Magnetic Field Equations Research Articles

Time-Stepping FEM-Based Multi-Level Coupling of Magnetic Field–Electric Circuit and Mechanical Structural Deformation Models Dedicated to the Analysis of Electromagnetic Actuators

The present paper introduced a framework for multi-level coupling transient electromagnetic fields (EMF) and mechanical structural dynamics based on the finite element method (FEM). This framework was dedicated to predicting, with better accuracy, the wave magnetic force density for obtaining the mechanical deformation occurring in electromagnetic actuators (EMAs). The first-level EMF transient model coupling is related to the magnetic field equations that are strongly coupled with the electric circuit input voltage equations. This is done by considering the magnetic saturation through the Newton–Raphson (N–R) method. The time-stepping solution of the EMF model resulted in the magnetic force densities being computed from the Lorentz force (LZ) expressions, based on the space–time variation of the induced eddy current. For the second coupling level, the EMF model was sequentially coupled with the mechanical structural deformation equations (MDef) through the local magnetic force density to achieve minimal material dynamic displacement and deformation. The developed multi-physics EMF–MDef time-stepping (FEM) model tools were implemented using the Matlab software.

Improved <italic>RLMC</italic>-Circuit HF-Dependent Parameters Using FE-EM Computation Dedicated to Predict Fast Transient Voltage Along Insulated Windings

The aim of the present paper is to develop methods that would be able to predict transient voltage distribution along the insulated-multiconductor windings of electric equipment fed by frequency converters. Several electromagnetic (EM) challenges are presented with this desire. First, the resistance ( R ) and self-/mutual inductances ( L , M ) frequency-dependent parameters are accurately computed using the enforced electric total current process based on the finite-element (FE) formulation of the strongly coupled time-harmonic magnetic field and total current density equations model including skin and proximity effects. Second, the coupling capacitances ( C ) are computed from the FE electrostatic analysis using the floating electric scalar potential process. Finally, a realistic high-frequency electric circuit of the distributed ( R, L, M, C ) EM parameter matrices is formed in order to predict the turn-to-ground and turn-to-turn voltage transients. The numerical results obtained from the proposed models implemented under the MATLAB code are successfully verified by the corresponding laboratory test measurements.

Fractional magneto-hydrodynamics: Algorithms and applications

We present two discretization methods for solving the fractional magneto-hydrodynamic (FMHD) equations with the fractional Laplacian defined in bounded domains. In the first method, we add a pseudo-pressure in the magnetic field equation to enforce that the magnetic field is divergence-free. In the second method, the magnetic field satisfies the divergence-free condition automatically with the no-slip boundary condition for the velocity and the perfectly conducting boundary condition for the magnetic field equation. In addition, we propose a discretization method for solving the modified fractional magneto-hydrodynamic (M-FMHD) equation. The divergence-free condition is preserved by employing the stream function in the M-FMHD system. Numerical experiments with fabricated solutions show that all three methods exhibit second-order accuracy in time for the velocity and magnetic fields. The pseudo-pressure and pressure also exhibit second-order convergence in time for small values of time step, however the pressure exhibits 1.5-order for convergence for large values of time step size when the fractional order α=1. We use the spectral decomposition method for spatial discretization, and we demonstrate that exponential convergence is achieved for all fields and for any fractional order. We also applied these methods to solve the lid-driven cavity flow with and without magnetic field, and we observe new flow patterns emerging as the fractional order varies.

Loss Simulation by Finite-Element Magnetic Field Analysis Considering Dielectric Effect and Magnetic Hysteresis in EI-Shaped Mn–Zn Ferrite Core

Loss Simulation by Finite-Element Magnetic Field Analysis Considering Dielectric Effect and Magnetic Hysteresis in EI-Shaped Mn–Zn Ferrite Core

Mathematical modelling of combined pressure driven and electrokinetic effect in a channel with induced magnetic field: An exact solution

This article presents exact solution for pressure driven flow formation of electrically conducting fluid in a parallel plate channel formed by two horizontal parallel plates with electrokinetic effects and induced magnetic field. Using the Poisson–Boltzmann, Navier-Stokes equations and induction equation, the governing electric potential, momentum, induced magnetic field and energy equations for the present article are presented and transformed to their corresponding dimensionless form using suitable parameters. The governing dimensionless equations are solved exactly and graphical representation are presented. During the cause of graphical illustration, it is found that the role of electrokinetic effects and Hartmann number is to decrease the electric potential, fluid velocity, induced magnetic field and fluid temperature. A special case is found and discussed when the value of Hartmann number equals the Debye-Hückel parameter. It is interesting to note that heat transfer is independent on governing parameters for large value of Debye-Hückel parameter.

