XR-EE-EME_2006_004

EVALUATION OF MOTOR ONLINE DIAGNOSIS
BY FEM SIMULATIONS
Thanis Sribovornmongkol
Master’s Thesis
XR-EE-EME 2006:04
Electrical Machines and Power Electronics
School of Electrical Engineering
Royal Institute of Technology
Stockholm 2006

I
ABSTRACT
Early detection of abnormal conditions during motors operation would eliminate
consequential damages on motors, so outage time and costs of repairing can be reduced.
Due to unique fingerprints from faults in line currents, it is possible to detect faults by
extracting fault information from line currents, which is so-called, Motor Current Signature
Analysis.
In this thesis, Finite Element Method has been implemented to simulate three main kinds of
faults; rotor asymmetry, airgap asymmetry and stator asymmetry. Time-Stepping FEM
simulation models have been developed for induction motors, and the various faults have
been implemented to the models. Then, three different approaches; i.e. Motor Current
Signature Analysis, Instantaneous Power Approach and The Extended Park’s Vector, based
on simple measurement have been applied to extract fault information from the FEM
simulation results, and the evaluation of three approaches has been presented.
Besides, two cases of operating conditions, which are unsymmetrical supplied voltages and
oscillating loads, have been described. In addition, the evaluated approach has been applied
to real measurement. The possible root causes of the inexplicable components in the real
measurement have been described.
Keywords: Condition Based Monitoring, Online Monitoring, Fault Diagnosis, Finite Element
Method, Motor Current Signature Analysis, Instantaneous Power Approach, The
Extended Park’s Vector, Frequency Analysis, Rotor Asymmetry, Airgap
Asymmetry, Stator Asymmetry

II
SUMMARY
In this study, FEM simulations have been implemented to study fault phenomena in induction
motors. Unique signatures in the electrical supply measurements are identified for Online
Motor Diagnosis. The four main ideas presented in this study are;
• Theoretical derivations of harmonics in the stator currents generated by the
most common faults
• Implementation of FEM simulation for fault studies
• Evaluation of three different approaches for Online Motor Diagnosis by simple
measurement on the supply side
• Application in a case study with the aim to find possible root causes for
abnormal harmonic sidebands in the stator currents
The six most common faults are Rotor broken bars, Rotor broken end rings, Static
eccentricity, Dynamic eccentricity, Mix of static and dynamic eccentricities and Inter-turn
short circuit. In most cases, each has a unique fault signature; however, there is a risk of
confusion in some cases as presented in the report.
The various faults have been studied both individually and in combination. The simulation
results agree well with the theory, but some unexpected harmonics are still present. The
conclusion is that FEM simulation is a powerful tool to study fault characteristics, but it is
time-consuming.
Three analysis approaches, Motor Current Signature Analysis (MCSA), Instantaneous Power
Approach (IPA) and The Extended Park’s Vectors (EPVA), have been applied to the
simulation results and are evaluated. The conclusion is that EPVA is recommended if
reference measurements are available, otherwise IPA is preferred.
Finally, a set of real measurement data is analyzed, resulting in the conclusion that the most
likely root cause for the inexplicable sidebands is the effect from load oscillations since no
other fault can generate such harmonics.

III
ACKNOWLEDGEMENTS
This master thesis has been carried out at Department of Electrical Machine and Intelligent
Motion, ABB Corporate Research, Västerås, Sweden, during September2005-April2006.
Firstly, I would like to express my heartfelt gratitude to my supervisor, Christer Danielsson for
his guidance and encouragement, which he gave me throughout my works. Without his
advices, I believe that it would have been impossible to finish this thesis.
I would also like to show my deep gratitude to my supervisor and examiner from KTH, Prof.
Chandur Sadarangani, for his advices and supports. I always felt enthusiastic after having a
conversation with him.
Many thanks also go to Sture Erixon for his supports on information for this project. My
special thanks would be presented to Robert J Anderson for his assistance on FLUX2D,
which is one of the most difficult software I have ever used before.
I am obliged to Dr. Heinz Lendenmann, who is the Manager at Department of Electrical
Machine and Intelligent Motion, for giving me a chance to do my thesis at ABB Corporate
Research.
Lastly, without them, I would have had today. I would like to present my loves and profound
gratitude to my parents, Dad & Mom. Thank you for their moral supports, cares and
encouragement.
Thanis Sribovornmongkol
Västerås, Sweden
April 2006
“There is nothing that perseverance cannot Win”

IV
TABLE OF CONTENTS
1 INTRODUCTION.............................................................................................................................. 1
2 TIME HARMONICS AND SPACE HARMONICS IN INDUCTION MOTORS ................................. 3
2.1 INTRODUCTION............................................................................................................................ 3
2.2 TIME HARMONICS AND SPACE HARMONICS................................................................................... 5
2.3 TIME HARMONICS ON AIRGAP MMF FROM TIME HARMONICS IN SUPPLY VOLTAGES........................ 5
2.4 MMF SPACE HARMONICS............................................................................................................ 6
2.4.1 Stator MMF ........................................................................................................................ 6
2.4.2 Rotor MMF......................................................................................................................... 7
2.4.3 Total MMF.......................................................................................................................... 8
2.5 AIRGAP PERMEANCE ................................................................................................................... 8
2.6 AIRGAP FLUX DENSITY ................................................................................................................ 9
2.6.1 Stator Flux Density ............................................................................................................ 9
2.6.2 Rotor Flux Density ........................................................................................................... 11
2.7 CONCLUSION ............................................................................................................................ 11
3 MOTOR CURRENT SIGNATURE ANALYSIS (MCSA)................................................................ 12
3.1 INTRODUCTION.......................................................................................................................... 12
3.2 ROTOR ASYMMETRY.................................................................................................................. 12
3.2.1 Cause for Rotor Breakage............................................................................................... 12
3.2.2 Detection of Rotor Breakage ........................................................................................... 13
3.3 AIRGAP ASYMMETRY ................................................................................................................. 16
3.3.1 Static eccentricity............................................................................................................. 17
3.3.2 Dynamic eccentricity........................................................................................................ 18
3.3.3 Mix eccentricity ................................................................................................................ 20
3.4 STATOR ASYMMETRY ................................................................................................................ 21
3.4.1 Cause for Inter-Turn Short Circuit.................................................................................... 21
3.4.2 Detection of Inter-Turn Stator Winding Fault................................................................... 22
3.5 BEARING DAMAGES ................................................................................................................... 23
3.5.1 Cause for Bearing Damages ........................................................................................... 23
3.5.2 Detection of Bearing Damages........................................................................................ 24
3.6 MECHANICAL COUPLING ............................................................................................................ 25
3.7 OSCILLATING LOADS.................................................................................................................. 25
3.8 CONCLUSION ............................................................................................................................ 26
4 INSTANTANEOUS POWER APPROACH & THE EXTENDED PARK’S VECTOR APPROACH27
4.1 INTRODUCTION.......................................................................................................................... 27
4.2 INSTANTANEOUS POWER APPROACH.......................................................................................... 27
4.3 THE EXTENDED PARK’S VECTOR APPROACH .............................................................................. 29
4.4 CONCLUSION ............................................................................................................................ 31
5 FEM SIMULATION MODELS........................................................................................................ 32
5.1 INTRODUCTION.......................................................................................................................... 32
5.2 FINITE ELEMENT MODELING........................................................................................................ 32
5.3 PHYSICAL MODEL OF INDUCTION MOTORS.................................................................................. 32
5.4 ROTOR ASYMMETRY.................................................................................................................. 36
5.4.1 Broken Rotor Bar ............................................................................................................. 36
5.4.2 Broken Rotor End Ring.................................................................................................... 38
5.5 AIRGAP ASYMMETRY ................................................................................................................. 40
5.5.1 Static Eccentricity ............................................................................................................ 40
5.5.2 Dynamic Eccentricity ....................................................................................................... 43
5.5.3 Mix of Static and Dynamic Eccentricities......................................................................... 44
5.6 STATOR ASYMMETRY ................................................................................................................ 48
5.6.1 Inter-turn Short Circuit on Stator Windings...................................................................... 48
5.7 CONCLUSION ............................................................................................................................ 53
6 FEM SIMULATION – MOTOR CURRENT SIGNATURE ANALYSIS .......................................... 54

V
6.1 INTRODUCTION.......................................................................................................................... 54
6.2 FEM MODEL.............................................................................................................................. 54
6.3 SINGLE FAULT ........................................................................................................................... 55
6.4 FAULT COMBINATIONS ............................................................................................................... 61
6.4.1 Combination of Rotor and Airgap Asymmetry ................................................................. 61
6.4.2 Combination of Rotor and Stator Asymmetry.................................................................. 64
6.4.3 Combination of Airgap and Stator Asymmetry ................................................................ 65
6.4.4 Combination of Rotor, Airgap and Stator Asymmetry ..................................................... 68
6.5 UNSYMMETRICAL SUPPLY VOLTAGE & OSCILLATING LOADS ........................................................ 69
6.6 CONCLUSION ............................................................................................................................ 71
7 FEM SIMULATION (CONTINUED) – INSTANTANEOUS POWER APPROACH........................ 73
7.1 INTRODUCTION.......................................................................................................................... 73
7.2 SINGLE FAULT ........................................................................................................................... 73
7.4 UNSYMMETRICAL SUPPLY VOLTAGE & LOAD OSCILLATION .......................................................... 84
7.5 CONCLUSION ............................................................................................................................ 85
8 FEM SIMULATION (CONTINUED) – THE EXTENDED PARK’S VECTOR APPROACH........... 87
8.1 INTRODUCTION.......................................................................................................................... 87
8.2 SINGLE FAULT ........................................................................................................................... 87
8.3.1 Combination of Rotor and Airgap Asymmetry ................................................................. 93
8.3.2 Combination of Rotor and Stator Asymmetry.................................................................. 96
8.3.3 Combination of Airgap and Stator Asymmetry ................................................................ 97
8.3.4 Combination of Rotor, Airgap and Stator asymmetry.................................................... 101
8.4 UNSYMMETRICAL SUPPLY VOLTAGE & LOAD OSCILLATION ........................................................ 102
8.5 CONCLUSION .......................................................................................................................... 103
8.6 EVALUATION FOR MCSA, IPA AND EPVA ................................................................................ 104
9 IMPLEMENTATION..................................................................................................................... 106
9.1 INTRODUCTION........................................................................................................................ 106
9.2 RAW MEASUREMENT DATA ...................................................................................................... 106
9.3 SIMPLE INVESTIGATION............................................................................................................ 107
9.4 DATA PREPARATION ................................................................................................................ 108
9.4.1 Number of Sample......................................................................................................... 108
9.5 INVESTIGATION........................................................................................................................ 108
10 CONCLUSION AND FUTURE WORKS .................................................................................. 115
10.1 CONCLUSION....................................................................................................................... 115
10.2 FUTURE WORKS .................................................................................................................. 116
11 REFERENCES ......................................................................................................................... 117
LIST OF SYMBOLS............................................................................................................................ 122
APPENDIX .......................................................................................................................................... 123
I. COMPARISON OF THE AIRGAP LENGTH FOR VARIOUS TYPES OF ECCENTRICITIES. ................................ 123
II. MATLAB SCRIPT FOR THE AIRGAP CALCULATION.............................................................................. 124
III. FEM SIMULATION.......................................................................................................................... 125
IV. DYNAMIC MODEL OF AN INDUCTION MOTOR ................................................................................... 145

