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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 [email protected],
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
Recent Advances in Electrical Engineering
ISBN: 978-960-474-318-6 98
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.
Recent Advances in Electrical Engineering
ISBN: 978-960-474-318-6 99
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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Harmonic order
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ISBN: 978-960-474-318-6 101
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