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Considering the influence of iron core saturation and complex boundary conditions, a novel no-load magnetic field analytical method and system multi-objective optimization model for interior permanent magnet synchronous motor (IPMSM) are proposed. Firstly, an analytical method based on magnetic potential permeance method is proposed. The complex cogging effect and equivalent air gap permeability function are fully considered in this method. Furthermore, on the basis of Taguchi optimization design based on Fuzzy theory, an improved optimization model considering multi-variable interaction is proposed. Among them, in order to comprehensively consider the performance of electromagnetic, vibration, and temperature rise, five optimization objectives including torque, pulsation, cogging torque, loss, and flux density have been optimized. The sensitivity of design variables and the interaction between variables are not ignored. Finally, a 6-pole 36-slot IPMSM is manufactured for prototype experiments. A large number of prototype experiments, finite element, and computational fluid dynamics simulation analysis including no-load magnetic field, back electromotive force, temperature rise, electromagnetic torque, and so on are carried out. The accuracy and rationality of the proposed no-load magnetic field analysis method and global optimization model have been well verified.

This paper investigates the air-gap field distribution of the permanent-magnet vernier machine (PMVM) using nonlinear exact conformal model (NECM) to account for slotting effect, flux modulation effect and iron nonlinearity. The exact conformal model based on the region of one-slot and one-flux-modulation-pole (OSECM) are introduced to show the effectiveness of linear analytical model for PMVM. It can keep high calculation accuracy and significantly reduce the computational burden. Then, the NECM is developed from OSECM by introducing the equivalent saturation current into the air region and coil region. The lumped parameter magnetic circuit model (LPMCM) model is used to obtain the magnetic potential of iron region and therefore calculate the equivalent saturation current. The NECM which combines LPMCM and OSECM can essentially improve the accuracy of linear analytical model. The harmonic analysis of air-gap field is performed to theoretically explain the component of electromagnetic torque. Both finite element model (FEM) simulation and test results are presented to validate the NECM.

The article presents a novel design of the axial active magnetic bearing with six poles. An analytical magnetic bearing model was developed to provide the axial magnetic induction distribution in 3D. The model utilizes magnetic vector potential formulation and Schwarz-Christoffel mapping. The paper covers in-depth deliberations on the end effect influence and the conjugate complex permeance function. Numerical model and simulations were provided in 3D mode with support of COMSOL Multiphysics software. The six pole axial active magnetic bearing was manufactured and investigated experimentally. Identification of the magnetic field distribution was carried out with a single axis magnetic induction sensor using custom automatic field scanner. High convergence of the modeling results and experimental research was demonstrated. The main advantage of the proposed analytical method is significantly shorter computation time compared to the numerical one that is useful from the modeling and controller study point of view. The configuration, modeling, simulation and experimental investigation results are well illustrated for the better overview.

This article proposes a new hybrid analytical model combining subdomain model and equivalent magnetic network for permanent magnet vernier machines, which are easy to get saturated due to the air-gap field modulation effect. The key is representing saturation effect by iterating equivalent surface current as part of interface conditions in the proposed subdomain model. First, five subdomains are divided to calculate magnetic field distribution. The equivalent surface current is utilized as the boundary constraint between subdomains and iron core. Second, the equivalent magnetic network is established based on stator structure, and the fluxes flowing into flux modulation poles are equivalent as the magnetic flux sources. Node magnetic potentials can be achieved with the application of Kirchhoff's law of current. Then, the equivalent surface current is updated for convergence by node magnetic potential to accurately predict saturated performance. Finally, a prototype is manufactured, and the effectiveness of the proposed hybrid model is verified by the finite-element analysis (FEA) and experimental results. The hybrid model combines the subdomain model and the equivalent magnetic network model to complement each other and it is demonstrated to be more efficient than the FEA model while maintaining high accuracy.

In this article, a new and general three-dimensional (3-D) analytical frame for evaluating the performance characteristics of the permanent magnet (PM) eddy-current coupling is presented using a magnetic vector potential formulation. Based on the 3-D Cartesian coordinate system, the actual coupling geometry is equated to a linear structure by applying the mean radius (for cylindrical conductor and PM rotors) and the equivalent cuboid (for PMs of various shapes) assumptions. The torque contribution from the overhangs of conductor back iron and the effects of PM and air gap thicknesses are all taken into considered. A verification process is implemented and it is shown that, in any case, the proposed 3-D analytical model can yield the results very close to those obtained by both the 3-D finite-element analysis and experimental measurement.

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation { − ( ∇ + i A ( x ) ) 2 u + ( λ V ( x ) + 1 ) u = ( 1 | x | μ ⁎ | u | p ) | u | p − 2 u x ∈ R N , u ∈ H 1 ( R N , C ) , where N ≥ 2 , λ > 0 is a real parameter, 0 μ 2 , i is the imaginary unit, p ∈ ( 2 , 2 ⁎ ( 2 ( N − μ ) 2 N ) ) , where 2 ⁎ = 2 N N − 2 if N ≥ 3 , 2 ⁎ = + ∞ , if N = 2 . The magnetic potential A ∈ L l o c 2 ( R N , R N ) and V : R N → R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω 1 , ⋯ , Ω k such that the interior of Ω j is non-empty and ∂ Ω j is smooth, then for λ > 0 large enough, the above equation has at least 2 k − 1 multi-bump solutions.

The inhomogeneous magneto-electro-elastic (MEE) coupling element-free Galerkin method (IMC-EFGM) is proposed for solving static behaviors of MEE structures. Plates made of homogeneous or inhomogeneous MEE materials are analyzed by IMC-EFGM, and the mechanical behaviors in different temperature fields are simulated. Displacement, magnetic potential and electric potential are studied using moving least squares (MLS) approximation. For inhomogeneous MEE materials, the constitutive equation in the MEE-thermal environment is first obtained by using Gibbs energy, and then used together with the Hamiltonian principle to determine the system governing equation. Since the MLS approximation does not satisfy the principle of Kronecker delta, the penalty function method is used to impose the natural boundary. The value of material properties at the point of integration is taken into account when calculating inhomogeneous materials. Compared with the analytical solution and finite element method (FEM), IMC-EFGM is straightforward and has greater precision than FEM. Some numerical examples are shown for the static FG-MEE plate with different hole shapes. Homogeneous MEE materials and inhomogeneous MEE materials are mainly used in intelligent structures and have good application perspectives. This research will promote the future use of inhomogeneous MEE materials.