1
1 INTRODUCTION
Robust induction motors are the most widely used electrical machines in industry. The
unexpected down time of induction motors can cause production and revenue losses. It is,
therefore, important to prevent unscheduled downtime, which can help not only to reduce
maintenance costs but also to gain up income of industry.
In the survey report by EPRI [58], it presented the percentage failure for the wide range of
induction motors. As can be seen in Figure 1.1, the survey found that 37% of motor failures
were caused by stator winding failures, 10% by rotor failures, 41% by bearing failures and
12% by miscellaneous failures.
This master thesis was initiated from an
inexplicable measurement of one induction
motor. That initiated an idea to implement
FEM simulations to study fault phenomena in
induction motors, so failures could be
identified before machines would be
dismantled. This can enable the application
on Online Motor Diagnosis.
The purposes of the thesis are to study influences from the most common faults on electrical
measurement based on the stator side by implementing FEM simulations, and to apply three
different approaches, which are Motor Current Signature Analysis (MCSA), Instantaneous
Power Approach (IPA) and The Extended Park’s Vector (EPVA), to identify faults. Besides,
the other purpose is to find the possible root causes for the inexplicable sidebands found in
the measurement.
This thesis is composed of three main parts. The first part consisting of Chapter 2, 3 and 4
discusses knowledge in diagnostic fields. The theoretical studies on influences from faults on
stator currents are introduced. Chapter 5, 6, 7 and 8 are the second part presenting
Implementation of FEM simulations on fault studies and implementation of three diagnostic
methods on simulation results. Then the inexplicable measurement has been investigated in
Chapter 9. The objective in each chapter is generally presented below.
Chapter 2 provides general knowledge of time and space harmonics in induction motors.
This is the main background for understanding influences from faults presented in Chapter 3.
Chapter 3 introduces causes of faults and influences on stator currents. Fault signatures in
stator currents called Motor Current Signature Analysis (MCSA) are presented. The
background from Chapter 2 is used to derive analytical expressions for fault indicators in
stator currents.
Chapter 4 introduces the other two approaches that are Instantaneous Power Approach
(IPA) and The Extended Park’s Vector (EPVA).
Figure 1.1 Percentage failure by component [58]

2
Chapter 5 presents FEM simulation
models. The knowledge on fault
characteristics from Chapter 3 is applied
to build up FEM simulation models.
Chapter 6, 7 and 8 present FEM
simulation results according to Chapter 5.
The implementations of three different
approaches presented in Chapter 3 and 4
are discussed.
Chapter 9 discusses the inexplicable
measurement. The knowledge gained
from the previous chapters is applied to
investigate the measurement.
Chapter 10 presents the conclusion and
future works.
Figure 1.2 Thesis Structure

3
2 TIME HARMONICS AND SPACE HARMONICS IN
INDUCTION MOTORS
2.1 Introduction
Normally, in the study of induction machines, some simplified assumptions are made. For
example, purely sinusoidal supply voltages, uniform airgap, infinite numbers of rotor and
stator slots etc. These simplified assumptions are useful when the principle operation of
induction machines is studied. However, they are not applicable for diagnostic problems
since unsymmetrical conditions must be considered.
This chapter introduces time and space harmonics in the airgap flux density, which are
influenced from non-sinusoidal quantities. The approach is to analyze the airgap flux density
using analytical expression for the airgap MMF and permeance. Figure 2.1 summarizes the
general process of harmonic generation in the airgap of induction motors.
This chapter is composed of five main sections. The purpose of the first section in 2.2 is to
present how different time and space harmonics are. The characteristics of time and space
harmonics are discussed. The second in 2.3 explains influences on the airgap MMF from
time harmonics in supply voltages. Then, the section 2.4 discusses the airgap MMF space
harmonics influenced from stator and rotor currents. Next, the expression of the airgap
permeance influenced from non-constant airgap length is presented in the section 2.5. The
description and expression in the section 2.4 and 2.5 will be applied to determine the airgap
flux density in the section 2.6. The frequencies of induced stator and rotor quantities are
discussed. The conclusion is presented in the section 2.7.

4
Figure 2.1 Schematic diagram of harmonics in induction machines

5
2.2 Time Harmonics and Space Harmonics
Any quantity that changes when time is varied is a time dependent quantity; on the other
hand, any quantity that distributes in distance is a space dependent quantity. In Figure 2.2,
time and space dependent waveform are shown.
According to Fourier analysis, any waveform or dependent quantity can be represented by a
series of sinusoidal terms. The fundamental frequency is called for the base frequency of the
waveform. The terms, of which the frequency is multiples of the fundamental frequency, are
called as harmonics.
(a) Space Harmonic (b) Time Harmonic
Figure 2.2 Time Dependent and Space Dependent Waveform
At a specific position in space, space harmonics pulsate at the fundamental frequency ω but
move with the angular speed kω . On the other hand, time harmonics pulsate at the
frequency ωk and move with the angular speed ωk , where k is a harmonic order.
In induction machines, space harmonic quantities can exist both in the airgap MMF due to
winding distributions in slots and in permeance waves due to non-constant airgap length.
Besides, time harmonic quantities can take place in the airgap MMF due to time harmonics in
the supply voltages.
The approach to analyze the airgap flux density is to determine the airgap MMF and
permeance functions. The airgap flux density can be expressed as below.
0 01
( , ) ( , ) ( , )
( , ) ( , )
( , ) ( , )
A
B t F t F t
t A t
t F t
μ μ
θ θ θ
δ θ δ θ
θ θ
= ⋅ ⋅ = ⋅
= Λ ⋅
(2. 1)
Where, ),( tθΛ is an airgap permeance per unit area, ),( tF θ is airgap MMF and ),( tθδ is
airgap length.
2.3 Time Harmonics on Airgap MMF from Time Harmonics in
Supply Voltages
Purely sinusoidal supply voltages feeding to an induction motor can create purely sinusoidal
currents in windings, so the fundamental airgap MMF arises. If the supply voltages are not
purely sinusoidal, such as when a motor is fed from an inverter, currents are also not purely
sinusoidal, but contain some time harmonics. Time harmonics in the airgap MMF can be

6
expressed as shown in Equation 2.2. The stator MMF generated from the magnetizing
currents in each phase winding can be obtained. As the phase windings are placed at 2 3π
from each other, the space shift will also contribute to the stator MMF [3].
)
3
4
cos(
)
3
2
cos(
)cos(
1,,
1,,
1,,
π
φω
π
φω
φω
−−=
−−=
−=
tkII
tkII
tkII
kmkc
kmkb
kmka
)
3
4
(cos)
3
4
cos(
2
)
3
2
(cos)
3
2
cos(
2
)cos()cos(
2
1,,
1,,
1,,
π
θ
π
ω
π
θ
π
ω
θω
−−=
−−=
=
ptkI
N
m
ptkI
N
m
ptkI
N
m
km
se
kc
km
se
kb
km
se
ka
(2. 2)
Therefore; )cos(
22
3
1
,
,,,. tkp
IN
mmmm
kmse
kckbkakm ωθ −=++= (2. 3)
Where, k is a time harmonic order and p is a number of pole pairs.
From Equation 2.3, it can be seen that time harmonics in the stator MMF can be obtained
from time harmonics in the supply voltages. In addition, the space shift of phase windings
can contribute to space harmonics in the stator MMF as well.
2.4 MMF Space Harmonics
The airgap MMF is generated from stator and rotor currents. Due to the distribution of stator
and rotor windings, space harmonics in the airgap MMF can take place. The total airgap
MMF can be expressed by the sum of the stator and rotor MMF as below.
( , ) ( , ) ( , )s rF t F t F tθ θ θ= + (2. 4)
Consider a three-phase induction motor with 2p poles, q slots per pole per phase, with
phase-belts displaced in space by 2 3π radians around the airgap. Harmonics in the airgap
MMF due to stator and rotor slots are considered [1,2].
2.4.1 Stator MMF
, 1 1
1
ˆ( , ) cos( )s s v
v
F t F vp tθ θ ω ϕ
∞
=
= − −∑ (2. 5)
Where, )16( 1 += gv with 1g = ,...3,2,1,0 ±±±
1,
23ˆ Ikk
v
qN
F dvpvvs
π
=
Where, 1ω is a fundamental angular frequency, 1ϕ is a fundamental phase angle, N is a
number of conductors per slot, 1I is stator phase currents, pvk is a pitch factor and dvk is a
winding distribution factor, and v is a space harmonic order.
From Equation 2.5, a mechanical angular speed of the v -th space harmonic is given by;
pgvp
v
)16( 1
11
+
==
ωω
ω (2. 6)
A sign of the mechanical angular speed of the v -th space harmonic expresses the direction
of the space harmonic wave. The positive sign; i.e. 1,7,13,...,v = means the harmonic wave

7
rotates as the same direction as the fundamental wave, but the negative sign,
i.e. -5,-11,-17,...,v = means the harmonic wave rotates in the backward direction. In addition,
stator slot harmonics and phase-belt space harmonics have to take into consideration due to
their significance. For the stator slot harmonics, they can be found at the v -th harmonic order
as below.
1( 1)sQ
v g
p
= + or )16( 1 +sqg (2. 7)
Where, sQ is a number of stator slots, and sq is a number of stator slots per pole per phase.
For the phase-belt harmonics, they are the harmonics, 5,7,11,13,...v = to the first stator slot
harmonics.
2.4.2 Rotor MMF
The rotor MMF can be considered from two different sources, which are due to the
fundamental rotor current and rotor harmonic currents.
2.4.2.1 Rotor MMF due to the Fundamental Rotor Current
,1 ,1 1 ,1
1
ˆ( , ) cos( )r rF t F p tμ μ
μ
θ μ θ ω ϕ
∞
=
= − −∑ (2. 8)
Where, 1 2( 1)rQ
g
p
μ = + With 2g = ,...3,2,1,0 ±±±
,1 1 1 1
3 2ˆ ( 1) cosr
qN
F Iμ
μξ ξ ϕ
πμ
= − −
Where, rQ is a number of rotor slots, μ is a space harmonic order, 1ϕ is a phase angle of
stator currents, μϕ is a phase angle of rotor MMF harmonics, the parameters 1ξ and μξ are
winding factors for the fundamental component and the μ -th harmonics respectively.
Hence, the angular speed of the airgap flux density generated by the rotor MMF due to the
fundamental rotor current in the stator reference frame can be calculated as;
1 2
,1 1 1
1
(1 )
( ) 1r rs g Q s
p
p p p
μ
ω ω
ω μ ω
μ
⎡ ⎤−
= + = +⎢ ⎥
⎣ ⎦
(2. 9)
Where, rω is an electrical rotor angular frequency, and s is slip.
2.4.2.2 Rotor MMF due to Rotor Current Harmonics
,2 ,2, 2 ,2
1
ˆ( , ) cos( )r rF t F p tμ μ μ
μ
θ μ θ ω ϕ
∞
=
= − −∑ (2. 10)
Where, 2 2( )rQ
g v
p
μ = + With 2g = ,...3,2,1,0 ±±±
,2,
ˆ ( 1) 2r rv v
v
F Iμ
μ ξ
μ
= − −

