Special Issue "Mathematical Models for the Design of Electrical Machines"

A special issue of Mathematical and Computational Applications (ISSN 2297-8747). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (31 October 2020).

Special Issue Editors

Dr. Frédéric Dubas
Website
Guest Editor
Département ENERGIE, FEMTO-ST, CNRS, University Bourgogne Franche-Comté, F90000 Belfort, France
Interests: applied mathematics; partial differential equations; separation of variables method; principle of superposition; (semi-)analytical modeling; subdomain technique; magnetic equivalent circuit; electrical machines
Special Issues and Collections in MDPI journals
Prof. Dr. Kamel Boughrara

Guest Editor
Laboratoire de Rcherche en Electrotechnique (LRE-ENP), 16200 Algiers, Algeria
Interests: electromagnetic field; electrical machines; analytical methods; numerical methods

Special Issue Information

Dear Colleagues,

Electrical machines are used in many electrical engineering applications, viz., transports (e.g., electric/hybrid/fuel cell vehicles, railway traction, aerospace, etc.), energy harvesting (e.g., flywheel, etc.), renewable energy (e.g., wind power turbine, hydroelectric power plant, etc.), magnetic refrigeration device, etc. For decades, numerical methods (i.e., the finite-element, finite-difference or boundary-element analysis) have been widely used in R&D departments for their accuracy as compared to measurements. Nevertheless, mainly in 3-D, these approaches are time-consuming and not suitable for the optimization problems. Nowadays, in order to reduce the computation time, R&D engineers must develop full computer-aided-design for electrical machines with accurate and fast models in simulations. Hence, the main objective of this Special Issue is to bring the latest advances and developments in mathematical modeling and design of electrical machines for different applications. The main models discussed will be based on the:

  • Equivalent circuits (e.g., electrical, thermal, magnetic, etc.);
  • Schwarz-Christoffel mapping method;
  • Maxwell-Fourier (i.e., multi-layers models, eigenvalues models, subdomain technique).

The interest topics in the mathematical models include, but are not restricted to:

  • 2-D, quasi 3-D and 3-D;
  • Global/local saturation, slotting and/or eddy-current effects;
  • Adaptive Generic models;
  • Multi-physic modeling with new materials;
  • Hybrid models.

The numerical method as well as the experimental tests will be used as comparisons or validations.

Assoc. Prof. Dr. Frédéric Dubas
Prof. Dr. Kamel Boughrara
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematical and Computational Applications is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (13 papers)

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Editorial

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Open AccessEditorial
Mathematical Models for the Design of Electrical Machines
Math. Comput. Appl. 2020, 25(4), 77; https://doi.org/10.3390/mca25040077 - 09 Dec 2020
Viewed by 336
Abstract
Electrical machines are used in many electrical engineering applications [...] Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)

