4th World Conference on
Applied Sciences, Engineering & Technology
24-26 October 2015, Kumamoto University, Japan
Model Based Analysis of Three Phase Squirrel Cage Induction Motors
BINDU. S, VINOD V. THOMAS
Department of Electrical and Electronics Engineering, Manipal Institute of Technology,
Manipal University, Manipal, Karnataka, India
Email: bindu.s@manipal.edu
Abstract: Studies in the field of condition monitoring and fault diagnoses of induction motors demand suitable
models to analyse internal fault conditions of the machine. An implementation of detailed multiple coupled
circuit based model of three phase squirrel cage induction motor with m-stator circuits and n-rotor bars in
MATLAB/SIMULINK® platform is presented. The model parameters are calculated based on geometry and
winding layout of the ac machine. The transient and steady state behaviour of the machine for symmetric and
asymmetric operating conditions are analysed. The simulation results in both time and frequency domain are
presented. Behavioural study of induction motors under various faulty situations through asymmetric machine
modelling supports in signature extractions for online condition monitoring.
Keywords: Asymmetric machine modelling, Fault signatures, Condition monitoring, Fault diagnoses, Motor
current signature analysis
Introduction:
Suitable mathematical models are very much
essential to conduct behavioural studies on induction
motor in various faulty conditions and to extract
signatures of internal faults using advanced signal
processing tools. These studies will certainly assist in
online condition monitoring of induction motors
which has great significance in recent years due to
the necessity to reduce machine downtime and to
improve reliability of these highly popular machines
in industries. Sensitivity of various machine
parameters to fault has to be studied and set of
parameters for fault characterization need to be
identified.
Mathematical models such as single phase equivalent
circuits with linear algebraic equations can be used
for steady state analysis of symmetrical machines.
Conventional d-q model formed with nonlinear
differential equations can be employed for the
dynamic analysis. It is a conceptually simple model
with fictitious two phase representation because the
model is obtained with two sets of windings, one on
the stator and the other on the rotor. D-q model
assumes sinusoidal distribution of windings. It is
incapable of representing a general machine with
arbitrarily connected windings and thus not flexible
enough to incorporate various asymmetric conditions.
Finite element based and circuit based simulation
studies are in use for the analysis of asymmetric
operation of machine. Finite element modelling
provides an exact evaluation of magnetic field
distribution inside the machine. Perturbations in field
distribution give the indication of the presence of
fault.
Multiple Coupled Circuit Model (MCCM)[1] is
grounded on basic geometry and winding layout of
an arbitrary N-phase machine and machine is
modelled as a set of coupled coils. It provides wide
range of simulation options for incorporating stator
turn, rotor bar or air-gap deformities. In this model
mutual inductance between stator and rotor windings
are considered to be time varying since it depends on
rotor position and are evaluated in real time.
Secondary parameters such as leakage inductance are
considered as constants by placing values from
machine design data. The coupling inductances are
derived using winding function approach. Machine
modelling and implementation with this approach is
emphasized in this paper. An implementation of
complete model of three phase squirrel cage
induction motor with m-stator circuits and n-rotor
bars using MATLAB/SIMULINK® is presented.
System Modelling:
The generalized model consists of a set of m+n+3
linear state equations with time varying coefficients.
The modelling is grounded on basic geometry and
winding layout of the machine.
Considering
generalized model with following assumptions,
i. Effect of saturation, eddy current losses, friction,
and windage losses are neglected.
ii. m identical windings on stator with axes of
symmetry.
iii. n rotor bars , uniformly distributed cage or rotor
windings with axes of symmetry.
iv. Insulated rotor bars.
v. Uniform air-gap.
vi. Iron of infinite permeability.
The steady voltage equations in vector matrix form
can be expressed by,
Vs = Rsis +
(1)
s=
Lssis + Lsrir
(2)
Where Vs = [Vs1, Vs2 . . . Vsm]' , stator voltage
is = [is1, is2, . . . . ism]' , stator current
WCSET 2015089 Copyright © 2015 BASHA RESEARCH CENTRE. All rights reserved
BINDU S., VINOD V. THOMAS
The stator is considered as symmetrical with a
balanced three-phase circuit. Rs is the resistance
square matrix of stator winding of the order of m. Lss
matrix contains self and mutual inductance of stator
windings themselves. Lss is a square matrix of order
m . The mutual and magnetizing inductances are
calculated by applying winding function theory [1].
