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Dynamic Model of Induction Machine with Faulty
Cage in Rotor Reference Frame
Vanja Ambrozic, Klemen Drobnic, Rastko Fiser, and Mitja Nemec
University of Ljubljana, Faculty of Electrical Engineering, Trzaska 25, 1000 Ljubljana, SLOVENIA
E-mail: vanjaa@fe.uni-lj.si
Abstract- This paper presents a simplified model of an induction machine with broken rotor bars. The model is founded on the fact that broken bars cause asymmetries in the normally homogeneous cage’s resistance and inductance. Thus, the rotor becomes pseudo-salient. This effect could be best comprised through the rotor time constants that differ in the orthogonal axes of a rotor reference frame. A method for their evaluation is also presented. Consequently, a choice of rotor co-ordinates for the model of a faulty machine to be developed in, is obvious. Experiments show good agreement with simulation results obtained from the developed model.
I. INTRODUCTION
Operation of a healthy induction motor (IM) can be
adequately described by a set of ordinary differential
equations. In this way good results in both steady-state as
well as transient modes can be achieved therefore avoiding
the use of partial differential equations which form a
foundation for finite elements analysis (FEA) [1-3]. FEA
is a very time consuming process, especially when
transients and asymmetrical geometry have to be taken
into consideration [4].
As induction motor squirrel cage is essentially a system
of intertwined windings located in space, one can model
an IM by describing particular loop with an electrical
equation. Knowing actual resistance, inductance and
capacitance of electrical circuit, a very accurate insight
into behavior of the machine can be obtained. This
approach is known as winding function. Unfortunately,
this type of modeling inevitably produces complex
mathematical problem and demands adequate
computational power to solve it [5]. Moreover, a common
drawback of both FEA and winding function approach is
the necessity of number of design data that could be
unavailable. When IM model forms a part of larger
simulation model (e.g. controlled electrical drive) the non-
complex IM scheme is advantageous for the analysis of the
control system as a whole. Logical action towards less
complex model is reducing the number of differential
equations by various types of transformations which are
based on specific (symmetrical) properties of IM [4, 6].
Ultimate step of this simplification is a well known two-
axis model of IM [7].
There are two fundamental effects that develop in the
machine due to rotor broken bars. In current spectrum two
distinctive components at (1-2s)f0 and (1+2s)f0 arise
which are acknowledged as basic signs of rotor asymmetry
(s, f0 denote per-unit slip and electrical supply frequency,
respectively) [8]. In addition, alternating component with
frequency of 2sf0 adds to the electromagnetic torque of
machine. In order to model these two characteristic effects
it is sufficient to use models that are taking into account
only fundamental component of stator magnetomotive
force, i.e. two-axis representation of motor windings.
Fault condition of IM degrades its symmetrical
characteristics and puts all simplified models under
question. For example, breakage of rotor bars is a typical
fault that distorts existing magnetic field by changing
electrical parameters of a machine [9]. Recent papers from
the authors [10-12] tackle this issue by devising new ways
to measure/evaluate changing parameters and by studying
different reference frames in the light of their suitability to
describe broken rotor bars scenario.
II. PSEUDO-SALIENCY OF A FAULTY ROTOR
The basic premise adopted in this paper is the awareness
that broken rotor bars cause asymmetry in the rotor cage
[13]. Consequently, rotor resistance and inductance differ
depending on the number of broken bars. Usually, when
the fault occurs it starts to propagate by damaging bars in
immediate proximity of the original fault location. In this
paper, measurements on a machine with a healthy and
faulty rotor (7 broken out of 44 bars) are presented (Fig.
1,2). The severity of a fault in question could be
considered exaggerated; nevertheless, it was the only one
available as the same rotor was used for previous
measurements and analysis. The sinusoidal dependence of
both parameters on rotor angle is clearly visible, thus
justifying the future modeling.
0 20 40 60 80 100 120 140 160 1800,30
0,40
0,50
0,60
0,70
0,80
0,90
healthy rotor
7 broken bars
R (
Ω)
ε (°)
Fig. 1: Measured resistance versus rotor position for healthy rotor and
rotor with 7 broken rotor bars.
IEEE PEDS 2011, Singapore, 5 - 8 December 2011
978-1-4577-0001-9/11/$26.00 ©2011 IEEE 77
0 20 40 60 80 100 120 140 160 1801,20
1,30
1,40
1,50
1,60
1,70
1,80
healthy rotor
7 broken bars
L (
mH
)
ε (°)Fig. 2: Measured inductance versus rotor position for healthy rotor and
rotor with 7 broken rotor bars.
III. MODEL OF A FAULTY INDUCTION MACHINE IN ROTOR
REFERENCE FRAME
As demonstrated, the almost sinusoidal distribution of
rotor resistance allows for establishment of a rotor
reference frame whose alignment is determined by the
number and arrangement of broken rotor bars in normally
symmetric rotor (Fig. 3). Therefore, the choice of so
defined rotor reference frame – RRF as a basis for a model
development, seems to be the most obvious one.
b
a
dq
Fig. 3: Definition of rotor reference frame.
