Magnetic Vector Potential Research Articles

Closed-Form Spectral-Domain Green's Functions for Infinitesimal Current Source in Multilayer Soil

This article deals with an infinitesimal current source (ICS) in a multilayer conducting medium that consists of air and multilayer soil. Closed-form kernel functions of the spectral-domain Green's function for scalar electric and vector magnetic potentials (Lorentz potentials) were obtained for every layer of the multilayered medium as analytic solutions of the systems of linear equations. Validation of the proposed closed-form expressions is performed by numerical solving of the system of linear equations and by numerical comparison with expressions derived from existing closed-form expressions developed for the source layer of the multilayer dielectric medium. For an arbitrary-oriented ICS, scalar electric potential of its horizontal component is chosen as a unique scalar electric potential. Accordingly, dyadic Green's function of vector magnetic potential is corrected.

Active Demagnetization Fault Compensation for Axial Flux Permanent-Magnet Synchronous Machines Using an Analytical Inverse Model

In this article, a fault tolerant deadbeat controller is proposed, which is able to compensate for both partial and uniform demagnetization faults in axial flux permanent-magnet synchronous machines with double rotor, even with asymmetric defects in both rotors. For this purpose, the system model used by the deadbeat controller is equipped with a look-up table of the back-emf, in function of the mechanical rotor position. This look-up table is generated by means of an analytical model based on the magnetic vector potential. With this information, the deadbeat controller can eliminate both the additional bias and ripple in the stator current components caused by demagnetization faults. The proposed control strategy is validated on a 4 kW test set-up of an axial flux permanent-magnet synchronous machine with yokeless and segmented armature topology fed by a three-phase two-level voltage source inverter, both by simulations and experiments. Key performance indicators concerning bias and ripple of the current components illustrate the superior performance of the proposed control strategy in comparison to a standard deadbeat controller without an up-to-date look-up table of the back-emf.

Mathematic Modelling and Numerical Analysis for a Novel Inner-Type Nutation Magnetic Drive

This paper presents an inner-type of magnetic gears for non-contact driving, based on the nutation gearing principle. An improved three-dimensional analytical model, based on the magnetic vector potential of rectangular permanent magnets, via the equivalent current, is deduced for the proposed nutation magnetic gears. The transformation of coordinates is applied for a global field solution. The analytical model and the finite element model have analyzed the output torque of the nutation magnetic drive, respectively. The results show that the values calculated by the two models are consistent. The proposed analytical model can provide a reference for the design and optimization of nutation magnetic gears and other permanent magnetic mechanisms with rectangular magnets.

An Efficient and Robust Mixed-Gauge Scheme Based on Current-Splitting for Nonlinear Magnetostatic Field Computation

In the edge element discretization for magnetostatic problems, a gauge scheme, such as tree gauge, Lagrangian multiplier (LM) gauge, and auto gauge, is usually adopted to handle the singular curl-curl equation in terms of the magnetic vector potential. However, tree-gauge and LM-gauge schemes are not very efficient as a nonlinear problem with many increments has to be resolved. In order to overcome those difficulties, a mixed-gauge scheme based on current-splitting is proposed in this article. First, we make an improvement for LM-gauged formulation and adapt it to calculate the initial increment with a specially designed CG solver; then split the discrete source current density into a compatible part and incompatible part based on discrete Helmholtz decomposition; auto gauge cooperates with the compatible part to calculate the following increments. The mixed-gauge scheme combines the advantages of LM-gauge and auto-gauge schemes, which avoids the step to determine the source term, and at the same time, has high computational efficiency. The mixed-gauge scheme is tested in some examples against some other gauge schemes and found to be efficient and robust, where others possibly converge slowly or fail to produce a reliable solution.

