Magnetic Vector Potential Research Articles

Electromagnetic force calculation within finite volume framework

ABSTRACT In this research, the comparison of force calculation methods in magnetostatic simulations within a cell-centered finite volume framework is presented. Various procedures for postprocessing of the magnetic vector potential field are laid out to obtain the surface and volume integrals of Maxwell’s stress tensor and the Lorentz force density. Two different magnetostatic problems with multiple operating points were modeled with the AVL FIRETM M. The force results have been verified by comparing them to the Ansys Maxwell and were validated on a benchmark case from a Testing Electromagnetic Analysis Methods (TEAM) workshop. The interpolation of the magnetic flux density on a multi-material interface is presented, and it is used in conjunction with the surface integral of Maxwell’s stress tensor for an efficient global force computation. For all observed cases, the calculated results correspond well to the results found in the literature. It was found that the accuracy of the volume integral of Maxwell’s stress tensor is dependent on the mesh quality and gradient calculation scheme. Employing the semi-structured mesh with Gauss’s gradient scheme results in force discrepancies of less than 3% between the surface and volume integral of Maxwell’s stress tensor. The convergence of the iterative solver is observed to be slower in the case of the materials with permeability jump, where discontinuities between the tangential component of the magnetic flux density and the normal component of the magnetic field intensity occur.

Spin-dependent conductance and magnetoresistance ratio in two magnetic-barrier structures

An improved transfer matrix method is developed with modular matrices for rectangular electric and magnetic vector potentials, as well as the δ-function like potentials. As an example, the spin-dependent conductance in two-dimensional electron gas under modulation of two magnetic barriers are recalculated and compared with those illustrated in (Wang et al. Chin. Phys. B 19, 037301(2010)). The previous results are found inaccurate, their transfer matrix was derived for slowly varying magnetic vector potential, which is inappropriate for δ-function like Zeeman interaction potentials. Furthermore, the last transfer matrix of eq. (4) is also found incorrect, and the Zeeman interaction is wrongly expressed and mistakenly enhanced about (1/0.067 ≈ 15) times. Our improved method here is general and is demonstrated very fast forevaluating various spin-dependent transport properties in complex hybrid magnetic–electric structures.

On Finite-Element Modeling of Large-Scale Magnetization Problems with Combined Magnetic Vector and Scalar Potentials

On Finite-Element Modeling of Large-Scale Magnetization Problems with Combined Magnetic Vector and Scalar Potentials

A Method for Calculating Critical Current of High Temperature Superconducting Machine Based on Magnetic Vector Potential

A Method for Calculating Critical Current of High Temperature Superconducting Machine Based on Magnetic Vector Potential

Structure-preserving and efficient numerical simulation for diffuse interface model of two-phase magnetohydrodynamics

In this paper, we propose a novel diffuse interface model of two-phase magnetohydrodynamics (MHD) based on a magnetic vector potential formulation in the three-dimensional case. This model ensures an exact divergence-free approximation of the magnetic field by introducing a magnetic vector potential A and defining the magnetic field by B=curlA. The resulting framework constitutes a highly coupled, nonlinear saddle point system consisting of the Cahn–Hilliard system and MHD potential system. To solve the model efficiently, we present two fully decoupled, first-order, linear, and unconditionally energy-stable schemes and strictly prove their well-posedness and energy stability. Finally, we present several numerical examples that demonstrate the stability and effectiveness of our schemes.

Enhancing electric field calculation in HTS tape simulations for currents exceeding the critical limit using full HTS tape modeling

Superconducting devices are an interesting technological option for improvement of efficiency of energy systems. High-temperature superconducting tapes are among the best choices for power applications. Because of their high non-linearity, the design of devices with superconducting tape requires specific simulation models and methods to correctly represent the superconducting behavior. The J-A formulation has been investigated as an efficient tool to represent the electromagnetic behavior of such devices. It is based on the current density (J) and magnetic vector potential (A), and able to represent both superconducting materials and ferromagnetic materials. With regards to the tapes, the use of thin-film approximation can be successful with the J-A formulation. Usually, only representations considering the superconducting layer of tapes have been investigated. This becomes a problem for operations where the current in the superconductor surpasses the critical current, for example in transient analysis. This paper presents an analysis of the effects the inclusion of all of the tapes’ layers have on simulations with the J-A formulation with thin-film approximations. By considering all layers of the tape, one represents the transition between superconducting and high-loss states of the tape. A simple system, consisting of a single tape with applied current, has been studied as benchmark and simulated with J-A and the more established T-A formulation in two cases: considering all layers; and only the superconducting layers. Results show that results for the J-A and T-A formulations are compatible and the inclusion of all layers improves the stability of the simulation and provides correct assessment of the electric field in the tapes.

