Analytical and numerical
computation of the no-load
magnetic field in induction
motors
No-load
magnetic field in
induction motors
225
Dan M. Ionel
University of Glasgow, Glasgow, Scotland, UK and
Mihai V. Cistelecan
Research Institute for Electrical Machines, Bucharest
Keywords Electromagnet, Induction motor, No-load
Abstract In the paper a comparison between the analytical and numerical method of no-load
computation will be made taking into account the initial hypothesis and computational effort face
to the concrete obtainable results. It is concluded that the reduced fixed-mesh FEM analysis of the
induction motors is very suitable for no-load computation because it relates directly to the well
known equivalent circuit parameters. The symmetry of the machine allows for the computation
only on a small part of the magnetic circuit so that the time of computation can be reduced even
when a relatively fine mesh is used. However, for the optimal design of the motor it appears that
analytical methods are very suitable owing to the simplicity and accuracy of the design.
1. Introduction
It is well known that a sinusoidal mmf in the air gap of the induction motor
leads to a non-sinusoidal air gap flux density. Associated harmonics are called
saturation harmonics. These harmonics are rotating at the synchronous speed
and their amplitude depends on the degree of the iron saturation, that means
the iron mmf of the teeth and the yokes. For a long time the air gap flux density
shape was characterised by the flattening factor α defined as the ratio of the
pole average flux density to the maximum one and some additional curves were
used in the magnetic circuit computation to determine the flattening factor as
function of the teeth saturation factor[1]. The increase of the flattening factor α
from the unsaturated value 2/π to values up to 0.8-0.85 is related to the high
degree of the teeth saturation. Later it was observed that the saturation of the
yokes implies the decrease of the flattening factor[2], so that there may exist
induction motors with high degree of saturation having the flattening factor
close to the unsaturated value[3].
Another problem is related to the computation of the yoke mmf taking into
account the non-uniformity of the field on both radial and tangential directions.
In the classical method some additional curves for correction coefficients are
used depending on the maximum flux density in the neutral axis of the rotating
field. As it was demonstrated[3], this coefficient is depending on the shape of
the flux density itself and also on the quality of the electrical sheet. Having
COMPEL – The International
Journal for Computation and
Mathematics in Electrical and
Electronic Engineering,
Vol. 17 No. 1/2/3, 1998, pp. 225-231.
© MCB University Press, 0332-1649
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these in mind the analytical method should offer the possibility to compute the
shape of the air-gap flux density in its interdependence with the teeth and
yokes’ mmfs without above mentioned additional curves.
2. Analytical method for no-load computation
At no-load there are no currents in the rotor and, because of the symmetry, only
a half pole of the machine will be considered. In the developed analytical
method only one additional curve is necessary, the magnetising curve of the
magnetic material expressed in a.c. maximum values at rated frequency. The
space harmonics of the air gap mmf are neglected and also the tooth pulsation
due to the air gap reluctance variation. As it is stated in the literature[2] the
saturation affects mainly the fundamental space harmonic because it has the
highest amplitude. The teeth mmf is simply computed taking into account the
actual shape and cross section of the teeth and the value of the air gap flux
density in the front of the actual tooth. For high values of the teeth flux density
the longitudinal flux in the slot should be considered[3].
For the computation of the yoke mmf one can have in mind that the flux in the
yoke is not constant and the flux density at the point θ depends on the shape of
the flux density from 0 to θ, the reference frame being the synchronous one (the
air gap flux density is maximum at θ = 0 and vanishes at θ = π/2p). If the yoke
flux density is considered only by its tangential component the computation
can be simplified without a significant decrease of accuracy of the results. If the
variable is the angular co-ordinate θ the functional equation of the air-gap flux
density can be derived from the magnetic circuit law applied to a curve which
passes across the air gap in the point characterised by the angle θ :
(1)
where the iron mmf Fi (θ ) represents the sum of the stator and rotor teeth and
yokes’ mmfs corresponding to the co-ordinate θ. The maximum value of the air
gap flux density is noted Bg (θ = 0). The air gap flux density equation can be
solved iteratively by turning the angular continuous variable into a discrete one
and dividing the interval (0, π/2p) into n intervals. In this case the yoke mmf
computed at the average diameter Dyav of the yoke can be written for the stator
or the rotor as:
(2)
where hy is the height of the yoke, Da is the diameter of the armature at the air
gap side, p is the number of the pole pairs and ki is the lamination filling factor.
