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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) | Viewed by 54307

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Special Issue Editors

Dr. Frédéric Dubas

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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, Collections and Topics in MDPI journals

Prof. Dr. Kamel Boughrara

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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

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Published Papers (13 papers)

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Editorial

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4 pages, 187 KiB

Open AccessEditorial

Mathematical Models for the Design of Electrical Machines

by Frédéric Dubas and Kamel Boughrara

Math. Comput. Appl. 2020, 25(4), 77; https://doi.org/10.3390/mca25040077 - 9 Dec 2020

Viewed by 2316

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

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21 pages, 10020 KiB

Open AccessFeature PaperArticle

Steady State and 2D Thermal Equivalence Circuit for Winding Heads—A New Modelling Approach

by Julien Petitgirard, Tony Piguet, Philippe Baucour, Didier Chamagne, Eric Fouillien and Jean-Christophe Delmare

Math. Comput. Appl. 2020, 25(4), 70; https://doi.org/10.3390/mca25040070 - 18 Oct 2020

Cited by 2 | Viewed by 3100

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 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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17 pages, 3935 KiB

Open AccessFeature PaperArticle

Investigation of Volumic Permanent-Magnet Eddy-Current Losses in Multi-Phase Synchronous Machines from Hybrid Multi-Layer Model

by Youcef Benmessaoud, Daoud Ouamara, Frédéric Dubas and Mickael Hilairet

Math. Comput. Appl. 2020, 25(1), 14; https://doi.org/10.3390/mca25010014 - 4 Mar 2020

Cited by 4 | Viewed by 2925

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 (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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25 pages, 3937 KiB

Open AccessFeature PaperArticle

Reduced-Order Model of Rotor Cage in Multiphase Induction Machines: Application on the Prediction of Torque Pulsations

by Abdelhak Mekahlia, Eric Semail, Franck Scuiller and Hussein Zahr

Math. Comput. Appl. 2020, 25(1), 11; https://doi.org/10.3390/mca25010011 - 29 Jan 2020

Cited by 5 | Viewed by 2942

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 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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13 pages, 2058 KiB

Open AccessArticle

2D Hybrid Steady-State Magnetic Field Model for Linear Induction Motors

by Samuil R. Aleksandrov, Timo T. Overboom and Elena A. Lomonova

Math. Comput. Appl. 2019, 24(3), 74; https://doi.org/10.3390/mca24030074 - 25 Jul 2019

Cited by 1 | Viewed by 3141

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 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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22 pages, 7810 KiB

Open AccessFeature PaperArticle

Combining the Magnetic Equivalent Circuit and Maxwell–Fourier Method for Eddy-Current Loss Calculation

by Youcef Benmessaoud, Frédéric Dubas and Mickael Hilairet

Math. Comput. Appl. 2019, 24(2), 60; https://doi.org/10.3390/mca24020060 - 4 Jun 2019

Cited by 7 | Viewed by 4052

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 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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27 pages, 5744 KiB

Open AccessFeature PaperArticle

Study of a Hybrid Excitation Synchronous Machine: Modeling and Experimental Validation

by Salim Asfirane, Sami Hlioui, Yacine Amara and Mohamed Gabsi

Math. Comput. Appl. 2019, 24(2), 34; https://doi.org/10.3390/mca24020034 - 27 Mar 2019

Cited by 21 | Viewed by 5861

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 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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25 pages, 5252 KiB

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

by AmirAbbas Vahaj, Akbar Rahideh, Hossein Moayed-Jahromi and AliReza Ghaffari

Math. Comput. Appl. 2019, 24(1), 24; https://doi.org/10.3390/mca24010024 - 17 Feb 2019

Cited by 6 | Viewed by 4381

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 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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16 pages, 1740 KiB

Open AccessFeature PaperArticle

Magnetic Field Analytical Solution for Non-homogeneous Permeability in Retaining Sleeve of a High-Speed Permanent-Magnet Machine

by Gabriel A. Mendonça, Thales A. C. Maia and Braz J. Cardoso Filho

Math. Comput. Appl. 2018, 23(4), 72; https://doi.org/10.3390/mca23040072 - 10 Nov 2018

Cited by 5 | Viewed by 4463

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 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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8 pages, 999 KiB

Open AccessFeature PaperArticle

Memory Efficient Method for Electromagnetic Multi-Region Models Using Scattering Matrices

by C. H. H. M. Custers, J. W. Jansen and E. A. Lomonova

Math. Comput. Appl. 2018, 23(4), 71; https://doi.org/10.3390/mca23040071 - 9 Nov 2018

Cited by 1 | Viewed by 2735

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 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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21 pages, 11385 KiB

Open AccessFeature PaperArticle

Two-Dimensional Exact Subdomain Technique of Switched Reluctance Machines with Sinusoidal Current Excitation

by Mohammed Ben Yahia, Kamel Boughrara, Frédéric Dubas, Lazhar Roubache and Rachid Ibtiouen

Math. Comput. Appl. 2018, 23(4), 59; https://doi.org/10.3390/mca23040059 - 11 Oct 2018

Cited by 11 | Viewed by 3519

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 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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17 pages, 3811 KiB

Open AccessArticle

An Analytical Expression for Magnet Shape Optimization in Surface-Mounted Permanent Magnet Machines

by Ali Jabbari

Math. Comput. Appl. 2018, 23(4), 57; https://doi.org/10.3390/mca23040057 - 5 Oct 2018

Cited by 7 | Viewed by 3379

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 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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18 pages, 1160 KiB

Open AccessFeature PaperReview

Permanent-Magnet Eddy-Current Losses: A Global Revision of Calculation and Analysis

by Daoud Ouamara and Frédéric Dubas

Math. Comput. Appl. 2019, 24(3), 67; https://doi.org/10.3390/mca24030067 - 9 Jul 2019

Cited by 22 | Viewed by 9853

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) 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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