Hence the stator voltage and current vectors can be
expressed as Vs = [Vsa, Vsb, Vsc]' and is = [isa, isb, isc]'
Each rotor circuit formed by two rotor bars and end
ring segments. n+1 rotor circuits are considered as
shown in Fig. 1. Rotor voltage equations in vector
matrix form can be expressed by,
Vr = Rrir +
=0
t
r = Lsr is + Lrrir
(3)
(4)
Rr is the rotor resistance square matrix of order (n+1)
which also includes the resistance of end ring
segments. Lsr is the mutual inductance matrix
between stator windings and rotor bars. The order of
Lsr matrix is m x (n+1). Lrr is the inductance matrix of
rotor which includes self and mutual inductance
between rotor-bars themselves. The order of Lrr
square matrix is n+1. The squirrel cage rotor can be
taken as n identical and uniformly spaced rotor
loops. Hence, n+1 rotor currents can be specified . n
number of rotor loop currents and a circulating
current flowing in one of the end rings, ie .
Obviously, the end ring current would be equal to
zero in cage rotor. The equivalent circuit of the
induction motor with three phase stator windings and
n rotor bars is presented in Fig.1. [2]
The mechanical equations of the machine are,
= (Te- TL)
(5)
=
(6)
rm
Where Te is the electromagnetic torque developed in
the motor. TL is the load torque. θrm is the spatial
position of rotor and ωrm is the angular speed. The
electrical torque Te can be found from magnetic coenergy Wco as given in equation (7)
Figure 1: Multiple Coupled Circuit representation
of Induction Motor
Te =
(7)
A magnetic system which is linear has co-energy
equal to the stored energy or field energy as,
Wco =
(8)
Determination of Parameters:
Machine parameters used in the model includes,
stator/rotor resistances and inductances. Resistances
and leakage inductances can be obtained from
machine design data. Magnetizing and mutual
inductances are calculated by applying winding
function theory. Since this model does not assume
symmetry in placement of any of the motor coils in
the slots, it is suitable for modelling of asymmetrical
operations.
The mutual inductance between any two windings i
and j in an electric machine can be computed using
equation (9) according to winding function theory.
[1,2]
Lij ( ) =
(9)
Where θ is the angular position of the rotor with
respect to a stator reference, ϕ is a particular position
along the stator inner surface, L is the stack length, r
is the mean radius of air-gap, and
is inverse
air-gap function and its value is reciprocal of
uniform air-gap length lg. To calculate mutual
inductance between stator and rotor circuits, the term
is the winding distribution of stator circuit i
and
is the winding function of rotor circuit j.
Winding function signifies the MMF distribution
along the air-gap for a unit current flowing in
corresponding winding.
The inductance can be
calculated by using modified winding function theory
[3] as given below,
Lij = 2
rL
Proceedings of the 4th World Conference on Applied Sciences, Engineering and Technology
24-26 October 2015, Kumamoto University, Japan, ISBN 13: 978-81-930222-1-4, pp 394-398
(10)
200
180
160
200
150
100
50
0
0
4
theta
6
8
1
0.5
0
-4
4
120
0
2
4
theta
6
8
2
4
theta
6
8
-3
x 10
4
x 10
100
2
2
60
dLsr
80
Lsr
Stator winding distribution
140
2
Rotor winding distribution
Where P is the air gap permeance , ni and nj are the
turn functions of i and j windings respectively. Fig.2
gives the winding distribution of stator circuits
(phases) A, B, and C respectively, and Fig. 3 gives
the stator to rotor mutual inductance with respect to
rotor position , obtained by simulation.
Stator winding distribution
Model based analysis of three phase squirrel cage induction motors
0
0
40
-2
-2
20
0
0
1
2
3
4
5
6
7
theta
Figure.2 The winding distribution of A, B and C
phase windings
-4
0
2
4
theta
6
8
-4
0
Figure. 4. Stator winding distribution of stator
circuit-A , Rotor circuit-1 , Stator to rotor mutual
inductance and its derivative w.r.t. rotor position.
Model Implementation and Result Analysis:
Fig.5 shows the SIMULINK block schematic of the
machine. The motor used for simulation studies is 5.5
kW, 60 Hz, 460V, four pole three phase squirrel cage
induction motor.
Its parameters are given in
Appendix.
Figure3 Stator to rotor mutual inductances with
respect to rotor position
The 4 pole machine considered for simulation study
has number of stator slots = 36, and number of turn
per phase = 90. Winding distribution of each stator
circuit changes at each 2п/36 radians which is the
angle between each slot. For each phase circuit
winding distribution shifts by 1200.
Fig. 4 shows the stator winding distribution of one
circuit in stator and rotor. Stator to rotor mutual
inductance and its derivative w.r.t. rotor position,
obtained from simulation is also shown in Fig. 4.