8
Where, rvI is rotor end-ring harmonic current and vξ is a winding factor. The speed of the
airgap flux density generated by rotor harmonic currents can be analyzed from harmonics of
the stator flux density. The harmonics of the stator flux density rotate at the speed of 1 vpω
and induce currents in the rotor windings with the frequency as;
1
2 1 1( ) (1 )r
vp v s
vp p
ω ω
ω ω ω= − = − − (2. 11)
Thus, the speed of the airgap flux density generated by rotor harmonic currents relative to
the stator frame can be calculated as;
1 1 2
,2 2 1
2
(1 ) (1 )
( ) 1r rv s g Q s
p
p p p
μ
ω ω ω
ω μ ω
μ
⎡ ⎤− − −
= + = +⎢ ⎥
⎣ ⎦
(2. 12)
2.4.3 Total MMF
From Equation 2.5 to 2.11, the total MMF in Equation 2.4 can be expressed as below.
1 1
, 1 1 ,
1 1
0 1 1
1 1 1 1 2 1 1 1
0,1,2,... 1, 2,...
3
ˆ ˆ( , ) cos( ) cos( )
ˆ cos( )
ˆ ˆcos((6 1) ) cos((6 1) )
ˆ cos((
s v r
v
g g
s
F t F vp t F p t
F p t
F g p t F g p t
F Q
μ μ μ
μ
θ θ ω ϕ μ θ ω ϕ
θ ω ϕ
θ ω ϕ θ ω ϕ
∞ ∞
= =
= =− −
= ± − + ± −
= − −
+ + − − + + − −
+
∑ ∑
1 1 4 1 1
5 1 5 1
ˆ) ) cos(( ) )
cos(( )( ) ) cos(( )( ) )
s
r m r m
p t F Q p t
F Q p t s t F Q p t s t
θ ω ϕ θ ω ϕ
θ ω ω θ ω ω
− − − + + − −
+ − − + + + − −
(2. 13)
It should be noted that the stator MMF harmonics are influences from stator windings and
slots, but the rotor MMF harmonics are influences from only rotor slots. Equation 2.13 can be
simplified if only the most significant harmonics, such as the fundamental, the first order slot
harmonics, phase-belt harmonics and winding harmonics, are considered.
2.5 Airgap Permeance
To determine an airgap permeance, the airgap length has to be considered. At a specific
location in space, when a rotor is rotating, the airgap length is not constant, but it is
changing. Consider the airgap shown in Figure 2.3, the function of the airgap length can be
determined as shown in Equation 2.14. [1]
021 ),()(),( δθδθδθδ −+= tt (2. 14)
The airgap length can be separated into three
parts, which are the constant airgap length 0δ ,
the space dependent airgap length 1δ and the
time and space dependent airgap length 2δ .
Therefore, the airgap permeance can be
expressed as shown in Equation 2.15.
Figure 2.3 Airgap with both stator slots and
rotor slot

9
[ ]
0
1 2 0
1 2 0
0
1, 2,
1 10 1 2
1,2,
1 1
0 1 2 1,2
1
( , )
1 1 1( ) ( , )
1
cos( ) cos ( )
cos ( )
k s m r m
k mc c
k m s r r m
k m
t
t
kQ mQ t
k k
kQ mQ mQ t
μ
θ
δ θ δ θ δ
μ
θ θ ω
δ
θ ω
∞ ∞
= =
∞ ∞
±
= =
Λ = =
+ − + −
Λ Λ Λ
= + Λ + Λ −
⎡ ⎤+ Λ ± −⎣ ⎦
′ ′ ′= Λ + Λ + Λ + Λ
∑ ∑
∑∑
(2. 15)
Where, mω is mechanical rotational speed ( 1(1 )m r p s pω ω ω= = − )
It can be seen from Equation 2.15 that the airgap permeance is composed of four terms. The
first term is the constant term that results from the constant airgap length. The second term
represents the influences from stator slots. The third term can describe the effect from rotor
slots related to numbers of rotor slots and rotor speed. Lastly, the influences from both stator
and rotor slots are presented in the terms of sum and difference between numbers of stator
and rotor slots.
2.6 Airgap Flux Density
The airgap flux density can be obtained by substituting Equation 2.13 and 2.15 into Equation
2.1, which gives the results in infinitely many terms. Nevertheless, the airgap flux density can
be approximated by taking only the simplified MMF and the first three terms in the
permeance expression into account as shown in Equation 2.16. The results are summarized
as in Table 2.1. More details can be found in [1].
[ ]{
[ ] [ ]
0 1 2 1,2 ,1 , ,1 ,
0 1 2
0 1 1 1
2 1 3 1
( , ) ( , ) ( , )
( ) ( )
( cos cos ( ))
cos( ) cos 5
cos 7 cos ( )
s s v r r
s r r
s
B t t F t
F F F F
b b Q b Q w t
a p t a p t
a p t a Q p t
μ
θ θ θ
θ θ
θ ω θ ω
θ ω θ ω
= Λ ⋅
′ ′ ′= Λ + Λ + Λ + Λ ⋅ + + +
= + + −
− + + +
− + − + +
[ ] [ ]
[ ]}
4 1 5 1
6 1
cos ( ) cos ( )( )
cos ( )( )
s r m
r m
a Q p t a Q p t s t
a Q p t s t
θ ω θ ω ω
θ ω ω
+ − + − − + +
+ − −
(2. 16)
2.6.1 Stator Flux Density
The influence from the airgap flux density generated from the stator currents is to induce
voltages in the rotor windings. The expressions for the frequency of induced voltages can be
determined as below;
2.6.1.1 Fundamental Stator Flux Density
,1 ,1 1 1( , ) cos( )s sB t B p tθ θ ω ϕ= − −
)
(2. 17)
Due to the difference between the rotational angular speed r pω and the synchronous
speed of the fundamental airgap flux density 1 pω , voltages and currents can be induced in
the rotor windings. The frequency of the induced rotor voltages and currents can be

10
determined from the relative angular speed between the rotational speed and the
synchronous speed as shown in Equation 2.18.
1
2 1
1
( )
2
rf p sf
p p
ω ω
π
= − ⋅ = (2. 18)
2.6.1.2 Harmonics of Stator Flux Density
, 1 1
ˆ( , ) cos( )sw s vB t B vp tθ θ ω ϕ= − − (2. 19)
Harmonics of the airgap flux density rotate at an angular speed of 1 vpω and pulsate at the
fundamental frequency 1ω . This means the speed of harmonics decreases vp times of the
fundamental component, but the pulsation is still constant, or the number of pulsations is vp
times of the fundamental component. Therefore, the frequency of induced rotor voltages and
currents from harmonics of the stator flux density is determined as shown in Equation 2.20.
[ ]1
2, 1
1
( ) (1 ) 1
2
r
vf vp s v f
p vp
ω ω
π
= − ⋅ = − − (2. 20)
Forward Harmonic order
Frequency of
induced stator
voltage and
currentω
1 0 0 1cos( )a b p tθ ω− 1 1ω
2 ( )1 1
1cos
2
s
a b
Q p tθ ω⎡ ⎤+ −⎣ ⎦ 1sQ
p
+ 1ω
3
[ ]2 0 1cos 7a b p tθ ω− 7 1ω
4
( )4 0 1cos sa b Q p tθ ω⎡ ⎤+ −⎣ ⎦ 1sQ
p
+ 1ω
5
( )6 0 1cos ( )r ma b Q p t s tθ ω ω⎡ ⎤+ − −⎣ ⎦ 1rQ
p
+ 1( 1)r
r
Q
s
p
ω ω+ +
6 ( )4 0 1cos cos ( )s r ra b Q p t Q tθ ω θ ω⎡ ⎤+ − −⎣ ⎦
7 ( )6 1 1cos ( ) cosr r r sa b Q p Q t Q tθ ω ω⎡ ⎤+ − +⎣ ⎦
Backward
1 ( )1 1
1cos
2
s
a b
Q p tθ ω⎡ ⎤− +⎣ ⎦ 1sQ
p
−
1ω
2
[ ]1 0 1cos 5a b p tθ ω+ 5 1ω
3
( )3 0 1cos sa b Q p tθ ω⎡ ⎤− +⎣ ⎦ 1sQ
p
− 1ω
4 ( )5 0 1cos ( )r ma b Q p t s tθ ω ω⎡ ⎤− − +⎣ ⎦ 1rQ
p
− 1( 1)r
r
Q
s
p
ω ω− −
5 ( )3 2 1cos cos ( )s r ra b Q p t Q tθ ω θ ω⎡ ⎤− + −⎣ ⎦
6 ( )5 1 1cos ( ) cosr r sa b Q p t s t Q tθ ω ω⎡ ⎤− − +⎣ ⎦
Table 2.1 Airgap flux density considering a finite number of space harmonics

11
2.6.2 Rotor Flux Density
Similar to the stator flux density, the flux density created by the rotor currents also induce
voltages in the stator windings. The expression for the frequency of induced voltages can be
determined as below;
2.6.2.1 Fundamental Rotor Flux Density
,1 ,1 1
ˆ( , ) cos( )r rB t B p t μθ θ ω ϕ= − − (2. 21)
Due to the space-fixed stator windings, the rotor flux density also induces voltages in the
stator windings as;
1
, 1( 0)s induced p
p
ω
ω ω= − = (2. 22)
2.6.2.2 Harmonics of Rotor Flux Density
, ,
ˆ( , ) cos( )r rB t B p tμ μ μ μθ μ θ ω ϕ= − − (2. 23)
Similar to the fundamental rotor flux density, harmonics of the rotor flux density rotate with
the speed μω in the airgap. Thus, the frequency of induced voltages is μω .
2.7 Conclusion
In this chapter, the effect of time and space harmonics in the airgap flux density on the stator
currents is discussed. Due to finite numbers of stator and rotor slots as well as distribution of
stator windings, a non-sinusoidal airgap flux density will arise in the airgap. The influences
from space harmonics of the airgap flux density can cause time harmonics in the supply
voltages and currents, which cause pulsating torque. By determining the airgap MMF
generated from stator and rotor currents and airgap permeance, the frequency of induced
quantities in the rotor, airgap and stator, such as rotor voltages and currents, airgap flux
density and stator voltages and currents can be determined. The details presented here are
the main background for determining influences from faults on stator currents in the next
chapter.

12
3 MOTOR CURRENT SIGNATURE ANALYSIS (MCSA)
3.1 Introduction
In this chapter, a technique to detect faults in induction machines by frequency analysis of
stator currents is presented. Thanks to a unique consequence of each fault, unique
harmonics in line currents are produced. By locating the specific harmonic components,
which are called as Motor Current Signature Analysis (MCSA), faults in induction motors can
be detected.
In this chapter, four fault types are presented. First, rotor asymmetry, which is the
consequence of breakage in rotor bars or end rings, is introduced. Next, airgap asymmetry
that is static, dynamic and mix of static and dynamic eccentricities is discussed. Then, stator
asymmetry that is an inter-turn short circuit fault is described, and bearing faults are also
reviewed. Moreover, influences from mechanical couplings in line currents are introduced. In
addition, two operating conditions; unsymmetrical supply voltages and oscillating loads, of
which consequences are similar to that of some faults, are presented.
The aim of this chapter is to study influences in stator currents from various faults. Causes,
phenomena and characteristics of each fault type are presented. These will be applied to
build FEM simulation models presented in Chapter 5. The background from the previous
chapter is used to determine analytical expressions of fault indicators.
3.2 Rotor Asymmetry
According to the failure survey [58], it stated that about 10% of total failure cases related to
rotor failures. One of rotor failures found frequently is rotor breakages. Breakages in a rotor
can take place in rotor bars or rotor end rings. A broken piece of rotor bars or end rings can
move along the airgap between the stator and rotor, and it can disrupt surfaces of stator
windings leading to a sudden failure. This can result in high repairing costs and outage time.
For this reason, the detection of rotor breakages at an early state is advantageous.
3.2.1 Cause for Rotor Breakage
Rotor breakages can be caused by many reasons. However, they can be summarized as;
1. Thermal stresses due to thermal overload and unbalance, hot spots or excessive
losses, sparking (mainly fabricated rotor type)
2. Magnetic stresses caused by electromagnetic forces, unbalanced magnetic pull,
electromagnetic noise and vibration
3. Residual stresses due to manufacturing problems
4. Dynamic stresses arising from shaft torque, centrifugal forces and cyclic stresses
5. Environmental stresses caused by contamination and abrasion of rotor material
due to chemicals or moisture
6. Mechanical stresses due to loose laminations, fatigued parts, bearing failures and
etc
7. Operating condition; pulsating load leading to rapid changes on the shaft torque

