Download 5. numerical and experimental results

ISSN 1843-6188
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13)
THREE-PHASE INDUCTION MOTOR’S MAGNETIZING
CHARACTERISTIC BASED ON FINITE ELEMENT METHOD
Aurel. I. CHIRILĂ, Constantin GHIŢĂ, Valentin NĂVRĂPESCU, I. Dragoş DEACONU
Electrical Engineering Department, University Politehnica of Bucharest
Splaiul Independenţei, 313, Sector 6, Bucharest, Romania
E-mail: [email protected]
where Lm0 represents the unsaturated value of the
magnetizing inductance, im0 is the current threshold value
of the magnetizing current over which the machine can be
considered to become saturated, and  is a coefficient
specific to each electrical machine. Expression (1) is
nonlinear and its determination supposes some
approximations. These approximations are more or less
precise depending on the currents flowing through the
induction machine’s windings. Relation (1) does not offer
entirely a sufficient approximation especially in the cases
where the machines are working in hard loading
conditions such as start-ups or electrical breaks. In these
cases it is necessary to know more precisely the
dependency between the magnetizing inductance and the
currents flowing through the machine’s windings. The
approximations errors for the dependency described by (1)
can be diminished if the magnetic field inside the machine
for different values of the windings currents is known.
Starting from this point, the magnetic energy stored in the
statoric and rotoric ferromagnetic armatures can be
computed and so the magnetizing inductance or the
magnetizing reactance (by multiplying the inductance
with the statoric current pulsation).
The magnetic field computation inside the machine can
be performed with various methods, the most precise and
used in the last period being the finite element method
[7], … , [10]. In the paper a numeric method to obtain the
magnetizing reactance of an induction motor starting
from the magnetic field computation found within the
armatures and the magnetic energy stored inside these
armatures is presented. The experimental measurements
validate the obtained numerical results based on the finite
element method.
Abstract: The paper presents a 2D numeric method for computing
the magnetizing reactance Xm of a three phase induction machine,
for various values of the magnetizing current im. In other words,
the magnetizing characteristic is determined Xm = f(im). The
magnetizing reactance is computed starting from the magnetic
energy found within the ferromagnetic armatures of the machine
and its windings. It is concluded that a high saturation state of the
machine leads to a decreasing magnetizing reactance. The results
are validated by experiments, based on no-load and short-circuit
trials. It is noted that the measured values and the calculated ones
are different by a few percents.
Keywords: induction machine, magnetizing reactance, finite
element method
1. INTRODUCTION
The magnetizing reactance of a machine is an important
parameter and corresponds to a main magnetic flux that
links the machines stator and rotor, passing two times over
the air-gap. This reactance is present within the equivalent
circuit and can be determined experimentally by a no-load
trial at synchronous speed. The magnetizing reactance is
influenced by the saturation state of the machine [1], i.e. a
saturated machine has a lower magnetizing reactance. In
technical literature various papers on the parameters
saturation influence can be a found [2], …, [5].
In order to take into account the saturation effect of the
ferromagnetic armatures of the induction machine, the
variation law of the magnetizing reactance Xm with the
magnetizing current im is needed, i.e. Xm = f(im), within the
mathematical model expressed with space vectors [6].
This dependency is input into expression relating the flux
linkages and machine’s currents. If the magnetizing
reactance Xm is divided by the stator’s currents pulsation ω
then it is found the magnetizing inductance Lm computed
with the following expression Lm = Xm / ω.
An empirical analytical expression for the magnetizing
inductance Lm variation with the induction machine
magnetizing current im is pointed out in [1]:
 Lm0 , im  im0