The present paper investigates the scattering phenomenon of horizontally polarized Shear wave on an irregular surface of the piezomagnetic stratum lying over a piezomagnetic substrate. The expressions of frequency relation and group velocity of scattered Shear wave on the considered piezomagnetic stratum-substrate structure are deduced for magnetically open and magnetically short cases by inducing appropriate boundary conditions. A closed form expressions of reflected mechanical displacement and reflected magnetic potential function related to stratum and substrate are also determined for three different shaped irregularities (rectangular shaped, parabolic shaped and triangular shaped) by employing perturbation technique and residue theorem for both magnetically open and short cases. The efficacy of piezomagnetic coupling parameter, vertical irregular parameter due to presence of surface irregularities is portrayed on the phase velocity, group velocity, reflected mechanical displacement and reflected magnetic potential function related to the stratum for both magnetically open and short conditions graphically. A comparative study is also made to examine the impact of different shaped irregularities on the reflected mechanical displacement and reflected magnetic potential function of the stratum.

Purpose This study aims to accurately calculate the magnetic field distribution, which is a prerequisite for pre-design and optimization of electromagnetic performance. Accurate calculation of magnetic field distribution is a prerequisite for pre-design and optimization. Design/methodology/approach This paper proposes an analytical model of permanent magnet machines with segmented Halbach array (SHA-PMMs) to predict the magnetic field distribution and electromagnetic performance. The field problem is divided into four subdomains, i.e. permanent magnet, air-gap, stator slot and slot opening. The Poisson’s equation or Laplace’s equation of magnetic vector potential for each subdomain is solved. The field’s solution is obtained by applying the boundary conditions. The electromagnetic performances, such as magnetic flux density, unbalanced magnetic force, cogging torque and electromagnetic torque, are analytically predicted. Then, the influence of design parameters on the torque is explored by using the analytical model. Findings The finite element analysis and prototype experiments verify the analytical model’s accuracy. Adjusting the design parameters, e.g. segments per pole and air-gap length, can effectively increase the electromagnetic torque and simultaneously reduce the torque ripple. Originality/value The main contribution of this paper is to develop an accurate magnetic field analytical model of the SHA-PMMs. It can precisely describe complex topology, e.g. arbitrary segmented Halbach array and semi-closed slots, etc., and can quickly predict the magnetic field distribution and electromagnetic performance simultaneously.

In this paper, the improved and the new analytical and semi-analytical expressions for calculating the magnetic vector potential, magnetic field, magnetic force, mutual inductance, torque, and stiffness between two inclined current-carrying arc segments in air are given. The expressions are obtained either in the analytical form over the incomplete elliptic integrals of the first and the second kind or by the single numerical integration of some elliptical integrals of the first and the second kind. The validity of the presented formulas is proved from the particular cases when the inclined circular loops are addressed. We mention that all formulas are obtained by the integral approach, except the stiffness, which is found by the derivative of the magnetic force. The novelty of this paper is the treatment of the inclined circular carting-current arc segments for which the calculations of the previously mentioned electromagnetic quantities are given.

In this article, a novel pre-processing method is proposed for magnetic field approximation based on neural networks. The process of calculating magnetic fields based on neural networks is presented first. In order to save the training time and reduce the estimation error, an excitation distance function, which is inspired by an analytical formula of magnetic potential, is proposed to integrate the input data, such as the geometry, excitations, and boundaries. This pre-processing or roughly reduces the dimension of the input layer by three quarters, while the training process is accelerated by a more integrated input layer. Preliminary experiments show that the predicted results by the proposed pre-processing method are close to the ground truth, with a significant reduction of computation time compared with the traditional finite element method (FEM).

We prove that the spectrum of a periodic 3D magnetic Schrodinger operator whose electric potential $$V=d\mu/dx$$ is the derivative of a measure is absolutely continuous provided that the distribution $$d|\mu|/dx$$ is $$(-\Delta)$$ -bounded in the sense of quadratic forms with bound not exceeding some constant $$C(A)\in(0,1)$$ , and the periodic magnetic potential $$A$$ satisfies certain conditions, which, in particular, hold if $$A\in H^q_{\mathrm{loc}}(\mathbb R^3;\mathbb R^3)$$ for some $$q>1$$ or $$A\in C(\mathbb R^3;\mathbb R^3)\cap H^q_{\mathrm{loc}}(\mathbb R^3;\mathbb R^3)$$ for some $$q>1/2$$ .

To solve the problem that coupled magnetic flux and saturation of active magnetic bearing (AMB) cannot be taken into account in the traditional magnetic circuit calculation method together, this paper proposes a novel analytical calculation method based on distributed magnetic circuit method (DMCM). First, a magnetic circuit model with multiple magnetic circuits is built, we can obtain the initial flux density of each section by magnetic circuit calculation. Next, the magnetomotive force (MMF) of each magnetic circuit is calculated by the B-H curve of the ferromagnetic material. Then, the air-gap flux density under each magnetic pole center is obtained by further iterative calculations according to the magnetic potential error. On the basis, the flux density distribution along the air-gap circumference is obtained by one-dimensional (1D) relative permeance function, and thus the bearing capacity is derived and a loss calculation method is introduced. Finally, the finite element method (FEM) and experimental results show that the proposed method is feasible and effective.

A method proposed earlier, relying on the use of harmonic Cartesian polynomial and rational functions, is extended here to find a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a system of long parallel, non-intersecting, transversely isotropic elastic cylindrical electrical conductors. Results are presented for two conductors of equal circular normal cross-sections carrying currents of equal densities flowing along the same direction, subjected to Robin-type thermal boundary conditions. Quantities of practical interest are represented graphically and discussed. Consideration of a system of electrical conductors is of practical importance in power plants and in various technological instruments, where it is required to assess the interaction between conductors. The obtained formulas for the magnetic vector potential may be of importance for the determination of the coefficients of self- and mutual inductance of long electric conductors, otherwise difficult to calculate by standard methods. Comparing the results with those of a single conductor allows us to assess the interaction between conductors.

