Calculation and evaluation of the magnetic field air gap in permanent

magnet synchronous machine

BELLAL ZAGHDOUD1, SAADOUN ABDALLAH

2

Department of electrical engineering

University of Annaba, ALGERIA 1b_zaghdoud@yahoo.com,

2saadoun_a@yahoo.fr

Abstract: - The best way to understand the phenomena in any investigated motors is to get inside and to see

magnetic field distribution. In this paper we will determine and evaluate the air gap field in a permanent magnet

synchronous machine (PMSM) using finite element method (FEM). At first a numerical calculation of the

magnetic field distribution is applied. Then a harmonic analysis of the air gap flux density waveform is carried

out. The results are presented by diagrams. They discussed and compared with experimentally obtained ones,

under no load and full load conditions. They show a very good agreement.

Key-Words: - Finite element, harmonic analysis, magnetic flux density, air gap, permanent magnet synchronous

machine.

1 Introduction Prediction and performance analysis of electrical

machines depend mainly on the accuracy in the

evaluation of the magnetic field linking the different

parts of the machine [1-2]. During the last century

several approaches have been used to solve this

problem. The formulation of the magnetic field by

Maxwell's equations using the vector potential is

described by the Poisson differential equation [1-3].

Although its formulation is relatively easy to obtain,

resolving the equation is virtually impossible in the

case of electrical machines, mainly because of the

complexity of the geometry and the nonlinearity of

the various media of the domain’s solution. In the

case of permanent magnet machines the problem

becomes insurmountable because of the lack of an

analytical formulation of the magnetomotive force

(mmf) magnets. The only alternative to solve this

problem is to use numerical methods [4-6]. During

the last two decades the finite element method

proved to be the most appropriate numerical method

in terms of modeling, flexibility and accuracy to

solve the nonlinear Poisson’s equation governing

the magnetic field in electric machines [5,6].

Currently, several modeling of electrical machines

softwares are available. The finite element package

FEMM 4.2 (Finite Element Method Magnetic)

developed by D. Meeker available for free on its

website was used for the modeling of the PMSM

model and to solve the Poisson’s equation

governing the magnetostatic field.

2 Notations

A(x,y) magnetic potential vector

Ω(Г) domain solution bounded by the contour Г

vx reluctivity of the media in the x direction

vy reluctivity of the media in the y direction

J current density of the carrying current

conductors

Jm equivalent current density of the magnets

v peripheral speed of the rotor

B flux density

H field intensity

M magnetization

M0 magnetization constant

µ permeability of medium

µ0 permeability of free space

3 Poisson equation of magnetic field The formulation for magnetic field in quasi-static

regime formulated using the magnetic vector

potential is represented by Maxwell's equations:

The relation (2) states that the magnetic field is

solenoidal, while the relationship (3) which

represents the Ampere in differential form defines

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the sources of the magnetic field in a medium of

permeability.

The equation of the magnetic field expressed in

terms of its sources is using:

[ ] [ ]

[ ] [ ]

The permanent magnet is represented by a surface

current density which equivalent intensity is

calculated using Stokes' theorem [4,7].

∯ ∮ [ ]

∮

In Two-dimensional Poisson equation (9) becomes:

[ ] [

]

Finally the formulation of the magnetostatic field

expressed using the vector potential is:

(

)

(

)

Where and are reluctivity of the medium and

take the value of and within the permanent

magnet.

4 Resolution of the magnetic field

problem with FEM The finite element method is a numerical procedure

designed to obtain an approximate solution to a

variety of field problems governed by differential

equations. The solution domain is replaced by the

problem of the subdomains of simple geometric

shapes, called elements, in order to reconstruct the

original domain by their assembly. The unknown

variables of the considered field are then expressed

by an approximate function called interpolation

function [1,2]. These functions are defined on each

element using the values that the variable takes from

the field on each node. Therefore the knowledge of

nodal values and interpolation functions allow to

define completely the behavior of the variable field

on each element. Once the nodal variables, which

are actually the unknown factors of the problem, are

calculated, the values of the variables of the field on

any point of the field can also be determined using

interpolation function. The precision of the method

depends not only on the dimensions of elements and

their number but also the type of the interpolation

function. As for the numerical method, the finite

element method converges to the exact solution

provided to increase the number of subdivisions of

the solution domain and to ensure continuity of the

interpolation function of its first derivatives along

the borders of adjacent elements [5,1].

