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Showing papers in "IEEE Transactions on Magnetics in 1982"


Journal ArticleDOI
Sadamichi Maekawa1, U. Gafvert1
TL;DR: In this article, a spin-polarized tunneling of electrons through Ni-NiO-Ni, Co, and Fe junctions is discussed, and the hysteresis of the tunneling resistance in a magnetic field originates in the magnetization process and demonstrates a new interplay of electronic and magnetic properties in ferromagnetic metals.
Abstract: Spin-polarized tunneling of electrons through Ni-NiO-Ni, Co, and Fe junctions is discussed. The hysteresis of the tunneling resistance in a magnetic field originates in the magnetization process and demonstrates a new interplay of electronic and magnetic properties in ferromagnetic metals.

303 citations


Journal ArticleDOI
O. Kubo1, Ido Tadashi1, H. Yokoyama1
TL;DR: In this paper, fine Ba ferrite particles, suitable for coated perpendicular magnetic recording media, have been prepared, about 0.08 μm in average diameter, are thin hexagonal platelets with easy magnetization axes normal to their planes.
Abstract: Fine Ba ferrite particles, suitable for coated perpendicular magnetic recording media, have been prepared. The particles, about 0.08 μm in average diameter, are thin hexagonal platelets with easy magnetization axes normal to their planes. Coercivity H c is controllable in a wide range, without significant reduction in magnetization, by Co and Ti substitution. The measured temperature dependences of H c and σ s showed stable characteristics. Good squareness ratio was obtained from an orientation capability measurement.

288 citations


Journal ArticleDOI
TL;DR: A method is proposed to solve eddy-current problems in three dimensions based on a special blend of FEM and BIEM techniques, and can be applied to non-linear situations.
Abstract: A method is proposed to solve eddy-current problems in three dimensions. It is based on a special blend of FEM and BIEM techniques, and can be applied to non-linear situations. It has been implemented in the linear harmonic case and used in particular for studies on non-destructive testing. We first expose the mathematical foundation, next the dicretization technique. The current possibilities of the code are described and illustrated by computer graphic displays.

201 citations


Journal ArticleDOI
TL;DR: A cubic magnetically shielded room has been constructed in this paper, which consists of three concentric cubic shields, and the side length of the smallest shell is 2A5 m. It utilizes ferromagnetic and eddy current shielding, active noise compensation, and shaking to achieve a shielding factor of about 400 000 between 1 Hz and 100 Hz.
Abstract: A cubic magnetically shielded room has been constructed. The room consists of three concentric cubic shields, and the side length of the smallest shell is 2A5 m. It utilizes ferromagnetic and eddy current shielding, active noise compensation, and shaking to achieve a shielding factor of about 400 000 between 1 Hz and 100 Hz. At lower frequencies, the shielding drops gradually and is about 16 000 for a 0.1-Hz disturbance. The residual magnetic induction in the room is less than 5 nT, and the magnetic noise above 0.5 Hz is lower than that of the superconducting quantum interference device (SQUID) magnetometer used in the measurements. When the room was completed, its performance in shielding was better than that of any other room made before.

196 citations


Journal ArticleDOI
A. Konrad1
TL;DR: A novel integrodifferential approach to steady-state skin effect problems is applied to two-dimensional geometries using traditional triangular finite elements.
Abstract: A novel integrodifferential approach to steady-state skin effect problems is applied to two-dimensional geometries using traditional triangular finite elements. The forcing function in the new formulation is the total measurable current in the conductors. An explanation of why the new approach works is included. The application of Galerkin's criterion to the integrodifferential equation is outlined and it is shown how to proceed with the finite element discretization. The methods used in the computation of performance measuring quantities useful to designers are described in detail. Simple examples illustrate the method. Results are validated by comparison to known solutions.

181 citations


Journal ArticleDOI
L. Urankar1
TL;DR: In this paper, the vector potential and magnetic field of a circular arc segment of a current-carrying conductor of arbitrary cross section and arbitrary azimuthal length were derived, based on known functions such as Jacobian elliptic functions, complete and incomplete elliptic integrals of the first, second, and third kind.
Abstract: Analytical expressions for the components of the vector potential and magnetic field of a circular arc segment of a current-carrying conductor of rectangular cross section and arbitrary azimuthal length are derived. Components of the field gradient are also calculated. All the expressions developed consist of known functions such as Jacobian elliptic functions, complete and incomplete elliptic integrals of the first, second, and third kind, and thus permit a new timesaving efficient computation algorithm.