This paper mainly studies two-dimensional analytic method to analyze and calculate a short-secondary type slotless permanent magnet linear synchronous motor (SST-SPMLSM) performance. Due to the large air-gap and more magnetic flux leakage of electric machine, this paper proposes the groove-by-groove method to analyze the winding equivalent current density. Meanwhile, the volume current density is used to equivalent the permanent magnet magnetization, which is obtained the permanent magnet equivalent current density. According to the electric machine structure, the divided layers model of the SST-SPMLSM is built. Based on the hierarchical analytical model and equivalent current density of the electric machine, the two-dimensional magnetic vector potential of each region in the SST-SPMLSM is derived by the separated variable method. Combined with the boundary conditions of the electric machine, the magnetic flux densities of the SST-SPMLSM with different regions are derived. In the meantime, a 30 N SST-SPMLSM is designed. On this basis, the magnetic flux density, detent force, and electromagnetic thrust force of the SST-SPMLSM are analyzed and calculated, which of the results are verified by finite element method. From the results, it satisfies the design requirements of the SST-SPMLSM, and verifies the validity and correctness of analytical method.

Electrodynamic suspension with a permanent magnet Halbach array is suitable for ultrahigh-speed vacuum pipeline transportation. The optimization of a Halbach array is helpful to save cost and increase carrying capacity. Intricate analytical expressions of electromagnetic forces or time-consuming numerical simulation show inherent defects in the structural optimization. To solve the abovementioned problem, this article develops an analytical optimization by two stages. First, a brief approximate analytical expression of electromagnetic forces is established by omitting high-order harmonic components for both the scalar magnetic potential and vector magnetic potential. Second, the lift-to-weight ratio is taken as the optimization index, and the optimal wavelength-to-gap ratio and thickness-to-gap ratios are put forward by seeking extreme values. The numerical simulation is conducted to verify the validation of the analytical optimization, and the result shows that the optimal structural parameters are highly reliable. An experimental rotatory device for electrodynamic suspension with optimal sizes for a Halbach array is built to verify the validation of the analytical expressions of electromagnetic forces.

We study the uniform weighted resolvent estimates of Schrodinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on R 2 , we prove some new uniform weighted resolvent estimates for this 2D Schrodinger operator and, as applications, we show local smoothing estimates for the Schrodinger equation in this setting.

The accumulated large amount of satellite magnetic data strengthen our capability of resolving the electrical conductivity of Earth’s mantle. To invert these satellite magnetic data, accurate and efficient forward modeling solvers are needed. In this study, a new finite-element based forward modeling solver is developed to accurately and efficiently compute the induced electromagnetic field for a realistic 3D Earth. Firstly, the nodal-based finite element method with linear shape function on tetrahedral grid is used to assemble the final system of linear equations for the magnetic vector potential and electric scalar potential. The FGMRES solver with algebraic multigrid (AMG) preconditioner is used to quickly solve for the final system of linear equations. The weighted moving least-square method is employed to accurately recover the electromagnetic field from the numerical solutions of magnetic vector and electric scalar potentials. Furthermore, a local mesh refinement technique is employed to improve the accuracy of estimated electromagnetic field. At the end, two synthetic models are used to verify the accuracy and efficiency of our newly developed forward modeling solver. A realistic 3D Earth model is used to simulate the induced magnetic field at 450 and 200 km altitudes which are the planned flying altitudes of Macau’s geomagnetic satellites. The simulation indicates that (1) amplitude of the mantle-induced magnetic field can reach 10–30 nT at 450 km altitude, which is 10–30% of the primary magnetic field. The induced magnetic field at 200 km altitude has larger amplitudes. These mantle-induced magnetic fields can be measured by Macau geomagnetic satellites; (2) amplitude of the ocean-induced magnetic field can reach 5–30 nT at satellite altitudes, which needs to be carefully considered in the interpretation of satellite magnetic data. We are confident that our newly developed forward modeling solver will become a key tool for interpreting satellite magnetic data.

As the performance of an E-Core homopolar linear synchronous motor (HLSM) is affected by the air-gap reluctance, it can be enhanced by shaping its pole adroitly. Therefore, the primary design choice relates to the pole geometry, which can be simply studied by ignoring saturation and considering only the field excitation, This paper presents the theoritcal optimisation of three different track-pole shapes. This optimisation is based on a magnetic field study using the 3-dimensional finite-difference method (FDM) of scalar magnetic potential as well as the average of zero and infinite permeability boundaries. The shape which maximises the armature flux per pole will be thought of as the optimum shape. So, the flux per pole in the middle limb of the E-core is taken as a "goodness factor" for pole shape. It is found that the l-shape was the optimum track pole shape. The results of the experimental investigations which are carried out on a static model of this motor confirm the computed results.

In this article, a model based on finite element–boundary element methods (FEM–BEM) coupling is developed for the magnetic field computation in thin magnetic shells submitted to external magnetic fields. The model is formulated in terms of the reduced magnetic vector potential. Surface fictitious currents are used for the FEM–BEM coupling. Only the active arts are discretized by using edge elements. The model is validated by comparing with analytical results for a magnetic hollow sphere in a homogeneous magnetic field. An application to the evaluation of ship magnetization is then considered.

In the absence of wave propagation, transient electromagnetic fields are governed by a composite scalar/vector potential formulation for the quasi-static Darwin field model. Darwin-type field models are capable of capturing inductive, resistive, and capacitive effects. To avoid possibly non-symmetric and ill-conditioned fully discrete monolithic formulations, here, a Darwin field model is presented, which results in a two-step algorithm, where the discrete representations of the electric scalar potential and the magnetic vector potential are computed consecutively. Numerical simulations show the validity of the presented approach.

Skew is taken into account in single- and multi-slice 2-D finite-element machine models by skew interface conditions applied at a cylinder hull in the center of the air gap. A set of biorthogonal basis functions is constructed to discretize the tangential magnetic field strength at the interface and to weakly enforce the tangential continuity of the magnetic vector potential. The harmonic filtering as a result of skew is already observed in a single-slice model. Multi-slice models with different numbers of stator and rotor slices increase the modeling accuracy.