Since most control schemes adopt rotor field reference
frame, the model could be also developed in field co-
ordinates. In that case, more complicated terms involving
slip angle appear due to different angular frequencies
between RRF (in which the anomalies of the rotor are
defined) and field reference frame. The latter model will
not be discussed in this paper.
In RRF, the stator and rotor equation of cage IM would
normally be [14]
( ) ( )( )
( ) ( )RRF
SRRF RRF RRF
S S S S
d tt R t jp t t
dtω= + +
ψu i ψ (1)
( )( )
0
RRF
RRRF
R R
d tR t
dt= +
ψi (2)
However, in a case of a faulty rotor, the rotor resistances
differ in both rotor axes (indexes d and q) [12].
Consequently, uniform RR in (2) should be replaced with
appropriate resistances RRd and RRq after defining the
vectors in RRF.
( )( )
0Rd Rq
Rd Rd Rq Sq
d jR i jR i
dt
ψ ψ+= + + (3)
From (3), derivatives of rotor fluxes in both co-ordinates
can be separately expressed as functions of their respective
currents. Rotor currents (in d, q co-ordinates) can be
obtained from (4). From both equations, we can also
obtain the term for rotor flux without using rotor currents
(L0 denotes mutual inductance).
0S S S RL L= +ψ i i (4)
0R R R SL L= +ψ i i (5)
After this manipulation, rotor flux derivative is
calculated and, together with calculated rotor currents,
inserted into (3). Consequently, a full set of equations for a
voltage model of an IM with a faulty rotor is obtained.
Sd
Sd S Sd Sq
Sq
Sq S Sq Sd
du R i p
dt
du R i p
dt
ψωψ
ψωψ
= + −
= + +
(6)
Sd Sd
S Rd S Sd Rd Sd
Sq Sq
S Rq S Sq Rq Sq
di dL L i
dt dt
di dL L i
dt dt
ψσ τ τ ψ
ψσ τ τ ψ
+ = +
+ = +
(7)
In the presented equations common symbols for motor
parameters are used. The most important achievement is
the elimination of separate resistances and inductances in
both rotor axes. Instead, they are represented through
different time constants (τRd and τRq). In such a way, there
is no need for separate measurements of resistance and
inductance.
Another important issue is the assumption of fault-
independent total leakage factor σ. Although this
parameter depends on the inductance, due to its definition,
which involves ratio of inductances, it is assumed
constant. This simplification, in our opinion, does not
influence heavily the model accuracy [12].
Being the cross-product of flux and current vectors, the
torque can be expressed in arbitrary reference frame as
( )1.5el Sd Sq Sq Sd
T p i iψ ψ= − (8)
Note that the main goal of this whole manipulation is to
obtain the dependence of the stator current components in
rotor reference frame (as outputs) on the stator voltages (as
inputs). If necessary, current components can be further
easily transformed into arbitrary reference frame (e.g.
stator co-ordinates). Fig. 4 shows the block diagram of IM
in RRF.
IV. MEASUREMENT OF ROTOR TIME CONSTANTS
As already mentioned, different rotor time constants in
both rotor axes are the basis for modeling the faulty rotor.
Therefore, a simple method for their determination has
been developed. It is based on producing only pulsating
field without torque generation (Fig. 4) [15]. In order to
get the dependence of resistance and inductance (thus also
a time constant) on the rotor angle ε (Fig. 1,2) the
78
measurement procedure described below has to be repeated at different angles.
Fig. 4: Block diagram in RRF of an IM with a faulty rotor.
Fig. 5: Single phase measurement of machine parameters at standstill
(assuming 2-pole machine).
In order to simplify the calculation, equation for the
squared impedance has been appropriately modified (9).
Impedance is determined at steady state by impressing the
stator voltage at different frequencies ω1 and then dividing
by the measured stator current.
( ) ( )2 2 2 2
12 2 2
1 2 2
1
2 1 1
1
S R S RS
S S S
S R
R LVZ R L
I
σ τ σ τ ωω
τ ω
− + + = = +
+ (9)
Stator parameters, as well as the total leakage factor
(taking its value for the healthy rotor, for simplification
[15]), are already known, so the rotor time constant(s) can
be calculated by fitting (9) to the measured impedance
frequency characteristics, shown in Fig. 6. In this case,
only the impedance for the healthy rotor (ZS2) and
impedances at the extremities of the co-ordinate system
(ZSd2 and ZSq
2), are shown. Note that the measurement is
performed only up to the pull-out slip frequency (say 10
Hz). At higher frequencies, skin effect causes modification
of (9) [15].
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0 10 20 30 40 50 60
Fig. 6: Squared impedance vs. frequency for healthy and faulty (in d- and
q-axis) rotor.
Measurement and fitting described above gave the
results summarized in Table I. Although both the
inductance and resistance in a faulty rotor increase
throughout the whole rotor angle range, it is obvious that
the increase of the resistance is much higher (as could be
expected). Consequently, rotor time constants of a faulty
rotor are considerably lower in both axes than the one of a
healthy rotor. The latter does not depend on the rotor
angle, except for very small change due to the rotor
asymmetry caused by the manufacturing process.
