Rev. Roum. Sci. Techn.– Électrotechn. et Énerg.
Vol. 61, 1, pp. 18–21, Bucarest, 2016
NUMERICAL MODELING APPROACHES
FOR THE ANALYSIS OF SQUIRREL-CAGE INDUCTION MOTOR
CRISTINA MIHAELA GHEORGHE1, LEONARD MARIUS MELCESCU, TIBERIU TUDORACHE, EMIL MIHAI
Key words: Induction motor, Magnetoharmonic model, Transient magnetic model, Experimental validation.
This paper presents a comparative analysis in terms of results accuracy and computational effort of two numerical
models used for the 2D finite element analysis of the three-phase squirrel-cage induction motor: the
magnetoharmonic model and the transient magnetic model. The operating characteristics of an induction machine
obtained by these models are presented in comparison with the experimental ones. The limitations and applicability
of each numerical model are detailed and discussed.
1. INTRODUCTION
One of the most important electric energy consumers in
industry is the three-phase squirrel-cage induction motor
(TPSCIM) [1]. This motor is characterized by simple
construction, robustness, long lifespan and low manufacturing
and maintenance costs.
TPSCIMs are built in a large range of rated powers from
hundreds of watts to hundreds of kilowatts and they could
be found in various applications as machine tools, fans,
industrial or agricultural pumps, cranes etc. Fed from inverters,
TPSCIMs could be used in electrical drive systems where
the speed varies over a wide range. Being a reliable and a
cheap device, the cost of the electricity consumed by a
TPSCIM during its entire lifespan may be up to one
hundred times larger than its acquisition price [1].
Improvements of TPSCIM performance require finding
out strategies to accurately model and analyze the machine
operation regimes with a reasonable computation effort [2].
The analysis of TPSCIM is typically done by using
circuit models, field models or field-circuit models. Circuit
models are easier to implement, the numerical results requiring
less computation effort since these models are usually based
on simplifying hypothesis. The circuit models are not able
to take into account accurately special effects such as teeth
harmonics, skin effect in eddy current regions, iron losses etc.
Field models, usually based on finite element (FE)
method, are more complex, more accurate and they are able
to take into account more sophisticated phenomena but with
the price of a higher computational effort. These models are
implemented starting from electromagnetic field equations
specific to the operation of the TPSCIM. Depending on the
state of the studied machine, there are two types of numerical
models that are typically used: (a) magnetoharmonic model
used for the steady state ac magnetic analysis of TPSCIM
and (b) transient magnetic model used for the transient
analysis of the machine [2, 3].
Field-circuit models represent an extension of the field
models and they allow the study of TPSCIM when supplied
from a voltage source (when currents are apriori unknown).
These models suppose to add an associated circuit model of
the machine to its field model. This operation assumes that
circuit equations are added to the existing field model equations,
all of them being assembled in a single matrix [4−10].
This study aims at highlighting the applicability, limitations
and performance features in terms of accuracy and com-
putational effort of two numerical models used for the FE
analysis of TPSCIM: magnetoharmonic model and transient
magnetic model.
2. NUMERICAL MODELS
FOR THE ANALYSIS OF TPSCIM
A detailed analysis of TPSCIM able to evaluate the
running characteristics of the machine cannot be done using
circuit models. The machine efficiency, the electromagnetic
torque ripples, the teeth harmonics, the magnetic non-linearity
and skin effect in rotor bars are some aspects difficult or
impossible to be considered by simple circuit models. To
take into account all these effects the machine should be
studied by field or field-circuit models, usually based on FE
method. The FE analysis of radial flux TPSCIMs is carried
out typically by 2D plane approaches. The most used models
for such studies are the: the magnetoharmonic model and
the transient magnetic model [2, 3].
2.1. MAGNETOHARMONIC MODEL
The magnetoharmonic model (steady state ac magnetic
harmonic model) uses the complex magnetic vector potential
as state variable [2]. In 2D applications the partial differential
equation governing this model, expresses in complex
magnetic vector potential A is the following [3]:
curl[(1 / µ )curl A] = J s − jωσ A ,
(1)
where µ is the magnetic permeability, A[0, 0, A(x, y )] is the
complex magnetic vector potential, J s [0, 0, J (x, y )] is the
complex image of source current density, σ is the electric
conductivity of solid conductor regions, ω = 2πf is the angular
frequency and the term ( − jωσ A ) represents the induced
current density in solid conductor regions.
The features of the magnetoharmonic model are as follows:
(a) all material properties are considered linear, magnetic
non-linearities of materials can be taken into account by
energy equivalence methods (e.g. Flux by Cedrat) [2],
(b) time-harmonic physical quantities (electrical and magnetic)
are represented by their complex image for a given frequency
value (i.e. supply voltage frequency), (c) current harmonics,
teeth harmonics and electromagnetic torque ripples cannot
be taken into account [2, 3]. This model is however eco-
1
“Politehnica” University of Bucharest, Faculty of Electrical Engineering, 313 Splaiul Independentei, 060042 Bucharest, Romania, E-mail:
cristina.gheorghe@upb.ro
2
19
Numerical analysis of three-phase squirrel-cage induction motor
nomical from computation point of view and it can be
applied to estimate rapidly the TPSCIM running characteristics
in steady state such as: torque-slip curve, current-slip curve,
efficiency characteristic etc. Such a model may be ten times
faster than a transient magnetic model used for the same
application.