The function f(*) represents the magnetisation curve expressed by H = f(B).
The λm is given by additional notation:
No-load
magnetic field in
The tooth mmf depends only on the value of the air gap flux density in front of induction motors
(3)
the respective tooth. The yoke mmf depends on all of the values Bgk since the
distribution of the field in the yokes depends entirely on the air gap flux density.
Hence the necessity that the method of determining the yoke mmf should be
applicable to any shape of the air gap flux density. The equation (2) represents
a simplified Simpson approximation of the integral of the yoke mmf. Whenever
the yokes are unsaturated the system of equation derived from (1) turns into n
independent equations having Bgk as unknowns. If the discrete variable is θk =
kπ /(2pn) the new system of equation can be expressed by:
(4)
Using the mentioned discretization the functional equation of the air gap flux
density becomes a non-linear system of equation, the unknown being the values
Bgk = Bg(θk). The system can be solved iteratively starting from a supposed
shape of the air gap flux density. The resultant shape, after the convergence is
obtained, leads to the value of the no-load induced voltage and to the no-load
current. Two successive computational cycles are needed, one for determining
of the air gap flux density shape at the given maximum value and the second to
determine the maximum value of the air gap flux density at given phase
voltage. The resulting shape of the gap flux density can be analysed from the
point of view of the saturation harmonics.
3. FEM analysis of no-load induction motor
The numerical analysis starts from the stator in-slot currents supposed to be
time sinusoidal. The rotor currents are supposed to be negligible. For no-load
computation usually a reduced fixed mesh model is used covering one pole of
the machine. The assumption is made that all currents and field variables vary
sinusoidally in time[5]. This assumption allows for using the complex variables
in the two dimensional field equation:
(5)
in which the over-bars indicate complex quantities. The magnetic vector
potential A and the current density J are supposed to have only components
normal to the cross section. The local reluctivities ν are used in the same
manner as in the analytical method as ratio between the time maximum values
of the field quantities.
The current density J is given from the supposed phase current, the three
phase winding arrangement and the moment of time in which the computation
is performed. In the particular case the computation may be repeated for
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Figure 1.
One-pole mesh
Figure 2.
Enlarged mesh
(detailed)
different values of the currents (time variation) and/or different relative position
of the rotor. The mesh is automatically performed (Figure 1) and should be
viewed in detail when necessary (Figure 2). Of course the density of the finite
elements should be appropriate to local conditions and the mesh should have
more elements in the air gap and also in the places where there are big
variations in the electric or magnetic properties.
After the field problem is solved by known methods the field pattern may be
analysed in order to obtain the useful values of the flux densities in the teeth,
yokes and air gap and also the corresponding no-load current. If the end
winding inductance is supposed or computed by classical methods the phase
leakage reactance may be obtained. Because there is not relative motion
No-load
between stator and rotor meshes the model will lead to the neglecting of the magnetic field in
effects produced by the movement of the stator and rotor teeth past each other. induction motors
This does not necessarily mean that such effects must be altogether neglected in
the performance computation. Fourier analysis of the air gap flux distribution
will enable the magnitudes of slot order fields to be determined. These may be
229
used in standard ripple torque and high frequency loss expressions in place of
field amplitudes obtained in the analytical (classical) manner by multiplying
mmf and air gap permeance harmonics[4]. However, it may be claimed that this
approach has the advantage face to classical approach in that the flux density
waves are determined taking into account the tooth tip saturation.
4. Computation results
Both analytical and numerical method for no-load computation were applied to
a 3kW, four-poles squirrel cage induction motor in the frame 112. The electrical
silicon sheet had 3.6W/kg at 1 T and 50Hz and the magnetisation characteristic
had B25 = 1.58 T[6]. The shaft was considered to be made from non-magnetic
material. The rotor and the stator teeth have a constant cross section, that
means they are with parallel sides. The number of stator/rotor slots was 36/28.