146 citations


Journal ArticleDOI
TL;DR: In this article, an analytical solution of the Laplace equation in the air-gap is presented, combined with the classical finite element solution, and the results of field calculation obtained using such a method are more precise than those calculated when modelling the air gap with classical elements.
Abstract: In the design of electric machine working under unbalanced conditions or supplied by non sinusoidal current, a detailed knowledge of the magnetic field distribution is required to predict the machine performance. Under such conditions this field distribution must be calculated using a dynamic model for the machine air-gap: A particular element constituted by the uniform part of the air-gap (u.p.a.g.) allows such a dynamic model in the present paper. An analytical solution of the Laplace equation in the air-gap is done. The field distribution in a saturated machine can be obtained within this air-gap model, combined with the classical finite element solution. The results of field calculation obtained using such a method are more precise than those calculated when modelling the air-gap with classical elements. Examples are given in the paper, showing this fact for both the cases of static and dynamic calculations.

139 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed two new methods, called the "finite element method taking account of external power source" and the "fine element method with shape modification" to calculate the magnetic flux distribution of a magnetic circuit by using the conventional finite element method.
Abstract: When the flux distribution of a magnetic circuit is analyzed by using the conventional finite element method, the magnetizing currents must be given. Therefore, if the flux distribution is specified, it is difficult to obtain the distributions of magnetomotive forces or configuration of magnets producing the specified field distribution by the conventional finite element method. New methods which are called the "finite element method taking account of external power source" and the "finite element method with shape modification" have been developed. The processes of calculation in these methods are contrary to the conventional technique. These new methods have the following advantages: (a) If there are many unknown independent magnetizing currents, these currents are directly calculated by the new method. (b) When a flux distribution is specified, the optimum shapes of the magnets can be directly calculated. (c) As these new methods need no repetition, computing time can be considerably reduced. The principles and the finite element formulations of these new methods are described, and a few examples of application of these methods are shown. These new methods make it possible to design the optimum magnetic circuits by using the finite element method.

127 citations


Journal ArticleDOI
TL;DR: In this paper, several approaches that could be used for eddy current computation in three dimensions are compared, using vector potentials or stream functions, and results from computer programs that use these methods are given for steady state alternating current problems.
Abstract: In this paper several approaches that could be used for eddy current computation in three dimensions are compared Only partial differential equation formulations are considered, using vector potentials or stream functions Results from computer programs that use these methods are given for steady state alternating current problems The finite element method was used for this study, and improved economy was achieved by ensuring symmetry of the linear equations

98 citations



Journal ArticleDOI
TL;DR: In this paper, a three-dimensional eddy current problem is formulated as a boundary value problem for electric and magnetic strengths and the boundary integral equations for the densities of these surface sources are derived, these integral equations are of minimum order.
Abstract: A three-dimensional eddy current problem is formulated as a boundary value problem for electric and magnetic strengths. Some peculiarities of this boundary value problem are emphasized. The most important of them is the possibility to split this boundary value problem into two boundary value problems which can be solved in succession one after another: 1) the boundary value problem for the magnetic strength in the whole space and 2) the boundary value problem for the electric strength in outer air space. From the practical point of view it is enough to find the solution only of the first boundary value problem. The principle of separate surface imaginary sources is proposed for the solution of this boundary value problem and the boundary integral equations for the densities of these surface sources are derived, These integral equations are of minimum order.

Journal ArticleDOI
TL;DR: In this article, the current and magnetic field distributions in the rails and armature of an electromagnetic launcher are obtained in closed form for the steady state for both solid and laminated armatures.
Abstract: The current and magnetic field distributions in the rails and armature of an electromagnetic launcher are obtained in closed form for the steady state. These solutions assume that the armature moves with a steady velocity and account fully for the two-dimensional skin effect caused by the relative motion between the rails and the armature. Both solid and laminated armatures are considered. It is found in the case of the laminated armature that the phenomenon can be described by a single dimensionless parameter, \frac{\ell}{w}\frac{\sigma_{o}}{\sigma_{r}}\sqrt{\frac{u\ell}{\pi\eta_{r}}} .