Time‐varying magnetic field analysis using an improved meshless method based on interpolating moving least squares

In this study, analysis of 2D time-varying magnetic field using an improved meshless method based on interpolating moving least squares (IMLS) is proposed. In this study, solving the time-domain magnetic field equation in a meshless method carried out by selection of magnetic field intensity as a variable instead of magnetic vector potential which is widely used as a variable. Using this approach, the post-processing process will be fast and the boundary condition is implemented simply. For performance evaluation of the proposed method in eddy current analysis, two other meshless techniques, i.e. moving least squares (MLS) and radial point interpolation method (RPIM) have been considered. The results of improved IMLS are compared with MLS, RPIM and finite element method (FEM) results. Verification of improved meshless results is performed by FEM.

Interplay of non-conducting and conducting walls on magnetohydrodynamic (MHD) natural convection flow in vertical micro-channel in the presence of induced magnetic field

In this research paper, exact solution for fully developed magnetohydrodynamic (MHD) natural convection flow of viscous, incompressible, electrically conducting fluid in vertical micro-channel formed by electrically non-conducting and conducting infinite vertical parallel walls in the presence of velocity slip and temperature jump at the micro-channel walls is investigated. The influence of induced magnetic field (I.M.F) arising due to the motion of an electrically conducting fluid in the presence of transverse magnetic field is taken into consideration. Due to the presence of induced magnetic field, the momentum and induction equations are coupled. The exact solutions in dimensionless form have been obtained under relevant boundary conditions. The expressions for velocity field, the induced magnetic field, temperature field, induced current density and skin friction have been obtained. The effects of the various physical parameters appearing into the model such as rarefaction, fluid wall interaction, Hartmann number and the magnetic Prandtl number are demonstrated through graphs and table. The results indicate that, increase in Hartmann number and magnetic Prandtl number causes a pronounced reduction in volume flow rate.

Path integrals for mean-field equations in nonlinear dynamos

Mean-field dynamo equations are addressed with the aid of the path integral method. The evolution of magnetic field is treated as a three-dimensional Wiener random process, and the mean magnetic-field equations are obtained with the Wiener integrals taken over all the trajectories of the fluid particles. The form of the equations is just the same as the conventional mean-field equations, but here the equations are derived with the velocity field realisation affected by the force exerted by the magnetic field. In this sense, we derive nonlinear dynamo equations.

Problem of stability of multidimensional solutions of the BK class equations in space plasma

The problem of stability of the multidimensional solutions of the BK class equations describing the nonlinear waves which are forming on the low-frequency branch of oscillations in plasma for cases when β≡4πnT/B2≪1 and β>1 is studied. In first case, for ω<ωB=eB/Mc,kλD≪1 the FMS waves are excited, and their dynamics under conditions kx2≫k⊥2, vx≪cA near the cone of θ=arctan(M/m)1/2, is described by the equation of the BK class known as the GKP equation for magnetic field h=B∼/B with due account of the high order dispersive correction defined by values of plasma parameters and angle θ=(B,k). In another case, the dynamics of the finite-amplitude Alfvén waves propagating near-to-parallel to B is described by the equation of the same class known as the 3-DNLS equation for h=(By+iBz)/2B|1-β|. To study the stability of multidimensional solutions in both cases the method of investigation of the Hamiltonian bounding with deformation conserving momentum by solving the variation problem is used. As a result, we have obtained the conditions of existence of the 2D and 3D soliton solutions in the BK system for cases of the GKP and 3-DNLS equation (i.e. for the FMS and Alfvén waves, respectively) in dependence on the equations’ coefficients, i.e. on the parameters of both plasma and wave.

Modeling and Investigation on Electromagnetic Noise in PM Motors With Single- and Double-Layer Concentrated Winding for EV and HEV Application

The radial force and the torque pulsation are identified as two main causes of electromagnetic noise and vibration in permanent magnet (PM) motors for electric vehicle (EV) and hybrid EVs systems. The aim of this paper is to systematically investigate the influence of pole and slot number combinations on noise and vibration characteristics of fractional-slot PM motors with either single- or double-layer concentrated windings. Preferentially, the explicit magnetic field equations are developed for analyzing the radial vibration force and the torque pulsation by the aid of Schwarz-Christoffel conformal mapping. Subsequently, an in-depth investigation of pole ( $2p$ ) and slot ( $Q_{s}$ ) number combinations is carried out to provide a detailed finding of noise behaviors due to the electromagnetic origins. The investigation on the electromagnetic noise is performed depending on the several major $S_{\mathrm{ pp}}$ families. Finally, the relationship between the vibration mode order and the torque variation is established to provide the comprehensive characteristics of the electromagnetic noise.