Marching On-In-Time Unstructured PEEC Method for Electrically Large Structures with Conductive, Dielectric, and Magnetic Media

The Marching On-In-Time (MOT) unstructured Partial Element Equivalent Circuit (PEEC) method for time domain electromagnetic problems is presented. The method allows the transient analysis of electrically large electromagnetic devices consisting of conductive, dielectric, and magnetic media coupled with external lumped circuits. By re-formulating PEEC following the Coulombian interpretation of magnetization phenomena and by using electric and magnetic vector potentials, the proposed approach allows for a completely equivalent treatment of electric and magnetic media and inhomogeneous and anisotropic materials are accounted for as well. With respect to the recently proposed Marching On-In-Time PEEC approach, based on the standard (structured) discretization of PEEC, the method presented in this paper uses a different space and time MOT discretization, which allows for a reduction in the number of the unknowns. Analytical and industrial test cases consisting in electrically large devices are considered (e.g., the model of a Neutral Beam Injector adopted in thermonuclear fusion applications). Results obtained from the simulations show that the proposed method is accurate and yields good performances. Moreover, when rich harmonic content transient phenomena are considered, the unstructured MOT–PEEC method allows for a significant reduction of the memory and computation time when compared to techniques based on Inverse Discrete Fourier Transform applied to the frequency domain unstructured PEEC approach.

Rapid 3-D Magnetic Integral Field Computation of Current-Carrying Finite Arc Segments With Rectangular Cross Section

The computation of 3-D magnetic fields is a demanding task in the analysis of electrical machines and other electromagnetic devices. In this context, integral field calculation provides a smooth solution, high precision and resolution, “on-demand”-calculation, and an origin-based formulation of the magnetic field and the magnetic vector potential. However, conventional elliptic methods lead to huge parallelizable computing efforts and significant errors. In this article, a 3-D generic current-carrying arc segment with rectangular cross section is studied. A new analytic formulation is proposed to speed up the computation of magnetic fields and reduce the error by more than three orders of magnitude. In addition, the proposed magnetic vector potential expression has a similar accuracy as numerical integration. In fact, a significant reduction of the error level has been showcased clearly with respect to the existing approaches. This article is promising for improving the design methodology and optimization of large superconducting dipole magnets or arched end-winding geometries of large electrical machines.

Calculation of Active Material’s Torque Contributions for a Flux Switching Machine

A new calculation method for the torque contribution of an arbitrary point or element in an electric machine's geometry is developed. The local torque contributions (LTCs) and their visualizations provide insights into the iron utilization and possible machine geometry improvements. The flux tubes created from the magnetic vector potential's equipotential lines are interpreted as a set of local magnetic circuits (LMCs). Each LMC that crosses the air gap contributes to the machine's torque, which is quantifiable by applying the Maxwell stress tensor to the LMC's air-gap intersections. The torque contributions are mapped to all finite elements belonging to each LMC, for each transient calculation's rotor step. Furthermore, the torque contribution data are briefly discussed and a harmonic analysis is performed, focusing on the mean and ripple components of the torque. Finally, a brief example of a topological optimization with the calculated torque contributions as optimization criteria is presented and evaluated.

Introduction and Validation of the APEC Code for the Simulation of Plasma Equilibrium and Evolution in the Presence of 3-D Passive Structures

Recently, a novel code for the simulation of the plasma equilibrium and its evolution has been developed and fully implemented in ANSYS. We named it ANSYS Plasma Equilibrium and Evolution Code (APEC). It is based on the solution of the Grad–Shafranov equation and exploits the finite-element method using the magnetic potential vector formulation. In this article, we present a further development on the coupling of the axisymmetric plasma equilibrium equations, describing the plasma behavior, with the full 3-D eddy current equations, defining the surrounding 3-D structures. One test case, related to the International Thermonuclear Experimental Reactor (ITER) tokamak, has been fully analyzed and benchmarked with a validated code showing the validity of the proposed approach. In particular, the benchmark has concerned the comparison of the initial equilibrium and plasma evolution.