The main steps for implementation of the finite

element method [8] are described below.

4.1 Pre-Processing After the problem geometry is defined we have to

complete the entire domain of the electric machine

by defining the material properties and boundary

conditions. By imposing the vector potential zero at

the outside diameter of the machine the magnetic

field will be confined to the solution domain, while

the periodicity conditions can restrict

the solution domain to a double pole pitch. The

basic idea of FEM application is to divide that

complex domain into elements small enough, under

assumption to have linear characteristics and

constant parameters. Usually, triangular elements

are widely accepted shapes for 2D FE models. After

this step is completed, the output is always

generation of finite element mesh. It is

recommended to make mesh refinements in the

regions carrying the interfaces of different materials,

or with expected or presumed significant changes in

the magnetic field distribution.

4.2 Processing In order processing part to be executed and output

results to be obtained system of Maxwell’s equation

should be solved.

4.3 Post-Processing Using the post-processor software, various results

can viewed as the mapping of field lines and the

intensity of the magnetic induction in different parts

of the machine as shown in Figure 1, however the

main result of the simulation is the field distribution

in the air gap which is given either in graphical form

as shown in figure 2 or in the form of a table that

provides the magnetic induction at a point as a

function of the air gap of its curvilinear abscissa.

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Fig 1: magnetic flux distribution

The symmetry of the magnetic flux lines change as

a function of the rotor position.

The orientations of the flux lines depend on the

magnets position witch are the origin of the flux.

Fig 2: magnetic field distribution along the air gap

5 Comparison between calculated and

measured results Figure 3 shows the measured [9] and computed air

gap magnetic field of the permanent magnet

synchronous machine at no load and full load.

computed at no load

measured at no load

computed at full load

measured at full load

Figure 3: computed and measured flux density

along the air gap at no load and full load

Examination of these curves shows that rapid

variations of the magnetic field are mainly due to

the presence of teeth and slots of the stator area.

Such variations are predictable because the

calculation of the field using an interpolation

function of the first order results in the constancy of

the field [4,2] at each element of mesh. Moreover,

the finite element solution is incomplete because the

model is unable to take into account electromagnetic

phenomena accompanying the rotation of the rotor.

Indeed, in a real machine, the effect of oscillation of

the field between the slots and teeth with induced

eddy currents in the inner surfaces of the teeth,

resulting in a local saturation of the interface zone

air gap magnetic circuit [2]. For these reasons the

field variations in a real machine are less

pronounced.

6 Harmonic analysis of the magnetic

flux distribution Like most software on the market do not have

specific processing functions for the harmonic

analysis of the field distribution: The MATLAB

software was used to perform harmonic analysis of

the magnetic field from the results obtained by the

FEMM software.

0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Length, mm

B.n

, Te

sla

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The following figure (figure 4) shows the measured

[9] and computed harmonic analysis of the magnetic

field distribution at no load and full load.

computed at no load

measured at no load

computed at full load

measured at full load

Figure 4: measured and computed harmonic

analysis of the magnetic field distribution at no load

and full load

It can be seen that the flux density waveforms are

heavily polluted by the high order spatial harmonic.

These harmonics are due to a slight asymmetry

between the consecutive pair poles [10,11]. Their

amplitudes are relatively small, but they still affect

the shape of measured air gap waveform as far as

comparison of the computed and measured flux

density is concerned. Although the computed and

measured value of the fundamental closely, some

uncertainties remain concerning the evaluation of

the high order harmonics because of the undesirable

but inevitable slots effects, field oscillation and local

saturation of the teeth [12-14]. These side effects

have not only an impact on the amplitude spectrum

leading to discrepancies between the measured and

computed harmonics but also to tend to introduce an

additional phase shift [10]. The later effect provides

an indication on the magnetic state of the machine;

the greater the phase shift is, the greater is the

departure of the measured air gap flux density

distribution from the computed ones.

7 Conclusion As part of the determination of the magnetic field in

the gap of permanent magnet synchronous

machines, solving the problem of the magnetostatic

field formulated using the Poisson equation by the

finite element method was presented. FEMM

software was used to obtain its numerical solution

including the analysis required the development of

specific processing functions. Moreover the

experimental results obtained confirm the accuracy

of the results of the numerical solution. Therefore,

the determination of various characteristics and

parameters that directly depend on its assessment

can be made with confidence

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

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