Journal ArticleDOI
TL;DR: Three-dimensional vector potential solution methods for linear and nonlinear diffusion and magnetostatic field problems are presented and illustrated by numerical examples where feasible.
Abstract: Modern electrical plant and machinery have to be designed to operate at high power densities at minimum cost and optimal efficiency with a high degree of reliability during operation. These stringent requirements necessitate accurate performance prediction at the design stage. As a first and important step in this process, the magnetic field distribution must be evaluated taking full account of the geometrical complexity of the field region, magnetic saturation of the iron parts, and circulating currents in conducting media. The advent of digital computers has spurred the development of sophisticated numerical techniques to accomplish this task with a high degree of precision. In this paper, three-dimensional vector potential solution methods for linear and nonlinear diffusion and magnetostatic field problems are presented. The methods are illustrated by numerical examples where feasible. This research has been partially funded by the Electric Power Research Institute, Palo Alto, California under EPRI contract RP 1288-1.

Journal ArticleDOI
TL;DR: Barium-ferrite fine-particle-coated perpendicular magnetic recording tapes have been experimentally fabricated as mentioned in this paper, and the relationship between perpendicular squareness ratio and output was also investigated to show the importance of perpendicular orientation for attaining a higher recording density.
Abstract: Barium-ferrite fine-particle-coated perpendicular magnetic recording tapes have been experimentally fabricated. Typical tapes have M s = 120 - 130 emu/cc, H c = 600 - 1500 Oe and perpendicular squareness ratio M_{r\perp}/M_{s} larger than 0.9. Tape recording performances were measured with a perpendicular head as well as With a ring-shaped head. A good short wavelength response, D 50 = 76k FRPI, which has never been attained by longitudinally oriented particulate tapes, was obtained; The relationship between perpendicular squareness ratio and output was also investigated to show the importance of perpendicular orientation for attaining a higher recording density.

Journal ArticleDOI
TL;DR: In this article, a composite medium of a Co-Cr film with a soft magnetic back layer was shown by Lorentz microscopy to have a horseshoe magnetization structure.
Abstract: In a perpendicular recording system, a Co-Cr film as a medium is capable of storing very high density signals. Lorentz microscopy of 1000 kV TEM was used to observe the structure of recorded magnetizations in Co-Cr films having perpendicular anisotropy. A composite medium of a Co-Cr film with a soft magnetic back layer was shown by Lorentz microscopy to have a horseshoe magnetization structure. The stable antiparallel magnetization of transition in the Co-Cr layer determined the head-on magnetization structure of the soft magnetic back layer, which consists of a new straw-rope domain structure. The perpendicular magnetization structure of the Co-Cr film was found to consist of small domains magnetized through the film thickness which correspond to the columnar microstructure of the film. Since the intrinsic hysteresis loop of a Co-Cr film was shown to essentially have an ideal rectangular shape, it can be concluded that the Co-Cr layer of a composite film can be recorded by an ideal magnetizing process with negligible demagnetizing field at the transition.

Journal ArticleDOI
TL;DR: In this article, a method for the solution of the two-dimensional eddy current problem in terms of the magnetic vector potential is presented, where any region in the problem may be represented by either finite elements or a boundary integral.
Abstract: A method is presented for the solution of the two dimensional eddy current problem in terms of the magnetic vector potential. In the formulation presented, any region in the problem may be represented by either finite elements or a boundary integral. This information is specified through use of an interactive graphics pre-processor which automatically generates the boundary elements and the finite elements. Examples are given.

Journal ArticleDOI
TL;DR: In this article, a-SiO 2 is used for the preparation of Ba-ferrite films by means of targets-facing type of sputtering method which is very useful to prepare magnetic films at high rate without any bombardment of high energy particles such as γ-electrons and negative ions emitted from the targets and is favorable to obtain stoichiometric films of good quality.
Abstract: Ba-ferrite films have been prepared by means of Targets-Facing type of sputtering method which is very useful to prepare magnetic films at high rate without any bombardment of high energy particles such as γ-electrons and negative ions emitted from the targets and is favorable to obtain stoichiometric films of good quality. C-axis orientation of the films depends strongly on substrate temperature and crystal structure of substrate. C-axis well oriented BaM films are deposited on amorphous materials such as a-SiO 2 . The deposited films of 3000 A in thickness have ΔΘ 50 less than 1 degree, magnetic anisotropy constant of 3.2\times10^{6} erg/cc, saturation magnetization of 380 emu/cc and coercive force of 1.3 kOe. These films reveal much better c-axis orientation and surface smoothness than the films deposited by DC diode type of sputtering method and have almost the same composition as that of the targets.