Research

Jump to: Editorial, Review

Open AccessFeature PaperArticle
Steady State and 2D Thermal Equivalence Circuit for Winding Heads—A New Modelling Approach
Math. Comput. Appl. 2020, 25(4), 70; https://doi.org/10.3390/mca25040070 - 18 Oct 2020
Cited by 1 | Viewed by 509
Abstract
The study concerns the winding head thermal design of electrical machines in difficult thermal environments. The new approach is adapted for all basic shapes and solves the thermal behaviour of a random wire layout. The model uses the nodal method but does not [...] Read more.
The study concerns the winding head thermal design of electrical machines in difficult thermal environments. The new approach is adapted for all basic shapes and solves the thermal behaviour of a random wire layout. The model uses the nodal method but does not use the common homogenization method for the winding slot. The layout impact can be precisely studied to find different hotspots. To achieve this a Delaunay triangulation provides the thermal links between adjoining wires in the slot. Voronoï tessellation gives a cutting to estimate thermal conductance between adjoining wires. This thermal behaviour is simulated in cell cutting and it is simplified with the thermal bridge notion to obtain a simple solving of these thermal conductances. The boundaries are imposed on the slot borders with Dirichlet condition. Then solving with many Dirichlet conditions is described. Some results show different possible applications with rectangular and round shapes, one ore many boundaries, different limit condition values and different layouts. The model can be integrated into a larger model that represents the stator to have best results. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Investigation of Volumic Permanent-Magnet Eddy-Current Losses in Multi-Phase Synchronous Machines from Hybrid Multi-Layer Model
Math. Comput. Appl. 2020, 25(1), 14; https://doi.org/10.3390/mca25010014 - 04 Mar 2020
Cited by 1 | Viewed by 651
Abstract
This paper investigates the permanent-magnet (PM) eddy-current losses in multi-phase PM synchronous machines (PMSM) with concentric winding and surface-mounted PMs. A hybrid multi-layer model, combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method [...] Read more.
This paper investigates the permanent-magnet (PM) eddy-current losses in multi-phase PM synchronous machines (PMSM) with concentric winding and surface-mounted PMs. A hybrid multi-layer model, combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series), performs the eddy-current loss calculations. First, the magnetic flux density was obtained from the 2-D generic MEC and then subjected to the Fast Fourier Transform (FFT). The semi-analytical model includes the automatic mesh of static/moving zones, the saturation effect and zones connection in accordance with rotor motion based on a new approach called “Air-gap sliding line technic”. The results of the hybrid multi-layer model were compared with those obtained by three-dimensional (3-D) nonlinear finite-element analysis (FEA). The PM eddy-current losses were estimated on different paths for different segmentations as follow: (i) one segment (no segmentation), (ii) five axial segments, and (iii) two circumferential segments, where the non-uniformity loss distribution is shown. The top of PMs presents a higher quantity of losses compared to the bottom. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Reduced-Order Model of Rotor Cage in Multiphase Induction Machines: Application on the Prediction of Torque Pulsations
Math. Comput. Appl. 2020, 25(1), 11; https://doi.org/10.3390/mca25010011 - 29 Jan 2020
Cited by 2 | Viewed by 713
Abstract
For three-phase induction machines supplied by sinusoidal current, it is usual to model the n-bar squirrel-cage by an equivalent two-phase circuit. For a multiphase induction machine which can be supplied with different harmonics of current, the reduced-order model of the rotor must be [...] Read more.
For three-phase induction machines supplied by sinusoidal current, it is usual to model the n-bar squirrel-cage by an equivalent two-phase circuit. For a multiphase induction machine which can be supplied with different harmonics of current, the reduced-order model of the rotor must be more carefully chosen in order to predict the pulsations of torque. The proposed analysis allows to avoid a wrong design with non-sinusoidal magnetomotive forces. An analytical approach is proposed and confirmed by Finite-Element modelling at first for a three-phase induction machine and secondly for a five-phase induction machine. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessArticle
2D Hybrid Steady-State Magnetic Field Model for Linear Induction Motors
Math. Comput. Appl. 2019, 24(3), 74; https://doi.org/10.3390/mca24030074 - 25 Jul 2019
Cited by 1 | Viewed by 1056
Abstract
This paper presents a 2D hybrid steady-state magnetic field model, capable of accurately modeling the electromagnetic behavior in a linear induction motor, including primary slotting, finite yoke length, and longitudinal end-effects by primary motion. This model integrates a complex harmonic modeling technique with [...] Read more.
This paper presents a 2D hybrid steady-state magnetic field model, capable of accurately modeling the electromagnetic behavior in a linear induction motor, including primary slotting, finite yoke length, and longitudinal end-effects by primary motion. This model integrates a complex harmonic modeling technique with a discretized magnetic equivalent circuit model. The Fourier model is applied to regions with homogeneous material properties, e.g., air regions and the track of the motor, while the magnetic equivalent circuit (MEC) approach is used for the regions containing non-homogeneous material properties, e.g., the primary of the linear induction motor (LIM). By only meshing the domains containing highly-permeable materials, the computational effort is reduced in comparison with the finite element method (FEM). The model is applied to a double-layer single-sided LIM, and the resulting thrust and normal forces show an excellent agreement with respect to finite element analysis and measurement data. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Combining the Magnetic Equivalent Circuit and Maxwell–Fourier Method for Eddy-Current Loss Calculation
Math. Comput. Appl. 2019, 24(2), 60; https://doi.org/10.3390/mca24020060 - 04 Jun 2019
Cited by 1 | Viewed by 1128
Abstract
In this paper, a hybrid model in Cartesian coordinates combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and [...] Read more.
