Vector Potential Equation Research Articles

Analytical prediction of electromagnetic performance for surface-embedded permanent magnet in-wheel machines considering iron’s nonlinearity

Accurate magnetic field calculation is the premise of electromagnetic performance prediction. Conventional subdomain (SD) techniques assume that the iron’s relative permeability is infinite, leading to falsely overestimated flux density. We propose an accurate magnetic field analytical model for permanent magnet (PM) in-wheel machines considering iron’s magnetization nonlinearity and saturation. Specifically, according to the excitation source and topology, the entire solution domain of the machine is divided into sub-regions such as stator slots/teeth, stator slot-openings/tooth-tips, air-gap, and rotor slots/teeth, etc. Poisson’s or Laplace’s magnetic vector potential (MVP) equations are solved using Maxwell’s electromagnetic theory and complex Fourier series methods in each sub-region. Specifically, in our approach, The Cauchy product theorem addresses the discontinuous magnetic permeability change at the slot and tooth interface. The machine’s magnetic saturation effect is considered by combining the actual magnetization characteristics of iron with an iterative algorithm. The general solution for the MVP is solved using the boundary conditions between adjacent subregions. Subsequently, electromagnetic properties such as air-gap flux density, back electromotive force (EMF), and electromagnetic torque are obtained. The accuracy of the analytical model is verified by finite element analysis (FEA) and prototype tests, which proved that the proposed analytical model can consider the iron’s nonlinearity and the magnetic saturation. In addition, the inaccurate overestimation of electromagnetic torque and air-gap magnetic flux density by the conventional SD techniques has also been proven.

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.

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.

Modeling and investigation of crack evaluation in metal tube based on magnetic field analysis method

In this paper, an analytical model of eddy current testing of a metal tube has been proposed to evaluate circumferential cracks from the signals of the magnetic field. First, based on Maxwell’s equations in time-harmonic form, magnetic vector potential equations for each region of space are built under two different configurations of the bobbin probe and the encircling probe respectively. Next, the magnetic vector potentials are solved by imposing conditions at the interface. Then, series expressions of the radical and axial magnetic induction intensities have been obtained. In addition, based on the proposed modeling, the influences of cracks at different locations on the surface magnetic field have been analyzed. The results have shown that the radial magnetic induction intensity increases while the axial magnetic induction intensity decreases for the internal and external cracks with the same shape and size on the surface of the metal tube. Moreover, the magnetic induction intensity has shown much more sensitive property to external cracks than it has done to internal cracks for the encircling probe. However, for the bobbin probe, the magnetic induction intensity has demonstrated almost the same sensitivity in detection for both internal and external cracks in the metal tube. The researching results have important theoretical and practical significance for understanding the physical process of eddy current testing, optimizing parameters of the experimental system and building the inversion model.

Analytic Preconditioners for Decoupled Potential Integral Equations and Wideband Analysis of Scattering From PEC Objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

Theoretic impetuses and lengths of Feinberg-Horodecki equation

The exact solution of the Feinberg-Horodecki equation for time-dependent harmonic vector potential has been investigated under a one-dimensional system. The quantized momentum and its corresponding un-normalized wave functions were explicitly obtained. The Fisher information (for time and momentum) and variance (for time and momentum) were calculated using expectation values of time and momentum via Hellman-Feynman Theory (HFT). The time and momentum Shannon entropy were obtained using an existing formula. Numerical results were for time and momentum Fisher information to confirm the Cramer-Rao inequality. Another numerical results were obtained for time and momentum Shannon entropy to confirm Bialynick-Birula, Mycielski (BBM) inequality. The effects of the potential parameters such as mass of the spring and the frequency on the theoretic quantities were fully examined. The new variance inequality was established using the inequalities of Fisher information. The established inequality was confirmed by numerical results which also satisfied the popular Cramer-Rao inequality. The theoretic impetuses for Fisher information, variance and Shannon entropy respectively, were calculated and their variation with some potentials were studied.

An exact, time-dependent analytical solution for the magnetic field in the inner heliosheath

We derive an exact, time-dependent analytical magnetic field solution for the inner heliosheath, which satisfies both the induction equation of ideal magnetohydrodynamics in the limit of infinite electric conductivity and the magnetic divergence constraint. To this end, we assume that the magnetic field is frozen into a plasma flow resembling the characteristic interaction of the solar wind with the local interstellar medium. Furthermore, we make use of the ideal Ohm’s law for the magnetic vector potential and the electric scalar potential. By employing a suitable gauge condition that relates the potentials and working with a characteristic coordinate representation, we thus obtain an inhomogeneous first-order system of ordinary differential equations for the magnetic vector potential. Then, using the general solution of this system, we compute the magnetic field via the magnetic curl relation. Finally, we analyze the well-posedness of the corresponding Dirichlet-type initial-boundary value problem, specify compatibility conditions for the initial-boundary values, and outline the implementation of initial-boundary conditions.

