Maxwell's Tensor Method Research Articles

Halbach structure parameter optimization by Maxwell tensor method for PML in manned hybrid maglev

This paper presents an optimization of the Halbach structure parameters for the permanent magnetic levitation (PML) component in manned hybrid maglev system using Maxwell tensor method. The focus is on enhancing the levitation force and reducing the lateral stiffness, which are important for improving the system’s load capacity and lateral stability. Based on the analysis of the magnetic flux density formula as presented by Halbach (1985), seven structural parameters were reduced to four independent variables. The Maxwell tensor formulation enabled the derivation of analytical expressions for the levitation force and stiffness, leading to identification of parametier conbinations that maximize magneitc properties. Contour plots were constructed to visually demonstrate the impact of parameter variations on magnetic characteristics, including magnetic flux density, levitation force, and lateral stiffness. Ultimately, a series of optimization cases for structural parameters were proposed in accordance with the design requirements of the manned hybrid Maglev system, and these cases were comprehensively evaluated through contour plot analysis. Furthermore, the finite element software FEMM was utilized to model and validate the optimal cases, thereby confirming the effectiveness of the analytical expressions for magnetic properties and the optimization strategy. This research not only provides theoretical support for the design of PML components but also offers new insights into the optimization of magnetic characteristics for hybrid maglev systems.

The influence of ITSF on vibration of high voltage Line-Start permanent magnet synchronous motor

PurposeThe purpose of this paper is to present the influence of inter-turn short circuit faults (ITSF) on electromagnetic vibration in high-voltage line-starting permanent magnet synchronous motor (HVLSPMSMS).Design/methodology/approachIn this paper, the ampere–conductor wave model of HVLSPMSM after ITSF is established. Second, a mathematical model of the magnetic field after ITSF is established, and the influence law of the ITSF on the air-gap magnetic field is analyzed. Further, the mathematical expression of the electromagnetic force density is established based on the Maxwell tensor method. The impact of HVLSPMSM torque ripple frequency, radial electromagnetic force spatial–temporal distribution and rotor unbalanced magnetic tension force by ITSF is revealed. Finally, the electromagnetic–mechanical coupling model of HVLSPMSM is established, and the vibration spectra of the motor with different degrees of ITSF are solved by numerical calculation.FindingsIn this study, it is found that the 2np order flux density harmonics and (2 N + 1) p order electromagnetic forces are not generated when ITSF occurs in HVLSPMSM.Originality/valueBy analyzing the multi-harmonics of HVLSPMSM after ITSF, this paper provides a reliable method for troubleshooting from the perspective of vibration and torque fluctuation and rotor unbalanced electromagnetic force.

Analytical Calculation and Low-Order Component Optimization of Axial Electromagnetic Force for an Axial Flux Motor

The low-order high-amplitude component of axial electromagnetic force is the leading cause of electromagnetic vibration and noise of axial flux permanent magnet (AFPM) motors. In this paper, based on the analytical calculation of the axial electromagnetic force for a 20-pole 24-slot AFPM motor, the low-order axial electromagnetic force component is weakened by optimizing the motor’s electromagnetic structure. First, the three-dimensional (3-D) structural model of the motor is equated to a two-dimensional (2-D) analytical model, which is used to calculate the magnetic field generated by each of the armature winding and permanent magnet. Then, combined with the Maxwell tensor method, the axial electromagnetic force waveform is further obtained, and the target low-order high-amplitude component is extracted by 2-D Fourier analysis. Finally, taking this axial electromagnetic force component minimization as the target function of optimization, considering the output torque and ripple as constraints, the genetic algorithm is used to search for the optimal point by varying the structural parameters of the motor. The results show that the target low-order high-amplitude component of the axial electromagnetic force of the optimized motor is effectively suppressed, and the output torque ripple decreases with slightly increased torque amplitude, which provides a practical method for vibration and noise reduction of AFPM motors.

Accurate Modeling and Optimization of Electromagnetic Forces in an Ironless Halbach-Type Permanent Magnet Synchronous Linear Motor

