Research on Finite Permeability Semi-Analytical Harmonic Modeling Method for Maglev Planar Motors

Ship magnetic field modeling is not only beneficial to understanding the characteristics of ship magnetic field, but also can predict the space distribution of ship magnetic field, which has an important application in ship protection and underwater weapons. Aiming at the problems of low modeling accuracy and poor stability in establishing the ship magnetic field hybrid model, a method of establishing a high precision stablity model is proposed in this paper. A hybrid model of magnetic field of a ship is established by using a uniformly magnetized rotating ellipsoid and a magnetic dipole array. Since the number and positions of magnetic dipoles in the hybrid model have an important effect on the modeling accuracy and stability, the fitting error function representing the modeling accuracy and the coefficient matrix condition number function representing the stability of the model are constructed by taking the magnetic dipole parameters as unknown variables. The multi-objective function is constructed by combining the fitting error function with the coefficient matrix conditional number function, which indirectly transforms the modeling problem into a multi-objective optimization problem. The multi-objective function is solved by using the multi-objective particle swarm optimization algorithm, and an optional set of modeling solution results is obtained. In order to select the best results from the optional set, the corresponding selection rules are designed based on the modeling accuracy. The proposed method is validated by the measured data of three kinds of ship models, the modeling results show that the relative error of the model is less than 3%, and the conversion error is less than 6%, which verifies that the proposed method can effectively model the ship magnetic field. Though the measurement data error exists, the modeling solution results from the proposed method have the best stability, which verifies that the modeling method proposed in this work has good stability. Compared with the two existing modeling methods, the proposed method has very good modeling accuracy and stability. Finally, the actual data of a ship on the sea are used for modeling, and the modeling results further verify that the proposed method has high modeling accuracy and conversion accuracy, and can be effectively applied to the relevant projects.

Halbach array has unilateral magnetic effect, so the eccentric harmonic magnetic gear (EHMG) with Halbach array has higher torque density than the traditional EHMG with radial magnetization. An exact analytical model for the air-gap magnetic field of EHMG with Halbach array based on hyperbolic cotangent transformation is proposed. The relative permeability function is solved and used to correct the air-gap magnetic field caused by the stator and rotor permanent magnets acting separately when concentric. The analytical solution of the air-gap magnetic field of EHMG with Halbach array is obtained by superposition of the corrected eccentric air-gap magnetic field. The static torque of low-speed inner rotor is calculated by Maxwell tensor method. The analytical solutions of the air-gap magnetic field and the static torque are consistent with the finite element solutions, which verifies the effectiveness and correctness of the analytical model.

In order to introduce the magnetic field into micro electrical machining technology to explore the influence of magnetic field on micro electrical machining, the development of a precision controllable magnetic field-assisted platform is particularly important. This platform needs to precisely control the spatial magnetic field. This study first completes the hardware design and construction of the magnetic field generation device, using electromagnetic coils with soft iron cores as the sources of the magnetic field. Mathematical models of the magnetic field are established and calibrated. Since the magnetic dipole model cannot effectively describe the magnetic field generated by the electromagnetic coil, this study adopts a more precise description method: the spherical harmonic function expansion model and the magnetic multipole superposition model. The calibration of the magnetic field model is based on actual excitation magnetic field data, so a magnetic field sampling device is designed to obtain the excitation magnetic field of the workspace. The model is calibrated based on a combination of the theoretical model and magnetic field data, and the performance of the constructed setup is analyzed. Finally, a magnetic field-assisted platform has been developed which can generate magnetic fields in any direction within the workspace with intensities ranging from 0 to 0.2 T. Its magnetic field model arrives at an error percentage of 2.986%, a variance of 0.9977, and a root mean square error (RMSE) of 0.71 mT, achieving precise control of the magnetic field in the workspace.

As we know, the analytical calculation of magnetic field and magnetic force has important guiding significance for the optimal design of the size and structure of a magnet. At present, the magnetic field and magnetic force between two permanent magnet (PM) monomers have been described by effective analytical models. However, the magnetic field strength and magnetic force generated by two PM monomers are limited and cannot meet the requirements of a PM levitation system with high magnetic field strength and large levitation force. To this end, this article establishes analytical calculation models of magnetic field and levitation force between two Halbach PM arrays. First, on the basis of Ampere molecular circulation hypothesis, the 2-D analytical model of magnetic field was established based on the surface current method; on the basis of magnetic charge theory, the 3-D analytical model of magnetic field was established based on the magnetic charge method. Second, based on the analytical models of magnetic field, the 2-D and 3-D analytic models of levitation force were derived by virtual work method, respectively. Numerically, the results of two analytical models of magnetic field and two analytical models of levitation force have been compared to verify the accuracy of the models. At the same time, the 2-D and 3-D finite element simulation models of magnetic field and levitation force were established, respectively, and the accuracy of the analytical models were verified again. The results show that the analytical models established in this article is effective and can be used to guide the design and optimization of the magnet composed of two Halbach arrays.