On the Low-Frequency Behavior of Vector Potential Integral Equations for Perfect Electrically Conducting Scatterers

Low-frequency behavior of vector potential integral equations (VPIEs) for perfect electrically conducting (PEC) scatterers is investigated. Two equation sets are considered: the first set (VPIE-1) enforces the tangential component of the vector potential on the scatterer surface to be zero and uses the fundamental field relationship on its normal component. The second set (VPIE-2) uses the same condition as VPIE-1 for the tangential component of the vector potential but enforces its divergence to be zero. In both the sets, unknowns are the electric current and the normal component of the vector potential on the scatterer surface and are expanded using the Rao–Wilton–Glisson (RWG) and pulse basis functions, respectively. To achieve a conforming discretization, RWG, scalar Buffa–Christiansen (BC), and pulse testing functions are used. Theoretical and numerical analyses of the resulting matrix systems show that the electric current obtained by solving VPIE-1 has the wrong frequency scaling and is inaccurate at low frequencies.

An Algorithm for Fast Simulation of CW Illuminator

The continuous wave (CW) illuminator is aimed to simulate high-altitude nuclear electromagnetic pulse. In this article, a simulation algorithm is proposed covering the method of moments and finite difference method (FDM), with the high-speed simulation of the CW illuminator with arbitrary ellipse size, arbitrary loading, and arbitrary polarization direction. Based on the surface electric field equation of the CW illuminator, the mathematical model is established with the vector potential equation and the boundary conditions. In this model, the pulse function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> function are set as the basis function and the weight function, respectively, for the establishment of the current equation, with the current distribution determined by the simplification equation with image theory and nonuniform FDM. Finally, as the field distribution is determined by the superposition theorem. The simulation algorithm is verified by the self-built CW illuminator.

Analytical model of a fractional slot double‐layer‐winding vernier permanent magnet machine

AbstractThis study presents an analytical model (ANA) for predicting magnetic field and performances of a fractional slot (FS) double‐layer‐winding (DLW) vernier permanent magnet (VPM) machine. Magnetic vector potential equations of a FS DLW VPM machine are analytically solved by the technique of variable separation in three subdomains, that is, regions of permanent magnets (PM), air‐gap, and slots in 2‐D polar coordinate system. Field distributions and performances, such as flux linkage, electromotive force (EMF), torque and inductance of a FS DLW VPM machine are developed. Results from ANA of an 18‐slot/2‐stator‐pole‐pair FS DLW VPM machine prototype have been compared with those from finite element analysis (FEA), which indicates that ANA has a huge advantage in the computing speed in addition to a good accuracy. In the end, a prototype is built and its EMF and torque are tested in experiment platforms in which the results verify ANA. The necessity and importance of the research is that the model is an analytical method which provides a close solution of magnetic field and performances of a FS DLW VPM machine and its calculation time is much less than FEA on the base of a good precision.

Design and test validation of a novel permanent magnet eddy current brake

Aiming at slippage and belt breakage of large mechanical conveyor belts, a new type of water-cooled permanent magnet brake (WPMB) is proposed. In order to determine the braking performance of WPMB, a multi-physics coupling model and analysis method of magneto-thermal-structure are built. The braking torque and temperature characteristics of WPMB under different working conditions are obtained. Based on the magnetic vector potential equation, an electromagnetic eddy current model is established, and the variation trend of braking torque with speed is predicted. The temperature field model is built based on the equivalent thermal network method. Using multi-field coupling analysis method and 3-D FEM, the influence factors of WPMB braking torque are analyzed. Two test prototypes with rated speed of 78 rpm, braking torque of 3000 N⋅m are designed and developed. Braking torque characteristic tests and magnetic shielding characteristic tests at different speeds and continuous braking characteristic test are carried out. Experiments show that the proposed method provides very practical results to ensure the continuous and stable operation of WPMB.

Three-Dimensional Analytical Calculation and Optimization of Plate-Type Permanent Magnet Electrodynamic Suspension System

With the development of rare earth permanent magnet (PM) materials, more and more attention has been paid to the permanent magnet electrodynamic suspension (PM-EDS) structure. In this paper, the plate-type PM-EDS structure with an adjustable deflection angle of PM is researched, and the size of the PM is optimized. First, the second-order vector potential (SOVP) equation is established, and the analytical solutions of the electromagnetic forces and magnetic fields of the PM-EDS structure are obtained by analytical calculation. Secondly, the accuracy and reliability of the analytical model are verified by comparing the analytical calculation results with the finite element analysis (FEA) calculation results. Finally, the multi-objective particle swarm optimization (MOPSO) algorithm is used to optimize the size of the PM. The optimization results prove that the lift-to-drag ratio and lift-to-weight ratio of the suspension system are improved. The research results can provide a theoretical basis for the design of a PM-EDS train.