Identifying Coronal Sources of L1 Solar Wind Disturbances Using the Fisk Heliospheric Magnetic Field and Potential Field Extrapolations during Three Solar Minima

The solar minima between solar cycles 22–23, 23–24, and 24–25 are the best observed minima on record. In situ solar wind and interplanetary magnetic field measurements by the Wind and ACE spacecraft at L1 with 1 hr cadence are explored using wavelet analyses for the most quiescent year during each minimum. Times of local peaks in periodicities are identified in the solar wind velocity, magnetic field components, and proton number densities. The measured radial velocities at these times are used to trace magnetic field lines to the photosphere using two models. The first is the Fisk heliospheric magnetic field that traces field lines from L1 to the photosphere. They connect exclusively to solar poles and in 88% of instances to locations of polar coronal holes (PCHs). The second model uses the Parker spiral to trace from L1 to the solar source surface and potential-field extrapolations from the source surface to the photosphere. These field lines terminate at equatorial and midlatitude coordinates, of which some are located close to coronal holes (CHs). This study connects for the first time CH signatures in the ecliptic plane at L1 with PCHs using the Fisk field. It shows how sources from both the solar equator and poles influence the solar wind at L1 and how the two models complement each other to identify these sources.

Analysis of pole pair number impact on single-sided linear induction machine performances

This paper presents a parametric study of linear induction motor for design purpose. The chosen mathematical model uses a 2D formulation with magnetic vector potential A. The implementation of the model is carried out with the finite element method on the free platform Gmsh-GetDP. Circuit model is coupled to FE model so that constant voltage supply mode can be considered. This work aims to highlight the effect of the pole pair number on the characteristics and the performances of the linear induction machine through two numerical models associated to an analytical one. Furthermore, the study shows the effect of the pole pair number on the phase imbalance and the spatial harmonic spectrum of the machine. Reducing this imbalance and higher order harmonics presence will increase machine performances.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

МАТЕМАТИЧНА МОДЕЛЬ МАГНІТОЕЛЕКТРИЧНОЇ МАШИНИ

A mathematical model is proposed for the calculation of electromagnetic parameters of magnetoelectric machines using an analytical method. Permanent magnets are abstracted as equivalent solenoids with constant current represented as current loops. The model differs from known models by the presence of double linear current loops in the stator for the analysis of magnetoelectric machines with a two-layer winding. Additionally, the proposed model considers linear current loops of equivalent solenoids instead of point current loops for a more accurate calculation. The current loops of the stator and rotor are defined at the boundaries of the air gap of the electric machine, i.e., on the smooth slotless surfaces of the magnetic cores. The current loops are expressed for the first time as a product of three variables: current strength, linear current density coefficient, and spatial distribution coefficient. The inductance of stator slots and the resistance of the winding are taken into account using known analytical expressions from classical electric machine theory. The rotor rotation frequency is presented as a function of time. For magnetoelectric machines of specified dimensions and a given variable rotor rotation frequency, the distribution of magnetic field induction, vector magnetic potential, currents, winding spatial distribution coefficients, and electric field intensity of permanent magnets are calculated using the mathematical model. References 7, table 1, figures 4.

Electromagnetic lensing using the Aharonov–Bohm effect

We demonstrate theoretically and experimentally an electromagnetic lensing concept using the magnetic vector potential—in a region free of classical electromagnetic fields—via the Aharonov–Bohm (AB) effect. This toroid-shaped lens with poloidal current flow allows for electromagnetic lensing which can be tuned to be convex or concave with a spherical aberration coefficient of opposite polarity to its focal length. This field-free lens combines the advantages of traditional electromagnetic and electrostatic field-based lenses and opens up additional possibilities for the optical design of charged-particle systems. More generally, these results demonstrate that the AB effect can shape charged particle wavefronts beyond simple step shifts if topologies beyond simple flux lines are considered and further supports the physical significance of the magnetic vector potential.