In this paper, a hybrid model in Cartesian coordinates combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series) is developed. This model coupling has been applied to a U-cored static electromagnetic device. The main objective is to compute the magnetic field behavior in massive conductive parts (e.g., aluminum, magnets, copper, iron) considering the skin effect (i.e., with the eddy-current reaction field) and to predict the eddy-current losses. The magnetic field distribution for various models is validated with 2-D and three-dimensional (3-D) finite-element analysis (FEA). The study is also focused on the discretization influence of 2-D generic MEC on the eddy-current loss calculation in conductive regions. Experimental tests and 3-D FEA have been compared with the proposed approach on massive conductive parts in aluminum. For an operating point, the computation time is divided by ~4.6 with respect to 3-D FEA. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Study of a Hybrid Excitation Synchronous Machine: Modeling and Experimental Validation
Math. Comput. Appl. 2019, 24(2), 34; https://doi.org/10.3390/mca24020034 - 27 Mar 2019
Cited by 7 | Viewed by 1269
Abstract
This paper deals with a parallel hybrid excitation synchronous machine (HESM). First, an expanded literature review of hybrid/double excitation machines is provided. Then, the structural topology and principles of operation of the hybrid excitation machine are examined. With the aim of validating the [...] Read more.
This paper deals with a parallel hybrid excitation synchronous machine (HESM). First, an expanded literature review of hybrid/double excitation machines is provided. Then, the structural topology and principles of operation of the hybrid excitation machine are examined. With the aim of validating the double excitation principle of the topology studied in this paper, the construction of a prototype is presented. In addition, both the 3D finite element method (FEM) and 3D magnetic equivalent circuit (MEC) model are used to model the machine. The flux control capability in the open-circuit condition and results of the developed models are validated by comparison with experimental measurements. The reluctance network model is created from a mesh of the studied domain. The meshing technique aims to combine advantages of finite element modeling, i.e., genericity and expert magnetic equivalent circuit models, i.e., reduced computation time. It also allows taking the non-linear characteristics of ferromagnetic materials into consideration. The machine prototype is tested to validate the predicted results. By confronting results from both modeling techniques and measurements, it is shown that the magnetic equivalent circuit model exhibits fairly accurate results when compared to the 3D finite element method with a gain in computation time. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessArticle
Exact Two-Dimensional Analytical Calculations for Magnetic Field, Electromagnetic Torque, UMF, Back-EMF, and Inductance of Outer Rotor Surface Inset Permanent Magnet Machines
Math. Comput. Appl. 2019, 24(1), 24; https://doi.org/10.3390/mca24010024 - 17 Feb 2019
Cited by 3 | Viewed by 1357
Abstract
This paper presents a two-dimensional analytical model of outer rotor permanent magnet machines equipped with surface inset permanent magnets. To obtain the analytical model, the whole model is divided into the sub-domains, according to the magnetic properties and geometries. Maxwell equations in each [...] Read more.
This paper presents a two-dimensional analytical model of outer rotor permanent magnet machines equipped with surface inset permanent magnets. To obtain the analytical model, the whole model is divided into the sub-domains, according to the magnetic properties and geometries. Maxwell equations in each sub-domain are expressed and analytically solved. By using the boundary/interface conditions between adjacent sub-regions, integral coefficients in the general solutions are obtained. At the end, the analytically calculated results of the air-gap magnetic flux density, electromagnetic torque, unbalanced magnetic force (UMF), back-electromotive force (EMF) and inductances are verified by comparing them with those obtained from finite element method (FEM). One of the merits of this method in comparison with the numerical model is the capability of rapid calculation with the highest precision, which made it suitable for optimization problems. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Magnetic Field Analytical Solution for Non-homogeneous Permeability in Retaining Sleeve of a High-Speed Permanent-Magnet Machine
Math. Comput. Appl. 2018, 23(4), 72; https://doi.org/10.3390/mca23040072 - 10 Nov 2018
Cited by 3 | Viewed by 1573
Abstract
This work presents a novel solution for magnetic field calculation in two-dimensional problems in which one region is defined with space-varying magnetic parameter. The proposed solution extends the well-established Maxwell–Fourier method for calculating magnetic fields in surface-mounted cylindrical high-speed permanent-magnet machines. This contribution [...] Read more.
This work presents a novel solution for magnetic field calculation in two-dimensional problems in which one region is defined with space-varying magnetic parameter. The proposed solution extends the well-established Maxwell–Fourier method for calculating magnetic fields in surface-mounted cylindrical high-speed permanent-magnet machines. This contribution is effective to evaluate more realistic magnetic parameters, where measurements of a high-speed permanent-magnet generator prototype indicate saturation in the retaining sleeve due to pole-to-pole leakage flux. The saturation profile is a function of mechanical angle and can be modeled with the aid of a space-varying relative permeability, expressed in terms of a Fourier series. As an example, the presented solution has been applied to a surface-mounted PM machine at no-load condition. Magnetic field calculations show that a simple saturation profile, with low order space-varying permeability in the retaining sleeve significantly affects the magnetic flux density distribution in the air-gap. The analytical solution is confronted with finite-element method, which confirms validity of the proposed methodology. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Memory Efficient Method for Electromagnetic Multi-Region Models Using Scattering Matrices
Math. Comput. Appl. 2018, 23(4), 71; https://doi.org/10.3390/mca23040071 - 09 Nov 2018
Cited by 1 | Viewed by 1125
Abstract