A robust numerical method for drawing the fluid flux-barrier geometry in the inner and outer rotor radial flux synchronous reluctance machines based on the finite difference method

PurposeRotor shape optimization is crucial in designing synchronous reluctance machines (SynRMs) because the machine performance is directly proportional to the rotor’s magnetic saliency ratio. The rotor geometry in synchronous reluctance machines is complex, and many geometrical parameters must be optimized. When fluid flux-barrier geometry is desirable, using analytic equations to prepare the rotor geometry for finite element analysis could be tedious. This paper aims to provide a robust numerical procedure to draw the fluid flux-barrier geometry in transversally laminated radial flux inner and outer rotor SynRMs by directly solving the magnetic vector potential equation using the finite difference method..Design/methodology/approachIn this paper, the goal is to have a robust procedure for drawing the rotor geometry for an arbitrary number of slots (Ns), poles (p) and flux-barrier layers (Nfb). Therefore, this paper targeted several combinations to investigate the performance of the proposed algorithm. The MATLAB software is used to implement the proposed algorithm. The ANSYS Maxwell software is used for counterpart finite element simulation to check the correctness of the results derived by the proposed method.FindingsSeveral inner and outer rotor SynRMs considering a different number of poles and a different number of flux-barrier layers per pole are studied to investigate the performance of the proposed algorithm. Results corresponding to each case are presented, and it is shown that the method is robust, flexible and fast enough, which could be used for the generation of the rotor geometry for the finite element analysis effectively.Originality/valueThe value of the proposed algorithm is its simplicity and straightforwardness in its implementation for the preparation of the rotor geometry with the desired fluid flux-barrier layer curvature resolution suitable for the finite element analysis. The procedure presented in this paper is based on the ideal magnetic loading concept, and in future works, a similar idea could be used for linear and axial flux SynRMs.

Electromagnetic Fields around Black Holes in Einstein Æther Gravity

Axial symmetry and stationary properties of spacetime allow to find exact analytical solutions of differential equations describing fields and particles in a gravitational background. The present work is mainly devoted to derivation of exact solutions of Maxwell’s equations for magnetic fields generated by current loops around static black holes (BHs) in Einstein-aether gravity based on the spacetime symmetries in both regions: (i) interior and (ii) exterior to the current loop for a proper observer. The spacetime symmetries are applied in separating variables to solve the second order ordinary differential equation for vector potential of electromagnetic field and the equations of motion of test particles around the aether BH. We also study effects of the aether field on innermost stable circular orbits (ISCOs) of the test particles assuming the current loop position is placed there. It is obtained that the ISCO radius, as well as dipole magnetic moment of the current loop decrease with the increase of the aether parameter c14. Moreover, the performed analysis indicates that the aether field causes a decrease in the magnetic field inside and outside the current loop due to the change of its position.

Effect of various physical properties on the reflection coefficients of inhomogeneous waves at the stress-free surface of partially saturated soils induced by obliquely incident fast P-wave

Ghasemzadeh and Abounouri [1] developed a mathematical model of partially saturated soils that is solved using the potential method, which decomposes elastodynamics equations into two standard wave equations, a scalar wave equation for scalar potential and a vector wave equation for vector potential. In such a medium, four waves exist three longitudinal and one shear. Each fluid phase tortuous path is taken into account in this model. The inertial coupling between solid and fluid particles is considered. Furthermore, both open-pore and sealed-pore boundaries are explored to investigate the reflection phenomenon at the surface of partially saturated soils. For both boundaries, the reflection coefficients of inhomogeneous waves at a partially saturated soil surface are found as a non-singular set of linear equations. All waves (both reflected and incident) in partially saturated soils are pronounced as inhomogeneous due to viscosity in pore fluids (i.e., distinct directions of attenuation and propagation). The energy shares of reflected waves are determined using an energy matrix. A numerical example is used to determine the reflection coefficients and the distribution of incident energy among the various reflected waves. The effect of different physical features on reflection coefficients and incident energy partitioning is illustrated graphically. The conservation of incident energy at the surface of partially saturated soils is mathematically confirmed at all angles of incidence.