13
Risks of rotor failure can be reduced if these stresses can be kept under control. Designing,
Building and Installation as well as Maintaining should be considered and done properly in
order to reduce the stresses in a motor.
3.2.2 Detection of Rotor Breakage
There are many techniques to detect rotor breakages. In addition to the stator current
method, negative sequence impedance or negative sequence current[35], zero sequence
current[37], axial flux[36,40], torque [23], instantaneous power[21], the extend park’s vectors
[24,25] , injection of low frequency signal [39] or vibration[36], are able to detect rotor
breakages. However, determination of rotor breakages from stator currents is emphasized in
this study.
Ideally, no sideband component exists around the fundamental component in stator currents.
In case a rotor is asymmetrical due to a bar or end ring breakage or a non-oval shape, it
causes asymmetrical rotor MMF, and this leads to backward traveling rotor MMF. The
backward rotor MMF cause induced voltages in the stator windings with particular
frequencies. The comparison between the healthy machine and the asymmetrical rotor
machine is presented below;
Healthy Condition
Consider an ideal induction motor. The rotor is perfectly balanced. Rotor bar currents are
given by [2];
Figure 3.1 Rotor bar current
2 ( 1)
( ) 1Re( ) cos( ( 1) )r
p n
j
Q
b n b bI I e I s t n
π
ω φ
−
−
= = − − (3. 1)
Where,
2
r
p
Q
π
φ =
Consequently, the rotor MMF generated by each rotor bar current can be expressed as;
1cos( ( 1) ) cos( ( 1) )bn bF NI s t n nω φ θ φ= − − ⋅ − − (3. 2)
It should be noted that the distribution of rotor bars has been considered in Equation 3.2.
Consider only 1 pole pair of the motor in 360 electrical degrees. If there is no breakage, the
total rotor MMF at an angle θ can be expressed as;

14
, 1 1
1
1
1
2 2
[cos( )cos( ) cos( )cos( )
2 2 2 2
cos( )cos( ) ...
( 1) 2 ( 1) 2
cos( )cos( )]
1
cos( )
2
b res b
r r
r r
r r
r r
r
b
p p
F NI s t s t
Q Q
p p
s t
Q Q
Q Q
p p
p p
s t
Q Q
Q
NI s t
p
π π
ω θ ω θ
π π
ω θ
π π
ω θ
θ ω
= + − −
⋅ ⋅
+ − − +
− ⋅ − ⋅
+ − −
= ⋅ −
(3. 3)
It can be seen from Equation 3.3 that only the forward MMF is generated by the rotor
currents. Assume the airgap length is constant, and slotting effects are neglected. The rotor
flux density and induced voltages can be obtained through the following steps;
• The frequency of the rotor MMF; 2 1f sf=
• The electrical rotational frequency; 1(1 )rf s f= −
• The frequency of the rotor airgap flux density;
, 1 1 1(1 )B rf s f sf f= − + =
• The frequency of induced voltages in the stator
windings is 1f or the fundamental frequency.
Rotor Asymmetry
Consider an induction motor with one broken rotor bar. The position of the broken bar is
placed at the second rotor bar. Due to the defect, currents induced in each rotor bar are not
symmetrical. To simplify the analytical expression, each rotor bar current is assumed to be
unchanged. (The analytical expression for rotor currents due to broken bars is presented in
[32].) Therefore, the total rotor MMF can be determined by subtracting the rotor MMF
induced by the second rotor bar current from the ideal rotor MMF;
, , 1
1 1
1 1
2 2
cos( )cos( )
1 2 2
cos( ) cos( )cos( )
2
1 1 2 2
( 1) cos( ) cos( )
2 2
b res b res b
r r
r
b b
r r
r
b b
r
p p
F F NI s t
Q Q
Q p p
NI s t NI s t
p Q Q
Q p
NI s t NI s t
p Q
π π
ω θ
π π
θ ω ω θ
π
θ ω θ ω
′ = − − −
= ⋅ − − − −
⋅
= ⋅ − − − + −
(3. 4)
It can be seen from Equation 3.4 that the additional backward MMF exists in the airgap. The
frequency of the backward MMF is equal to the forward one. Due to the backward MMF, the
induced voltages corresponding to the twice slip frequency is generated around the
fundamental component. The frequency of rotor flux density and induced voltages due to the
backward MMF can be obtained through the following steps;
Figure 3.2 Speed and direction of rotor
MMF for healthy condition
(Red; rotor reference frame,
Black; stator reference
frame)

15
• The airgap flux density due to the backward MMF
can induce voltages in the stator windings with
the frequency as;
1 1 1 1(1 ) 2brbf s f sf f sf= − − = −
• Due to this component, speed and torque
oscillation at the frequency 12sf will be present.
From these, the upper sideband component
corresponding to the twice slip frequency 12sf
above the fundamental component will arise
[21,32].
According to [4-9, 15, 19, 21-25], in summary, the indicator of rotor asymmetry is the
sideband components around the fundamental. The frequency of the sideband components
corresponds to;
1(1 2 )brbf ks f= m , 1,2,3,...k = (3. 5)
In addition, in [4, 5, 10, 19, 20, 32], it has been presented that the consequences of rotor
asymmetry can be detected by the components following the equation below;
1[( )(1 ) ]brb
k
f s s f
p
= − ± , 1,5,7,11,13,...
k
p
= (3. 6)
Where, 1f is the fundamental frequency, and s is slip. The analytical explanations in
different approaches can be found in [11-13, 31-32].
Unfortunately, if broken bars are located at 180 electrical degrees away from each other,
sideband components do not exist [5]. The reason is that the rotor MMF is still symmetrical,
so only the forward rotor MMF is generated. Consider an induction motor with two broken
bars placed at 180 electrical degrees away from each other. The positions of the broken bars
are at the second and the ( 1)rQ p + -th bars. The total rotor MMF can be determined as
below.
, , 1
1
1 1
1
2 2
[cos( )cos( )
( 1) 2 ( 1) 2
cos( )cos( )]
1
cos( ) [cos( )]
2
1
( 2) cos( )
2
b res b res b
r r
r r
r r
r
b b
r
b
p p
F F NI s t
Q Q
Q Q
p p
p p
s t
Q Q
Q
NI s t NI s t
p
Q
NI s t
p
π π
ω θ
π π
ω θ
θ ω θ ω
θ ω
′′ = − − −
+ +
+ − −
= ⋅ − − −
= ⋅ − −
(3. 7)
It can be seen from Equation 3.7 that only the forward rotor MMF is produced. Therefore, the
sideband components do not exist in this case.
In practice, an uneven rotor bar resistance or rotor asymmetry can possibly exist due to
manufacture, so sidebands around the fundamental component can be observed even
though the machine is healthy. In addition, the amplitude ratio of the first sideband
components to the fundamental components is usually chosen as the fault feature to detect a
Figure 3.3 Speed and direction of rotor
MMF for rotor asymmetry

16
defected rotor through determining whether the ration exceeds a certain threshold or not.
However, there is no standard value for the threshold level.
1f s k
1(1 2 )brbf ks f= m
[Hz]
k
p
1[( )(1 ) ]brb
k
f s s f
p
= − ±
[Hz]
50 0.01 1 49 1 50
51 49
2 48 5 248
52 247
3 47 7 347
53 346
Table 3.1 Detected frequencies on stator current for rotor asymmetry
Furthermore, it is possible to estimate a number of broken rotor bars as stated in [5];
sin
2 (2 )
brb
b
I
I p
α
π α
≅
−
(3. 8)
And
2
r
np
Q
π
α =
Where, brbI = amplitude of the first lower sideband frequency
bI = amplitude of the fundamental component
n = numbers of broken bars
Another expression for estimation of defected rotor bars presented in [7] is;
20
2
10
r
N
Q
n
p
=
+
(3. 9)
Where; n = estimate of the number of broken bars
N = average dB difference value between the upper and lower sideband and
the fundamental component.
p = number of pole pair
3.3 Airgap Asymmetry
Airgap eccentricity is a condition of an unequal airgap that exists between a stator and a
rotor. It results in an unbalance magnetic pull (UMP) or unbalance radial forces, which can
cause damages in a motor by rubbing between a stator and a rotor. In addition, the radial
magnetic force waves can act on a stator core and subject to stator windings unnecessarily
and potentially harmful vibration. Therefore, it would be advantageous if airgap eccentricity
can be detected before machines are deteriorated. There are three types of eccentricities
called static, dynamic and mix of static and dynamic eccentricities. All can be distinguished
by the characteristic of the airgap.
The subjects of on-line detection of airgap eccentricities in three-phase induction motors
have been proposed by many researchers. In addition to the stator line current, other
techniques, such as negative sequence impedance[35], the extend park’s vector [26,27],

17
instantaneous power[5], axial flux[40], inject low frequency signal[39] ,or vibration[38], have
been presented for detecting airgap eccentric faults.
3.3.1 Static eccentricity
In the case of static eccentricity, a position of a
minimum radial airgap is fixed in space. It causes a
steady unbalanced magnetic pull (UMP) in one
direction. This can lead to a bent rotor shaft or
bearing wear and tear. It can also lead to some
degree of dynamic eccentricity.
Static eccentricity can occur when a rotor is displaced
from a bore center, but it is still turning upon its bore
center [29,30,33]. It can be simplified as shown in
Figure 3.4.
3.3.1.1 Cause for static eccentricity
Static eccentricity can be caused by;
1. The oval shape of the stator core due to manufacture
2. Misalignment bearing position due to assembly
3. Bearing wear
4. Misalignment of mechanical couplings
3.3.1.2 Detection of static eccentricity
If a rotor and a stator are assumed to be smooth, the airgap permeance can be expressed of
two terms, which are the constant permeance term and the time dependent permeance term
due to the rotor rotation as [2,28,29];
0( , ) cos( )t tε ε εθ θ ω ϕΛ = Λ + Λ − − (3. 10)
Where, 0Λ represents a constant airgap permeance, εΛ is a peak of the permeance
influenced by rotor eccentricity, εω is an angular frequency of a rotor center relative to a
stator and εϕ is a phase angle.
For static eccentricity, the angular frequency εω is zero because the rotor does not rotate
around the motor center but spins around its own center. Thus, the airgap permeance
function for static eccentricity can be expressed as;
0( , ) cos( )t εθ θΛ = Λ + Λ (3. 11)
By Equation 2.10 and 3.11, the airgap flux density can therefore be determined as shown in
Equation 3.12.
Figure 3.4 Static Eccentricity