Lm0

, im  im 0
Lm (im )  
2
1    L  i   1  1 
m0 m 


 im 0 im 

2. MOTOR DESCRIPTION
The induction motor used for numeric simulations and
experimental measurements is low power and low
voltage, squirrel-cage type. Its main characteristics are
shown in Table 1.
The magnetizing characteristic of the statoric and rotoric
ferromagnetic armatures is given in Fig. 1.
In Fig. 2 is presented a cross-section of the induction
machine. The machine has sheets, with parallel wall teeth
both for stator and rotor, and the slots are trapezoidal type
with rounded bottoms.
(1)
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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13)
field strength. After the magnetic field strength H is
computed then the magnetic flux density B and the current
density J are obtained with the following expressions:
Table 1. Rated data
Quantity
Value
3 kW
50 Hz
220 / 380 V
11 / 6.3 A
1427 rpm
0.8
Power
Frequency
Voltage
Current
Speed
Power factor
B  H
J  curl H
(3)
In non-conducting mediums the magnetic field is
computed as follows: from successive iterations the
magnetic field strength Hb generated by the currents
flowing through the motor windings (that cause the
current density J b ), is computed using the equation:
2
1.8
curl Hb  J b
1.6
1.4
(4)
Then, using Hb the scalar potential  is obtained as
follows:
1.2
[T]
B [T]
ISSN 1843-6188
1
div  grad  Hb   0
0.8
0.6
(5)
while the magnetic field strength H in non-conducting
mediums is given by the equation:
0.4
0.2
0
0
2000
4000
6000
8000
10000
H  grad  Hb
12000
H [A/m]
For non-conducting mediums, based on already known
magnetic field strength H, the magnetic flux density B and
current density J are obtained with (3) in a similar manner
as for conducting mediums. The finite element methods as
a numerical analyzing method is applicable for any
engineering field giving an approximate solution. The
computation error is imposed by the set tolerances within
the iterative solving algorithm of the equations system.
Taking into account that the electrical machines have quite
long ferromagnetic armatures, the magnetic field
computation is precise enough when supposing a 2D
numerical simulation case. In order to perform the
computation of the magnetic field based on the finite
element method, 2D CAD techniques have been applied
in order to draw the cross-section of the motor. Then, the
physical material properties of the model’s mediums have
been allocated for each one (electrical conductivities,
magnetic permeabilities). Further, the sources of the
magnetic field are imposed (motor’s windings currents)
along with specific boundary conditions. The final step
before solving is the mesh generation obtained by a
programmable operation. Once the values of the magnetic
field are obtained the magnetic energy is obtained and
with it the magnetizing reactance corresponding to the
imposed windings’ current values.
Figure 1. The magnetizing characteristic of the
statoric and rotoric ferromagnetic armatures.
Figure 2. Induction machine cross-section.
3. MAGNETIC FIELD COMPUTATION
The magnetic field for the considered motor can be
computed in 2D or 3D using numeric methods. The most
used numerical method is the finite element method that
is also applied in this paper using specialized software
described in [8].
In conducting mediums, the equation that has to be
solved with the finite element method by neglecting the
displacement current is:
curl curl H   
H
t
(6)
4. MAGNETIZING REACTANCE COMPUTATION
It is supposed that the induction machine’s rotor is
driven at synchronous speed and so the machine’s rotor
is synchronous with synchronous magnetic field
generated by the statoric three-phase winding. This
hypothesis reflects in the T-equivalent circuit by the
elimination of the rotoric branch. At synchronous speed
the magnetic field seen by the rotor is constant with
time, and so the back-emf of the rotoric windings is zero.
(2)
where  and  represent the electrical conductivity and the
magnetic permeability respectively, and H is the magnetic
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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13)
ISSN 1843-6188
In consequence, the rotoric windings have no currents
and can be considered as air (infinite resistivity or zero
electrical conductivity). From magnetic point of view the
spinning of the rotor has no importance whatsoever.
All the hypotheses previously mentioned justify the use
of a harmonic regime for solving the magnetic field
problem with the finite element method. The harmonic
regime supposes that the mediums are linear. However it
is possible to use the same equations in case of nonlinear mediums by the piecewise linearization of the
domain. Thus, each mesh element has its own magnetic
permeability, but globally the problem is nonlinear. The
software manufacturers do not provide the computation
algorithms for such problems but some information can
be found in [11], [12]. In this case the solving tolerances
have to be set to tight enough values.
The source of the magnetic field is determined by the
currents flowing through the stator’s windings, i.e. their
current densities. The three currents generate a
symmetrical three-phase system. The next step is to
numerically solve the problem and to obtain the magnetic
flux density B. Based on this quantity the magnetic vector
potential A is deducted, with the following expression:
B  curl A
current Im is expressed in [A], and the magnetizing
reactance Xm is expressed in [Ω].
5. NUMERICAL AND EXPERIMENTAL RESULTS
The simulations and experiments are performed on the
motor described in paragraph 2. The obtained mesh, after
some adaptations, is shown in Fig. 3. It is constructed so
that it is more dense in the air-gap where it is stored an
increased amount of magnetic energy and less dense in
the rest where the magnetic energy is not so dense.
(7)
Figure 3. The cross-section mesh.
The magnetizing reactance relates to the magnetic energy
stored within the computation domain D , where Σ
represents a closed surface that contains the entire
machine, its armatures and windings. The surface Σ can be
extended towards the infinite, without any change in the
total magnetic energy. The magnetic energy found within
the domain D is the volume integral of the magnetic
energy density, given by the expression [10]:
BH*
dv 
D
D 4