The eccentric harmonic magnetic gear (EHMG) has the characteristics of high torque density, high transmission ratio and high efficiency. The EHMG with Halbach array has better performance than the EHMG with traditional radial magnetization. Accurate calculation of its air‐gap magnetic field is of great significance for design and optimization. Based on the boundary perturbation method, a 2D analytical model of the air‐gap magnetic field of EHMG with Halbach array is established in this paper. The eccentric air‐gap magnetic field is calculated due to the permanent magnets of the eccentric rotor and the stator acting alone, and then the air‐gap magnetic field of the EHMG is obtained according to the superposition principle. The results of the air‐gap magnetic field are compared and evaluated by the regression evaluation indexes. The analytical results are consistent with the finite element results, which verifies the validity and accuracy of the analytical model. Comparing the EHMG with Halbach array and radial magnetization, the maximum static and dynamic electromagnetic torques of EHMG with Halbach array are increased by 59.1% and 55.6% compared with the EHMG with radial magnetization, which verifies the superior performance of EHMG with Halbach array. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

Due to the particular arrangement of permanent magnets, a Halbach array has an significant effect of magnetism and magnetic self-shielding. It can stretch the magnetic lines on one side of the magnetic field to obtain an ideal sinusoidal unilateral magnetic field. It has a wide application range in the field of energy harvesting. In practical applications, magnetic induction intensity of each point in magnetic field is not only related to the induced current and conductor but also related to the permeability of the medium (also known as a magnetic medium) in the magnetic field. Permeability is the physical quantity that represents the magnetism of the magnetic medium, which indicates the resistance of magnetic flux or the ability of magnetic lines to be connected in the magnetic field after coil flows through current in space or in the core space. When the permeability is much greater than one, it is a ferromagnetic material. Adding a ferromagnetic material in a magnetic field can increase the magnetic induction intensity B. Iron sheet is a good magnetic material, and it is easy to magnetize to generate an additional magnetic field to strengthen the original magnetic field, and it is easy to obtain at low cost. In this paper, in order to explore the influence of ferromagnetic material on the magnetic field and energy harvesting efficiency of the Halbach array energy harvesting structure, iron sheets are installed on the periphery of the Halbach array rotor. Iron sheet has excellent magnetic permeability. Through simulation, angle between iron sheet and Halbach array, radian size of iron sheet itself and distance between iron sheet and Halbach array can all have different effects on the magnetic field of the Halbach array. It shows that adding iron sheets as a magnetic medium could indeed change the magnetic field distribution of the Halbach array and increase energy harvesting efficiency. In this paper, a Halbach array can be used to provide electrical power for passive wireless low-power devices.

The effective mixing process is critical in biological and chemical processes. The main objective of the present study is to investigate the influence of normal and Halbach magnet arrays on the mixing performance of a three-inlet micromixer numerically. In this microdevice, ferrofluid is injected into the center inlet, and water is injected into two other inlets. The influence of Remanent Flux Density Norm (RFDN), number of magnets, magnet distance from the main microchannel entrance, and inlet flow rate is considered. It is revealed that the micromixer with magnets exhibits a 165% improvement in the mixing efficiency compared to the one with no magnetic field. The results show that increasing the magnetic field does not always increase the mixing quality. Even in some cases, it has a negative effect. It is demonstrated that the mixing efficiency is strongly influenced by the magnet arrangement. An optimal position is found for the magnet arrangement to achieve the maximum mixing efficiency of 91%. Contrary to the normal configuration, Halbach magnet array creates a parabolic profile for flux density. Halbach array can improve the mixing performance, depending on all magnets’ RFDN. The proposed microchannel can be used as a useful device for biological applications.