Fluid dynamics of the magnetic field dependent thermosolutal convection and viscosity between coaxial contracting discs

Abstract The effects of magnetic field dependent (MFD) thermosolutal convection and MFD viscosity of the fluid dynamics are investigated between squeezing discs rotating with different velocities. The unsteady constitutive expressions of mass conservation, modified Navier-Stokes, Maxwell and MFD thermosolutal convection are coupled as a system of ordinary differential equations. The corresponding solutions for the transformed radial and azimuthal momentum as well as solutions for the azimuthal and axial induced magnetic field equations are determined, also the MHD pressure and torque which the fluid exerts on the upper disc is derived and discussed in details. In the case of smooth discs the self-similar equations are solved using Homotopy Analysis Method (HAM) with appropriate initial guesses and auxiliary parameters to produce an algorithm with an accelerated and assured convergence. The validity and accuracy of HAM results is proved by comparison of the HAM solutions with numerical solver package BVP4c. It has been shown that magnetic Reynolds number causes to decrease magnetic field distributions, fluid temperature, axial and tangential velocity. Also azimuthal and axial components of magnetic field have opposite behavior with increase in MFD viscosity. Applications of the study include automotive magneto-rheological shock absorbers, novel aircraft landing gear systems, heating up or cooling processes, biological sensor systems and biological prosthetic etc.

Why fast magnetic reconnection is so prevalent

Evolving magnetic fields are shown to generically reach a state of fast magnetic reconnection in which magnetic field line connections change and magnetic energy is released at an Alfvénic rate. This occurs even in plasmas with zero resistivity; only the finiteness of the mass of the lightest charged particle, an electron, is required. The speed and prevalence of Alfvénic or fast magnetic reconnection imply that its cause must be contained within the ideal evolution equation for magnetic fields, $\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B})$, where $\boldsymbol{u}(\boldsymbol{x},t)$ is the velocity of the magnetic field lines. For a generic $\boldsymbol{u}(\boldsymbol{x},t)$, neighbouring magnetic field lines develop a separation that increases exponentially, as $e^{\unicode[STIX]{x1D70E}(\ell ,t)}$ with $\ell$ the distance along a line. This exponentially enhances the sensitivity of the evolution to non-ideal effects. An analogous effect, the importance of stirring to produce a large-scale flow and enhance mixing, has been recognized by cooks through many millennia, but the importance of the large-scale flow $\boldsymbol{u}$ to reconnection is customarily ignored. In part this is due to the sixty-year focus of recognition theory on two-coordinate models, which eliminate the exponential enhancement that is generic with three coordinates. A simple three-coordinate model is developed, which could be used to address many unanswered questions.

Derivation for Electric Current Regulation Equation of a Gradient Magnetic Field to Control Suspending Magnetic Particles Inside Dialysate Solution

To design a better adsorption performance in a novel magnetic adsorption device used for hemodialysis (HD), the mechanical properties of magnetic absorbents trapped inside a two-phase system are studied in this paper. A gradient magnetic coil field is assumed to produce the magnetic driving force that balances other hydraulic forces for the adsorbents. For this field, a related winding equation for the solenoid coil is obtained in our previous work; and a complement practical form of the winding equation is derived in this paper. Case studies are also described in this paper to explore the design aspects of the field.

Modified magnetomechancial model in the constant and low intensity magnetic field based on J–A theory**Project supported by the Major Program of Sichuan Province Science and Technology Plan, China (Grant No. 2015SZ0010) and the Scientific Research Foundation of Sichuan Province, China (Grant No. 2014GZ0121).

The existing magnetomechancial models cannot explain the different experimental phenomena when the ferromagnetic specimen is respectively subjected to tension and compression stress in the constant and low intensity magnetic field, especially in the compression case. To promote the development of magnetomechancial theory, the energy conservation equation, effective magnetic field equation, and anhysteretic magnetization equation of the original Jiles–Atherton (J–A) theory are elucidated and modified, an equation of the local equilibrium status is employed and the differential expression of the modified magnetomechancial model based on the modified J–A theory is established finally. The effect of stress and plastic deformation on the magnetic parameters is analyzed. An excellent agreement is achieved between the theoretic predictions by the present modified model and the previous experimental results. Comparing with the calculation results given by the existing models and experimental results, it is seen indeed that the modified magnetomechanical model can describe the different magnetization features during tension-release and compression-release processes much better, and is the only one which can accurately reflect the experimental observation that the magnetic induction intensity reverses to negative value with the increase of the compressive stress and applied field.