18
{ }
0 ,
1
0 ,
1
,
1
ˆ( cos( )) cos( )
ˆ cos( )
ˆ cos[( 1) ] cos[( 1) ]
2
R
R
R
B F p t
F p t
F p t p t
ε ε μ μ
μ
μ μ
μ
ε
μ μ μ
μ
θ μ θ ω
μ θ ω
μ θ ω μ θ ω
∞
=
∞
=
∞
=
= Λ + Λ ⋅ −
= Λ − +
Λ
+ − + − +
∑
∑
∑
(3. 12)
From Equation 3.12, it can be seen that there are two additional terms in the function of the
airgap flux density due to the static eccentricity. The frequency of induced voltages
influenced from the static eccentricity can be expressed as;
1 2
(1 )
[ 1]
2
ind r
s
f f g Q
p
μω
π
−
= = + (3. 13)
Consider influences from time harmonics in supply voltages. From Equation 3.13, the
frequency of induced voltages and currents in the stator windings can be expressed as given
by [6, 7, 9, 20, and 28];
, 1
(1 )
[ ]s ecc r
s
f kQ n f
p
−
= ± ⋅ (3. 14)
Where; k = 1, 2, 3…
n = order of stator time harmonics present in the power supply feeding the
motor ( 1,3,5...n = )
It can be seen that the expression in 3.14 is actually for rotor slot harmonics. On the other
word, the static eccentricity results in a rise of the rotor slot harmonic components. Moreover,
the experiments in [28] have shown that the amplitude of the components calculated from
Equation 3.14 does not change significantly, when an induction motor is applied by only the
static eccentricity. However, the static eccentric variations can result in the introduction of
dynamic eccentricity.
3.3.2 Dynamic eccentricity
Dynamic eccentricity occurs when a rotor turns upon
a stator bore centre but not its own center. It causes
a minimum airgap which is always moving in the
airgap. For the case that the rotor center rotates
around the motor center with the rotational speed as
shown in Figure 3.5, it is called as dynamic
eccentricity [4,29,30,33].
In addition, there are other cases of dynamic eccentricity. One of them is that the revolving
speed of the rotor center is not equal to the rotational speed [54]. It should be noted that only
the type, which is that the rotor axis is parallel to the stator axis, is studied here.
Figure 3.5 Dynamic Eccentricity

19
3.3.2.1 Cause for dynamic eccentricity
As mentioned under the topic of static eccentricity, static eccentricity can lead to some
degree of dynamic eccentricity due to UMP. Hence, the causes mentioned in 3.2.1.1 should
be valid. In addition, mechanical resonance at critical speed can result in dynamic
eccentricity.
3.3.2.2 Detection of dynamic eccentricity
According to Equation 3.10, the revolving speed of the rotor center is equal to the rotational
speed for the dynamic eccentricity. That means the angular frequency εω is equal to the
mechanical rotational speed mω . Therefore, the airgap permeance function for dynamic
eccentricity can be determined as [2,28];
0( , ) cos( )mt tεθ θ ωΛ = Λ + Λ − (3. 15)
Hence, the airgap flux density can be derived from the permeance function in Equation 3.15
and the rotor MMF in Equation 2.10.
{ }
0 ,
1
0 ,
1
,
1
ˆ[ cos( )] cos( )
ˆ cos( )
ˆ cos[( 1) ( ) ] cos[( 1) ( ) ]
2
r
R
R
r r
R
B t F p t
p
F p t
F p t p t
p p
ε ε μ μ
μ
μ μ
μ
ε
μ μ μ
μ
ω
θ μ θ ω
μ θ ω
ω ω
μ θ ω μ θ ω
∞
=
∞
=
∞
=
= Λ + Λ − ⋅ −
= Λ − +
Λ
+ − + + − + −
∑
∑
∑
(3. 16)
The induced frequency of stator voltages and currents influenced from dynamic eccentricity
can be determined from two additional terms in the airgap flux density as shown in Equation
3.17.
1 2
1 (1 )
( ) [( 1) 1]
2
r
ind r
s
f f g Q
p p
μ
ω
ω
π
−
= ± = ± + (3. 17)
When time harmonics of supply voltages and dynamic eccentric orders are taken into
account, Equation 3.17 can be modified as given by [6, 7, 9, 20, and 28];
, 1
(1 )
[( ) ]d ecc r d
s
f kQ n n f
p
−
= ± ± ⋅ (3. 18)
Where; k = 1, 2, 3…
dn = dynamic eccentric order ( dn =1, 2, 3,)
n = Time harmonic order of supply voltages driving motors ( 1,3,5...n = )
According to [4,28], in the case when one of these harmonics influenced from static or
dynamic eccentricities is a multiple of three, it may not exist theoretically in the line currents
of a balance three phase machine. Besides, induction motors corresponding to the
relationship in Equation 3.19 are ascertained to generate principle slot harmonics, but they
will not give rise to these harmonics with only static or dynamic eccentricities.
2 [3( ) ]rQ p m q r= ± ± (3. 19)

20
However, only a particular combination of machine pole pairs and numbers of rotor slots will
give a significant rise of only static or dynamic eccentricities related to components. The
relationship for a three-phase integral slot and 60-degree phase belt machine is given by
[4,28];
2 [3( ) ]rQ p m q r k= ± ± ± (3. 20)
Where, 0,1,2,3,...m q± =
0 1r or=
1 2k or=
3.3.3 Mix eccentricity
In reality, both static and dynamic eccentricities tend
to co-exits in machines. With this condition, a rotor
turns around neither its bore center nor a stator bore
center, but it revolves around a point between the
stator and rotor centers. This condition can be
presented by Figure 3.6 showing that the rotational
center or the motor center can be anywhere
between the stator and rotor centers [29].
3.3.3.1 Detection of Mix eccentricity
According to Equation 3.10, 3.11 and 3.15, the permeance function of mix eccentricity can be
determined as three different terms that are influences from the constant permeance, static
eccentric and dynamic eccentric terms [29];
0 1 2( , ) cos( ) cos( )mt tθ θ θ ωΛ = Λ + Λ + Λ − (3. 21)
It can be seen from Equation 3.21 that the airgap permeance for mix eccentricity results from
both static and dynamic eccentricities. In addition to the influences corresponding to
Equation 3.14 and 3.18, the consequence from both can be found as the results of amplitude
modulation. The low frequency components corresponding to the rotational frequency will
exist around the fundamental frequency. The expression for the low frequency components
is shown in Equation 3.22. The analytical study to derive this expression can be seen in [29].
, 1 1
1
[1 ( )]mix ecc r
s
f f mf f k
p
−
= ± = ± (3. 22)
Where; , 1,2,3,...m k p =
It should be noted that the formula in Equation 3.22 is for the case that the revolving speed of
the rotor center is equal to the rotational speed. In addition, it can be observed that the low
frequency components are placed away with a multiple of the rotational frequency from the
fundamental component.
Figure 3.6 Mix eccentricity

21
Static Ecc. Dynamic Ecc. Mix Ecc.
1f p s rQ
k n
,s eccf
[Hz] k dn n
,d eccf
[Hz] m
,mix eccf
[Hz]
50 3 0.01 68 1 1 1172 1 1 1 1188.5 1 66.5
1072 1088.5 33.5
5 1372 1155.5 2 83
872 1055.5 17
1 2 1 1205
1105
1139
1039
Table 3.2 Detected frequencies on stator current for airgap asymmetry
3.4 Stator Asymmetry
According to the surveys [58], the majority of failure related to a motor stator is breakdown of
the turn-to-turn insulation. Although the induction motor can still run when some of the turns
are shorted, they can consequently lead to damages on adjacent coils and a stator core, so
that a ground fault can occur. To reduce repairing costs and outage time due to the stator
winding fault, the early detection of inter-turn short circuit is useful.
3.4.1 Cause for Inter-Turn Short Circuit
There are many reasons that can cause the degradation on the stator insulation. The causes
can be summarized as [4, 36];
1. Thermal stresses due to thermal ageing and thermal overloading: For the thermal
ageing, it is a result from the operating temperature. As known, the insulation life
gets half for every 10o
k increase in temperature. To cope with the thermal ageing
due to the temperature in the windings, reducing the operating temperature or
increasing the class of insulation materials can be applied. Thermal overloading
can be caused by the applied voltage variations, unbalanced phase voltage,
cycling overloading, obstructed ventilation, higher ambient temperature, etc. All of
these can increase the temperature and can initiate the thermal stress in the
machine.
2. Electrical stresses due to voltage stresses in the windings: The voltage stress in
the windings can be caused by having a void in the insulation, which can cause
the partial discharge. In addition, the surge on electrical supply system can initiate
the voltage stresses in the windings as well.
3. Mechanical Stresses: These stresses might be due to coil movement, which is a
result from the force inside the machine, and rotor striking the stator, which is
caused from many reasons, such as bearing failures, shaft deflection, rotor-to-
stator misalignment, etc.
4. Environmental stresses/Contamination: the winding insulation can be deteriorated
by chemicals, such as oil, moisture or dirt, etc.
5. Ageing: the winding insulation can be degraded by time.

22
3.4.2 Detection of Inter-Turn Stator Winding Fault
The early technique to detect the stator fault is the partial discharge technique [36]. Axial
leakage flux monitoring [42], negative sequence impedance or negative sequence current
[35,46] and zero-sequence component [37,41] have been presented as method of detection
for the inter-turn short circuit fault in an early state.
The effect of the inter-turn short circuit fault is that some turns from stator windings are
removed. This causes a small but finite effect on the airgap flux density. When a short circuit
happens, phase windings have less numbers of turns, so they produce less MMF. Moreover,
the currents that flow in the shorted windings also produce MMF, which is opposite to and
against the main MMF produced by the phase windings [42,43,44,54].
Figure 3.7 Diagram of inter-turn short circuit in one section of a single phase coil winding
For MCSA, the induced frequency resulting from inter-turn stator fault has been presented in
[6] as expressed in Equation 3.23.
1 (1 )st
n
f f s k
p
= ⋅ − ± (3. 23)
Where; n = 1, 2, 3…
k = 1, 3, 5…
In addition, the other two expressions considering influences from saturation in materials as
well as influences from different sources have been presented in [43]. The first expression
originating from the stator currents is as;
1
(1 )
2st r
s
f f kQ m n
p
⎧ ⎫−
= ± ±⎨ ⎬
⎩ ⎭
(3. 24)
The second expression originating from the rotor currents is as;
1
(1 )
( ) 2st r
s
f f kQ i m n s
p
⎧ ⎫−
= ± ± ± ⋅⎨ ⎬
⎩ ⎭
(3. 25)
Where; i = 1,2,3,…
n = 1,2,3,…
k = influences from the rotor slots = 1,2,3,…
m = influences from the saturation = 0,1,2,3,…

23
1f p s rQ n k
stf
[Eq(3.23)]
[Hz]
k m n
stf
[Eq (3.24)]
[Hz]
k i m n
stf
[Eq (3.25)]
[Hz]
50 3 0.01 68 1 1 66.5 1 0 -1 1072 0 1 0 1 17
1 1 33.5 1 0 1 1172 0 2 0 1 33.5
2 1 83 1 1 1 1272 0 3 0 1 50
1 3 17 1 2 1 1372 0 4 0 1 66.5
1 5 266.5 0 5 0 1 83
1 5 233.5
Table 3.3 Detected frequencies on stator current for inter-turn short circuit
It can be observed that the expression in 3.24 is similar to the expression for rotor slot
harmonics. In addition, the expressions in 3.23 are similar to Equation 3.25 if influences from
the saturation in material are not taken into account. These imply that the inter-turn short
circuit fault makes influences on line current by rises of rotor slot harmonics and the
components corresponding to the rotational frequency. However, these are not sufficient to
identify the mix eccentricity and the inter-turn short circuit fault.
To separate both faults, current amplitude of each phase and phase shifts between each
phase current are required. In the healthy condition, impedances of each phase winding are
normally balanced, so the current amplitude of each phase is also balanced. The phase shift
between each phase is 120o
. Due to defected turns, impedances of the three phase
windings become unbalanced, so the three phase currents will be unbalanced as well. The
phase shift between each phase will be also distorted from 120o
. Moreover, the third
harmonic contents will become dominant. Therefore, information on amplitude, phase shift
and the third harmonic contents is necessary for separating the inter-turn short circuit fault
from the mix eccentricity.
3.5 Bearing Damages
Bearing is the part used to hold a rotor shaft of induction motors. Faults on bearing may
result in increasing vibration and noise levels. Bearing faults can also cause some damages
on mechanical couplings that connect to a rotor shaft. To protect motors and mechanical
couplings, detection of bearing faults in an early state becomes useful.
3.5.1 Cause for Bearing Damages
The bearing damages can result from many causes; internal causes, such as induced
bearing currents due to an unbalanced rotor or external causes, such as grease. However,
they can be summarized as;
1. High vibration due to foundations, mechanical couplings or loads
2. Inherent eccentricities, which cause unbalance magnetic force
3. Bearing current which cause an electrical discharge or sparking in bearings
4. Contamination and corrosion which is caused by pitting and sanding action of hard
and abrasive minute particles or corrosive action of water, acid, dirt etc
5. Improper lubrication including both over and under lubrication causing heating and
abrasion