Purpose. Research and analysis trolley busbar’s parameters in condition of higher current harmonic actions, with taking into account the structural features of nonlinearity of magnetic and electrical properties of materials, proximity effects, surface and external surface effects. Methodology. The researches were carried out using the electromagnetic field theory methods, the electrical circuit theory, mathematical physics, finite elements, interpolation, approximation and regression analysis. Findings. The mathematical spatial model of electromagnetic processes in a steel trolley busbar in time statement of a problem of distribution of an electromagnetic field is developed. The dependences of the distribution of equipotential lines of the resulting z-component of the magnetic potential vector along the busbar, as well as the distribution of the resulting normal component of magnetic induction and magnetic field strength in the transverse (XY) cross section at non-sinusoidal current in busbar trolleys is obtained. Along the length of the busbar, in their cross section, the magnetic field tends to a plane-parallel shape it is proved. The error of the modulus of the vector magnetic potential along the length of the busbar does not exceed 0.9-1.2%. To reduce the dimension of the problem, computational resources and calculation time, a two-dimensional plane-parallel mathematical model in the frequency setting of the electromagnetic field distribution is proposed. To take into account the nonlinear magnetic properties of steel trolleys, to determine the effective magnetization curve for the nonlinear two-dimensional problem of the electromagnetic field of the busbar it is proposed. The verification results, according to the calculated voltage drop, confirm the high accuracy of the calculation and the reliability of the obtained results (error does not exceed 1.88% ÷ 2.06%) of the two-dimensional model in the frequency setting relative to the spatial model in the problem of time-dependent electrical -magnetic field is obtained. Originality. A mathematical two-dimensional model of electromagnetic processes in the frequency formulation of the problem of electromagnetic field distribution in a trolley busbar is proposed, which takes into account design features, nonlinearity of magnetic and electrophysical properties of materials, proximity effects, surface and external surface effects, influence of harmonic current components power transmission, which allows with high accuracy and efficiency of numerical implementation to determine the parameters of the bus trolls for the corresponding values of the amplitudes and frequencies of the frequencies harmonics of the current. Practical value. Verification of the calculated voltage drop confirms the high accuracy of the calculation and the reliability of the results (error does not exceed 1.88% ÷ 2.06%) of the two-dimensional model in the frequency reference relative to the spatial model in the problem of time-dependent electromagnetic field distribution is performed., Цель работы. Исследование и анализ параметров троллейных шинопроводов, в условиях действия высших гармонических составляющих тока, с учетом конструктивных особенностей, нелинейности магнитных и электрофизических свойств материалов, эффектов близости, поверхностных и внешних поверхностных эффектов. Методы исследования. Исследования проводились с применением методов теории электромагнитного поля, теории электрических цепей, математической физики, конечных элементов, интерполяции, аппроксимации и регрессионного анализа. Полученные результаты. Разработана математическая пространственная модель электромагнитных процессов в стальном троллейном шинопроводе во временной постановке задачи распределения электромагнитного поля. Получены зависимости распределения эквипотенциальных линий результирующей z-составляющей вектора магнитного потенциала вдоль шинопровода, а также распределения результирующей нормальной составляющей магнитной индукции и напряженности магнитного поля в поперечном (XY) сечении при несинусоидальном токе в троллеях шинопровода. Доказано, что вдоль длины шинопровода, в поперечном их пересечении, магнитное поле стремится к плоско-параллельной форме. Невязка модуля векторного магнитного потенциала вдоль длины шинопровода не превышает 0,9-1,2%. Для уменьшения размерности задачи, вычислительных ресурсов и времени на расчет, предложена двухмерная плоско-параллельная математическая модель в частотной постановке распределения электромагнитного поля. Для учета нелинейных магнитных свойств стальных троллеев предложено определение эффективной кривой намагничивания для нелинейной двумерной задачи электромагнитного поля шинопровода. Полученные результаты верификации, по рассчитанному падению напряжения, подтверждают высокую точность расчета и достоверность полученных результатов (погрешность не превышает 1,88% ÷ 2,06%) двумерной модели в частотной постановке, по отношению к пространственной модели, в постановке задачи зависимой от времени электромагнитного поля. Научна новизна. Предложена математическая двухмерная модель электромагнитных процессов в частотной постановке задачи распределения электромагнитного поля в троллейном шинопроводе, учитывающая конструктивные особенности, нелинейность магнитных и электрофизических свойств, эффекты близости, поверхностные и внешние поверхностные эффекты, влияние гармонических составляющих тока на падение напряжения и потери мощности в процессе электропередачи, которая позволяет с высокой точностью и эффективностью численной реализации определить параметры троллей шинопровода для соответствующих значений амплитуд и частот весших гармоник тока. Практическая ценность. Проведена верификация по рассчитанному падению напряжения подтверждающая высокую точность расчета и достоверность полученных результатов (погрешность не превышает 1,88% ÷ 2,06%) двумерной модели в частотной постановке. по отношению к пространственной модели. в постановке задачи зависимой от времени распределения электромагнитного поля., Мета роботи. Дослідження електромагнітних параметрів тролеїв шинопроводу, та верифікація отриманих результатів розрахунку за допомогою просторової (3D) та плоско-паралельної (2D) математичних польових моделей. Методи дослідження. Дослідження проводилися із застосуванням методів теорії електромагнітного поля, інтерполяції та апроксимації, математичної фізики, скінченних елементів. Отримані результати. Розроблена математична просторова модель електромагнітних процесів в сталевому тролейному шинопроводі у часовій постановці задачі розподілу електромагнітного поля. Отримані залежності розподілу еквіпотенціальних ліній результуючої z-складової вектору магнітного потенціалу уздовж шинопроводу, а також розподілу результуючої нормальної складової магнітної індукції і напруженості магнітного поля у поперечному (XY) перетині при несинусоїдальному струмі в тролеях шинопровода. Доведено, що уздовж довжини шинопроводу, у поперечному їх перетині, магнітне поле прагне до плоско-паралельної форми. Нев’язка модуля векторного магнітного потенціалу уздовж довжини шинопроводу не перевищує 0,9-1,2%. Для зменшення розмірності задачі, обчислювальних ресурсів та часу на розрахунок, запропонована двомірна плоско-паралельна математична модель у частотній постановці розподілу електромагнітного поля. Для врахування нелінійних магнітних властивостей сталевих тролеїв запропоновано визначення ефективної кривої намагнічування для нелінійної двовимірної задачі електромагнітного поля шинопровода. Отримані результати верифікації, за розрахованим падінням напруги, підтверджують високу точність розрахунку та достовірність отриманих результатів (похибка не перевищує 1,88%÷2,06%) двовимірної моделі у частотній постановці по відношенню до просторової моделі в постановці задачі залежної від часу електромагнітного поля. Наукова новизна. Запропоновано математичну двомірну модель електромагнітних процесів у частотній постановці задачі розподілу електромагнітного поля в тролейному шинопроводі, що враховує конструктивні особливості, нелінійність магнітних та електрофізичних властивостей матеріалів, ефекти близькості, поверхневі та зовнішні поверхневі ефекти, вплив гармонійних складових струму на падіння напруги та втрати потужності в процесі електропередачі, яка дозволяє з високою точністю і ефективністю чисельної реалізації визначити параметри тролей шинопроводу для відповідних значень амплітуд та частот віщих гармонік струму. Практична цінність. Проведена верифікація за розрахованим падінням напруги підтверджуює високу точність розрахунку та достовірність отриманих результатів (похибка не перевищує 1,88%÷2,06%) двовимірної моделі у частотній постановці по відношенню до просторової моделі в постановці задачі залежної від часу розподілу електромагнітного поля.

