and vector field problems is presented. Several applications and comparison of solutions with test results, where available, and other methods are included. New computational techniques employing mixed scalar and vector formulations are introduced and comparison of results in one case with those obtained by direct methods is presented.     

Reconstruction and assessment of the least-squares and slope discrepancy components of the phase

The concept of slope discrepancy developed in the mid-1980's to assess measurement noise in a wave-front sensor system is shown to have additional contributions that are due to fitting error and branch points. This understanding is facilitated by the development of a new formulation that employs Fourier techniques to decompose the measured gradient field (i.e., wave-front sensor measurements) into two components, one that is expressed as the gradient of a scalar potential and the other that is expressed as the curl of a vector potential. A key feature of the theory presented here is the fact that both components of the phase (one corresponding to each component of the gradient field) are easily reconstructable from the measured gradients. In addition, the scalar and vector potentials are both easily expressible in terms of the measured gradient field. The work concludes with a wave optics simulation example that illustrates the ease with which both components of the phase can be obtained. The results obtained illustrate that branch point effects are not significant until the Rytov number is greater than 0.2. In addition, the branch point contribution to the phase not only is reconstructed from the gradient data but is used to illustrate the significant performance improvement that results when this contribution is included in the correction applied by an adaptive optics system.     

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     

Geometric T–Ω approach to solve eddy currents coupled to electric circuits

This paper describes a systematic geometric approach to solve magneto‐quasi‐static coupled field–circuit problems. The field problem analysis is based on formulating the boundary value problem with an electric vector potential and a scalar magnetic potential. The field–circuit coupling and the definition of potentials are formally examined within the framework of homology theory. Copyright © 2007 John Wiley & Sons, Ltd.     

Magnetostatic field of a conductor of semicircular cross section in a highly permeable half-space: exact analytic solution for infinite permeability with perturbative correction

Poisson's equation for the magnetic vector potential is solved using complex Fourier (Laplace) transforms in bipolar coordinates, the natural system for the subject two-dimensional geometry. The source is a dc current uniformly distributed over the semicircular cross section of a long conductor that is buried in, and flush with, the otherwise planar boundary of an infinitely permeable material. Exact closed-form potentials are obtained in the conformal mapping of the Neumann boundary value problem that characterizes the case of an infinitely permeable magnetic medium. One term of a perturbative correction that accounts for finite permeability is constructed for both the uniform source distribution and for the associated Green's function.     

Scalar potentials in multiply connected regions

An algorithm, for use on a digital computer, is explained for cutting multiply connected regions in three-dimensional space. The algorithm is suitable for complex electromagnetic field problems where a magnetic scalar potential, affected by current loops, is to be calculated in a multiply connected region. Problems in fluid dynamics with vorticity affecting a scalar potential, could also be attacked with this algorithm. Since finite element methods are ubiquitous in field calculations, the cuts of the space constructed by the algorithm should ideally correspond to the given discretization; this can be arranged by an application of the Lebesgue Covering Theorem, for which a second algorithm is provided.     

An accurate scalar potential finite element method for linear,two-dimensional magnetostatics problems

We have assessed the accuracy of a commercially available computer software package for finite element method calculations of magnetostatic fields. The computer program, MSC/NASTRAN,

1 Available from the MacNeal-Schwendler Corporation, Los Angeles, CA 90041, U.S.A.

is well known for its wide applicability in structural analysis and heat transfer problems. We exploit the fact that the differential equations of magnetostatics are identical to those for heat transfer if the magnetic field problem is formulated with the reduced scalar potential.¹ Consequently, the powerful, optimized numerical routines of NASTRAN can immediately be applied to two- and three-dimensional linear magneto-statics problems. Application of the NASTRAN reduced scalar potential approach to a ‘worst case’ two-dimensional problem for which an analytic solution is available has yielded much better accuracy than was recently reported² for a reduced scalar potential calculation using a different finite element program. Furthermore, our method exhibits completely satisfactory performance with regard to computational expense and accuracy for a linear electromagnet with an air gap. Our analysis opens the way for large three-dimensional magnetostatics calculations at far greater economy than is possible with the more commonly used vector potential and boundary integral methods.     

Equations of local gradient electromagnetothermomechanics of dielectrics with regard for polarization inertia

The previously obtained relations of the local gradient theory of electromagnetothermomechanics of polarized nonferromagnetic bodies are generalized with regard for polarization inertia. We obtain the corresponding main system of equations of the model, written for the potentials of displacement vector and electromagnetic field vectors as well as the potential μ′_π, which takes into account the influence of local mass displacement on the internal energy of the body. The Lorentz gauge condition is generalized. We establish that taking the polarization inertia into account leads to the appearance of dispersion of the electromagnetic wave velocity in the body and additional dynamic components in differential equations connecting the potential μ′_π and scalar potentials of the displacement vector and electromagnetic field vectors.     

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     

3-D magnetic field computations using finite-element approach with localized functional

An extended application of a finite-element approach with localized functional to a three-dimensional magnetic field problem is described in this paper. The field region is modeled by a set of partial differential equations in terms of scalar potentials. The variational approach is used to obtain the system matrix. The localized functional is derived, which consists of the domain integral of the finite element region only and the boundary integral of the interfacial boundary between the finite and infinite element regions. The proposed approach is applied to a sample problem. The result has been compared with the standard finite element method and an analytic solution. The numerical solutions obtained by the proposed approach are in good agreement with the analytic solutions and show better accuracy than those of the standard finite element method.     

Analytical and numerical solution of the eddy-current problem inspherical coordinates based on the second-order vector potentialformulation

The three-dimensional (3-D) eddy-current problem, described in spherical coordinates, is studied both analytically and numerically. Since the vector field equation is not separable in the spherical coordinate system, the second-order vector potential (SOVP) formulation is used to treat the problem by reducing it to the solution of the scalar field equation. While the analytical solution is expressed in terms of known orthogonal expansions, the numerical solution utilizes the finite difference method. Examples of engineering applications are provided, concerning computation of eddy-current distribution in a conducting sphere by a filamentary excitation of arbitrary shape