Analyses of the coupling model combining field and circuit equations based on solenoidal coils for wireless power transfer

A method using the coupling model combining the field and circuit equations is put forward to describe the wireless power transfer (WPT) system in this study. Hollow solenoidal resonant coils are usually excited by alternating currents in practice, doing studies related to the solutions to the solenoidal coils in alternating electromagnetic fields is important. In this study, the analytic expressions of the alternating magnetic vector potential are obtained from Maxwell's formulas for time-harmonic electromagnetic fields, and the mathematical expressions of the electric field intensity, the magnetic flux density, the induced voltage, and the mutual inductance are established for the coaxially symmetric solenoidal coils. The field-circuit coupling model is built for WPT, and the relationship between the input and output variables is obtained. The equations contain the dielectric information of the transmission space and the coils' radii and turn numbers, and can be used for analysing the dielectric effects of the space at lower or higher frequencies. The results are verified by experiments of 2-coil and 4-coil structures. The advantages of being able to analyse the distribution of the fields are illustrated by a comparison experiment with a laterally displaced coil, which is valuable to WPT design.

3-D Numerical Modeling of Claw-Pole Alternators With its Electrical Environment

This article describes a methodology for modeling a six-phase claw-pole alternator with its electrical environment. Magnetic nonlinearities, eddy currents, and rectifiers are taken into account. To solve magnetodynamic problems, we use the modified magnetic vector potential formulation. The complex structure of the machine requires a 3-D finite-element analysis. To limit the mesh size, we introduced a refinement strategy based on the calculation of the time derivative of the magnetic vector potential, a solution to the magnetostatic case. In addition, we propose to reduce the transient state by improving the initial solution from the solution of a magnetostatic problem. These different numerical techniques drastically reduce the computational time and memory resources. To validate the proposed approach, some results are compared with experimental ones.

Performance Improvement of a Direction Finding System Antenna Using Method of Moment (MOM)

This paper focuses on the Performance Improvement of a Direction Finding System Antenna Using Method of Moment (MoM) Approach. The work is developed to provide an approximate current distribution for a direction finding system antenna by employing the use of Method of Moment on an array of Yagi-uda antenna. The parameters of the experimental antenna are derived and analyzed via Magnetic Vector Potential (MVP) operator. The accurate current flowing through the radiating elements of the direction finding system is analyzed using combination of Method of Moment technique and Magnetic Vector Potential (MVP) operator. This helps to avoid the detection of false alarms and inability of the system to detect remote targets. A typical direction finding system Yagi antenna is designed and operated at a frequency range of 0.6- 0.8 GHz. The antenna has a single reflector, an active (driven) element and three (3) parasitic directors. The antenna parameters are simulated using MatLab R2010a software tool. The average pointing vector of the designed Yagi antenna was obtained as 3.73watt per square metre, and Radiation Intensity value of about 9.400 coulomb per kilogram. The simulation results indicate an appreciable increase in directivity of 9.03dBi, an enhanced directive gain compared to that of the equivalent dipole antenna of 1.76dBi, signifying 7.27dBi enhancement.

Performance Improvement of a Direction Finding System Antenna Using Method of Moment (MOM)

This paper focuses on the Performance Improvement of a Direction Finding System Antenna Using Method of Moment (MoM) Approach. The work is developed to provide an approximate current distribution for a direction finding system antenna by employing the use of Method of Moment on an array of Yagi-uda antenna. The parameters of the experimental antenna are derived and analyzed via Magnetic Vector Potential (MVP) operator. The accurate current flowing through the radiating elements of the direction finding system is analyzed using combination of Method of Moment technique and Magnetic Vector Potential (MVP) operator. This helps to avoid the detection of false alarms and inability of the system to detect remote targets. A typical direction finding system Yagi antenna is designed and operated at a frequency range of 0.6- 0.8 GHz. The antenna has a single reflector, an active (driven) element and three (3) parasitic directors. The antenna parameters are simulated using MatLab R2010a software tool. The average pointing vector of the designed Yagi antenna was obtained as 3.73watt per square metre, and Radiation Intensity value of about 9.400 coulomb per kilogram. The simulation results indicate an appreciable increase in directivity of 9.03dBi, an enhanced directive gain compared to that of the equivalent dipole antenna of 1.76dBi, signifying 7.27dBi enhancement.