The magneto-electro-elastic coupling static problem for a functionally graded piezoelectric-piezomagnetic (FGPEPM) cylinder/spherical shell is studied for various axial/spherical symmetric loads. On the basis of unified basic equations with geometric parameters and the material assumption of exponential inhomogeneous properties, the analytical solutions for the physical quantities of displacement, stress, electric potential, and magnetic potential of the FGPEPM hollow cylinder/spherical shell are obtained. In the numerical discussions, the static response of the FGPEPM cylinder/spherical shell is numerically solved under mechanical load on the inner surface or given an electric/magnetic potential difference between the inner and outer surfaces when the material on the inner surface is assumed to be BaTiO3-CoFeO4. The distribution of radial and circumferential stress and the electric and magnetic potentials under different functionally graded (FG) parameters are presented. The influence of FG parameters on the positive (converse) piezoelectric, positive (converse) piezomagnetic, and positive (converse) magnetoelectric potentials is discussed in detail.

This paper presents the comparison between two different geometries for the core of homo polar linear synchronous motor HLSM. The theoretical analysis estimated the airgap flux density , the attraction and traction forces This analysis is based on a magnetic field study using 3-dimensional finite difference method FDM of scalar magnetic potential. The average results of zero and infinite permeability boundaries is considered. The flux per pole, under a.c. limbs either in"C" or in " XI" core is taken for obtaining the majority of normal force. Results reveal that the " XI" core shape was the optimum form because it provides higher both of traction and attraction forces than "C" core . But, the extra iron required and attraction force s than "C" core. But, the extra Iron required and the cost of rail track poles were the weak points for unchoosing this form in practical application An experimental investigation is carried out on laboratory Static prototype motor LSPM of "C" core HLSM to verify the theoretical results. It is shown that the experimental and computed results are in good agreement.

In this work, a flexible discretization technique for the approximate solution of electromagnetic field problems with rotating parts, based on non-conforming (NC) interfaces of Nitsche-type, is investigated. This approach enables the use of edge elements of the first and second kind for the discretization of the magnetic vector potential without posing severe restrictions on gaps and overlaps of elements on both sides of the rotating interface. It improves the computational efficiency, compared to other NC interface formulations because no additional unknowns are introduced, and edge elements of different polynomial order can be coupled with this approach. The applicability is shown in two numerical examples, involving a cylindrical NC interface between a stationary and a rotating domain.

In this article, a rotor magnetic potential (RMP) model is proposed, which can accurately describe the rotor surface magnetic potential distribution under the interaction of multi-excitation sources. Furthermore, a new magnetic equivalent circuit (MEC) model is used to model the complex saturation in the bridge region, which employs an improved nonlinear reluctance modeling method. Then, the unknown parameters in the RMP model are obtained by solving the MEC model. In this article, the RMP model and MEC model are combined to consider the stator and rotor saturation effects, and the magnetic field is calculated accurately. Compared with the existing RMP model, the proposed one can take into account the effects of the stator saturation and PM magnetic field. Compared with the existing MEC and equivalent magnetic network (EMN), the area of the nonlinear reluctance in this paper varies with excitation conditions, which can accurately consider the saturation effects in the bridge region. This improvement can improve calculation accuracy on the basis of reducing the number of nonlinear reluctances. Moreover, the proposed method does not need to consider the change of the air-gap reluctances caused by rotor rotation. Finally, the proposed method is verified by both finite element analysis and experiments.

This article presents a nonlinear analytical (NA) solution based on subdomains for predicting magnetic field and performances of a spoke array vernier permanent magnet (SAVPM) machine which has spoke array type permanent magnet (PM) excitation. On one hand, equivalent current sheets replace tangentially magnetized PMs to make the magnetic vector potential in the region with PMs satisfy a Laplace's equation, which simplifies the problem and reduces the workload of numerical calculation. On the other hand, in contrast to the existing literature, regions of soft-magnetic material such as a tooth and a rotor pole are subdivided into small pieces, and are no longer a whole with a single permeability for considering local magnetic saturation. Poisson's and Laplace's equations with the given permeability of soft-magnetic material are analytically solved by separation technique of variables in seven subdomains, and nonlinear iteration is used to determine the solution of NA and update soft-magnetic material permeability. NA solves the magnetic field and performances of a SAVPM machine. Frozen permeability method (FPM) is used to determine PM flux linkage and d-axis inductance under load. Results from the NA model have been compared with those from FEA and experiments.

