Kamel Boughrara

Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using Schwarz–Christoffel Transformation

In this paper, we apply an original numerical Schwarz-Christoffel (SC) transformation to analyze magnetic field originating from permanent magnets and the armature winding currents in a slotted air gap of an inset permanent-magnet synchronous motor. We obtained the solution of the SC integral numerically using Matlab SC Toolbox. We used this field solution to calculate both cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. The case without inter-polar piece, which is equivalent to a surface-mounted permanent-magnet motor, is also treated. The accuracy of the developed method is verified by comparing its results with those obtained from the developed numerical finite-element models.

Analytical Model of Slotted Air-Gap Surface Mounted Permanent-Magnet Synchronous Motor With Magnet Bars Magnetized in the Shifting Direction

An analytical model is presented, which uses two-dimensional field theory in polar coordinates to determine the flux density distribution, cogging torque, back EMF and electromagnetic torque in the slotted air gap of permanent-magnet motors with surface mounted magnet bars which are magnetized in shifting direction. The magnet arc to pole pitch ratio in the motor is not necessarily equal to unity like in the case of Halbach array magnetization. The effect of stator slots is introduced by modulating the magnetic field distribution in the slotless stator by the complex relative air-gap permeance. With this complex permeance, the radial and tangential components of flux density are calculated. In the analytical and numerical study a finite number of magnet bars, which is considered sufficient to get a sinusoidal magnetization, is used. The influence of the number of magnet bars on magnetization is also investigated. The accuracy of the developed model is verified by comparing its results with those obtained from experimental measurement and previously validated linear and nonlinear numerical finite element code.

Magnetic Field Analysis of External Rotor Permanent-Magnet Synchronous Motors Using Conformal Mapping

This paper presents analytical and numerical conformal mapping (CM) to analyze magnetic fields originating from permanent magnets and armature winding currents in a slotted air-gap of a surface mounted radial permanent-magnet synchronous motor (SPM), taking into account the effect of arbitrarily curved motor surfaces. We also studied the slotless configuration of the external rotor permanent-magnet motor for the purpose of calculating the complex relative air-gap permeance. The model is able to predict the air gap field and torque with high accuracy, which cannot be achieved using the previously available analytical methods in both cases: internal and external rotor permanent-magnet motor. We obtained one of the conformal mappings used in this study, the numerical solution of the Schwarz-Christoffel (SC) integral, by using the Matlab SC Toolbox. We used this field solution to calculate cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. We verified the accuracy of the developed method by comparing its results with those obtained from the developed numerical finite-element (FE) model.

Analytical Prediction of Magnetic Field in Parallel Double Excitation and Spoke-Type Permanent-Magnet Machines Accounting for Tooth-Tips and Shape of Polar Pieces

This paper presents an analytical method based on subdomain method for the computation of open circuit, armature reaction, and on-load magnetic field distribution in integer slot winding parallel double excitation and spoke-type tangential permanent-magnet machines. The proposed model takes into account for stator and rotor slots tooth tips and shape of polar piece. A 2-D exact analytical solution of magnetic field distribution is established. It involves solution of Laplaces and Poissons equations in semi-closed stator and rotor slots, air-gap, buried permanent magnets into rotor semi-closed slots, and nonmagnetic region under magnets. Obtained exact analytical results of open circuit, armature reaction, and on-load magnetic field distribution are verified with those issued from the finite element method.

General Subdomain Model for Predicting Magnetic Field in Internal and External Rotor Multiphase Flux-Switching Machines Topologies

This paper presents a general analytical subdomain model for the computation of magnetic field distribution in any number of stator slots and rotor poles with and without electrically excited, permanent magnet-excited and hybrid-excited multiphase flux-switching machine (FSM) topologies. The goal of this work is to elaborate an analytical general method based on the subdomain model for predicting the magnetic field in any FSM topology with defining in advance the number of subdomains and affect the general form of vector potential in each subdomain. The presented general subdomain method is comparable to the finite-element method (FEM) where the mesh elements can be compared with the number of harmonic terms used in each subdomain.