24
6. Improper installation of bearing; by improperly forcing bearings onto a shaft or in a
housing (due to misalignment) indentations are formed in the raceways.
3.5.2 Detection of Bearing Damages
Bearing faults can be detected by the increased vibration in the high frequency spectrums
[36]. However, the cost of obtaining the vibration measurement is routinely high due to the
measurement equipment. Instead, the stator-current-based monitoring scheme is
inexpensive because it requires no additional sensors.
In [4,5,47,48], it has been suggested that bearing faults can be caused by mechanical
displacements in the airgap. They can manifest themselves as a combination of rotating
eccentricity moving in both directions. This can result in the increased bearing vibrations, and
the bearing vibrations can reflect themselves in the currents regarding the components as;
1mech vf f mf= ± (3. 26)
Where, m = 1,2,3,…
vf = the characteristic vibration frequencies based upon the bearing
dimensions.
Generally, the majority of electrical machines use ball or rolling element bearings. These
bearing types consist of two rings; inner and outer rings. Damages on these bearing types
can be categorized into four different damages [4,5,48];
1. Damage on an outer bearing race: the vibration
frequency is as;
( ) [1 cos( )]
2
v r
N BD
f f
PD
β= − (3. 27)
2. Damage on an inner bearing race : the vibration
frequency is as;
( ) [1 cos( )]
2
v r
N BD
f f
PD
β= − (3. 28)
3. Damage on a ball: the vibration frequency is as;
2
1 [ cos( )]r
v
PD f BD
f
BD PD
β
⋅ ⎧ ⎫
= −⎨ ⎬
⎩ ⎭
(3. 29)
4. Damage on a train: the vibration frequency is as;
2
( )
[1 cos( )]
2
r
v
f BD
f
PD
β= − (3. 30)
Where; N = number of bearing balls
BD = ball diameter
PD = ball pitch diameter
β = The contact angle of the ball with the races
Figure 3.8 Ball Bearing Dimension

25
3.6 Mechanical Coupling
It has been presented in [7] that mechanical equipments, such as gearbox etc. can influence
on stator currents as sideband components around the fundamental frequency. Their
frequency corresponds to their rotational speed. Therefore, it is necessary to take the
influences from the mechanical couplings into consideration when an investigation on stator
currents is preformed. Information of mechanical systems is required in order to make a
proper investigation.
1 ,mech r mf f mf= ± (3. 31)
Where, ,r mf is the rotational frequency of the mechanical coupling equipment.
3.7 Oscillating Loads
Influences from oscillating loads on stator currents have been presented in [49]. The
experiment was performed by running an induction motor with the 10Hz periodical load
torque with the 50% duty cycle. The sidebands placed at 10Hz away from the fundamental
component, were found. It should be noted that influences from oscillating loads may lead to
the wrong conclusion. Therefore, this point should be considered when one makes a
diagnosis.
With the assumptions that induction motors are lossless and are fed by the perfect sinusoidal
supply voltages, it is possible to consider the effects of the oscillating loads in stator currents.
The input currents can be made up of the sum of the components from the fundamental
frequency and the influences from the oscillating loads, which can reflect themselves as the
sidebands as shown in Equation 3.32.
1 1 2
1 1 2
1 1 2
cos(2 ) cos(2 )
2 2
cos(2 ) cos(2 )
3 3
2 2
cos(2 ) cos(2 )
3 3
a sb
b sb
c sb
I I f t I f t
I I f t I f t
I I f t I f t
π φ π φ
π π
π φ π φ
π π
π φ π φ
= − + −
= − − + − −
= + − + + −
(3. 32)
Hence, the input power can be obtained from the product of the input currents and voltages.
[ ]1 1
3 3
cos cos 2 ( )
2 2
a a b b c c
sb sb
P V I V I V I
VI VI f f tφ π φ
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅ − −
(3. 33)
Where; 1I = Amplitude of the current from the fundamental frequency
2I = Amplitude of the current from the sideband components
1f = The fundamental frequency
sbf = The sideband frequency
V = Amplitude of the supply voltage
From Equation 3.33, it can be seen that the input power is not constant but pulsates at the
frequency 1 sbf f− . On the other word, if the loads pulsate with the frequency 1 sbf f− , the
frequency component at sbf can present in the stator currents.

26
3.8 Conclusion
The consequences and detection on stator currents for each fault type have been
summarized in the Table below;
Fault Consequences Detection
Broken Rotor Bars &
End Rings
• Backward rotor MMF due to
unsymmetrical rotor currents
Equation 3.5 and 3.6
Static eccentricity • Steady unbalanced magnetic pull due to
a space-fixed minimum radial airgap
Rotor Slot Harmonics
Equation 3.14
Dynamic eccentricity • Unbalanced magnetic pull due to
periodical minimum radial airgap length
Equation 3.18
Mix eccentricity • Unbalanced magnetic pull influenced
from static and dynamic eccentricities
Equation 3.14,3.18
and 3.22
Inter-turn Short Circuit • Unsymmetrical stator MMF and windings Equation 3.14, 3.23
Bearing Damages • Increased vibrations Equation 3.26-3.30
• Influences from Mechanical couplings can also reflect themselves in stator
currents as the components, of which frequency corresponds to their rotational
speed.
• Oscillating loads can cause sideband components in stator currents, of which
frequency corresponds to the load frequency.

27
4 INSTANTANEOUS POWER APPROACH & THE
EXTENDED PARK’S VECTOR APPROACH
4.1 Introduction
In this chapter, two alternative approaches are introduced. The first approach is
Instantaneous Power Approach (IPA). The advantage of IPA is that the harmonics can be
more easily separated from the fundamental component. The sideband components, which
are placed around the fundamental component in the stator currents, will instead appear
around DC and twice-fundamental frequency in the instantaneous power. With MCSA, it is
difficult to filter out the fundamental component without affecting any sideband components
in the stator currents. In contrast, separating the sidebands from DC is much easier by DC
compensation. In addition, the instantaneous power also contains more information and has
stronger tolerance on distortions than the stator currents since the instantaneous power is a
product of multiplying of the voltage and current. [5,9,49,50]
The other technique is the Extended Park’s Vector (EPVA). This technique is to consider the
three phase currents in the terms of the d-axis and q-axis components. By this, two
indicators, which are Lissajou’s curve and current modulus, will be obtained. By monitoring
deviations of an acquired Lissajou’s curve from an expected one, faulty conditions can be
easier detected without any profound knowledge requirement. However, the Lissajou’s curve
cannot clearly identify what a fault type is. The current modulus is required for fault
identification. Similar to IPA, the sideband components are also converted to appear around
DC in the current modulus. [14,24-27,45]
4.2 Instantaneous Power Approach
Healthy Condition
Begin with one phase instantaneous power of an ideal induction motor. The expression can
be derived from ideal supply voltages and currents as shown in Equation 4.1.
( ) ( ) ( )p t v t i t= ⋅ (4. 1)
Thus, instantaneous power can be expressed as;
1( ) [cos(2 ) cos( )]rms rmsp t V I tω ϕ ϕ= − + (4. 2)
Where, ( )v t = phase voltages (L-N or L-L) = 12 cos( )rmsV tω
( )i t = line currents = 12 cos( )rmsI tω ϕ−
ϕ = a load angle
It can be seen from Equation 4.2 that the instantaneous power consists of 2 terms; DC and
the sinusoidal term with the twice-fundamental frequency. The former represents the real
power, and the latter represents the apparent power. Moreover, some additional components
on the instantaneous power caused by interactions of the first three harmonics of supply
voltages and currents are also present at the frequencies 1 1 1 1 1, 3 , 4 , 5 , and 6f f f f f [49].

28
Faulty condition
According to the previous chapter, the stator currents contain some additional components
due to an abnormality in induction motors. For simplicity, it can be assumed that the
additional components generated by faults result from the amplitude. Thus, the stator
currents in the abnormal condition is expressed as shown in Equation 4.3 [9,49,50].
{ }1 1
( ) ( ) [1 cos( )]
( ) cos[( ) ( )] cos[( ) ( )]
2
M f
rms
f f
i t i t M t
I M
i t t t
ω β
ω ω ϕ β ω ω ϕ β
= ⋅ + −
= + + − + + − − −
(4. 3)
Where, M is a modulation index, fω is a modulation angular frequency caused by the
abnormality, and β is a modulation phase. Hence, the modulated instantaneous power can
be expressed as;
1
1
( ) ( ) {cos[(2 ) ( )]
2
cos[(2 ) ( )] 2cos( )cos( )}
rms rms
M f
f f
MV I
p t p t tω ω ϕ β
ω ω ϕ β ϕ ω β
= + + − +
+ − − − + −
(4. 4)
From Equation 4.4, it can be seen that the sideband components are still present in the
instantaneous power, but they are converted to place at fω around DC and 12 fω ω± around
the twice fundamental frequency. The components placed around DC, subsequently called
characteristic components, provide an extra piece of diagnostic information about the health
of the motor.
According to the previous chapter, the fault indicators for IPA can be derived from the
particular components influenced from faults on stator currents as shown in Table 4.1.
Condition mf IPA
Expected components
Rotor asymmetry 12ksf 12ksf , 1 12 2f ksf±
Mix eccentricity rmf rmf , 12 rf mf±
Inter-turn short circuit
[Eq (3.23) when k=1] 1(1 ) r
n
s f nf
p
− = rnf , 12 rf nf±
Table 4.1 Detected frequencies in the instantaneous power for different types of faults
From Table 4.1, it can be seen that the IPA cannot separate the mix eccentricity and the
inter-turn short circuit fault. However, the consequences of the latter, which results in
increasing the third harmonic contents in stator currents, have to be considered. These can
cause the components at 1 1 1 12 ,4 ,8 and 10f f f f to exist on the instantaneous power
spectrums.
However, IPA does not gain any advantage to detect static and dynamic eccentricities. In
addition, since IPA is based on information from the stator currents, noise on the stator
currents also still exist on the instantaneous power. This can cause some difficulty to detect
the particular components when fault severity is small. Nevertheless, IPA is still better than
MCSA for detecting small particular components since the power amplitude is normally much
higher than the current amplitude. By filtering out DC, the small particular components can
show themselves to be significant.