Direct, simple and efficient computation of all components of the virtual-casing magnetic field in axisymmetric geometries with Kapur–Rokhlin quadrature

Direct, simple and efficient computation of all components of the virtual-casing magnetic field in axisymmetric geometries with Kapur–Rokhlin quadrature

Analysis of DC bias characteristics of transformer by using fixed-point time-step FEM and dynamic J–A hysteresis model

The injection of the Direct Current (DC) bias component will lead to oversaturation of the transformer core and affect its safe operation. In this paper, the vector magnetic potential A and winding current I are used as the variables solved in the time-step finite element method to simulate the nonlinear transient magnetization characteristics of the transformer under DC bias, and the fixed-point reluctivity is introduced to deal with the nonlinear iterative problem. The field-circuit coupling relationship is established through the electromagnetic induction law, and the two-dimensional time-step finite element model of the transformer core is found. The dynamic J–A hysteresis model is used to deal with the hysteresis effect of iron cores. The fixed-point reluctivity of difference form is introduced to solve the problem of uneven transition and multi-value of magnetization curve with hysteresis in the iterative process. The determination scheme for the equivalent thickness of the core model and the initial iterative value are discussed. A laminated core transformer is made for experiments, and the accuracy and applicability of the model proposed in this paper are verified by comparing the calculation and experimental results. The influence of different voltages and different DC bias components on the electromagnetic characteristics of the transformer is analyzed.

Improving force characteristics of linear oscillatory generator with spring permanent magnet for Stirling engines based on subdomain method

In this paper, an electromagnetic analysis method is proposed to improve force characteristics by using spring permanent magnets (PMs) along the axial and circumferential directions of a linear oscillatory generator (LOG). For accurate electromagnetic analysis, a detailed analysis model of the LOG is developed, and the governing equations of each subdomain are derived based on Maxwell’s equations and electromagnetic theory. Analytical solutions of the magnetic vector potential in each subdomain are derived using boundary conditions. The reliability of the proposed method is verified through a comparison with the results of a two-dimensional finite element analysis (FEA). In particular, the force characteristics depending on the spring PM is effectively derived by considering the three-dimensional (3D) circumferential and axial end effects. The proposed analysis method is used to determine the thickness of spring PM that transforms the detent force of the LOG with the spring PM into restoring force. The reliability of the proposed analysis method is verified through comparison with the results of a 3D FEA.

Subdomain Analytical Modeling of a Double-Stator Spoke-Type Permanent Magnet Vernier Machine

This paper proposes an analytical model of the double-stator spoke-type permanent magnet vernier machine (DSSTVM) using the subdomain method (SDM), which can be used to calculate the magnetic field distribution and corresponding electromagnetic parameters of the DSSTVM. The whole field domain is divided into several subdomains according to the magnetic characteristics of each region, within which Laplace’s and Poisson’s equations are solved accordingly in terms of magnetic vector potential (MVP). Then, the corresponding magnetic flux density distribution, back electromotive force (EMF), and electromagnetic torque of the DSSTVM can be obtained. Ultimately, finite element analysis (FEA) is adopted to validate the proposed analytical model’s effectiveness for quickly predicting the no-load and on-load performances of the DSSTVM.

Model of an E-cored probe over layered conductor containing corrosion for eddy current testing

PurposeIn eddy current nondestructive testing, ferrite-cored probes are usually used to detect and locate defects such as cracks and corrosion in conductive materials. However, the generic analytical model for evaluating corrosion in layered conductor using ferrite-cored probe has not yet been developed. The purpose of this paper is to propose and verify the analytical model of an E-cored probe for evaluating corrosion in layered conductive materials.Design/methodology/approachA cylindrical coordinate system is adopted and the solution domain is truncated in the radial direction. The magnetic vector potential of each region excited by a filamentary coil is derived first, and then the expansion coefficients of the solution are obtained by matching the boundary and interface conditions between the regions and the subregions. Finally the closed-form expression of the impedance of the multi-turn coil is derived by using the truncated region eigenfunction expansion (TREE) method, and the impedance calculation is carried out in Mathematica. In the frequency range of 100 Hz to 10 kHz, the impedance changes of the E-cored coil and air-cored coil due to the layered conductor containing corrosion are calculated, respectively, and the influences of corrosion on the coil impedance change are investigated.FindingsAn analytical model for the detection and evaluating of corrosion in layered conductive materials using E-cored probe is proposed. The model can quickly and accurately calculate the impedance change of E-cored coil due to corrosion in layered conductor. The correctness of the analytical model is verified by finite element method and experiments.Originality/valueAn accurate theoretical model of E-cored probe for evaluating corrosion of multilayer conductor is presented. The analytical model can be used to detect the inhomogeneity of layered conductor, design ferrite-cored probe or directly evaluate the corrosion defects of layered conductors.