Targeted drug delivery using magnetic particles (MPs) and external magnets for focusing them at the diseased regions, called magnetic drug targeting (MDT), is a next-generation therapeutic method that is being continually improved. However, most existing magnetic systems cannot focus MPs in the targeted region due to there not being enough magnetic capturing force and absence of schemes to generate localized high magnetic field at the wall of the target region. This paper suggests a novel scheme to utilize half of a static saddle potential energy configuration generated using four electromagnets that not only enhances the pushing magnetic forces but also simultaneously generates pushing and attracting forces in the desired direction to help focus spherical MPs on the wall of the target region. Furthermore, by changing amplitudes or directions of the currents, the focal point in the target region can be changed. Through extensive simulations and in vitro experiments, we demonstrate that half of a static saddle magnetic potential energy configuration can be successfully utilized to attract and focus MPs at the wall of a target region.

We consider the Schrodinger periodic magnetic operator in an infinite flat straight strip. We prove that if the magnetic potential satisfies certain conditions and the period is small enough, then the lower part of the band spectrum has no inner lacunas. The length of the lower part of the band spectrum with no inner lacunas is calculated explicitly. The upper estimate for the small parameter allowing these results is calculated as a number as well.

One of the ways to carry out various technological processes of fine and ultrafine grinding is the electromechanical disintegrators of multifactor action using. The disintegrator consists of two flat inductors with three-phase windings, which form traveling magnetic fields with the opposite phase sequence, and a working chamber with ferromagnetic working bodies, located in the inter-inductor gap.The set of factors allowing to influence the processed substance with the required intensity and in the required direction requires knowledge of the physical essence of the phenomena occurring in the vortex layer. To determine the force acting on a ferromagnetic body, it is necessary to calculate the magnetic field in the working area of the electromechanical disintegrator. The quantity of interest is determined by numerically solving a field problem using the finite element method. This method makes it possible to accurately describe the configuration of the active part, as well as take into account the nonlinearity of material properties.In this paper, the pictures of the vector magnetic potential distribution in the active zone of a electromechanical disintegrator are obtained. The modeling results of the forces acting distribution on ferromagnetic bodies and a working electrically conductive chamber are presented. A consequence of the interaction of the running magnetic fields of inductors and eddy currents flowing in the walls of the working chamber is the appearance of characteristic deflections of the walls of the working chamber within each pole division. The calculation confirms the presence of localization of the acting electromagnetic forces on the walls of the electrically conductive chamber within the pole division of the inductors. The deformation of the working chamber is caused, first of all, by forces that are directed normally to the surface of the chamber wall.The results of numerical calculations can be used for choosing the geometric dimensions of the active part for obtaining the specified operating properties and characteristics.The results of numerical calculations can be used for choosing the geometric dimensions of the active part for obtaining the specified operating properties and characteristics.

In recent years, the $H$ formulation of Maxwell's equations has become the de facto standard for simulating the time-dependent electromagnetic behavior of superconducting applications with commercial software. However, there are cases where other formulations are desirable, for example for modeling superconducting turns in electrical machines or situations where the superconductor is better described by the critical state than by a power-law resistivity. In order to accurately and efficiently handle these situations, here we consider two approaches based on the magnetic vector potential: the $T$ - $A$ formulation of Maxwell's equations (with power-law resistivity) and the implementation of a quasi critical state model (QCSM) with a steep $E$ - $J$ relationship limited at $J_{\rm c}$ . In this article, we extend the $T$ - $A$ formulation to thick conductors so that large coils with different coupling scenarios between the turns can be considered. We also discuss the QCSM in terms of its ability to calculate ac losses: in particular, we investigate the dependence of the calculated ac losses on the frequency of the ac excitation and the possibility of using quick one-step (instead of full-cycle) simulations to calculate the ac losses.

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

Accurate calculation of magnetic field distribution is a prerequisite for motor design and optimization. This article develops an analytical subdomain model of external rotor in-wheel motor with surface-inset magnets (ERIWMSIMs). Besides, to weaken its torque ripple, an improved topology by integrating magnetic flux modulation slots (MFMSs) into tooth tips is proposed. First, the entire field problem is divided into five subdomains, viz., magnet rotor slots, air gap, stator slots, stator slot openings, and MFMSs. Then, this study solves Laplace’s or Poisson’s equation of magnetic vector potential in each subdomain according to the type of excitation source and use the boundary conditions of adjacent subdomains to determine the magnetic field solution. Consequently, the electromagnetic performances such as air gap flux density, back electromotive (EMF), cogging torque, and on-load output torque are accurately predicted. The analytical results are shown in line with those obtained from finite-element analysis (FEA) and experimental results. Furthermore, the influence of MFMS parameters on the torque is further investigated. When double MFMSs are integrated into each tooth tip, the depth of the MFMS is 0.7 mm, and the width is 1.05 times the slot opening; the results demonstrate that the torque ripple can be essentially weakened compared with other topologies.

Computational complexity, magnetic saturation, complex stator structure and time consumption due to repeated iteration compels researchers to adopt alternate analytical model for initial design of electric machines especially Switched Flux Machine (SFM). To overcomes the abovesaid demerits, In this article alternate analytical sub-domain model (SDM) for magnetic field computation in Segmented PM switched flux consequent pole machine (SPMSFCPM) with flux bridge and flux barriers accounting boundary and interface conditions, radial magnetized PMs (RM-PMs) and circumferential magnetized PMs (CM-PMs), interaction between stator slots and inner/outer rotor topologies is proposed. Overall field domain is divided into air gap, stator slots and Permanent Magnet (PM) accounting influence of CM/RM-PMs under no-load and on-load conditions. Analytical expression of field domain is obtained by solving magnetic vector potential utilizing Maxwell's equations. Based on the magnetic field computation especially no-load and on-load condition, Magnetic Flux Density (MFD) components, open-circuit flux linkage, mechanical torque and cogging torque are computed utilizing Maxwell Stress Tensor (MST) method. Moreover, developed analytical SDM is validated with globally accepted Finite Element Analysis (FEA) utilizing JMAG Commercial FEA Package v. 18.1 which shows good agreement with accuracy of ~98%. Hence, authors are confident to propose analytical SDM for initial design of SPMSFCPM to suppress computation time and complexity and eliminate requirements of expensive hardware and software tools.