2.2. TRANSIENT MAGNETIC MODEL
The transient magnetic model allows the study of the
TPSCIM in dynamic regimes such as: motor start up, motor
acceleration and deceleration etc. The partial differential
equation of the transient magnetic model in 2D applications,
expressed in magnetic vector potential A is as follows [2, 3]:
curl[(1/µ )curlA] = J s − σ
∂A
,
∂t
(2)
where µ is the magnetic permeability, A[0, 0, A(x, y, t )] is
the magnetic vector potential, J s [0, 0, A(x, y, t )] is the current
density, σ is the electric conductivity of solid conductors.
Some important features of the transient magnetic model
are as follows: (a) it is able to consider the rotor-stator
relative motion of the TPSCIM, the electromagnetic field
problem being solved step by step in time domain, (b) it can
take into account the magnetic non-linearity of the magnetic
cores, (c) the time variation of physical quantities (electrical
and magnetic) may be arbitrary (i.e. time-harmonic or not),
(d) it can take into account the current harmonics, the eddy
currents induced in the rotor bars and the electromagnetic
torque ripples [2, 3]. The major drawback of this model is the
important computational effort. The transient magnetic
model is used frequently for the analysis of electrical
machines when operating at constant speed, at constant
torque or in transient regimes.
In order to the model dynamic regimes of the TPSCIM the
kinematic equation should be added to the field-circuit
model of the machine:
J
dΩ
= Te − TL − FΩ ,
dt
(3)
where J is the inertia moment of the rotating parts, Ω is the
angular speed, Te is the electromagnetic torque of the
motor, TL is the load torque and F is the viscous friction
coefficient.
2.3. CIRCUIT MODEL ASSOCIATED
TO THE FE MODEL. FIELD-CIRCUIT COUPLING
In order to analyze a TPSCIM by using eq. (1) and (2)
we must know the current density values in the stator coils
( J s ) and rotor bars ( − jωσ A ). Unfortunately the quantities
that are typically known for a voltage supplied TPSCIM are
the voltage and frequency values, the currents being apriori
unknown since they depend on several factors such as:
machine geometry, load torque, material properties. Therefore in order to solve the under determination in eq. (1) or
(2) additional equations able to link the current density
values and the magnetic vector potential values are needed.
These supplementary equations are usually generated by
coupling the circuit model of the machine, Fig. 1, to the
field models presented in Sections 2.1 and 2.2, obtaining
thus field-circuit models. Such coupled models can be
therefore used to analyze the voltage supplied TPSCIM in
various operation regimes, the stator/rotor currents resulting
accordingly by solving the circuit and field model equations.
Fig. 1 – Circuit model of the TPSCIM
associated to the FE model.
In Fig. 1 we can identify the following circuit components:
(VU, VV, VW) – voltage sources supplying the machine, (RU,
RV, RW) – stator resistances per phase, (LσU, LσV, LσW) –
stator end winding inductances per phase analitically
calculated (LσU = 8.6 mH), (RFe-U, RFe-V, RFe-W) – phase
resistances modeling the iron losses, (BU, BV, BW) – coils of
the three-phase windings whose inductances are implicitly
computed by FE analysis, Q1 – macro-circuit used to model
the squirrel cage (each rotor bar is modeled as a solid
conductor and each interbar ring section is modeled by a
series connected resistor and inductor).
3. MAGNETOHARMONIC AND TRANSIENT
MAGNETIC ANALYSIS OF A TPSCIM
The analysis of the TPSCIM was carried out by using the
magnetoharmonic and the transient magnetic field-circuit
models implemented in the professional software package
Flux®. A comparison of the results obtained by the two
models with the experimental ones is made and comments
are added.
3.1. MAIN DATA OF STUDIED TPSCIM.
COMPUTATION DOMAIN
AND BOUNDARY CONDITIONS
The studied TPSCIM is characterized by the main technical
data presented in Table 1.
Table 1
Main technical data of the studied TPSCIM
Rated mechanical power [kW]
Rated phase voltage [V]
Rated phase current [A]
Rated supply frequency [Hz]
Rated torque [Nm]
Internal diameter of the stator core [mm]
External diameter of the stator core [mm]
ρcage (115°C) [Ωm]
Rated speed [rpm]
Rated rotor slip [%]
Rated power factor
Number of stator slots (Z1)
Number of rotor bars (Z2)
Length of stator/rotor core [mm]
Phase stator resistance [Ω]
Electric steel laminations type
2.2
220
5.4
50
15.15
100
160
5.07·10−8
1420
5.33
0.78
36
28
90
2.28
M600-50A
20
Cristina Mihaela Gheorghe et al.
Fig. 2 – Electromagnetic field computation domain
and regions with different physical properties.