In order to solve the electromagnetic force optimization problem of a high-power-density ironless Halbach-type permanent magnet synchronous linear motor, this paper adopts an electromagnetic force optimization method based on magnetic field analysis, electromagnetic force modeling, and genetic algorithm optimization: Firstly, the magnetic field of the Halbach permanent magnet array is solved by the combination of the equivalent magnetization strength method and the pseudo-periodic method, which takes into account the influence of the edge effect of the secondary magnetic field, and the magnetic field of the primary winding is solved by Fourier series expansion method. Secondly, the Maxwell tensor method is used to establish the functional relationship between the electromagnetic thrust and the main structural parameters of the unilateral motor. Finally, based on the parameter sensitivity analysis of the optimized variables and the response surface calculation, the optimal combination of the optimized variables to meet the optimization objective is found by a genetic algorithm. This method of the accurate modeling and optimization of an electromagnetic force can accurately calculate the motor air gap magnetic field and electromagnetic thrust, and the optimization speed is fast, which can greatly save time. The optimization results show that, under the premise of constant input power, the unilateral average thrust of the motor is increased by 18.75%, the peak value of thrust fluctuation is decreased by 30.27%, and the results match well with the finite element results, which verifies the correctness of the optimization results of the electromagnetic force and the reasonableness of the optimization method.

An improved equivalent magnetic network model of IPM motor in electric vehicles for dynamic simulation of transient effects

The existing electric motor model for electric vehicles cannot address the simulation of transient conditions. Moreover, it is difficult to balance the computation time and simulation accuracy. To address these shortcomings, an improved internal permanent magnet motor model containing nonlinear magnetic field factors was proposed based on the equivalent magnetic circuit theory. The effects of the magnetic circuit saturation, cogging torque, and magnetic field harmonics on the electromagnetic torque could be reflected. Furthermore, the electromagnetic vibration and rotor eccentricity could be considered. To improve the computational efficiency of the equivalent magnetic network method, lumped parameter or mesh-based methods were used for flux paths with different complexities. Second, the single-layer mesh-based method with local variable permeability parameters improved the efficiency of the air-gap dynamic analysis and computational efficiency while simulating 2D flux paths. The radial and tangential electromagnetic forces were calculated using the Maxwell tensor method and the action on the rotor. The accuracy of the nonlinear magnetic field modeling was verified by finite element analysis and experimental. In addition, the motion modeling method and dynamic characteristics of the model were studied. A reference was provided for dynamic simulations of electric vehicles and other using permanent magnet motors.

Design and analysis of concave‐core stator direct‐drive permanent magnet motor

AbstractTo solve the problems mechanically caused by parallel direct‐drive permanent magnet motors (DD‐PMSM). This paper proposes a concave‐core stator direct‐drive permanent magnet motor (CCSDD‐PMSM) structure. In this study, an auxiliary yoke structure is proposed to optimise the magnetic circuit and analyse the cogging torque to achieve the minimum detent force. Based on the unit motor and combined with the magnetomotive force to analyse the symmetry of different winding distributions, the stability characteristics of the motor are studied. The magnetic field distribution of the motor load air gap is calculated using the finite element method. The contribution of different spatial order harmonics to the motor output torque is calculated using the Maxwell tensor method. The torque calculation expression of the CCSDD‐PMSM is given. Finally, an experimental prototype is made to verify the accuracy of the analysis in this paper.

Study on unbalanced magnetic pull of permanent magnet toroidal motor with planet rotor eccentricity

Permanent magnet toroidal motor (PMTM) is a novel spatial motor that integrates toroidal drive system and permanent magnet synchronous motor. The unbalanced magnetic pull (UMP) with planet rotor eccentricity is the main cause of vibration in PMTM. In this paper, the analytical model of magnetic field with planet rotor eccentricity is established with the consideration of magnetic flux leakage. The calculation results of eccentric magnetic fields are compared with simulation results of finite element model. Based on the Maxwell tensor method, the analytical model of UMP with planet rotor eccentricity is established. The changes of UMP with initial eccentricity are obtained. The UMP is analyzed with the effects of eccentric distance and angle. The transient variations of UMP are discussed with different eccentric situations respectively. In addition, both simulation and calculation results for UMP have a good agreement, which verifies the validity of analytical model.

Mechanism model of suspension force for spherical bearingless flywheel machine

The flywheel energy storage system (FESS) has drawn more and more attentions in the past decades, and its operation performance is greatly dependent on the suspension characteristics of the flywheel machines. This paper proposed a new mechanism model of the suspension force for the spherical bearingless flywheel machine (S-BFM), which may be applied for the machine optimal design, performance analysis or efficiency evaluation. In the proposed model. the magnetic field partition method was used to solve the permeability of the machine in different diffused flux regions. The inherent characteristics of the rotor eccentricity and magnetic circuit saturation are considered by combining maxwell tensor method and equivalent magnetic circuit method. By considering the rotor eccentricity, magnetic circuit saturation and diffusion flux, the proposed model processes higher precision over the traditional model, and its outputs are quite close to that of the finite element analysis (FEA).