Melting Heat Transfer and Induced-Magnetic Field Effects on the Micropolar Fluid Flow towards Stagnation Point: Boundary Layer Analysis

In this article, the problem of a non-Newtonian fluid (micropolar) flow over a horizontal melting surface in the presence of internal heat source and dual stretching (i.e. at the wall and at the free stream) is presented. Since the magnetic-Reynold of the flow is substantial, the influence of induced magnetic field is properly accounted in the governing equation. The viscosity and thermal conductivity of the micropolar fluid are considered to vary linearly with temperature. Classical models of these thermophysical properties were modified to suit the case of melting heat transfer. A similarity transformation is applied to reduce the governing partial differential equation to coupled ordinary differential equation corresponding to dimensionless momentum, angular momentum, energy and induced magnetic field equation. These equations along with the boundary conditions are solved numerically using shooting method along with Runge-Kutta-Gill method together with quadratic interpolation. The results of the present study indicate that due to the formation of boundary layer on melting surface (region of low heat energy) in the presence of induced magnetic field, space and temperature dependent internal heat generation enhances the heat transfer rate.

Influence of electromagnetic stirrer position on fluid flow and solidification in continuous casting mold

• Numerical investigation of solidification behavior with EMS position is proposed. • Breaking of solidified layer was observed by shifting the EMS position downward. • With EMS application, liquid fraction values were decreased in mold core region. • Tangential velocity values were found to be maximum near solidified shell. A computational study of the effect of stirrer position on fluid flow and solidification in a continuous casting billet mold with in-mold electromagnetic stirring has been carried out. The numerical investigation uses a full coupling method in which alternating magnetic field equations are solved simultaneously with the governing equations of fluid flow and heat transfer. An enthalpy-porosity technique is used for the solidification analysis while the magnetohydrodynamics technique is used for studying the fluid flow behavior under the electromagnetic field. The streamline, liquid fraction, and solid shell thickness at the mold wall have been predicted with and without EMS application at different positions along the length of the mold. Recirculation loops are seen to be formed above and below the stirrer position when fluid flow and electromagnetic field equations were solved, without incorporating the solidification model. Application of the solidification model interestingly resulted in the reduction of the size of the recirculation loops formed. The tangential component of velocity of the fluid near the solidification front, stirring intensity and the effective length of stirring below the stirrer decrease as the stirrer position is moved downwards. Significant changes in characteristics of solid shell formation like delay in initiation of solidification at the mold wall and formation of a gap in the re-solidified shell have been observed with change in stirrer position.

Dynamic motion analysis of magnetic particles in microfluidic systems under an external gradient magnetic field

To gain an insightful understanding of motion behavior of paramagnetic particles suspended in a nonmagnetic fluid under a gradient magnetic field, a coupled fluid–structure model based on a direct numerical scheme is developed in this work. The governing equations of magnetic field, fluid flow field and particle motion are simultaneously solved using an Arbitrary Lagrangian–Eulerian method, taking into account magnetic and hydrodynamic interactions between particles in a fully coupled manner. The accuracy of the proposed method is validated using the magnetic particulate flows of two particles under a uniform magnetic field as the test problem and is then applied to investigate effects of magnetic and hydrodynamic interactions between particles on the particle motion behavior. Results show that neighboring magnetic particles are easy to form chain-like clusters along field direction due to magnetic interactions between particles and then move together toward the surface of magnetic source under the action of gradient magnetic force. More importantly, it has been found that both magnetic and hydrodynamic interactions between particles are conducive to the acceleration of particles and the chain formation of particles. The present method and results could help in understanding the basic mechanism underlying the low-gradient magnetophoretic separation process and designing magnetic aggregate-based microfluidic devices.

回転機の始動特性解析のための運動方程式を考慮した時間領域並列有限要素法

This paper proposes the time domain parallel finite-element method (TDPFEM) coupling with equation of motion to efficiently obtain transient solutions of electric machines using parallel computing. The proposed TDPFEM is based on an iterative procedure combined with the Newton-Raphson iteration for a nonlinear magnetic field problem and equation of motion for a rotor. The scalability of the proposed method is examined in the starting transient analysis of a cage induction motor on large-scale parallel computing environment.