Study of Permanent Magnet Electrodynamic Suspension System with a Novel Halbach Array

In this paper, a permanent magnet (PM) electrodynamic suspension (EDS) system with a novel Halbach array is presented, to overcome the uncontrol of the levitation force in the PM EDS system. The novel Halbach array consists by winding normal conductor coils on permanent magnets surface. The 2-D analytical expression of the space static magnetic field is calculated using Biot-savart law and surface current method. When the vehicle-mounted novel Halbach array moves to cut the conductive plate, the space magnetic field equations are obtained by establishing equations of magnetic vector potentials using complex Fourier transform of space static magnetic field. The levitation force and drag force are derived using Maxwell stress tensor method. And a 2-D finite-element model is built to validate the accuracy of this proposed analytical model. Furthermore, the effects of the main characteristic parameters on the levitation force are analysed.

CFD and Experimental Investigation of Desulfurization of Rejected Electrolytic Manganese Metal in Electroslag Remelting Process

An innovative electroslag remelting furnace with a water-cooled electrode was introduced to recycle the rejected electrolytic manganese metal (EMM) scrap. To clarify the desulfurization process in the rejected EMM scrap, a transient three-dimensional comprehensive numerical model was elaborated. Using the magnetic potential vector approach, the respective electromagnetic fields were calculated via the Maxwell equations. The Lorentz force and the Joule heating fields were derived as phase distribution functions and interrelated via the momentum and energy conservation equations as source terms, respectively. The molten manganese metal droplet motion, as well as the fluctuation of the slag–metal interface, was described by the volume-of-fluid (VOF) approach. Besides, the solidification was modeled via the enthalpy-based technique. A thermodynamic module was established to estimate the sulfur mass transfer rate between the molten manganese metal and the molten slag. Furthermore, a factor related to the magnitude and frequency of the alternating current and the physical properties of the melt was introduced to include the electro-emulsification phenomenon. An experiment has been carried out with a commercial-scale ESR device. The predicted values of the slag temperature and sulfur content in the final manganese ingot were found to agree reasonably with the corresponding measured data. Under continuous melting of the rejected EMM scrap, molten manganese metal droplets are formed at the domain inlet, grow, and fall down. Highly conductive molten manganese metal droplets significantly change distributions of the current streamline, the Joule heating, and the Lorentz force around and within it. Moreover, droplets are inclined to rotate and move inside the mold. With the renewal of the slag–manganese interface, sulfur in the molten manganese metal is constantly transferred to the molten slag. With the applied current ranging from 3000 to 4000 A, the average sulfur content of the manganese ingot dropped from 0.0447 to 0.0291 pct, and thus, the desulfurization rate rose from 55.3 to 70.9 pct.

Modeling and stabilization of current-controlled piezo-electric beams with dynamic electromagnetic field

Piezoelectric materials can be controlled with current (or charge) as the electrical input, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect on properties of the control system such as stabilizability. Besides the consideration of current control, there are several new aspects to the model. Most importantly, a fully dynamic magnetic model is included. Also, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Hamilton’s principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a current (or charge) source at the electrodes. Two sets of decoupled system of partial differential equations are obtained; one for stretching of the beam and another one for bending motion. Since current (or charge) controller only affects the stretching motion, attention is focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb type gauge condition is used. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the natural energy space is obtained. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models.

An improved low magnetic Reynolds magnetohydrodynamic method based on computing induced magnetic vector potential by integrating induced current