Journal ArticleDOI
J.J. Croat1
TL;DR: In this paper, the potential of producing permanent magnets by rapid quench processing has been demonstrated, either by crystallization of an amorphous or rapidly quenched precursor or by direct quenching.
Abstract: Recent studies have demonstrated the potential of producing rare earth-iron permanent magnets by rapid quench processing. High coercivity, unachievable by traditional powder-metallurgy methods, has been obtained either by crystallization of an amorphous or rapidly quenched precursor or by direct-quenching. Results obtained by both techniques on a variety of rare earth-iron alloys are discussed. In particular, melt-spun Nd-Fe and Pr-Fe alloys develop an appreciable maximum (7-9 kOe) in room temperature coercivity (H ci ) as a function of quench rate, which is controlled by varying the surface velocity of the melt-spinner substrate. Even higher H ci (>20 kOe) has been observed in Sm-Fe. Magnetic and crystallization properties suggest that the coercive force of these materials is related to the formation of one or more metastable rare earth-iron phases.

Journal ArticleDOI
TL;DR: Diamagnetic behavior of white cells in plasma was confirmed and the experimental conditions of this study are more physiologic than the Saline dilution and chemical reducing agents employed by others in previous studies.
Abstract: Magnetic behavior of human blood cells in plasma was studied in High Gradient Magnetic Separation (HGMS) with the single wire technique. Human venous blood was oxygenated or deoxygenated by passing it through a gas-permeable tubular membrane with oxygen or nitrogen gases on the outside. The rate of collection of red blood cells in the deoxygenated blood havinz Hematocrit 9 - 45% was determined by observing the increase in radius of cell collection on the wire. Effects of hematocrit and velocity of blood on buildup were determined. At low red cell concentration, the magnetic susceptibility of individual red blood cells was determined using trajectory analysis. Particles of unknown type were found with susceptibility values different from red blood cells. Diamagnetic behavior of white cells in plasma was confirmed. The experimental conditions of this study are more physiologic than the Saline dilution and chemical reducing agents employed by others in previous studies.

Journal ArticleDOI
TL;DR: In this article, a model for the effect of demagnetizing field caused by closed pores and grain boundaries in ferrites was developed and the expressions to transform the magnetic properties free from the demagnetic field into the apparent magnetic properties were presented.
Abstract: A model for the effect of demagnetizing field caused by closed pores and grain boundaries in ferrites was developed This paper presents the expressions to transform the magnetic properties free from the demagnetizing field into the apparent magnetic properties The expression for the apparent permeability is \mu_{app} = \frac{(1 - P)\mu_{1}}{(1 + \frac{P}{2})(1 + 075 \frac{t}{D}\frac{\mu_{1}}{\mu_{b}}} , where p is porosity, D is average grain size, t is effective thickness of the grain boundary region, μb is the permeability of the grain boundary region and μl corresponds to the permeability free from the demagnetizing field, it was found that the demagnetizing field caused by the pores is proportional to the magnetic field, whereas that caused by the grain boundaries is proportional to the flux density, and that the grain boundary region is important for the control of the magnetic properties of ferrites

Journal ArticleDOI
TL;DR: The magnetization of the CoCr recording medium has been investigated by several methods as discussed by the authors, including the perpendicular hysteresis loops in the thickness range from 500 to 20,000 A. This provided evidence that the magnetization process is typefied by domain wall motion.
Abstract: The magnetization of the CoCr recording medium has been investigated by several methods. First the perpendicular hysteresis loops are analysed in the thickness range from 500 to 20,000 A. This provided evidence that the magnetization process is typefied by domain wall motion. Second the dependence of the coercivity on the film thickness has been determined. The dependence found can be explained if it is assumed, that the coercivity is caused by domain walls, impeded by the crystallite boundaries. Finally stand-still recording experiments have been performed, which confirm that magnetization takes place by the displacement of domain walls. The switching criterion in the writing process is best met by taking the field averaged over the film thickness.