Optimal design of Axial-Flux Induction Motors based on an improved analytical model

This paper presents an improved analytical model for axial-flux induction motors (AFIMs) based on solving the magnetic vector potential equations. The advantage of the proposed approach is its fast simulation time (fractions of a second against minutes/hours in FEA) while maintaining over 95 % accuracy compared to FEA that makes it suitable for an optimization purpose. This method applies a uniform distribution of the current density in the rectangular shaped rotor/stator slots. The equivalent circuit parameters of the studied AFIMs are extracted using the magnetic energy analysis. The proposed model allows the saturation effect in calculation of the motor performance parameters (power factor and efficiency). The accuracy of analytical model is validated using FEA results. A comprehensive design optimization is presented to show its capability for an optimization purpose. The objective functions in this study are maximum efficiency, maximum power factor, maximum efficiency times power factor, minimum mass, minimum price, maximum efficiency over slip, and maximum power factor over slip. The proposed analytical model can be easily used for AFIMs with different number of poles and phases.

A Numerical Study on the Influence of an Axial Magnetic Field (AMF) on Vacuum Arc Remelting (VAR) Process

A comprehensive numerical model is proposed to study the influence of an axial magnetic field (AMF) on the solidification behavior of a Titanium-based (Ti–6Al–4V) vacuum arc remelting (VAR) ingot. Both static and time-varying AMF are examined. The proposed 2D axisymmetric swirl model includes calculating electromagnetic and thermal fields in the entire system composed of the electrode, vacuum plasma, ingot, and mold. A combination of vector potential formulation and induction equation is proposed to model the electromagnetic field accurately. Calculations of the flow in the melt pool and solidification of the ingot are also carried out. All governing equations are presented in cylindrical coordinate. The presence of a weak AMF, such as the earth magnetic field, can dramatically influence the flow pattern in the melt pool. The “Electro-vortex flow” is predicted ignoring AMF or in the presence of a time-varying AMF. However, the flow pattern is “Ekman pumping” in the presence of a static AMF. The amount of side-arcing has no influence on the pool depth in the presence of an AMF. Modeling results are validated against experiments.

The formulation of vector potential evolution equation on rapidly rotating and accreting neutron star in the frame of Zero Angular Momentum Observers (ZAMO)

The equation for the dynamics of the magnetic field can explain the decrease in the magnetic field due to accretion. Maxwell’s relativistic equations formulate these equations. The formulation of the solution to the Maxwell equation is said to be correct if it is by the Schwarzschild space-time case by formulating the vector potential evolution equation. Maxwell’s second relativistic equation, the covariance tensor of the electromagnetic field, and the vector potential are used to obtain the equation for the evolution of the vector potential. All equations are calculated in the Zero Angular Momentum Observers (ZAMO) framework. The purpose of this research is to formulate the vector potential evolution equation for each component in a rapidly rotating and accreting neutron star. The results obtained have covariant terms with the slow rotating neutron star. However, the equation for a rapidly rotating neutron star has a more complex description. There are terms that cannot be simplified, such as in slowly rotating neutron stars.

Three-Dimensional Two-Temperature Modeling of Ar Loop-Type Inductively Coupled Thermal Plasma for Surface Modification

In this paper, numerical calculations were made for Ar loop-type inductively coupled thermal plasma (loop-ICTP). The loop-ICTP was developed originally by the authors’ group for rapid surface modification of large areas. Loop-ICTP is sustained with a unique three-dimensional (3D) configuration inside a circular loop quartz tube and on the substrate. A 3D and two-temperature thermofluid thermal plasma model was adopted for this calculation. Mass, momentum, and energy conservation equations were solved using a Maxwell equation for vector potential, an electron energy transport equation, and Saha’s equation in the 3D space. Results indicate that Ar loop-ICTP can be sustained and formed in the loop tube and also on the substrate. Moreover, the heavy particle temperatures reaches 1800–2000 K, whereas the electron temperature is about 10,000 K. Loop size effects on the gas temperature and gas flow field were also investigated using the developed model. Results show that adoption of a larger loop tube can be expected to improve the plasma uniformity on the substrate when applied to rapid surface modification.

ON THE SIMILARITY OF PLANE PULSED MAGNETIC FIELDS CONTINUED FROM DIFFERENT COORDINATE AXES

Purpose. The purpose of this work is formulation of similarity conditions for plane magnetic fields at a sharp skin-effect continued in non-conducting and non-magnetic medium from different axes bounding plane surfaces of conductors. Methodology. Classic formulation of Cauchy problem for magnetic vector potential Laplace equations, mathematic physics methods and basics similarity theory are used. Two problems are considered: the problem of initial field continuation from one axis and the problem of similar field continuation form other axis on which magnetic flux density or electrical field strength in unknown. Results. Necessary and sufficient similarity conditions of plane pulsed or high-frequency magnetic fields continued from different axes of rectangular coordinates are formulated. For the given odd and even magnetic flux density distributions on axis of initial field corresponding the distributions on axis and solution of continued similar field problem are obtained. Originality. It is proved that for similarity of examined fields the proportion of corresponding vector field projections represented by dimensionless numbers in similar points of axes is necessary and sufficient.