Fractional-slot windings linear phase-shifting transformers (FW-LPSTs) are a new type of transformer that can be used as an important component of multi-rectifier and multi-inverter systems. The FW-LPSTs are proposed for the first time, and few scholars have studied their magnetic field. However, accurate calculation of magnetic field distribution of the FW-LPSTs is the premise of the optimal design of the ontology, and the establishment of an accurate analytical model which remains a challenging task is the key to solving the magnetic field distribution. In view of the limitation of traditional analytical theory in solving the magnetic field, the FW-LPST model based on the exact subdomain method is proposed to obtain the exact magnetic field distribution. Considering the influences of the permeability and the structural parameters of the slot subdomain and the interaction of all slots on the distribution of the magnetic field, the governing equations of the air gap and slot subdomains are established using magnetic vector potential as the variable. According to the boundary conditions of each subdomain, all equations of the boundary condition are listed based on the Fourier series method. Finally, the vector magnetic potential and open-circuit magnetic flux density distribution of each subdomain are solved. The model accurately analyzes the influence of the cogging effect on the magnetic field distribution. The inductances of the primary side and the induced potentials of the secondary side are solved using the magnetic flux linkage method. The accuracy of the proposed analytical model is verified by comparison of the results obtained with those obtained by the finite element method.

The method of path integrals is used to average the magnetic-induction equation written for the vector potential over the velocity field. This approach avoids the assumption that the random field is two-scale, replacing it with the assumption of short time correlations. As a result of averaging, the classical equation of the mean field is obtained, at least for a homogeneous and isotropic medium. Based on the developed approach, the problem of the average field solenoidality is discussed, as well as a possible generalization of the method to the case of a statistically inhomogeneous and anisotropic flow.

In order to improve the reliability of the electrical machine, a dual-redundancy brushless DC motor is designed. two kinds of stator structure are designed, including one slot and two slots. The mathematical model of the motor is set up and the winding inductance is analyzed and calculated. The relationship between the winding inductance and mutual inductance is deduced. The magnetic potential of the motor is analyzed. Results and their analysis show preliminarily that: (1) the reduced magnetic potential is equal to the increased magnetic potential with same axes; (2) the reduced magnetic potential is larger than the increased magnetic potential with different axes. According to the mathematical model, the parameters of the motor are designed, and the rotational speed and torque are simulated. The motor is tested on the MAGTROL testing system. Mechanical characteristic curves of both kinds of stator structure under different working modes are compared. Both the simulation and the experiment results show that: (1) the motor of each kind can achieve good dynamic characteristics, and the torque is increased by 100% with two sets of windings; (2) two sets of winding did not affect motor performance, but it did indeed greatly improve the system reliability.

A coreless hybrid magnetic coupling mechanism has been studied and designed in the dynamic wireless charging system, through the mathematical model building, the parameter analysis and the model simulation comparison. The mechanism can improve the endurance of the electric vehicles, reduce the times of the power supply of the electric vehicles, and increase the offset ability. First of all, the dynamic model of the magnetic coupling mechanism is established based on the Kirchhoff’s law of the voltage and current, and the output power and the system efficiency are deduced considering the driving speed of the electric vehicle. Then, the influence of the mutual inductance between coils on the output power and the system efficiency is deeply analyzed, and the mutual inductance between coils is increased by adding the magnetic potential at a proper position, to propose and analyze a coreless hybrid magnetic coupling coil structure. Finally, the co-simulations are implemented through the finite element analysis simulations to compare and analyze the coil structure performance before and after the optimization. The results show that the designed coil structure can effectively improve the coupling between magnetic fields, and increase the offset ability. The mechanism has the characteristics of the small dynamic output power fluctuation range and high efficiency, and then can realize the stable power supply for the mobile power load.

Harmonic Cartesian polynomial and rational functions are used to obtain a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a long, transversely isotropic elastic cylinder of nearly-circular normal cross-section carrying an axial, steady electric current. Numerical results and plots are provided and discussed. Comparison with the case of a circular cross-section allows to assess the in uence of imperfection of the cross-sectional shape on the quantities of practical interest. The effect of dependence of the magnetic permeability on strain is investigated. We use the magnetic vector potential which is parallel to the axis of the cylinder. The mathematical problem reduces to the determination of ve harmonic functions, interrelated by a set of boundary conditions. Two of these harmonic functions combine to yield Airy's stress function in the cross-sectional domain in case the mechanical problem is solved in stresses. A third harmonic function enters in the formulation of the thermal problem for temperature distribution. The two remaining harmonic functions enter in the description of the vector potential distribution inside the conductor and in the surrounding region. The results nd application in calculating the deformation of straight central sections of electric conductors in various instruments, in particular busbars in electric power stations, and the stresses induced in them due to the electric current.

A model of an axisymmetric probe with a new T-core coil that can be used in eddy current testing is developed. The truncated region eigenfunction expansion (TREE) method is used to analyze the coil impedance problem of the T-core coil located above a multi-layer conductive material. First, the magnetic vector potential expressions of each region are formulated, then the coefficients of the magnetic vector potential are derived by using the boundary conditions, finally, the closed form expression of the coil impedance is obtained. The normalized impedance changes of the T-core coil caused by the presence or absence of the multi-layer conductor are calculated using Mathematica. The presented T-core coil is compared with I-core coil and air-core coil. The effects of the T-core parameter a2, relative permeability μf and the thickness of the top section on the change in the coil impedance are discussed respectively. The analytical calculation results are compared with the results of finite element method, and the two agree very well, which verifies the correctness of the proposed T-core coil model.

A physical and mathematical model is considered to support the investigation of the prospects of using the effect of metal’s “magnetic memory” for the nondestructive testing of items made from ferromagnetic materials located in the Earth’s magnetic field. Based on the finite-element method, an algorithm and a computer code for the 3D computations of the magnetic potential distribution in a medium with nonuniform magnetic permeability are worked out. The developed methods and software tools are used to model the leakage field near the clamped heating pipes of steam boilers. Relationships demonstrating the interdependence between the change in magnetic induction and the magnetic leakage field are obtained. Satisfactory agreement is observed between the calculated and experimental data on the distribution of the field strength of the magnetic leakage on the surface of the heating pipe.