We have performed three-dimensional magnetohydrodynamical calculations of stream accretion in cataclysmic variable stars for which the white dwarf primary possesses a strong, complex magnetic field. These calculations were motivated by observations of polars: cataclysmic variables containing white dwarfs with magnetic fields sufficiently strong to prevent the formation of an accretion disk. In this case, an accretion stream flows from the L1 point and impacts directly onto one or more spots on the surface of the white dwarf. Observations indicate that the white dwarfs in some binaries possess complex (non-dipolar) magnetic fields. We performed simulations of ten polars, with the only variable being the azimuthal angle of the secondary with respect to the white dwarf. These calculations are also applicable to asynchronous polars, where the spin period of the white dwarf differs by a few percent from the orbital period. Our results are equivalent to calculating the structure of one asynchronous polar at ten different spin-orbit beat phases. Our models have an aligned dipolar plus quadrupolar magnetic field centered on the whitedwarf primary. We find that, with a sufficiently strong quadrupolar component, an accretion spot arises near the magnetic equator for slightly less than half our simulations, while a polar accretion zone is active for most of the remaining simulations. For two configurations, accretion at a dominant polar region and in an equatorial zone occurs simultaneously. Most polar studies assume that the magnetic field is dipolar, especially for single-pole accretors. We demonstrate that, with the orbital parameters and magnetic-field strengths typical of polars, the accretion flow patterns can vary widely in the case of a complex magnetic field. This may make it difficult formany polars to determine observationally whether the field is pure dipolar or is more complex, but there shoulid be indications for some systems. In particular, a complex magnetic field should be suspected if there is an accretion zone near the white dwarf’s equator (assumed to be in the orbital plane) or if there are two or more accretion regions that cannot be fitted by dipolar magnetic field. Magnetic-field constraints are expected to be substantially stronger for asynchronous polars, with clearer signs of complex field geometry due to changes in the accretion flow structure as a function of azimuthal angle. These indications become clearer in asynchronous polars because each azimuthal angle corresponds to a different spin-orbit beat phase.

A Halbach array is a specialized arrangement of permanent magnets designed to generate a strong, uniform magnetic field in the designated region. This unique configuration has been widely utilized in various applications, including magnetic levitation (maglev) systems, electric motors, particle accelerators, and magnetic seals. The advantages of Halbach arrays include high efficiency, reduced weight, and precise directional control of the magnetic field. Halbach arrays are commonly categorized into two configurations: linear and cylindrical. A linear Halbach array produces a concentrated magnetic field on one face and is frequently employed in maglev trains and conveyor systems to ensure stable and efficient operation. In contrast, a cylindrical Halbach array consists of magnets arranged in a ring, generating a uniform magnetic field within the cylinder while suppressing the external field. This configuration is particularly advantageous in applications such as brushless electric motors and magnetic resonance imaging (MRI) systems. Traditionally, the design of electromagnetic systems incorporating Halbach arrays relied on engineers’ expertise and intuition due to the complexity of the permanent magnet configuration. However, advancements in numerical methods, particularly topology optimization, have introduced a systematic approach to optimizing the shape and distribution of permanent magnets within a given design domain. In the context of Halbach array design, topology optimization aims to maximize the total magnetic flux within a designated region while simultaneously determining the optimal material distribution to achieve a specified design objective. This approach enhances the performance and efficiency of Halbach arrays, providing a more precise and automated framework for their development. In this paper, we propose a Cardinal Basis Function (CBF)-based level-set method for designing a circular Halbach array capable of generating a uniform magnetic field within a designated region. The CBF-based level-set method offers significant computational advantages by reducing the computational cost and accelerating the convergence process. This approach enhances the efficiency of the optimization process, making it a promising technique for the systematic design of Halbach arrays.

The Halbach array permanent magnet (PM) spherical motor (PMSM) consists of a spherical rotor and a spherical-shell stator. The magnetic field distribution surrounding the rotor PMs is closely related to the spherical structure of the Halbach array PMSM. The Halbach array PM configuration characteristics lead to the end leakage magnetic field in the latitudinal direction of the spherical structure, which is end effect. Establishment of an accurate 3-D magnetic field analytical model containing the end leakage magnetic field of the Halbach array PMSM is the foundation of magnetic field distortion research with its influence on eddy-current loss. The magnetic field distribution of the Halbach array PMs in the air gap is calculated using the scalar magnetic potential. Laplace and Poisson equations of the scalar magnetic potential in the spherical coordinate system are derived. To analyze the end leakage magnetic field of the Halbach array PMSM in the latitudinal direction, concept of mechanical pseudo-cycle in the motor latitudinal direction is proposed, and the cycle of Fourier series is redefined. A completed 3-D analytical expression of the Halbach array PMSM magnetic field distribution is derived by the new Fourier series and spatial integral technology, in which the end leakage magnetic field is contained. The results obtained by the proposed analytical method are compared with the numerical results, i.e., the results obtained by the finite-element method, which verify the effectiveness of the analytical method. Magnetic field distribution containing the end leakage magnetic field in different magnetization methods is compared and analyzed, which shows that the Halbach array magnetization has a larger main air-gap magnetic field and a smaller end leakage magnetic field. The effect of motor structure parameters on the end leakage magnetic field is analyzed, and the effect of end leakage magnetic field on eddy-current loss is preliminarily discussed, which shows that eddy-current loss caused by end effect almost accounts for half of the total eddy-current loss. The research on end effect and the effect of motor parameters on end effect are of great significance. In the end, the leakage magnetic field results obtained by the proposed analytical method are verified by the experimental results.