Analysis and inhibition of radial electromagnetic force in composite magnetic coupler

AbstractA new inhibition method was proposed for the problem of harmonic of radial electromagnetic force during the operation of the composite magnetic coupler. In order to suppress its radial electromagnetic force, a radial permanent magnet rotor eccentricity method is proposed, and a three‐dimensional finite element model is established. Based on the Maxwell tensor method, the radial electromagnetic force of the composite magnetic coupler is analysed, and the mathematical expression of its radial electromagnetic force is established, Maxwell stress tensor was used to analyse the radial electromagnetic force in a composite magnetic coupler and to establish the mathematical equation for the radial electromagnetic force. In order to suppress the radial electromagnetic force, a method based on rotor eccentricity in a radial permanent magnet was proposed. A prototype of a composite magnetic coupler with a rated power of 5 kW was designed to test its output torque under load. The results showed that the simulated curve was consistent with the tested curve, with a maximum numerical error of 8.9%. The tested values of the output torque before and after eccentricity were compared, and the results showed that the output torque remained basically unchanged after the eccentric structure was adopted and that the torque pulsation was significantly reduced.

Reduction of Pole-Frequency Vibration of Surface-Mounted Permanent Magnet Synchronous Machines With Piecewise Stagger Trapezoidal Poles

An optimization method of the piecewise stagger trapezoidal poles is proposed for the reduction of the pole-frequency vibration of the surface-mounted permanent magnet synchronous machines (SPMSMs) in this article. By offsetting the zero-crossing region of flux density, this method can weaken the pole-frequency force harmonic while ensuring the torque density. First, based on the Maxwell tensor method, the analytical model of concentrated force acting on the teeth is deduced. Taking a 10-pole/12-slot SPMSM as an example, the deep cause of the pole-frequency radial force harmonic is analyzed. Then, the weakening mechanism of the piecewise stagger trapezoidal poles for the pole-frequency radial force harmonic is revealed with the finite-element model (FEM). Based on the multiphysics coupling model, the key electromagnetic properties and vibration spectrum between the common and proposed motors are compared. The results show that the piecewise stagger trapezoidal pole structure has a significant suppression effect on the pole-frequency vibration. Finally, the vibration experiment of the two prototypes is carried out, and the effectiveness of the optimized structure is verified.

Accurate Modeling and Control System Design for a Spherical Radial AC HMB for a Flywheel Battery System

Different from common cylindrical magnetic bearings, the spherical radial AC HMB has obvious advantages in suppressing the gyroscopic effect, thus it is very suitable for application in flywheel battery systems. In this study, a precise mathematical model of suspension forces is deduced adopting wide-area and universal modeling theories in detail for the spherical radial AC HMB. Different from the traditional Maxwell tensor method based on a cylindrical two-dimensional coordinate system, the improved Maxwell tensor method based on a spherical coordinate system has a wider range and versatility. Then, the detailed explanation about the accuracy, wide-area and universality are given. Further, based on the model, the control system is designed, which includes the overall schematic, hardware and software designs. Finally, stiffness and performance tests are conducted. The good results of the stiffness tests indicate the error between the proposed model and the force-deflection stiffness test results is less than 2.8%. The good performance test results show that, based on the established model and control system, the rotor will soon return to equilibrium position under the disturbance of vehicle action.

Modeling and Multilevel Design Optimization of an AC–DC Three-Degree-of-Freedom Hybrid Magnetic Bearing

In this article, in order to improve the bearing capacity per unit area and reduce the nonlinearity and couplings of three-pole magnetic bearing, which is driven by a three-phase power inverter, a three-degree-of-freedom (3-DOF) six-pole hybrid magnetic bearing (HMB) is proposed. To realize the larger bearing capacity and smaller volume, a multilevel design optimization is proposed to conduct multiobjective optimization. At first, the structure and working principle of the ac–dc 3-DOF six-pole HMB are introduced. Then, the suspension force mathematical models of the 3-DOF HMB based on the Maxwell tensor method are established. Next, the design variables and optimization objectives are selected, and the comprehensive sensitivity analysis is adopted to divide the design variables into three levels; the response surface method and multiobjective particle swarm optimization algorithm are applied to realize a compromise among the three optimization objectives. Finally, the results of the simulation and experiment verify the validity and correctness of the modeling and optimization design.