29
4.3 The Extended Park’s Vector Approach
Park transformation is used to transform stator currents from the three-phase system (A-B-C)
to the two-phase system (D-Q). The expression for transformation is as presented by
[5,14,24,25,27,45];
2 1 1
3 6 6
1 1
2 2
d a b c
q b c
i i i i
i i i
= − −
= −
(4. 5)
In addition, the expression for the current modulus is as;
d qCurrent Modulus i ji= + (4. 6)
It should be noted that the transformation is based on the stator reference frame.
Healthy condition
Under a healthy condition, the three phase currents can be expressed as shown in Equation
2.2. Therefore the d-axis and q-axis currents can be determined as;
1
1
6
sin( )
2
6
sin( )
2 2
d
q
i I t
i I t
ω
π
ω
=
= −
(4. 7)
The Lissajou’s curve represents the function between the d-axis and q-axis components
as ( )q di f i= . From Equation 4.7, the Lissajou’s curve for the healthy induction motor has a
perfect circular shape with the center at the origin, and its diameter is equal to ( 6 / 2)I as
can be seen in Figure 4.1(a). Since the diameters of the Lissajou’s curve are proportional to
the current amplitude, the shape becomes thicker when motor loads are changing. In
addition, from Equation 4.6 and 4.7, the current modulus for the healthy condition contains
only DC.
Faulty condition
In a faulty condition, due to the particular components influenced from faults on stator
currents, the shape of Lissajou’s curve becomes distorted. In [14,24], detection of rotor
asymmetry by monitoring the Lissajou’s curve has been presented. The rim of the Lissajou’s
curve becomes thicker when the rotor is asymmetrical. For example, the Lissajou’s curve for
10-broken rotor bars shown in Figure 4.1(b). This is one of advantages, which allows the
detection of faulty conditions by monitoring the deviations of the acquired patterns. In
addition, the analytical expression for the rotor asymmetry has been derived by [25], and the
results have shown that the sideband components in the stator currents influenced from the
rotor asymmetry could be transformed to place at the frequency 1 12 ,4sf sf around DC in the
current modulus.
In [26, 27], it has presented that the Lissajou’s curve in the case of eccentricity is quite
similar to the healthy one. However, it becomes a bit thicker, when a high degree of

30
eccentricities takes place. This can imply that the Lissajou’s curve cannot detect the
eccentricities.
According to the previous chapter, the signature of mix eccentricity is the sidebands
corresponding to the rotational frequency in the stator currents. Similar to rotor asymmetry,
these components are also transformed to be placed at the frequency ,2r rf f around DC in
the current modulus. However, EPVA does not give any advantage to detect static and
dynamic eccentricities.
Figure 4.1 Lissajou’s curve for various conditions (FEM Simulation)
To detect the inter-turn short circuit fault, both the Lissajou’s curve and the current modulus
have to be determined. In the healthy condition, the stator current contains only the positive
sequence, so the circle shape of the Lissajou’s curve is still valid. However, under an
abnormal condition, since phase impedances are unbalance due to the defected windings,
they cause unbalanced supply currents and induces the negative sequence. Due to the
negative sequence, the Lissajou’s curve can show some distortion as an elliptical shape. For
example, Lissajou’s curve for the 6-inter-turn short circuit fault shown in Figure 4.1(d). In
addition, due to the existence of the negative-sequence, it manifests itself in the current
modulus by the component at the twice-fundamental frequency as shown in Figure 4.2
[45,52].
In table 4.3, the summarized table for fault indicators by EPVA is presented. It should be
noted that the Lissajou’s curve is not effective to recognize an abnormal condition, of which
the fault severity is small. Therefore, it is necessary to determine both the Lissajou’s curve
and the current modulus.

31
a) The Lissajou’s curve [45] b) The Park’s vector modulus [45]
Figure 4.2 Relationship between the symmetrical components and Park’s vector for stator asymmetry
Condition The Lissajou’s curve Spectrum of Park’s modulus
Healthy Circle DC
Broken Rotor Bars or End Rings Circle, Thicker DC, 12sf , 14sf
Mix eccentricity
Circle (Thicker for high
degree of eccentricities)
DC, rf , 2 rf
Stator Winding Fault Ellipse DC, rf , 2 rf , 12 f
Table 4.2 Detected Lissajou’s curves and spectrum of park’s modulus for each type of faults
4.4 Conclusion
The Instantaneous Power Approach and The Extended Park’s Vector Approach have been
presented in this chapter. The advantage of IPA is to convert the characteristic components
to appear around DC and twice-fundamental frequency. By easily filtering out DC, the
characteristic components can show themselves more clearly. However, the disadvantage of
this approach is that it is not sufficient for static and dynamic eccentricities. It also requires
additionally one phase voltage. The summary of fault indicators by IPA can be seen in Table
4.1.
There are two indicators for EPVA; Lissajou’s curve and current modulus. Faulty conditions
can be easily detected by monitoring deviation in the Lissajou’s curve. However, in order to
identify the fault type, both indicators have to be determined. The drawback of this approach
is that it is not effective for static and dynamic eccentricities either. It also requires two phase
currents additionally. The summary of fault indicators by EPVA can be seen in Table 4.2.
In addition, both approaches require information on three phase rms currents so that they are
able to separate the inter-turn short circuit fault from mix eccentricity.

32
5 FEM SIMULATION MODELS
5.1 Introduction
In this chapter, it is presented how the fault can be implemented in FEM models with
FLUX2D and what the simulation results are. Models of six fault types categorized in to three
fault groups have been built up. First, rotor asymmetry caused by broken rotor bars or broken
rotor end rings is described. Second, airgap asymmetry resulting from static, dynamic, and
mix of static and dynamic eccentricities is explained. Then, stator asymmetry caused by an
inter-turn short circuit fault is introduced.
The chapter begins with details about FEM models in general. Next, the physical modeling of
an induction motor is introduced. Then, the implementations of the six fault types are
discussed. In all cases, the characteristics of each fault type presented in Chapter 3 are
applied to form and verify FEM models.
5.2 Finite element modeling
A FEM model is formed by three main parts. The first part
is geometry of a studied induction motor. The second is
an electrical circuit, which represents connections,
couplings and electrical parameters. The last part
includes material properties, such as electric and
magnetic characteristics.
Usually, it is sufficient for a healthy induction motor to
consider only 1 pole due to its symmetry regarding the
electric and magnetic phenomena within each pole.
However, it is not valid for a faulty machine due to
asymmetry with regard to the electric and magnetic
phenomena. Hence, it requires studying a fully detailed
physical model of an induction motor. The studied
induction motor is HXR400LD6. Its specifications can be
found in Table 5.1.
5.3 Physical Model of Induction Motors
As mentioned above, three main parts for forming up a physical model are geometry, circuit
and material properties. However, the material properties for the studied motor are not
presented here. The geometry for the studied induction motor is shown in Figure 5.2.
Figure 5.1 Steps to model the
problem

33
(a) (b)
Figure 5.2 Geometry of the studied induction machine in the healthy condition
The geometry represents the real dimension of the
motor and also contains information on its windings;
i.e. pitch factor. The most important thing regarding
the geometry is the airgap.
Due to the Time-Stepping simulation, the so-called,
“Rotating airgap” is required. The rotating airgap
can be either equal to or less than the actual airgap.
In order to get better accuracy in the simulation
results, the real airgap should contain several
layers, but only one layer of the airgap is required to
be the rotating airgap. Figure 5.3 shows two types
of the rotating airgap; one airgap layer and several
airgap layers. Moreover, the rotating airgap is
required to be uniform in length around the origin
point (0, 0). [53]
The circuit shown in Figure 5.4(a) is composed of two main parts. The first part represents
the stator circuit as shown in Figure 5.4(b). The stator circuit is formed by three phase
voltage sources, coils and end winding resistances and inductances. The coils represent the
winding resistance and inductance as well as the coupling of the electric and magnetic
phenomena. In addition, the stator circuit also contains information about the winding
connection. Here is “Star-connection” for the studied induction motor.
The second part describes the rotor circuit as shown in Figure 5.4(c). It is composed of 2
main parts. The first part is all vertical components representing rotor bars. Each rotor bar is
modeled by a resistance, an inductance and a coil. The resistance and inductance describe
the electrical characteristics. The coil is for the electric and magnetic coupling. The other part
is all horizontal components representing rotor end rings. The coil is not required for the
model of rotor end rings because there is no coupling.
Figure 5.3 Two definitions for the
rotating airgap

34
Moreover, the magnetic coupling between the stator and rotor circuits is represented by the
line linking between both circuits. The details of the circuit parameters are described in Table
5.2 and 5.3.
Specification Induction Motor
Model HXR400LD6
Number of poles 6
Number of phase 3
Number of parallel paths 1
Number of stator slot 54
Number of rotor slots 68
Connection Star
Rated voltage [V] 6000
Rated frequency [Hz] 50
Rated current [A] 43.467
Rated power [kW] 350
Number of conductor in a half slot slotN
Pitch factor 8/9
Table 5.1 Specifications of the studied induction motor
Component Definition Parameter
Ua Supply Voltage, A Phase (rms)
Ub Supply Voltage, B Phase (rms)
Uc Supply Voltage, C Phase (rms)
3464.1 V
SS1AP1P1
Coil A (+), representing the winding turns in the A-phase
slot, Current direction (+)
SS1AM1P1
Coil A (-), representing the winding turns in the A-phase
slot, Current direction (-)
SS1BP1P1
Coil B (+), representing the winding turns in the B-phase
SS1BM1P1
Coil B (-), representing the winding turns in the B-phase
SS1CP1P1
Coil C (+), representing the winding turns in the C-phase
SS1CM1P1
Coil C (-), representing the winding turns in the C-phase
N = ,total windingsN turns
R = ,total windingsR Ohm
Stacking Factor =
. .S F
STAR1_1 End Winding Resistance, A Phase
STBR1_1 End Winding Resistance, B Phase
STCR1_1 End Winding Resistance, C Phase
R = ,total endR Ohm
STAL1_1 End Winding Inductance, A Phase
STBL1_1 End Winding Inductance, B Phase
STCL1_1 End Winding Inductance, C Phase
L = ,total endL H
Table 5.2 Descriptions of the components in the stator circuit

35
Component Definition Parameter
RBAREND(X) Rotor bar resistance of the rotor bar No. X R = barR Ohm.
LBAREND(X) Rotor bar inductance of the rotor bar No. X L = barL H.
RBAR(X) Solid Coil, the rotor bar No. X, (1 turns) -
RRING(X)_1 Rotor end ring resistance of the rotor end ring No. X R = ringR Ohm.
LRING(X)_2 Rotor end ring inductance of the rotor end ring No. X L = ringL H.
(X) Index of rotor bar and rotor end ring 1 to rQ
Table 5.3 Description of the components in the rotor circuit
Figure 5.4 Circuit model of the induction motor
(a) Complete circuit, (b) Stator circuit, (c) Rotor circuit
(a)
(b)
(c)

36
5.4 Rotor Asymmetry
5.4.1 Broken Rotor Bar
Figure 5.5 Circuit model of broken rotor bars
According to Chapter 3, broken rotor bars can cause the unsymmetrical rotor current
distribution. This results in the distortion in the magnetic field [10,13,17-19]. Ideally, rotor bar
resistance is low, so the currents resulting from induced voltages by the stator flux density
can flow through all rotor bars. Under the abnormal condition, cracks or breakages can
cause increased rotor bar resistance, so very little or no current can flow in defected bars.
The FEM model for rotor broken bars can be developed by 2 different approaches defined as
Model 1 and Model 2.
The first model is to eliminate the element, which
represent the broken rotor bar in the rotor circuit.
This also requires modifying the material properties
for the broken bar. Figure 5.6 and Table 5.4 show
the modified rotor circuit and material properties for
the defected bar.
The second model is to change only the rotor bar
resistance to a high value without modifying the
rotor circuit and material properties; i.e. 1 6R e= Ω .
This will force a low current flow in the broken rotor
bar.
The rotor current density and the normal component of the airgap flux density with one
broken rotor bar are shown in Figure 5.7 and 5.8 respectively. It can be seen that the
induced current in the broken rotor bar is very low, and this cause the unsymmetrical rotor
current distribution. Consequently, the airgap flux density is distorted from the healthy one. In
addition, to compare the degree of severity, the model with two-broken rotor bars has been
investigated. The normal components of the airgap flux density for the cases with one and
two broken rotor bars are compared in Figure 5.9.
Normal Rotor Bar Broken Rotor Bar
Iso Mu 1 1
Iso Rhm 2.72E-08 1.00E+6
Table 5.4 Material properties for rotor bars
Figure 5.6 Modified circuit for
1 broken rotor bar