1
1

A  H* n d 
AJ * dv
4 
4 D
Wm 


wm dv 


  
The magnetic flux density map for 1.9A, obtained using
the 2D finite element method is presented in Fig. 4,
obtained using the 2D finite element method. From Fig. 4
it can be seen that an increased magnetic field is found
near the magnetic poles at the yoke level. The teeth
corresponding to the magnetic poles are also
supplementary magnetically stressed. Due the symmetry
of the machine the magnetic field is also periodic. The
equations of the magnetic field have been integrated for 9
values of the magnetizing current (ranging between 1.1
and 4.3A), the magnetic potentials vector have been
computed along with the magnetic energy using (9).
Then, the values of the magnetizing reactance have been
obtained using (10) for each value of the stator current.
The results of the numerical simulations are shown in
Fig. 5, the curve with full line. Up to 2.2A the
magnetizing reactance is still constant, that is, the
machine is not saturated. Over this value the reactance is
decreasing almost linearly with about 10.5 Ω/A.
(8)
where B and H* are the amplitudes of the complex
magnetic flux density, the conjugated magnetic field
strength, and J* represents the complex conjugate of the
current density. The first term in (8) cancels when the
surface  is infinitely extended [10], [13], [14]. Thus, in
harmonic regime, the time-averaged value of the
magnetic energy is:
1
Wm 
AJ * dv
(9)
4 D
Based on the obtained magnetic energy Wm, the
magnetizing reactance for one-phase of the induction
motor is given by:
  
Xm 
2 Wm
2
3 Im
(10)
where the magnetic energy is expressed in [J], the
pulsation  is expressed in [rad/s], the magnetizing
Figure 4. The magnetic flux density map for 1.9A.
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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13)
ISSN 1843-6188
machine’s electrical currents is also non-linear, and in
consequence the mathematical model of the machine
based on space vectors becomes non-linear.
If the magnetizing reactance of the induction machine is
computed using a 2D numerical method and measured
experimentally it can be concluded that there is a good
agreement, but the computed values are less with a few
percents than the measured values. This happens due to
the fact that the finite element method is 2D type and
does not take into account the magnetic effects of the
end-windings, which contribute to the magnetic energy
and the magnetizing reactance. It is preferred to obtain
the magnetizing characteristic of the induction motor by
2D numerical method, which is simple and fast to apply,
eventually followed in the end by a small correction.
For the validation of the numerical results an
experimental trial under no-load conditions has been
conducted. The measurements have been performed with
advanced equipment: Fluke 434 power analyzer, HP
34401A digital multi-meter, DeLorenzo DL1059 threephase AC voltage regulator.
7. REFERENCES
[1]
Figure 5. Magnetizing reactance variation with the per
phase magnetizing current.
[2]
[3]
The variation of the magnetizing current has been
achieved by the variation of the stator’s winding supplying
voltage. For each value of the supplying voltage (and so
the magnetizing current) the leakage reactance has been
obtained (using a graphical-analytic method described in
[15]), then the no-load factor has been computed and in
the end the back-emf of the stator winding. Based on these
quantities the per-phase magnetizing reactance Xm is
deducted. The results of the experimental measurements
are also shown in Fig. 5 (dots line).
It can be concluded that the measured values are just few
percents higher than the computed ones with the finite
element method, because the employed numerical
method is only a 2D a can not take into account the endwinding magnetizing reactance (that is indeed very low).
It can be stated that the errors obtained between the
computed and the measured ones are systematic errors
and can be corrected. Thus, in order to get the values of
the magnetizing reactance with enough precision it is
necessary to increase the values with about 1 – 2
percents by energetic means, using 2D numerical
method. It has to be said that the 2D approach requires a
less computing effort, a less programming effort and a
not so high-performance computational system. This is
why it is preferred to compute the magnetizing reactance
of the induction motors using 2D finite element method,
and then a little positive correction.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
6. CONCLUSIONS
[13]
The magnetizing reactance of a three-phase induction
machine is an important parameter. This characteristic is
non-linear, that is the magnetizing reactance decreases
when the machine’s saturation increases. If this nonlinear variation is considered then the dependency
between the machine’s magnetic fluxes and the
[14]
[15]
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