In Figure 3 is presented the field pattern corresponding to a quarter of the cross
section of the motor. There is a relation between the shaded colours and the field
intensity in the magnetic circuit. One can observe the curvature of the field lines
in the neutral axis of the yokes, where the flux density has maximum values
and there exists some discharge of the yokes in the tooth basis. Starting from
the field computation the values of flux densities in the teeth and yokes were
determined and compared with those resulting from analytical method. The
comparison is presented in Table I and one can observe that the values are in a
close relation. Of course the advantage of the numerical method consists in the
fact that it can emphasise the variation of the flux density on the height of the
yokes or on the width of the teeth. In the analytical method only the average
values of these flux densities could be obtained.
Flux density [T]
Analytical method
Numerical (FE) method
Rotor yoke
Stator yoke
Rotor tooth
Stator tooth
1.44
1.40
1.75
1.81
1.71
1.73
1.66
1.69
In Figure 4 is represented the radial component of the air gap flux density as it
was computed by both methods on a half pole for the relative position of the
rotor as in Figure 3. Because of the teeth saturation one can see a strong
flattening of the resulting curve (the dotted line). The pulsation of the air gap
flux density owing to relative stator/rotor teeth position is pointed out from
FEM results. The no-load current computed by the two methods was also in a
Table I.
Analytical and
numerical (FEM)
computation results
(comparison)
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Figure 3.
Field pattern
Air-gap flux density [T]
1.4
Analytical results
FEA results
1.2
1
0.8
0.6
0.4
0.2
Figure 4.
Radial air gap flux
density
0
0
10
20
30
40
50
60
70
80
90
Angular coordinate [elect. deg]
close relation (4.34A for FEM, 4.54A analytical and 4.48A measured on the
actual motor).
5. Conclusions
In the paper a comparison has been made between the analytical and numerical
(FEM) method for investigation of the no-load running of induction motors. The
no-load current was the main object of comparison but some remarks were
made on the iron loss and the flux densities in the air gap and in the main parts
of the magnetic circuit. The main conclusion is that the fixed mesh FEM
computation offers much the same information about the induction motor no-
load running as analytical method. It is difficult to say that one or another of the
No-load
methods gives more accurate results for a large range of induction motors magnetic field in
knowing the widespread magnetic properties of the cold rolled electrical sheets. induction motors
From the researchers’ point of view, of course, the FEM method offers some
interesting results related to slot and tooth to tooth leakage flux, pulsation of the
air gap flux density and other local properties of the main or leakage field.
231
Applying the FEM method for different time varying currents in the slots one
can obtain useful information about differential reactance. As the optimisation
of the design is concerned, where the computation of the induction motor must
be performed for different loads and voltages for a thousand times, the
analytical method should be chosen because of its simplicity and speed.
References
1. Liwschitz, M., “Calcul des machines eletriques”, Spes Lausane-Dunod, Paris, 1967, pp. 1-53.
2. Weh, H., “Analitische Behandl.des mangetische Kreise von Asynchronmaschinen”, AfE,
1961, H1, pp. 27-40.
3. Cistelecan, M. and Onica, P., “Computation of the magnetic circuit and establishing of the
electrical sheet quality influence on the induction motor performance”, ICEM ’88, Pisa,
Italy, Vol. 1, pp. 237-42.
4. Williamson, S., “Induction motor modelling using finite elements”, ICEM ’95, Paris, Vol. 1,
pp. 1-8.
5. IEC Publication 404-8-4, “Magnetic materials. Spec. for colled rolled N/O magnetic steel
sheet and strip”, 1986.
Further reading
Eastham, J.F., Lai, H.C., Demeter, E., Ionel, D.M. and Postnikov, V.I., “Advanced finite element
analysis of rotating machines”, International Conference “OPTIM”, Vol. 4, Brasov, Romania,
May 1996, pp. 1081-90.