Effects of DC-Field Excitation on the Incremental Inductance of a Variable Flux Reluctance Machine

This paper presents a method for the computation of the incremental inductances in a 12/10 variable flux reluctance machine using the hybrid analytical modeling coupled with a fixed-point nonlinear solver. The variation of incremental and apparent inductance with respect to the dc-field excitation is investigated for both zero and non-zero ac-field excitations. The results show that the difference between both inductance values is not negligible after 25 A/mm2 dc-current density for the investigated benchmark without the ac field. Moreover, when a non-zero ac field is introduced in addition to the dc-field, the apparent inductance becomes misleading not only under magnetic saturation but also under low excitation in the linear region of the saturation curve. The results obtained with the proposed nonlinear hybrid model are compared with the finite element method in terms of magnetic flux density distribution and incremental inductance value. The root-mean-square discrepancy of magnetic flux density distribution is found to be 37.6 mT. Furthermore, the discrepancy between incremental inductance results of the proposed method and the finite element model is calculated as 1.43%, while the proposed approach requires less post-processing and necessitates ten times less number of degrees-of-freedom

Convergence analysis of the fixed-point method with the hybrid analytical modeling for 2-D nonlinear magnetostatic problems

This paper presents the convergence analysis of the fixed-point method (FPM) to model the nonlinear magnetic characteristics of a 2-D magnetostatic problem. In this study, FPM is used as the iterative nonlinear solver of the hybrid analytical modeling (HAM) technique for the accurate computation of the magnetic field distribution. The benchmark consists of a stator with excitation windings, an airgap, and a slotless mover. The relative errors between two successive iterations are calculated using different error estimators: the attraction force on the mover, the Fourier coefficients defined in the airgap, the magnetic flux density, and the magnetic scalar potential distributions. The effect of the number of mesh elements and harmonics on the accuracy and computational cost of the model is investigated for different levels of magnetic saturation. It is observed that the maximum rate of change in the relative difference of attraction force during the iterations is found to be 0.52 under the magnetic saturation. In addition, the absolute error of the attraction force between the developed hybrid model with FPM and the finite element method (FEM) is achieved to be 0.18%, while HAM has approximately three times less number of degrees-of-freedom compared to FEM

A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings

This study aims to improve the computational efficiency of the frequency domain analysis of medium-frequency transformers (MFTs) with the presence of large clearance distances and fine foil windings. The winding loss and magnetic energy in MFTs in the medium-frequency range are calculated utilizing a finite element method (FEM) using common triangular and alternative rectilinear mesh elements. Additionally, in order to improve the computational efficiency of the calculations, a spectral element method (SEM) is coupled with a FEM, thus creating a hybrid FEM–SEM formulation. In such a hybrid approach, the FEM is used to calculate the current density distribution in the two-dimensional (2D) cross-section of the foil conductors to achieve reliable accuracy, and the SEM is adopted in the nonconducting clearance distances of the winding window to reduce the system of equations. The comparative analysis of the calculated resistance and reactance of the under-study models showed that the FEM with rectilinear mesh elements and the FEM–SEM model outperformed the FEM with triangular mesh elements in terms of accuracy and computational cost. The hybrid FEM–SEM model enables a reduced system of equations for modeling the electromagnetic behavior of MFTs. This research provides valuable insights into both the computational approaches and meshing challenges in the analysis of MFTs and offers a foundation for future research on the design and optimization of MFT

Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems

This paper presents a comparison of different time- and frequency-domain solvers for the steady-state simulation of the eddy current phenomena, due to the motion of a permanent magnet array, occurring in the soft-magnetic stator core of electrical machines that exhibits nonlinear material characteristics. Three different dynamic solvers are implemented in the framework of the isogeometric analysis, namely the traditional time-stepping backward-Euler technique, the space-time Galerkin approach, and the harmonic balance method, which operates in the frequency domain. Two-dimensional electrical machine benchmarks, consisting of both slotless and slotted stator core, are considered to establish the accuracy, convergence, and computational efficiency of the presented solvers

Finite Element Analysis of Laminar Heat Transfer within an Axial-Flux Permanent Magnet Machine

Axial-Flux Permanent Magnet (AFPM) machines have gained popularity over the past few years due to their compact design. Their application can be found, for example, in the automotive and medical sectors. For typically considered materials, excessive heat can be generated, causing possible irreversible damage to the magnets, bonding, or other structural parts. In order to optimize cooling, knowledge of the flow and the consequent temperature distribution is required. This paper discusses the flow types and heat transfer present inside a typical AFPM machine. An Isogeometric Analysis (IGA) laminar-energy model is developed using the Nutils open-source Python package. The developed analysis tool is used to study the effects of various important design parameters, such as the air-inlet, the gap-length, and the rotation speed on the heat transfer in an AFPM machine. It is observed that the convective heat transfer at the stator core is negatively affected by adding an air-inlet. However, the heat dissipation of the entire stator improves as convective heat transfer occurs within the air-inlet

Framework for efficient dynamic analysis applied to a tubular generator for suspension applications