Magnets Shifting Design of Dual PM Excited Vernier Machine for High-torque Application

In this study, a novel dual permanent magnet excited vernier machine (DPMEVM) with magnets shifting in stator is proposed. Compared with the conventional permanent magnet synchronous machine (PMSM), the DPMEVM based on the bidirectional field modulation effect can operate in a wider torque range. However, the torque ripple of a conventional DPMEVM is high because of the superposition of the torque generated by the stator-side and rotor-side PMs. Consequently, a novel DPMEVM with magnets shifting is proposed to further reduce the torque ripple. First, the topologies and working principles of the baseline machine and proposed machines are introduced. Second, the torque-contribution harmonics are analyzed and calculated using the Maxwell tensor method. The calculation results reveal that the DPMEVM, benefiting from multiple working harmonics, can offer an enhanced torque capability compared to the PMSM. In addition, the torque ripple characteristics of the proposed machines are analyzed. It is verified that the torque ripple can be significantly reduced through magnets shifting. Third, the performances of the baseline machine and proposed machines are analyzed and compared in terms of flux density, open-circuit back-EMF, and torque characteristics. In addition, the proposed principle can be extended to machines with the same unit motor. Finally, a 120s-110p prototype machine is manufactured for validation.

Electromagnetic Design and Analysis of Permanent Magnet Linear Synchronous Motor

Since permanent magnet linear synchronous motors are widely used in fatigue testing, in this paper, the thrust of a permanent magnet linear synchronous motor (PMLSM) is amplified more than 10 times by the method of resonance. Firstly, the air gap magnetic field is analyzed by the equivalent magnetization current method (EMC), the electromagnetic thrust is calculated, and the expression is given by the Maxwell tensor method. The vibration analysis of the whole machine is used to obtain the conditions under which the motor resonates. The dynamic and static characteristics of the motor are analyzed through finite element simulation, the results of the motor design are judged to be reasonable, and the theoretical calculation results are compared with the simulation results to verify the accuracy of the theoretical calculation. The accuracy of the simulation results is verified by static force experiments. Finally, the rated thrust of the motor was enlarged more than 10 times by resonance experiments.

Research of a Six-Pole Active Magnetic Bearing System Based on a Fuzzy Active Controller

Magnetic bearings have a series of excellent qualities, such as no friction and abrasions, high speed, high accuracy, and so on, which have fundamentally innovated traditional forms of support. In order to solve the problems of the large volume, low power density and high coupling coefficient of three-pole magnetic bearings, a six-pole AC active magnetic bearing is designed. Firstly, the basic structure and working principle of a two-degree-of-freedom (2-DOF) six-pole active magnetic bearing is introduced. Secondly, a suspension force modeling method of a 2-DOF AC active magnetic bearing based on the Maxwell tensor method is proposed, and the mathematical model of active magnetic bearing is established. Considering the fact that AC active magnetic bearing is essentially a nonlinear system, a fuzzy active disturbance rejection control (ADRC) method is designed based on fuzzy control and ADRC theory. Its control algorithm and control block diagram are given, and the fuzzy ADRC method is simulated and verified. Finally, the control block diagram of an experimental system based on the 2-DOF six-pole active magnetic bearing is given, and the experimental platform is constructed. The experimental results show that the mechanical and magnetic circuit structure of the 2-DOF six-pole active magnetic bearing is reasonable, and the fuzzy controllers can realize the stable suspension of the rotor.

Radial Force and Vibration Analysis of Modular Fault‐Tolerant Permanent Magnet Motor with Unequal Span Windings

The low‐speed and high‐torque modular fault‐tolerant permanent magnet motor has the strong fault‐tolerant ability, and can make the motor output torque act on the load directly, and the system has the highest transmission efficiency, which is especially suitable for ship propulsion. In the electric ship direct‐drive propulsion system, the motor is the core of the power system. Because the motor is directly connected with the load, the intermediate buffer mechanism is canceled, so the smooth operation of the motor is put forward to higher requirements. Especially in some fault‐tolerant operations, the unbalanced radial force generated by the asymmetric operation may cause the motor to vibrate greatly, which will reduce the operational life of the motor and even bring new faults. In serious cases, the whole transmission system will be damaged. Therefore, it is necessary to study the vibration of the fault‐tolerant motor under different operating conditions. In this paper, the module combined stator fault‐tolerant permanent magnetic synchronous motor for ship direct‐drive propulsion is taken as the research object. Based on the Maxwell tensor method, the radial force wave expressions of the motor during regular operation and fault‐tolerant operation are derived, and the harmonic order and frequency, which have a significant influence on vibration are summarized. The radial force of the motor under different operating conditions with different control strategies is calculated by the finite element method, and the harmonic components are analyzed by the two‐dimensional Fourier transform. On this basis, the vibration of the motor and the mechanical strength of the rotor support frame are analyzed. The research results of this paper provide some reference for the subsequent vibration reduction strategies of fault‐tolerant motors. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Study of the Combined Effects of the Air-Gap Transfer for Maxwell Tensor and the Tooth Mechanical Modulation in Electrical Machines