TABLE I
DETERMINED ROTOR TIME CONSTANTS
Healthy rotor Faulty rotor
TR (s) TRd (s) TRq (s)
0.155 0.142 0.115
79
V. RESULTS
In order to check the validity of the model, simulation
results have been compared with experimental
measurements on a 3 kW induction machine (data in the
Appendix). Rotor cage has been used for previous tests,
when the bars have been gradually broken until reaching
seven. Therefore, the tests were performed only with this
severely damaged rotor.
The machine was fed by three-phase sinusoidal voltages
in an open loop to avoid suppression of oscillations that
occur under closed loop operation. Thus, the dynamic
behavior of the model can be examined.
Fig. 7 shows the speed response in steady state under
constant 10 Nm and 20 Nm load (50% and 100% of the
rated torque).
Time (0.1 s/div)
Speed (
2 r
pm
/div
)
measurement
model
Time (0.1 s/div)
Speed (
5 r
pm
/div
)
measurement
model
Fig. 7: Speed ripple at 10 Nm (above) and 20 Nm load (below).
As expected, speed oscillates whereas the magnitude
and frequency of the ripples show very good agreement.
Next, the torque ripple has been derived out of speed
measurement and compared to the simulation (Fig. 8).
Again, the shape, magnitude and frequency match very
well and in both cases differ from the purely sinusoidal
one.
Time (0.1 s/div)
Torq
ue (
0.2
Nm
/div
)
measurement
model
Time (0.1 s/div)
Torq
ue (
1 N
m/d
iv)
measurement
model
Fig. 8: Torque ripple at 10 Nm (above) and 20 Nm load (below).
In Figs. 9 and 10, the ripples obtained from
measurement and simulations are compared numerically.
Again, Fig. 9 shows rotor speed at two different loads as
above. Model and simulations were compared regarding
the average value and the peak value of the speed ripple. It
is clear that the difference is minimal.
Measurement Model1460
1465
1470
1475
1480
1485
1490
1495
1500
Speed [
rpm
]
average
peak ripple
Measurement Model1440
1445
1450
1455
1460
1465
1470
1475
1480
Speed [
rpm
]
average
peak ripple
Fig. 9: Speed: quantification of average value and peak ripple at 10 Nm
(above) and 20 Nm load (below).
The same form of numerical analysis has been
performed on the machine torque at different loads (Fig.
10). Again, the values match very well.
80
Measurement Model9
9.5
10
10.5
11
Torq
ue [
Nm
]
average
peak ripple
Measurement Model19
19.5
20
20.5
21
21.5
22
22.5
23
Torq
ue [
Nm
]
average
peak ripple
Fig. 10: Torque: quantification of average value and peak ripple at 10 Nm
(above) and 20 Nm load (below).
In the end, the response of the model to both torque
changes has been simulated throughout the whole time
range (Fig. 11). Again, the presence of torque ripple in a
faulty machine, as well as its absence in a healthy one,
show the validity of the approach.
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
Torque (Nm)
Fig. 11: Torque response of the proposed model at various loads (0 Nm,
10 Nm, and 20 Nm) for faulty (above) and healthy rotor (below).
Since the behavior of the mathematical model of the
faulty machine depends solely on the time constants
measured in both coordinates (assuming correct values of
the other, fault-independent parameters), it is interesting to
investigate how their erroneous determination influence
the comparison with measured data. Fig. 12 shows the
speed error between the simulations and measurements for
various values of both model rotor time constants. Value
of 1 p.u. corresponds to the measured time constants of a
faulty rotor summarized in Table I. In this case, a certain,
although very small error, as presented in previous figures,
exists. This is due to several causes: mismatch of the other
machine parameters, simplifications in the rotor time
constant measurement model or measurement error.
Important conclusion is that the error in either one or both
rotor time constants introduces a considerable speed error.
Fig. 12: Speed difference between measurements and simulation
depending on the rotor time constants’ error. Percents (%) are expressed
relatively to the rated slip (44 rpms).
VI. CONCLUSION
In this paper, a dynamic model of an IM with broken
bars has been presented. The fault causes parameter
asymmetry in the rotor circuit and is best represented
through different time constants in both rotor co-ordinates.
Consequently, current components in rotor reference
frame, as function of stator voltages, are calculated. The
response of the system shows good agreement with the
dynamics of the laboratory model.
The model could be used to analyze the behavior of
controlled IM for different states of rotor damage. Future
work involves modeling of various spatial combinations of
broken rotor bars, thus allowing for the development of
method for their early detection in servo drives. It is also
expected to retrieve the machine parameters using FEM at
different stages of faults and combinations of broken bars.
The simulation model is intended for analysis in closed-
loop systems, in which determination of rotor faults, due to
the intrinsic speed correction performed by the controller
is much harder to achieve.
81
APPENDIX
TABLE II
MACHINE PARAMETERS
Pmeh 3 kW RS 0.214 Ω
VS 177 V LS 35.4 mH
IS 14.8 A RR 0.231 Ω
ω 153,4 s-1 LR 35.0 mH
Tel 20 Nm M 34.1 mH
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