This paper describes the scattering matrix approach to obtain the solution to electromagnetic field quantities in harmonic multi-layer models. Using this approach, the boundary conditions are solved in such way that the maximum size of any matrix used during the computations is independent [...] Read more.
This paper describes the scattering matrix approach to obtain the solution to electromagnetic field quantities in harmonic multi-layer models. Using this approach, the boundary conditions are solved in such way that the maximum size of any matrix used during the computations is independent of the number of regions defined in the problem. As a result, the method is more memory efficient than classical methods used to solve the boundary conditions. Because electromagnetic sources can be located inside the regions of a configuration, the scattering matrix formulation is developed to incorporate these sources into the solving process. The method is applied to a 3D electromagnetic configuration for verification. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessFeature PaperArticle
Two-Dimensional Exact Subdomain Technique of Switched Reluctance Machines with Sinusoidal Current Excitation
Math. Comput. Appl. 2018, 23(4), 59; https://doi.org/10.3390/mca23040059 - 11 Oct 2018
Cited by 6 | Viewed by 1247
Abstract
This paper presents a two-dimensional (2D) exact subdomain technique in polar coordinates considering the iron relative permeability in 6/4 switched reluctance machines (SRM) supplied by sinusoidal waveform of current (aka, variable flux reluctance machines). In non-periodic regions (e.g., rotor and/or stator slots/teeth), magnetostatic [...] Read more.
This paper presents a two-dimensional (2D) exact subdomain technique in polar coordinates considering the iron relative permeability in 6/4 switched reluctance machines (SRM) supplied by sinusoidal waveform of current (aka, variable flux reluctance machines). In non-periodic regions (e.g., rotor and/or stator slots/teeth), magnetostatic Maxwell’s equations are solved considering non-homogeneous Neumann boundary conditions (BCs). The general solutions of magnetic vector potential in all subdomains are obtained by applying the interface conditions (ICs) in both directions (i.e., r- and θ-edges ICs). The global saturation effect is taken into account, with a constant magnetic permeability corresponding to the linear zone of the nonlinear B(H) curve. In this investigation, the magnetic flux density distribution inside the electrical machine, the static/dynamic electromagnetic torques, the magnetic flux linkage, the self-/mutual inductances, the magnetic pressures, and the unbalanced magnetic forces (UMFs) have been calculated for 6/4 SRM with two various non-overlapping (or concentrated) windings. One of the case studies is a M1 with a non-overlapping all teeth wound winding (double-layer winding with left and right layer) and the other is a M2 with a non-overlapping alternate teeth wound winding (single-layer winding). It is important to note that the developed semi-analytical model based on the 2D exact subdomain technique is also valid for any number of slot/pole combinations and for non-overlapping teeth wound windings with a single/double layer. Finally, the semi-analytical results have been performed for different values of iron core relative permeability (viz., 100 and 800), and compared with those obtained by the 2D finite-element method (FEM). The comparisons with FEM show good results for the proposed approach. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Open AccessArticle
An Analytical Expression for Magnet Shape Optimization in Surface-Mounted Permanent Magnet Machines
Math. Comput. Appl. 2018, 23(4), 57; https://doi.org/10.3390/mca23040057 - 05 Oct 2018
Cited by 4 | Viewed by 1053
Abstract
Surface-mounted permanent magnet machines are widely used in low and medium speed applications. Pulsating torque components is the most crucial challenge, especially in low-speed applications. Magnet pole shape optimization can be used to mitigate these components. In this research, an analytical model is [...] Read more.
Surface-mounted permanent magnet machines are widely used in low and medium speed applications. Pulsating torque components is the most crucial challenge, especially in low-speed applications. Magnet pole shape optimization can be used to mitigate these components. In this research, an analytical model is proposed to calculate the magnetic vector potential in surface-mounted permanent magnet machines. A mathematical expression is also derived for optimal the magnet shape to reduce the cogging torque and electromagnetic torque components. The presented model is based on the resolution of the Laplace’s and Poisson’s equations in polar coordinates by using the subdomain method and applying hyperbolic functions. The proposed method is applied to the performance computation of a surface-mounted permanent magnet machine, i.e., a 3-phase 12S-10P motor. The analytical results are validated through the finite element analysis (FEA) method. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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Review

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Open AccessFeature PaperReview
Permanent-Magnet Eddy-Current Losses: A Global Revision of Calculation and Analysis
Math. Comput. Appl. 2019, 24(3), 67; https://doi.org/10.3390/mca24030067 - 09 Jul 2019
Cited by 5 | Viewed by 1480
Abstract
Eddy-current analysis is an important research field. This phenomenon occurs in multiple areas and has several applications: electromagnetic braking, repulsive effects, levitation, etc. Thereby, this paper is limited to eddy-current study in rotating electrical machines. In the design process, if the permanent-magnet (PM) [...] Read more.
Eddy-current analysis is an important research field. This phenomenon occurs in multiple areas and has several applications: electromagnetic braking, repulsive effects, levitation, etc. Thereby, this paper is limited to eddy-current study in rotating electrical machines. In the design process, if the permanent-magnet (PM) loss calculation is very important, the overheating due to eddy-currents must be taken into account. The content of this paper includes sources, calculation methods, reduction techniques, and thermal analysis of PM eddy-current losses. This review aims to act as a guide for the reader to learn about the different aspects and points to consider in studying the eddy-current. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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