37
Figure 5.7 Rotor current density for 1-broken rotor bar
Figure 5.8 Normal components of airgap flux density for the healthy and 1- broken rotor bar condition

38
Figure 5.9 Normal components of airgap flux density for various fault severity
5.4.2 Broken Rotor End Ring
Figure 5.10 Circuit model of broken end ring
Ideally, there is no circulating current flowing in the rotor due to the complete end rings, all of
which are perfectly connected to make a short circuit in the rotor. However, when a part of
rotor end rings is broken, this causes non-zero circulating currents flowing in the rotor as
shown in Figure 5.10. To obtain the circulating currents, it is possible to modify the rotor
circuit by breaking the complete short circuit of the rotor end rings.
The approaches to create the FEM simulation model are similar to those of broken rotor
bars. As shown in Figure 5.11, the first approach is to remove the end ring resistance and
inductance, which represents the defected rotor end ring. However, its material properties do
not require changing. The second one is to increase the end ring resistance; i.e. 20R = Ω

39
The rotor current density and the normal
component of the airgap flux density with
one broken rotor end ring are shown in
Figure 5.12 and 5.13 respectively. It can be
seen that the rotor current density for one
broken end ring is distorted from the healthy
one due to the non-zero circulating currents.
This causes the distortion in the airgap flux
density.
Figure 5.12 Rotor current density for healthy and 1-broken end ring conditions
Figure 5.11 Modified circuit for 1-broken end ring

40
Figure 5.13 Normal components of airgap flux density for healthy and one-broken end ring conditions
5.5 Airgap Asymmetry
5.5.1 Static Eccentricity
The main characteristic of static eccentricity is the presence of a space-fixed minimum radial
airgap. This can cause a steady unbalanced magnetic pull (UMP) in one direction. The
model of static eccentricity can be developed by modifying only the geometry.
The geometry has three different centers; motor,
stator and rotor centers. Ideally, all the centers are
placed at the same position, so the airgap is
symmetrical. By shifting the stator center or the
stator geometry away from the motor and rotor
centers, the space-fixed minimum airgap will be
formed as shown in Figure 5.14.
Where, δ = symmetrical airgap length
ε = eccentric level
Figure 5.14 Geometry for static eccentricity

41
Figure 5.15 Geometry of an induction machine for static eccentricity
The normal component of the airgap flux density for the healthy and 40% static eccentric
conditions are compared in Figure 5.16. For the healthy condition shown in Figure 5.16(a),
the peak amplitudes of the airgap flux density are in the same level, but this is not valid in the
case of static eccentricity either. However, the position of the first highest peak as well as the
second peaks is fixed as can be seen from Figure 5.16(b) and (c) even time is varied. By
this, it is possible to conclude that the minimum radial airgap is fixed in space and time.

42
Figure 5.16 Normal components of
airgap flux density
(a) Healthy condition
(b) Static Ecc. at t1
(c) Static Ecc. at t2

43
5.5.2 Dynamic Eccentricity
The main characteristic of dynamic eccentricity is that a position of minimum airgap length
always revolves around a motor center. The studied dynamic eccentric case is that the
revolving speed of the rotor center around the motor center is equal to the rotational speed.
This will cause the periodically changing airgap. The model for dynamic eccentricity can be
achieved by shifting the rotor center away from the motor and stator centers as shown in
Figure 5.17.
Since the rotor center revolves around the motor
center with the rotational speed, the minimum
rotating airgap period can be calculated as;
1
2
(1 )r
p p
T
s f
π
ω
⋅
= =
−
(5. 1)
Where; p = number of pole pair
rω = electrical rotor angular speed
s = slip
1f = frequency of supply voltage
In Figure 5.18, the airgap at the particular position for varied time is presented. The
simulation of 40% dynamic eccentricity is performed at 1% slip, so the period time is about
0.06 seconds. It can be observed that at the specific position, the minimum airgap takes
place every 0.06 seconds. Moreover, at every half of the period time, the maximum airgap
will happen instead.
(a) t = 0.06 sec. (b) t = 0.08 sec.
(c) t = 0.10 sec. (d) t = 0.12 sec.
Figure 5.18 Airgap at varied time for dynamic eccentricity
Figure 5.17 Geometry for dynamic eccentricity

44
The normal component of the airgap flux density for varied time is compared in Figure 5.20.
It can be seen in Figure 5.20(a) and (b) that the airgap flux density at t = 0.06 and 0.12
seconds look similar. The positions of the first and second highest peaks are placed around
the same position. The airgap flux density at t = 0.09 and 0.15 seconds are also similar.
However, the highest peak changes be the lowest at every half of the period time as can be
seen by comparing Figure 5.20(a) and (c) or 5.20(b) and (d). Thus, it is possible to conclude
that the minimum radial airgap revolves around the motor center and the airgap length
changes periodically.
5.5.3 Mix of Static and Dynamic Eccentricities
The main characteristic of mix eccentricity is that the rotor turn around neither its bore center
nor the stator bore center, but it revolves around a point between the stator and rotor bore
centers. This results in an unsymmetrical airgap which changes periodically. The difference
between dynamic eccentricity and mix eccentricity is that the airgap length for the dynamic
eccentricity is changing from the maximum to minimum at every half of the period time, but
this is not valid for the mix eccentricity. This can be seen clearly in Appendix I.
Thus, the model for mix eccentricity can be obtained
by shifting both stator and rotor centers in order to
have the motor center placing between both as
shown in Figure 5.19.
Where, δ = symmetrical airgap length
1ε = static eccentric level
2ε = static eccentric level
In Table 5.5, the calculated airgap length for varied time for the 40% dynamic eccentricity
and the mix of 15% static and 25% dynamic eccentricities are presented. The Matlab code
for the calculation can be found in Appendix II.
Airgap length for dynamic ecc. [mm] Airgap length for mix of ecc. [mm]
Time [s] Zeta =0 Zeta = 180 Zeta =0 Zeta = 180
t = 0 1.0800 2.5200 1.0800 2.5200
t = 0.06 1.0814 2.5186 1.0809 2.5191
t = 0.12 1.0857 2.5143 1.0836 2.5165
t= 0.09 2.5168 1.0832 1.9780 1.6220
t= 0.15 2.5112 1.0889 1.9745 1.6256
Table 5.5 Calculated airgap length for dynamic and mix eccentricities
The normal component of the airgap flux density for the mix eccentricity at varied time is
shown in Figure 5.21. It can be seen from Figure 5.21(a) and (b) that the positions of the
highest peaks in both cases are placed around 0o
Zeta ≅ . In Figure 5.21 (c) and (d), the
positions of the highest peaks in both cases are located around 180o
Zeta ≅ . In addition, the
Figure 5.19 Geometry for mix eccentricity

45
different amplitude between the peaks placed around 0o
Zeta ≅ and 180o
Zeta ≅ in Figure
5.21(a) and (b) is bigger than that in Figure 5.21 (c) and (d). This agrees with the calculated
airgap length in Table 5.4. Therefore, the FEM model of the mix eccentricity is verified.

46
Figure 5.20 Normal components of airgap flux density for dynamic eccentricity
(a) t = 0.06 s., (b) t = 0.12 s., (c) t = 0.09 s., (d) t = 0.15 s.

47
Figure 5.21 Normal components of airgap flux density for mix eccentricity
(a) t = 0.06 s., (b) t = 0.12 s., (c) t = 0.09 s., (d) t = 0.15 s.

48
5.6 Stator Asymmetry
5.6.1 Inter-turn Short Circuit on Stator Windings
The inter-turn short circuit fault can result in some distortion in the stator MMF. This is due to
less numbers of the stator windings and the opposite MMF which is against the main MMF
as shown in Figure 3.7. To implement this fault in the FEM model, both the geometry and
circuit have to be modified.
Consider the stator circuit. According to [53], a coil conductor represents thin wires, in which
the induced currents are zero or negligible. The equation for the coil conductor used in the
FEM calculation is expressed as shown in Equation 5.2.
( ) ( )
d
V t R i t
dt
φ
= ⋅ + (5. 2)
Where; R = Resistance of the coil = strandn R⋅ = ( )strandn l Sρ⋅
φ = the flux embraced by the assembly of the coil strands
strandS = Cross section of a strand = strandF S n⋅
F = Stacking Factor
In the stator circuit, there are two coil conductors per phase. Each coil conductor represents
each phase winding in which the direction of the flowing current is taken into account. With
the inter-turn short circuit fault, the stator windings require separating into 2 parts as shown
in Figure 3.7.
Consider one coil conductor,
1
1 1 1
( ( ))
( ) ( ) ( ) B w
B B B w
d n L i td
V t R i t n R i t
dt dt
φ
= ⋅ + = ⋅ + (5. 3)
Consider 2 series coil conductors,
32
2 3 2 3
2 3
2 3
( ) ( ) ( ( ) ) ( ( ) )
[( ) ( )]
( ) ( )
BB
B B B B
B B w
B B w
dd
V t V t R i t R i t
dt dt
d n n L i t
n n R i t
dt
φφ
+ = ⋅ + + ⋅ +
+
= + +
(5. 4)
If 1 2 3( ) ( ) ( )B B BV t V t V t= + , therefore;
1 2 3B B Bn n n= + (5. 5)
Equation 5.5 shows that a coil conductor can be separated to several coil conductors.
The modified stator circuit is shown in Figure 5.22. One set composed of two coil conductors
representing the whole phase stator windings can be divided into three sets. The first and
second sets; i.e. Set1_A and Set2_A, represent the windings in a half of one slot, in which

49
the flowing current direction is taken into consideration. Set2_A stands for the winding turns,
in which the shorted circuit will be implemented. Set1_A describes the rest of windings in the
half slot excluding the defected windings. For the third set, Set3 represents the remaining
windings in the rest of the slots.
Figure 5.22 Modified stator circuit for the healthy condition
Since the windings in the stator slot are separated to
two parts, the geometry has to be divided as well. In
this study, a three-inter-turn short circuit fault is
modeled. According to the motor specification shown
in Table 5.1, the number of conductors in a half of
one slot is 15 turns. In order to keep the stacking
factor constant for all winding sets, the proper
dimensions in the geometry for the short circuit
windings are required.
Figure 5.23 shows the cross section of the winding
turns in a half of one slot. The defected turns are at 7,
8 and 9. Figure 5.24 shows the modified geometry
which has been created with the proper dimensions.
Moreover, the circuit parameters have to be also
adjusted as shown in Table 5.6.
To implement the three-inter-turn short circuit case, the circuit in Figure 5.22 has to be
modified to make a short circuit loop as shown in Figure 5.25.
Figure 5.23 The cross section of the
winding coil

50
Figure 5.24 Modified geometry for the inter-turn short circuit
Figure 5.25 Modified stator circuit for 3-inter-turn short stator circuit
In Table 5.7, the simulation results on the instantaneous currents for the modified circuit in
Figure 5.22 and the original circuit in Figure 5.4 are presented. In addition, the normal
component of the airgap flux density of both circuits is compared in Figure 5.26. The results
show clearly that both circuits are compatible.

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