3
because usually the load torque is considered as constant
while the motor speed may have ripples as a result of the
kinematic equation of the motor where several quantities
interferes (e.g. moment of inertia, electromagnetic torque).
However the computational effort for the imposed torque
model is much greater than for the imposed speed model,
since the first model supposes a transient mechanical and a
transient electromagnetic regime while the second one
assumes only a transient electromagnetic regime. A question
that still arises refers to the precision of these two models
and this aspect will be concluded by a comparative
numerical analysis for different speed and torque values
represented under the form of a mechanical characteristic of
the machine.
The obtained results in Fig. 4 show negligible differences
between the mechanical characteristics obtained by the two
transient magnetic models (the results are in good agreement
also with those obtained by the magneto-harmonic model).
That is why the transient magnetic analysis in the following
studies will be carried out using the imposed speed model
because it is more economical.
Fig. 3 – Finite element mesh.
3.2. COMPARISON BETWEEN MAGNETOHARMONIC
AND TRANSIENT MAGNETIC MODELS
The field-circuit analysis of TPSCIM using the magnetoharmonic model was carried out for different slip values in
the range s ∈[0 ÷ 0.08] (i.e. from ideal no-load to 150%
overload where the rated slip is sn = 0.053), and different
running characteristics were determined. The same characteristics of the machine were computed by the transient
magnetic model.
The transient magnetic analysis of the TPSCIM could be
performed in two ways: (a) by imposing the load torque
value (i.e. imposed torque model) or (b) by imposing the
rotational speed value (i.e. imposed speed model). If the
ripples of the electromagnetic torque are small (i.e. very
small speed ripples) both models should provide practically
the same results.
In most cases the first model (imposed torque model) is
closer to reality than the second one (imposed speed model)
Fig. 4 – Comparison between the imposed speed
and imposed torque models.
50
Transient Model
Harmonic Model
Experimental
45
40
35
Me [Nm]
In Fig. 2 is shown the 2D plan-parallel electromagnetic
field computation domain of the studied machine and in
Fig. 3 the finite element mesh.
Due to the symmetries the computation domain of
TPSCIM is reduced to ¼ of its cross-section (one magnetic
pole). The boundary condition imposed on the stator outer
surface is of Dirichlet type, A = 0 (i.e. nill normal
component of the magnetic flux density on that frontier,
B ⋅ n = 0 ). The physical symmetries are taken into account
by imposing anticyclic type boundary conditions on the two
radial boundaries of the computation domain.
30
25
20
15
10
5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
slip [-]
Fig. 5 – Torque-slip curves.
The comparison between the magnetoharmonic and the
transient magnetic (imposed speed variant) models of
TPSCIM is carried out for different motor speed and torque
values and the results are represented as electromagnetic
torque/slip characteristic, Fig. 5, and running characteristics
shown in Figs. 6−8. These figures, that include also
experimental results, illustrate the dependence of power
factor, the stator current and the efficiency on the level of
motor loading (P2/P2n output power/rated power).
4
Numerical analysis of three-phase squirrel-cage induction motor
computation effort associated to the magnetoharmonic model
that proved to be up to 60 times smaller than that of transient
magnetic model. Thus the magnetoharmonic model is well
adapted for a rapid evaluation of machine running characteristics. If particular aspects are studied such as teeth
harmonics, torque ripples, transient mechanical regimes of
the machine, the transient magnetic model is more adapted.
1
Transient Model
Harmonic Model
Experimental
0.9
0.8
0.7
cos( φ)
21
0.6
0.5
0.4
0.3
4. CONCLUSIONS
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
P2/P2n [-]
Fig. 6 – Power factor characteristics.
The main purpose of this study was to evaluate two FE
based modelling methods used for the analysis TPSCIM:
the magnetoharmonic model and the transient magnetic
model. The accuracy of results and the computational effort
were evaluated, the numerical results being also experimentally
validated.
From the perspective of results accuracy, both studied
models are suitable for the analysis of the TPSCIM above
the breakdown speed. In terms of computational effort, the
magnetoharmonic model is much more suitable because the
results are obtained much faster (up to 60 times). For the
analysis of the TPSCIM in transient regimes or around the
starting point (where high current harmonics may appear),
the transient magnetic model is more suitable, despite the
computational effort.
ACKNOWLEDGEMENTS
Fig. 7 – Input current characteristics.
The work of Ph.D. Eng. Cristina Mihaela Gheorghe was
supported by Project SOP HRD – PERFORM /159/1.5/S/
138963.
Received on August 7, 2015
REFERENCES
Fig. 8 – Efficiency characteristics.
The comparative analysis of the machine running characteristics obtained by the magnetoharmonic and transient
magnetic models highlights a relatively good agreement
between these results and a good correspondence with the
experimental ones. Relatively larger differences can be
noticed in case of efficiency curves at small load levels.
Some differences can be remarked also in case of power
factor characteristics. These differences could be due to the
poor estimation of mechanical and stray losses that cannot
be computed by an electromagnetic analysis of the machine.
Measurement inaccuracies could be also a cause for these
discrepancies.
Before formulating the concluding remarks about the two
models analyzed in the paper we should highlight the
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