Journal ArticleDOI
TL;DR: In this article, a torque transducer of the noncontact type employing the magnetostrictive effect is presented, where two stress-sensitive amorphous ribbons are glued to a torsion bar which is used as a part of the shaft.
Abstract: A new torque transducer of the noncontact type employing the magnetostrictive effect is presented. In this torque transducer, two stress-sensitive amorphous ribbons are glued to a torsion bar which is used as a part of the shaft. Unidirectional magnetoelastic magnetic anisotropy is created in each ribbon in such a manner that one of the anisotropies lies along 45° to the shaft axis and another along -45°. The applied torque is detected as a difference in permeability between the amorphous ribbons by an ac bias field. The amorphous ribbon used in experiments is of Fe-Si-B system whose saturation magnetostriction \lambda_{s} \simeq 24\times10^{-6} and I_{s} \simeq 1.5 T. With this torque transducer, instantaneous torque as well as static torque is detected with sufficient accuracy. Due to the easy-to-mount structure, this torque transducer is well suited for the torque control application.

Journal ArticleDOI
Y. Suganuma1, H. Tanaka, M. Yanagisawa, Fumio Goto, S. Hatano1 
TL;DR: In this article, a high density recording plated disk, which consists of a chemically plated Co-Ni-P medium and a spin-coated and lubricated SiO 2 overcoat, has been developed using techniques based on extensive studies on error defects, wear durability and wet durability throughout all the production processes.
Abstract: A high density recording plated disk, which consists of a chemically plated Co-Ni-P medium and a spin-coated and lubricated SiO 2 overcoat, has been developed using techniques based on extensive studies on error defects, wear durability and wet durability throughout all the production processes. The plated disks were evaluated by practical level tests. The average numbers of error defects were about 50 per surface at 620FRPM (15,700FRPI) linear density measured with 18.5 μm track width Mn-Zn ferrite heads. The plated disk wear durability was sufficient for 20,000 contact start/stop (CSS) cycle tests and 108times seek motion test at 0.2 μm flying height with Mn-Zn ferrite heads. Wet durability was determined for humidity tests under 40 °C, 80% R. H. for 7 months. Errors were not increased after the tests. Based on these excellent evaluation results, the plated disks are used in an 8 inch high performance 400 MB disk drive developed by NTT. Further, higher density recording characteristics of the plated disk were investigated using narrower gap length Mn-Zn ferrite heads at smaller head-medium spacings. As a result, 2,900 FRPM (74K FRPI) linear recording density was obtained as D 50 with a 0.03 μm thick Co-Ni-P medium disk and a 0.2 μm gap length head at 0.12 μm head-medium spacing.

Journal ArticleDOI
Paul H. Siegel1
TL;DR: A computer model of a peak detecting magnetic recording channel has been implemented and used for channel design and performance evaluation, and illustrative applications to RLL (d,k) code selection and pulse slimming equalizer design for a specific channel are discussed.
Abstract: A computer model of a peak detecting magnetic recording channel has been implemented and used for channel design and performance evaluation. The model predicts raw error rate, ontrack and off-track, as a function of linear density, run-length-limited (RLL) modulation code, write precompensation rules, and tapped-delay-line (TDL) equalizer. It assumes noise additivity and validity of linear superposition. and it bases calculations on a measured disk/electronics noise spectrum and digitized isolated transition readback signals from the data track and adjacent tracks. Details of the model are described, and illustrative applications to RLL (d,k) code selection and pulse slimming equalizer design for a specific channel are discussed.

Journal ArticleDOI
C. Tsang1, Robert E. Fontana1
TL;DR: In this paper, a 10μm × 15μm barber-pole biased unshielded MR sensors were designed, fabricated and tested and they showed that strong domain activities, inherent in these small elements, produce noisy MR responses without any observable biasing effects.
Abstract: We have designed, fabricated and tested 10μm × 15μm barber-pole biased unshielded MR sensors. Results show that strong domain activities, inherent in these small elements, produce noisy MR responses without any observable biasing effects. Furthermore, we found that these domains, caused by magnetostatic effects, can be suppressed by applying an adequate longitudinal bias field. These findings lead to the fabrication of MR sensors with longitudinal exchange-biasing and transverse barber-pole biasing. These sensors produce quiet and strongly biased MR responses.