Aming at the applicability of low magnetic Reynolds number method, in this paper we analyze the differences in the application of low magnetic Reynolds number condition and the limitation of full MHD method when it is applied to hypersonic flow. According to the low magnetic Reynolds number magneto-hydrodynamic control numerical simulation method, computing magnetic vector potential through integrating induced current, and considering the reduction of computation domain caused by truncation factors, we propose a low magnetic Reynolds number MHD computation method which is adjusted by the induced magnetic field, and the validation of this method is also presented. Through the numerical simulation of RAM-C blunt cone in flight test condition, we analyze the discrepancy caused by “neglecting induced magnetic field”, and also discuss the principle of the application of low magnetic Reynolds number assumption of hypersonic flow. The obtained results are as follows. (1) The adjusted computation method developed in this paper breaks through the limit of low magnetic Reynolds number, and expands the application range of low magnetic Reynolds number method to hypersonic flow, the numerical simulation result is reliable; Compared with direct integration of Biot-Savart law, the computation efficiency is considerably improved. (2) In the hypersonic flow, the influence of induced magnetic field is presented, thus weakening and distorting the applied magnetic field macroscopically, as a result weakening the effect of magnetic control to some extent. Under the condition of this paper, the low magnetic Reynolds number condition “<i>Re<sub>m</sub></i> < 0.1” is probably too conservative, and it is better to adopt <i>Re<sub>m</sub></i> < 1.0, and the characteristic conductivity and characteristic length should be chosen according to the actual plasma distribution.

Magnetic field and inductance of filament conductor segment model with current continuity

Traditional current-carrying model of filament conductor segment is the basic model for analyzing the magnetic field in a closed conductor loop. Although it does not satisfy the law of current continuity, it is still suitable for analyzing the magnetic field in the closed conductor loop. However, for the current distribution problem of some local multi-branch parallel-connection conductor segments in a closed conductor loop, the traditional model cannot fully reflect the interaction between magnetic fields of the local multi-branch parallel-connection conductor segments. For this reason, the displacement current model proposed in the existing literature satisfies the law of current continuity and solves the above problem well. But, there remain some problems to be solved, such as incomplete theory, inconsistency and complex formulae in the existing literature. In this paper, a conductive current-carrying model of filament conductor segment is proposed, which ensures current continuity inside, outside, and on two terminals of the filament conductor segment. The differential equation of total magnetic field with more profound physical connotation and the formulae of vector magnetic potential are derived in detail. Further, the formulae for calculating magnetic field energy and inductance of the conductive current-carrying model of filament conductor segment are proposed, which greatly reduces the computational complexity and makes up for the shortcoming of model in the existing literature. An example given in this paper verifies the correctness of the above theory and formulae.

Variational Principles of Micromagnetics Revisited

We revisit the basic variational formulation of the minimization problem associated with the micromagnetic energy, with an emphasis on the treatment of the stray field contribution to the energy, which is intrinsically non-local. Under minimal assumptions, we establish three distinct variational principles for the stray field energy: a minimax principle involving magnetic scalar potential and two minimization principles involving magnetic vector potential. We then apply our formulations to the dimension reduction problem for thin ferromagnetic shells of arbitrary shapes.

A novel efficient solver for Ampere's equation in general toroidal topologies based on singular value decomposition techniques

A new method is proposed to solve Ampere's equation in an arbitrary toroidal domain in which all currents are known, given proper boundary conditions for the magnetic vector potential. The novelty of the approach lies in the application of singular value decomposition (SVD) techniques to tackle the difficulties caused by the kernel associated by the curl operator. This kernel originates physically due to the magnetic field gauge. To increase the efficiency of the solver, the problem is represented by means of a dual finite difference-spectral scheme in arbitrary generalized toroidal coordinates, which permits to take advantage of the block structure exhibited by the matrices that describe the discretized problem. The result is a fast and efficient solver, up to three times faster than the double-curl method in some cases, that provides an accurate solution of the differential form of Ampere law while guaranteeing a zero divergence of the resulting magnetic field down to machine precision.

Air Gap and Edge Effect in the 2-D/1-D Method With the Magnetic Vector Potential ${A}$ Using MSFEM

Eddy currents (ECs) are simulated in a single laminate, representing the whole core of an electrical machine. Despite this drastic reduction in the complexity of the problem, a 3-D finite-element model turns out to be still too expensive for simulations. To overcome this difficulty, 2-D/1-D methods are used. This article presents a solution to consider both air gap and edge effect (EE) based on the multiscale finite-element method (MSFEM) using the magnetic vector potential (MVP) A. Linear material properties are assumed; therefore, this article is carried out in the frequency domain. The new 2-D/1-D MSFEM is discussed, and various simulation results are presented.