Simulation of forming process of Z-pinch dynamic hohlraum based on the program MULTI2D-Z

The radiation hydrodynamics code MULTI-2D, which was developed by Ramis et al. in 2009 (2009 Comput. Phys. Commun. 180 977) and adopted the single temperature fluid and unstructured lagrangian mesh, is modified into a radiation magnetohydrodynamics code MULTI2D-Z by adding the program module of evolution equation of magnetic field, and self-consistently considering the Lorentz force in the module of motion equation and the Ohmic heating in the module of energy equation. The newly developed module for magnetic field was validated to be reliable. The module is used to study the magnetic field diffusion process, and it is found that the diffusion is weakened due to the increasing of plasma temperature and density and the fluid convection, in which there is minus grads of velocity in radial direction. The new code MULTI2D-Z is used to simulate the formation process of dynamic hohlraum driven by tungsten wire-array Z-pinch at an 8 MA current level. The obtained results are that X-ray power and energy are, respectively, ～30 TW and ～300 kJ, radiation temperature in foam is ～120 eV, and the implosion trajectory of wire-array is also obtained. The calculated results reveal that the magnetic field is mainly distributed in the outside of tungsten plasma during the hohlraum formation. The foam expands due to the radiation heating from the shock wave created by the collision between wire-array plasma and the foam. The thermal radiation wave, which is characterized by radiation temperature, spreads towards the central axis faster than the plasma temperature. When the thermal radiation wave spreads to the central axis, the radiation temperature becomes comparatively uniform in space, and is almost equal to the plasma temperature except at the place of the shock wave. These results help the people to better understand the magnetic field diffusion and convection in Z-pinch, as well as the formation mechanism of dynamic hohlraum driven by wire-array Z-pinch. It is also indicated that the newly developed code MULTI2D-Z can be considered as a new tool for simulating Z-pinch and its applications, such as inertial confinement fusion and magnetically accelerated flyer plates.

Modeling plastic deformation effect on the hysteresis loops of ferromagnetic materials based on modified Jiles-Atherton model

Plastic deformation is one of the most important features that affect the hysteresis magnetic properties of steels, because it changes the dislocation density and affects domain-wall movement and pinning. In order to model the effect of plastic deformation on the magnetic properties, the prevailing Jiles-Atherton (J-A) theory is extensively used. However, the J-A models in a series of papers published by Jiles et al. are not completely consistent. As a result, there exists no uniform formula of magneto-plastic model established by different researchers, based on different J-A models, and various versions given by different mathematic expressions of magneto-plastic model often create difficulty in discriminating the accuracies and effectivenesses of the analyzed results. Therefore, it is necessary to establish an accurate and reasonable magneto-plastic model. In this paper, on the basis of magnetization mechanism of ferrimagnet and plastic deformation model, the effects of plastic deformation on the magnetic characteristic parameters adopted in magneto-plastic model, such as dislocation density, pinning coefficient and scaling constant, are analyzed and the relationship between them is first established. Then, by contrasting the fitting formula of the anhysteretic magnetization curve, the energy conservation equation and the effective magnetic field equation established by different researchers, several queries are proposed, and the irrationality and inaccuracy of the existing magneto-plastic model are elucidated, such as the mixing of anhysteresis magnetization and magnetization, the unreasonably regarding the irreversible magnetization energy as actual total magnetization energy. Thus, the energy conservation equation, the effective magnetic field equation and the anhysteretic magnetization equation are modified, and the differential expression of the magneto-plastic model is re-derived finally. Comparing the results calculated by the existing magneto-plastic models with the experimental results, it is seen indeed that a more sharp change of magnetization appears at small plastic deformation, then, the values of magnetization decrease more slowly with the increase of plastic deformation than those from the models respectively proposed by Li Jian-Wei, Leng Jian-Cheng and Wang Zheng-Dao; the saturation magnetization and residual magnetization decrease with the increase of plastic deformation, the coercive force is increased oppositely and the trend to reach the saturation magnetization becomes gentler, which is more exactly consonant with experiment observation than that calculated by the Sablik's model; additionally, the hysteresis loops of the plastically deformed carbon-steel samples calculated by the modified magneto-plastic model are also in better agreement with the experimental results than those from the existing models. Consequently, the modification is effective, and the modified magneto-plastic model is more accurate to simulate the plastic deformation effect on the magnetic property of ferromagnetic material.