Journal ArticleDOI
TL;DR: The metal in gap (MIG) heads as mentioned in this paper are glass-bonded ferrite cores with high saturation flux density Sendust pole tip tip and achieved an RMS signal-to-wideband noise ratio of 30dB on high squareness isotropic tapes.
Abstract: High density signal levels increase as the record head gap is reduced. When the gap is reduced, the optimum deep gap record field increases. With small gaps and high coercivity tapes, the wear resistant ferrites used to make standard heads saturate at the gap corners. This smears the record field gradient and performance suffers. Traditionally, the solution to this problem has been to make a ferrite core equipped with high saturation flux density Sendust pole tips. The record performance of tipped heads is excellent, but they are expensive and fragile because they are epoxied together, and their wear rate is much higher than ferrite heads. We have developed a process for vacuum depositing high permeability Sendust thin films on the gap faces of ferrite heads before gap spacer deposition. These metal in gap (MIG) heads can be glass bonded. The result is a long wearing, rugged, and inexpensive record head which gives optimum 80 KBPI performance on 860 Oe coercivity media using standard ferrite cores and gaps as small as 0.3μ. With a 250μ trackwidth, a tape speed of 19.1 cm/sec, and a frequency of 300 KHz (80 KFCI), MIG heads have achieved an RMS signal-to-wideband noise ratio of 30dB on high squareness isotropic tapes. Similar all-ferrite heads record only about half this signal. Construction and performance of MIG heads will be described.

Journal ArticleDOI
TL;DR: In this article, a method of providing bounded solutions to a wide range of magnetostatic field problems is outlined, which extends complementary and dual energy variational principles to encompass the T-Ω formulation of electromagnetic field problems.
Abstract: A method of providing bounded solutions to a wide range of magnetostatic field problems is outlined. The method extends complementary and dual energy variational principles to encompass the T-Ω formulation of electromagnetic field problems and shows how this leads to efficient finite element implementation of the technique. Examples are given that show clearly the bounded nature of the procedure, and indicate how it may be used to reduce the computational requirements necessary for a specific accuracy of solution.

Journal ArticleDOI
TL;DR: In this paper, the kinetics of the hydrolysis of polyester urethane binders of magnetic recording tape is described. Butler et al. used a linear, single product, reversible rate equation to predict proper storage conditions for magnetic tape.
Abstract: The kinetics of the hydrolysis of polyester urethane binders of magnetic recording tape is described. Kinetic data were generated from measurements of acetone-extractable hydrolyzed binder products versus time for various humidity-temperature environments. These data can be described by a linear, single product, reversible rate equation. This equation, coupled with measurements on the effect of hydrolysis on recorded tape performance, is used to predict proper environmental storage conditions for magnetic tape.

Journal ArticleDOI
TL;DR: In this paper, the authors calculated the crosstalk of shielded magnetoresistive (MR) heads with MR films much wider than the recorded track using a 2D transmission-line model.
Abstract: The crosstalk of shielded magnetoresistive (MR) heads with MR films much wider than the recorded track is calculated using a 2D transmission-line model. The head response versus the transverse displacement of head and recorded track can be expressed analytically in the form of a convergent series. Due to the intrinsic asymmetry of the 45° bias in the single domain MR film, the crosstalks from the two adjacent tracks are not equal. As the head is moved away from a positioned track, the crosstalk goes through a null in one direction but not the other. Results from experiments and calculations have been compared and show reasonable agreement. This effect can be employed to test whether the MR film of a shielded MR head is in a single domain state. Furthermore, for high track density applications, the MR head should be offset with respect to the recording head to achieve optimum performance.

Journal ArticleDOI
T. Morisue1
TL;DR: In this paper, the boundary conditions for the scalar potential and each of the components of the vector potential were determined for a two-dimensional eddy current problem with variable electric conductivity.
Abstract: In calculating eddy currents in a conductor by means of the vector potential for which the Coulomb Gauge is used, the scalar potential appears when the electric conductivity varies in the conductor, while it is not necessary for the case of the constant electric conductivity. As the field equation is a Helmholtz-type equation under the Coulomb Gauge condition, we must determine the boundary conditions for the scalar potential and each of the components of the vector potential. How to determine the boundary conditions is presented, together with an illustrative example of a two-dimensional eddy current problem with the variable electric conductivity.