We are concerned with the qualitative analysis of solutions for three classes of nonlinear problems driven by the magnetic Laplace operator. We are mainly interested in the perturbation effects created by two reaction terms with different structure. Two equations are studied on bounded domains (under Dirichlet boundary condition) while the third problem is on the entire Euclidean space. Our main results establish that if a certain perturbation is sufficiently small (in a prescribed sense) then the problems have at least two distinct solutions in a related magnetic Sobolev space. The proofs combine variational, topological and analytic methods.

Due to the multidisciplinary nature for the solution of magneto-electro-elastic (MEE) shell structures, developing a novel and accurate computational model is both essential and necessary for the practical engineering. The scaled boundary finite element method (SBFEM) is a semi-analytical technique in which only the surfaces or boundaries of the computational domain need to be discretized, while an analytical formulation can be derived in the radial direction of the surrounding area. These advanced features enable the spatial dimension to be reduced by one, while the accuracy of the proposed algorithm is maintained. In this paper, a novel semi-analytical numerical model based on the SBFEM is developed for the bending analysis of the laminated MEE cylindrical shells under the mechanical or electric/magnetic potential loads. According to the three-dimensional (3D) magneto-electro-elasticity theory, the magneto-electro-mechanical coupling equations and the associated boundary conditions in terms of the mechanical displacement as well as the electrical and magnetical potentials are derived in the scaled boundary coordinate system using the weighted-residual method. The analytical expressions for the generalized displacement and internal nodal force fields are determined by applying the state-space method and have been solved by means of the precise integration technique (PIT). Comparisons between the present numerical results for limiting conditions and solutions available in the published work have been carried out to demonstrate the convergence and accuracy of this approach. At the same time, by utilizing the proposed mechanics, the influences of the aspect ratio and stacking configuration on the through-thickness bending behaviors of the laminated MEE cylindrical shells are studied in detail.

An eccentric position diagnosis method of static eccentricity (SE) fault of external rotor permanent magnet synchronous motor (ER-PMSM) is presented. Firstly, an analytical model of no-load radial magnetic field of ER-PMSM is established. Analytical models of no-load Back-EMF of both unit motors and the whole motor are carried out and are verified by finite element method (FEM) and experimental measurements. Then, the influences of SE ratio, SE circumferential angle, winding distribution mode and number of parallel branches on no-load radial magnetic field and no-load Back-EMF are analyzed based on these analytical models. The results show that SE does not affect the frequency characteristics of no-load radial magnetic field, but changes space order characteristics. On one hand, for ER-PMSM, of which the number of unit motors is equal to 1, SE causes no-load Back-EMF distortion. On the other hand, for ER-PMSM, of which the number of unit motors is greater than 1, SE does not affect no-load Back-EMF of the whole motor, but it still leads to no-load Back-EMF distortion of unit motors. Therefore, based on total harmonic distortion (THD) of no-load Back-EMF of unit motor, a projection method of intersection lines for SE fault diagnosis of ER-PMSM is proposed finally.

3-D analytical model and direct measurement method (DMM) of ultra-thin open-circuit air-gap field (OCAGF) of interior permanent magnet synchronous motor (IPMSM) with multi-segmented skew poles (MSP) and multi-layered flat wire windings (MFWW) for electric vehicles (EV) are proposed. Basis functions of slotless OCAGF are derived through field reconstruction method (FRM). In order to consider MSP and slotting effects, a complex relative permeance function is extended from 2-D to 3-D. Based on the magnetic potential and permeance method, 3-D analytical model of OCAGF is obtained. Then the back electromotive force (BEF) is calculated through Faraday's Law and the cogging torque is obtained through Maxwell stress tensor method. Furthermore, spatial order and amplitude-frequency characteristics of OCAGF are revealed. Additionally, influence of local inhomogeneous saturation caused by PM materials on analytical accuracy is discussed. Finally, the proposed analytical model is verified through both the traditional indirect measurement methods (BEF measurement and cogging torque measurement) and a novel direct measurement method, in which ultra-thin OCAGF of 10−1 mm order of magnitude can be measured directly. Local microscopic characteristics of OCAGF in some important locations, such as tooth top and magnetic bridge, can be accurately detected by using of the direct measurement method.

The application of high-temperature superconductors to accelerator magnets for future particle colliders is under study. Numerical methods are crucial for an accurate evaluation of the complex dynamical behavior of the magnets, especially concerning the magnetic field quality and thermal behavior. We present a coupled $\textbf{A}$-$\textbf{H}$ field formulation for the analysis of magneto-thermal transients in accelerator magnets. The magnetic field strength $\textbf{H}$ accounts for the eddy current problem in the source regions containing the superconducting domains, while the magnetic vector potential $\textbf{A}$ represents the magnetoquasistatic problem in the normal and non-conducting domains. Furthermore, we include a slab approximation for the source regions, making the formulation suitable for large scale models composed of thousands of tapes. In this work, the relevant equations are derived and discussed, with emphasis on the coupling conditions. The weak formulation is derived, and numerical results are provided in order to both, verify the formulation and scale it to the size of an accelerator magnet., Comment: 10 pages, 11 figures

A new model for anisotropic magneto-electro-elastic Mindlin plates is developed by using an extended modified couple stress theory. The equations of motion and complete boundary conditions are simultaneously obtained by a variational formulation based on Hamilton’s principle. The new anisotropic magneto-electro-elastic plate model includes the models for orthotropic and transversely isotropic magneto-electro-elastic Mindlin plates and the model for isotropic Mindlin plates, all incorporating the microstructure effect, as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported transversely isotropic magneto-electro-elastic plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection, rotation, electric potential, and magnetic potential of the simply supported plate predicted by the current non-classical model are always smaller than those predicted by the classical elasticity-based model, and the differences are significant when the plate thickness is very small but is diminishing as the thickness increases. For the free vibration problem, it is found that the natural frequency predicted by the new plate model is higher than that predicted by the classical model, and the difference is quite large for very thin plates.

The mean magnetic field equation describes the process of generating a magnetic field on a large scale as a result the EMF arising on a small scale. We consider the case where the large-scale magnetic field is also random and determine the density function of magnetic helicity. This function is invariant under gauge transformations of the magnetic vector potential. We study the equation for the magnetic helicity flux of a large-scale field and introduce a correction term related to the quadratic magnetic helicity invariant.