Surface charge and surface current densities at material boundaries

In electromagnetism, materials with a polarization density P→ or a magnetization density M→ are known to exhibit a bound surface charge density σb=P→·n̂ or a surface current density κ→b=M→×n̂, respectively, where n̂ is the unit vector perpendicular to the material boundary surface, directed outward. These expressions can be obtained from volume integrations for the electric potential V, or the magnetic vector potential A→, in which the integrals are restricted to the material volumes delimited by their respective boundaries. In that case, applying the divergence theorem leads to surface integrals on material boundaries and to the above-mentioned surface quantities. In this paper, a simple derivation is presented, which shows that both σb and κ→b are included in the expressions for the volume charge or current densities, provided that the divergence and curl operators are evaluated at the boundary so as to account for discontinuities at interfaces.

Analytical Solution of Eddy Current in Parallel Conducting Strips for Low-frequency Shielding Purposes

In instrumentation systems, shielding is the main issue that judges the performance of the system. The electromagnetic (EM) noise may affect the performance of the instrumentation system if inadequate protection is reached. It is considered the main source of unprotectable interference that may affect these systems in many cases. In this paper, shielding is attained by wrapping the source carrying signal with periodic thin conductive strips separated by slots or openings. This arrangement will protect the sources from the outside EM fields. Shielding factor and shielding efficiency are studied by extracting magnetic fields. For this purpose, an analytical solution based on solving Laplace’s equation for the magnetic vector potential in the region of interest is presented. A closed form of the induced eddy current in the conductive strips is calculated based on Fourier series expansion. Furthermore, numerical simulation using the commercial software MWS CST is employed to validate the analytical solution. The performance of the proposed shielding structure is studied and analyzed in terms of shielding factor and shielding efficiency. The outcomes of both methods are showing very good agreement.

Error analysis of the linearized Crank-Nicolson FEM for the incompressible vector potential magnetohydrodynamic system

A linearized fully discrete Crank-Nicolson finite element scheme is proposed for solving the three-dimensional incompressible magnetohydrodynamic equations based on a magnetic vector potential formulation, where the magnetic induction is written to a rotation of magnetic vector potential. By using the MINI element and lowest order Nédélec edge element to approximate the velocity field and pressure of fluid and magnetic vector potential, respectively, the numerical solution of magnetic induction can preserve the exactly divergence-free condition in fully discrete level. Error estimates for the velocity field and magnetic vector potential are rigorously analyzed under some reasonable regularity assumptions of exact solution. Finally, numerical results are given to support the theoretical analysis.

Preliminary electromagnetic analysis of the COOL blanket for CFETR

The supercritical CO2 cOoled Lithium-Lead (COOL) blanket has been designed as one advanced blanket candidate for the Chinese Fusion Engineering Test Reactor (CFETR). This work focuses on the electromagnetic (EM) loads (Maxwell force and Lorentz force) acting on the COOL blanket, which are important mechanical loads in further structural analysis of the COOL blanket. A 3D electromagnetic analysis is performed using the ANSYS finite element method to obtain EM loads on the COOL blanket in this study. At first, the magnetic scalar potential (MSP) method is used to obtain the magnetic field and the Maxwell force on the COOL blanket. Then, the magnetic vector potential (MVP) method is performed during a plasma disruption event to get the eddy current distribution. At last, a multi-step method is adopted for the calculation of the Lorentz force and the torque. The maximum Lorentz forces of inboard and outboard blanket structural components are 5624 kN and 2360 kN respectively.