Figure 5. Develped SIMULINK model
Proceedings of the 4th World Conference on Applied Sciences, Engineering and Technology
24-26 October 2015, Kumamoto University, Japan, ISBN 13: 978-81-930222-1-4, pp 394-398
BINDU S., VINOD V. THOMAS
The dynamic responses of speed, torque and stator
current when the machine is fully loaded at 0.5
second is given in Fig. 6-8. At 0.5 second motor
speed falls from no lad to full load speed after a short
transient period. Motor torque rises from zero and
settles at full load torque after a short transient period
(Fig.6).
Figure 9. Frequency spectrum of stator current
Stator currents settles at no load currents after initial
transients and at 0.5 sec, when it is loaded, settles to
full load value after a transient period (Fig. 7, 8).
Spectrum of stator current in frequency domain
shows only fundumental component as the machine
data used was for healthy machine.(Fig.9). These
simulation results shows the expected behaviour of a
healthy ac motor, such as starting transients in speed,
torque and stator current, small transient period
before settling at steady state when loaded, and a
frequency spectrum with only the supply frequency
component. These results validates the model and
this detailed machine model can be used for further
studies in internal fault diagnoses.
Figure 7. Stator current, Ia
Scope of the model:
This model is flexible enough to incorporate stator
turn short circuit due to insulation failure or turn
open circuit since stator is represented as m coupled
circuits and rotor as n circuits [4,5] . Rotor bar
cracks can be detected by identifying the signatures
of rotor faults using this model [6,7]. The parameters
will change according to the fault. Air gap
eccentricities of static, dynamic , mixed [8] or
inclined in nature can be represented by making
airgap as a dependent function of rotor position and
thus mutual inductances as a function of airgap
length . In two dimentional winding function theory
(2D- MWFT) [9] mutual inductances are represented
as a function of axial length also .This can be used
for inclined eccentricity studies. Direct analysis of
bearing fault is not found to be done with the help of
MCCM but the secondary effect such as air gap
eccentricity can be analysed by MCCM.
Figure 8. Stator Currents Ia, Ib and Ic
Conclusion:
The model of three phase induction motor
implemented using MATLAB/SIMULINK® based
on Multiple couple circuit approach is capable to
incorporate asymmetrical operating conditions and
internal faults even at minor level .This model will
help in a detailed analysis of the machine which will
also help in condition monitoring and fault diagnoses
based investigations. The simulation results validate
the model.
Figure 6. Motor speed and torque
Proceedings of the 4th World Conference on Applied Sciences, Engineering and Technology
24-26 October 2015, Kumamoto University, Japan, ISBN 13: 978-81-930222-1-4, pp 394-398
Model based analysis of three phase squirrel cage induction motors
Appendix:
Parameters of induction motor used for
simulation study [2]
5.5kW, 60 Hz, 460V, 4 pole 3 phase Squirrel Cage
Induction Motor
Number of stator slots = 36
Number of rotor bars, n = 28
Number of turn per phase, N = 90
Stator resistance= 3.5332 Ω
Rotor bar resistance= 68.34*10^-6 Ω
Resistance of end ring segment= 1.56*10^-6 Ω
Leakage inductance of stator =0.028 H
Mean radius of air gap= 63.2968*10^-3 m
Length of stack= 102.4128*10^-3 m
Permeability in air= 4*pi*10^-7 Henry/m
Effective Air gap = 0.456438*10^-3 m
Leakage inductance of rotor bar=0.28*10^-6 H
Leakage inductance of rotor end ring segment =
0.03*10^-6 H
Stator inner radius= 63.525*10^-3 m
Rotor outer radius= 63.068*10^-3 m
Moment of inertia=0.012 Kgm2
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El-Antably and Thomas A.
Lipo, (1995)
”Multiple Coupled Circuit Modelling of
Induction Machines,” IEEE Transactions
on
industry applications, Vol.31, No.2, March-April
1995.
[2] Vinod V. Thomas, (2002) “Non-invasive
Techniques For Rotor Fault Detection of three
phase squirrel cage induction motor,” Ph.D.
thesis, IIT Madras, 2002.
[3] Jawad Faiz, Iman Tabatabaei, (2002) “Extension
of Winding Function Theory for Non-uniform
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[4] Subhasis Nandi and Hamid A. Toliyat, (2012)
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[5] Bindu S. and Vinod V. Thomas, (2014)
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squirrel cage induction motor-A review”, In
Proc. IEEE ICAECT,pp. 48-54, 2014.
[6] Vinod V. Thomas, K. Vasudevan, and V. J.
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[7] Hamid A. Toliyat and Tomas A. Lipo, (1995)
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[8] Nabil A. Al-Nuaim and Hamid A. Toliyat,(1998)
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pp.62-67, 2002
Proceedings of the 4th World Conference on Applied Sciences, Engineering and Technology
24-26 October 2015, Kumamoto University, Japan, ISBN 13: 978-81-930222-1-4, pp 394-398