This paper studies the feasibility of vibration energy harvesting using a Halbach array. A Halbach array is a specific arrangement of permanent magnets that concentrates the magnetic field on one side of the array while cancelling the field to almost zero on the other side. This arrangement can improve electromagnetic coupling in a limited space. The Halbach array offers an advantage over conventional layouts of magnets in terms of its concentrated magnetic field and low-profile structure, which helps improve the output power of electromagnetic energy harvesters while minimizing their size. Another benefit of the Halbach array is that due to the existence of an almost-zero magnetic field zone, electronic components can be placed close to the energy harvester without any chance of interference, which can potentially reduce the overall size of a self-powered device. The first reported example of a low-profile, planar electromagnetic vibration energy harvester utilizing a Halbach array was built and tested. Results were compared to ones for energy harvesters with conventional magnet layouts. By comparison, it is concluded that although energy harvesters with a Halbach array can have higher magnetic field density, a higher output power requires careful design in order to achieve the maximum magnetic flux gradient.

The magnetic focusing effect of Halbach array has a wide range of applications in many areas. In this paper, based on the general theory of static magnetic field, various factors affecting the energy harvesting output voltage of Halbach array are analyzed. After that, according to the simulation carried out by Comsol, the magnetic field distribution of the four module Halbach array with different numbers of magnets was obtained, which verified that the shape of the magnetic field was a periodic sinusoidal distribution converging on one side. Then a Halbach array rotor was designed, and the output voltage of the Halbach array energy harvesting structure with different numbers of magnets under the influence of different factors was simulated, and the optimal Halbach array energy harvesting structure was obtained. At the same time, the halbach array composed of 12 magnets was verified through experiments. The experimental results are consistent with the simulation results, and higher voltage and higher power can be obtained. The designed Halbach array with simple structure and small size, can be more suitable to the lower frequency vibration or motion.KeywordsLow frequencyHalbach arrayMagnetoelectric conversion

In the minimally invasive surgery and diagnoses such as wireless capsule endoscopy (WCE) or natural orifice transluminal endoscopic surgery (NOTES), accurate position and orientation information of the instrument is very important when it moves inside human body. Usually, magnetic tracking method is used to locate the target, in which a small magnet is enclosed. With the signals from the magnetic sensors, the position and orientation information of the magnet can be estimated based on an appropriate magnetic field model and optimization algorithm. Magnetic dipole model is the mostly used model in magnetic tracking, which can provide three dimensional position (3D) and two dimensional orientation (2D) information. The rotation information about the magnetic moment cannot be estimated. In order to utilize the six dimensional (6D) information, a specific mathematic model of magnetic field is needed rather than the magnetic dipole model. In this paper, we proposed a mathematic model for the annular magnet. This model can provide 3D position and 3D rotation information for tracking. The proposed mathematic model is based on the superposition principle and the Biot-Savart law. Finite element software is used to verify this model. Simulation result shows that the model is better than magnetic dipole model.

Problem. The implementation of strict requirements for magnetic silence of elongated energy-saturated technical objects – such as naval vessel and submarines is largely determined by the adequacy of mathematical models to the signatures of a real magnetic field. Aim. Simplification of mathematical modeling of the magnetic field of an uncertain extended energy-saturated object based on the development and application of a multispheroidal model of its magnetic field instead of the well-known multidipole model. Methodology. Coordinates of the geometric location and magnitudes of spatial extended spheroidal harmonics of spheroidal sources of multispheroidal model of magnetic field calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction far magnetic field magnitude of extended technical objects based on designed multispheroidal model of the magnetic field in the form of spatial prolate spheroidal harmonics in prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. For the first time the method for design of multispheroidal model of magnetic field of uncertain extended energy-saturated technical object based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties developed. Practical value. It is shown the possibility to reduce the number of spheroidal sources of the magnetic field for adequate modeling of the real magnetic field based on the developed multispheroidal model compared to the number of well-known dipole sources of the magnetic field in the multidipole model of the magnetic field. References 48, figures 4.

The end-effect of Halbach array affects the accuracy of the magnetic levitation (maglev) motor force model. This paper presents a general semi-analytical method to solve the magnetic force and torque considering the magnetic field end-effects in the whole 2-D plane. The new Fourier period of the Halbach array is defined according to the distance between the positions where the magnetic field decays to zero, and then an analytical magnetic field model of the array including the end-distorted area is derived by the new Fourier series and Maxwell equations. Benefited from the Gaussian quadrature, the complex Lorentz integral of the force and torque can be simplified into the form of summation while the high computation accuracy is still promised. Finally, taking the typical maglev planar motor introduced in the paper as an example, a prototype of the motor is manufactured, and the experiment is conducted to validate the proposed semianalytical method. Compared with the existing harmonic model, the force and torque predicted by the semi-analytical model are closer to the measurement results, and the decoupling results of the force and torque in all directions are more accurate based on the proposed model.