The Maxwell Tensor (MT) method is widely used to compute global forces or local surface forces for vibroacoustic design of electrical machines under electromagnetic excitation. In particular, the air-gap MT method is based on a cylindrical shell in the middle of the air gap. This article proposes to quantify the differences between the air-gap MT and the magnetic force wave experienced by the stator. In particular, the air gap to stator transfer and the tooth mechanical modulation effects are studied. A new formula is proposed to extend the tooth modulation effect to tangential forces. A numerical application is performed with a turbo-alternator to illustrate the respective and combined effects of both phenomena. This article highlights that the tooth mechanical modulation is relevant even for electrical machines with a high number of teeth. In addition, the combination of the two phenomena has a clear impact on the calculated surface force. Therefore, it is recommended to take into account the air-gap transfer for any study of the tooth mechanical modulation effect.

Comparative Study of Sideband Electromagnetic Force in Internal and External Rotor PMSMs With SVPWM Technique

The permanent magnet synchronous motors (PMSMs) with space vector pulsewidth modulation (SVPWM) technique will inevitably introduce sideband current harmonic components around the carrier frequency. The armature reaction field caused by the sideband current interacts with the permanent magnet field, generating sideband electromagnetic force and resulting in high-frequency noise. This paper provides a comparative study of the sideband electromagnetic force in internal and external rotor PMSMs with SVPWM technique. First, the analytical model of the electromagnetic force in internal and external rotor PMSMs is established based on the Maxwell tensor method. The source, spatial order, and frequency of the electromagnetic force is obtained. Then, the spatial and temporal characteristics of the sideband force in internal and external rotor PMSMs with different pole and slot combinations are derived and investigated. The results show that the frequency characteristics of the zeroth order sideband force in internal and external rotor PMSMs introduced by SVPWM are identical. However, the frequency characteristics demonstrate significant differences for the nonzeroth order sideband force. Finally, the theoretical analysis is validated through the finite element and experimental results, respectively. This study is benefit for the source identification and reduction of high-frequency acoustic noise for PMSMs.

Electromagnetic Vibration Characteristics Analysis of a Squirrel-Cage Induction Motor Under Different Loading Conditions

Electromagnetic vibration is an important excitation source for squirrel-cage induction motors. However, the electromagnetic vibration under various loadings has not been sufficiently analyzed. It is proposed in this paper that the electromagnetic vibration of motors under different loads can be obtained by analyzing the amplitude of the electromagnetic force wave. The Maxwell tensor method is employed to derive the spatial and temporal distributions of the radial force. This paper also calculates the radial force using the finite element method and decodes the calculated results using two dimensional fast Fourier transform (2D-FFT) to determine the amplitude of the electromagnetic force at the spatial order under different loads. In addition, through the modal analysis of the stator core, it can be concluded that in the case of nonresonance, the vibration response increases when the electromagnetic force of the first-order and second-order rotor slot harmonic increases. Finally, the conclusion is verified by separating the electromagnetic vibration of the motor using a vibration test rig.

Research on Operation Principle and Control of Novel Hybrid Excitation Bearingless Permanent Magnet Generator

Under the condition of load changing, the magnetic field of traditional permanent magnet generators (PMG) is hard to be adjusted, and the mechanical bearings are significantly worn. To overcome the drawbacks above, a novel hybrid excitation bearingless permanent magnet generator (HEBPMG) is proposed in this paper, which has integrated the merits of hybrid excitation permanent magnet generators and magnetic bearings. Firstly, the structure and winding configuration of the HEBPMG are introduced, and then the principles of radial suspension and power generation are presented. The suspension principle as well as power generation principle is analyzed in this paper. Then, the flux linkage and induced voltage equations are derived, and the accurate mathematical model of radial suspension force is built based on the Maxwell tensor method. Subsequently, by means of the finite element analysis software-ANSYS Maxwell, the corresponding electromagnetic characteristics are analyzed to verify the correctness of the mentioned models. In addition, a compensation control strategy based on flux-linkage observation is proposed to solve the problems of unstable suspension force and generating voltage under variable load condition in this paper. Meanwhile, the corresponding control system is constructed and its feasibility is validated by simulation results. Finally, an experimental prototype of a 2.2 kW HEBPMG is tested. Experimental researches show that the HEBPMG can operate steadily under variable load condition and possess good suspension performance and power generation quality.