This paper considers a slotless three-phase tubular permanent magnet generator located in an automotive suspension system for the application of vibration energy harvesting. A two-dimensional finite element method model of the harvester is produced and an experimental setup that contains the generator is constructed. Signal decomposition methods are applied to measured suspension displacement data and the resulting signal components are used as input for the model. Two approaches for signal decomposition are discussed, namely, the discrete Fourier transform and the continuous wavelet transform. The individual emf responses of the model are reconstructed to a single output, while a sideband prediction algorithm accounts for the non-linearities in the system. The simulation results are compared with the reference measurements conducted on the setup to determine the accuracy of each of the signal decomposition methods

Convergence analysis of the fixed-point method with the hybrid analytical modeling for 2-D nonlinear magnetostatic problems

This paper presents the convergence analysis of the fixed-point method (FPM) to model the nonlinear magnetic characteristics of a 2-D magnetostatic problem. In this study, FPM is used as the iterative nonlinear solver of the hybrid analytical modeling (HAM) technique for the accurate computation of the magnetic field distribution. The benchmark consists of a stator with excitation windings, an airgap, and a slotless mover. The relative errors between two successive iterations are calculated using different error estimators: the attraction force on the mover, the Fourier coefficients defined in the airgap, the magnetic flux density, and the magnetic scalar potential distributions. The effect of the number of mesh elements and harmonics on the accuracy and computational cost of the model is investigated for different levels of magnetic saturation. It is observed that the maximum rate of change in the relative difference of attraction force during the iterations is found to be 0.52 under the magnetic saturation. In addition, the absolute error of the attraction force between the developed hybrid model with FPM and the finite element method (FEM) is achieved to be 0.18%, while HAM has approximately three times less number of degrees-of-freedom compared to FEM

A new loop-based hybrid analytical modeling formulation and the selection of its nonlinear solver

This article presents a comprehensive comparative study between two nonlinear solvers for the hybrid analytical modeling (HAM) formulation. The Newton-Raphson method (NRM) and the fixed-point method (FPM) are compared in terms of their convergence rate, computational time, and accuracy. A new HAM formulation using the loop-based magnetic equivalent circuit (MEC) instead of the node-based one is proposed to improve the convergence and condition number. The loop-based formulation is coupled with both NRM and FPM nonlinear solvers to perform the magnetostatic analysis of a 12/10 variable flux reluctance machine (VFRM) under local magnetic saturation. It is shown that both methods can achieve convergence for various saturation levels, mesh, and harmonic refinements. However, FPM exhibits a 0.4 larger convergence rate than NRM. It is also observed that the accuracy of NRM decreases under deep magnetic saturation, and the number of required iterations of NRM increases with the model refinement. However, FPM is able to converge for all analyzed refinements with less than six iterations

Effects of DC-Field Excitation on the Incremental Inductance of a Variable Flux Reluctance Machine

This paper presents a method for the computation of the incremental inductances in a 12/10 variable flux reluctance machine using the hybrid analytical modeling coupled with a fixed-point nonlinear solver. The variation of incremental and apparent inductance with respect to the dc-field excitation is investigated for both zero and non-zero ac-field excitations. The results show that the difference between both inductance values is not negligible after 25 A/mm2 dc-current density for the investigated benchmark without the ac field. Moreover, when a non-zero ac field is introduced in addition to the dc-field, the apparent inductance becomes misleading not only under magnetic saturation but also under low excitation in the linear region of the saturation curve. The results obtained with the proposed nonlinear hybrid model are compared with the finite element method in terms of magnetic flux density distribution and incremental inductance value. The root-mean-square discrepancy of magnetic flux density distribution is found to be 37.6 mT. Furthermore, the discrepancy between incremental inductance results of the proposed method and the finite element model is calculated as 1.43%, while the proposed approach requires less post-processing and necessitates ten times less number of degrees-of-freedom

Analysis of motional eddy currents in the slitted stator core of an axial-flux permanent magnet machine

This article concerns the modeling and design of a slitted stator core for single-sided axial-flux permanent-magnet machine application. The stator core is specially designed to maximize the magnetic-flux density in the air gap and to minimize the eddy-current losses occurring at high rotational speeds. To reduce the effort needed for computing the motional eddy-current distribution in the presence of nonlinear material characteristics, a novel method is proposed. It combines the harmonic balance method, which is advantageous for simulating in the frequency domain the steady-state periodic response of a nonlinear system under harmonic excitation, together with a source description that introduces a complex magnetization that mimics the displacement of the permanent-magnet array. Following this method, time-domain distributions and losses can be reconstructed accurately with a low number of harmonics. A 3-D periodic model of the slotless axial-flux machine is built in the framework of isogeometric analysis (IGA) and a mixed formulation is employed, which relies on high-order Nédélec edge elements. The proposed model is embedded into a gradient-based optimization problem to determine the optimal shape of the slits in the stator core of the motor. This results in a novel cost-effective solution for improving the efficiency of axial-flux permanent-magnet machines