2-D Analytical Prediction of Eddy Currents, Circuit Model Parameters, and Steady-State Performances in Solid Rotor Induction Motors

This paper presents a 2-D analytical method in the complex domain for the computation of magnetic field distribution, eddy currents, circuit model parameters, and steady-state performances in solid rotor induction motors. The proposed static analytical model considers stator slotting with tooth-tips. The rotor motion is simulated by varying the slip. The analytical magnetic field distribution is computed in polar coordinates from 2-D subdomain method (i.e., based on the formal resolution of Maxwells equations applied in subdomain) in each region, i.e., semiclosed stator slots, air gap, solid rotor, and shaft. The electromagnetic torque is obtained from both the electrical equivalent circuit and Maxwell stress tensor that is given by the magnetic field distribution. Analytical results are validated by the static finite-element method.

Analytical Analysis of Cage Rotor Induction Motors in Healthy, Defective, and Broken Bars Conditions

This paper presents a 2-D static analytical method in the frequency domain for the calculation of magnetic field distribution, eddy-currents, circuit model parameters, and steady-state performances in multiphase/multipole cage rotor induction motors with integer and fractional stator winding. The complex model allows to study the healthy, defective, and broken bars conditions in these electrical machines. The proposed static analytical model considers stator and rotor slotting with tooth-tips. The method is based on the resolution of Poissons, Laplaces, and Helmholtzs equations in stator slots, air-gap, and rotor bars regions, respectively. The electromagnetic torque is obtained from both the electrical equivalent circuit and Maxwell stress tensor that is given by the magnetic field calculation. The analytical results are validated by those issued from time harmonic finite-element method.

Cogging Torque Minimization of Surface-Mounted Permanent Magnet Synchronous Machines Using Hybrid Magnet Shapes

This paper deals with the magnet pole shape design for the minimization of cogging torque in permanent magnet synchronous machines (PMSM). New shapes of permanent magnet are proposed. The magnet shape is modeled analytically by a set of stacked and well dimensioned layers relatively to the height and opening angle. The final shape of magnet is configured by using three models in view of obtaining lower magnitude of cogging torque. A 2-D exact analytical solution of magnetic field distribution taking into account the shape of magnet, the irregular mechanical thickness of air-gap and semi-closed stator slots is established. The influence of motors parameters such as the number of stator slots per pole and per phase and PMs magnetization on cogging torque is discussed. Analytical results are validated by the static finite-element method (FEM).

ANALYTICAL CALCULATION OF PARALLEL DOUBLE EXCITATION AND SPOKE-TYPE PERMANENT-MAGNET MOTORS; SIMPLIFIED VERSUS EXACT MODEL

This paper deals with the prediction of magnetic fleld distribution and electromagnetic performances of parallel double excitation and spoke-type permanent magnet (PM) motors using simplifled (SM) and exact (EM) analytical models. The simplifled analytical model corresponds to a simplifled geometry of the studied machines where the rotor and stator tooth-tips and the shape of polar pieces are not taken into account. A 2D analytical solution of magnetic fleld distribution is established. It involves solution of Laplaces and Poissons equations in stator and rotor slots, airgap, buried permanent magnets into rotor slots and non magnetic region under magnets. A comparison between the results issued from the simplifled model with those from exact model (EM) (which represents a more realistic geometry with stator and rotor tooth-tips and the shape of polar pieces) is done to show the accuracy of the simplifled geometry on magnetic fleld distribution and electromagnetic performances (cogging torque, electromagnetic torque, ∞ux linkage, back-EMF, self and mutual inductances). The analytical results are verifled with those issued from flnite element method (FEM).

Analytic calculation of magnetic field and electromagnetic performances of spoke type IPM topologies with auxiliary magnets

This paper presents a 2D analytical method based on subdomain model for the computation of magnetic field and electromagnetic performances in spoke-type interior permanent-magnet motor (IPM) topologies. The conventional spoke-type IPM rotor is constituted with ferrite permanent magnets magnetized in tangential direction. To maximize the torque density, auxiliary ferrite magnets magnetized in radial direction are inserted in the center of the conventional spoke-type IPM rotor in different way to get different topologies. Obtained analytical results are verified with those issued from finite element method (FEM) and a comparison is done between the topologies.