(Energy Science, Engineering and Technology) Mykhaylo v. Zagirnyak, Zhanna IV. Romashykhina, Andrii P. Kalinov - The Diagnostics of Induction Motor Broken Rotor Bars On The Basis of The Electromotive

ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY
THE DIAGNOSTICS OF
INDUCTION MOTOR BROKEN
ROTOR BARS ON THE BASIS OF
THE ELECTROMOTIVE
FORCE ANALYSIS
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ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY
THE DIAGNOSTICS OF
INDUCTION MOTOR BROKEN
ROTOR BARS ON THE BASIS OF
THE ELECTROMOTIVE
FORCE ANALYSIS
MYKHAYLO V. ZAGIRNYAK
ZHANNA IV. ROMASHYKHINA
AND
ANDRII P. KALINOV
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Library of Congress Cataloging-in-Publication Data

Names: Zagirnyak, Mykhaylo V., author.
Title: The diagnostics of induction motor broken rotor bars on the basis of the electromotive force
analysis / Mykhaylo V. Zagirnyak, Zhanna Iv. Romashykhina, and Andrii P. Kalinov (Kremenchuk
Mykhailo Ostrohradskyi National University 20, Pershotravneva ul, Kremenchuk, Ukraine).
Description: Hauppauge, New York : Nova Science Publishers, Inc., [2017] |
Series: Energy science, engineering and technology | Includes bibliographical references and index.
Identifiers: LCCN 2017045782 (print) | LCCN 2017055629 (ebook) | ISBN 9781536126846 H%RRN| ISBN
9781536126839 (softcover)
Subjects: LCSH: Electric motors, Induction--Testing. | Electric machinery--Monitoring. | Fault location
(Engineering)
Classification: LCC TK2785 (ebook) | LCC TK2785 .Z34 2017 (print) | DDC 621.46--dc23
LC record available at https://lccn.loc.gov/2017045782
Published by Nova Science Publishers, Inc. † New York
CONTENTS
Preface vii
List of Abbreviations xi
List of Symbols xiii
Chapter 1 The Contemporary State of the Problem of the
Diagnostics of Induction Motor Broken Rotor
Bars 1
Chapter 2 Theoretical Foundation for the Research of
Induction Motor Broken Rotor Bars in the Self-
Running-Out Condition 25
Chapter 3 Mathematical Models for the Research of the
Method of Induction Motor Broken Rotor Bars
Diagnostics 69
Chapter 4 The Method of Induction Motor Broken Rotor
Bars Diagnostics with the Use of Wavelet-
Transform 105
vi Contents
Chapter 5 The Experimental Verification of the Method

for the Diagnostics of Induction Motor Broken
Rotor Bars 133
Conclusion 149
References 153
Appendices 163
Authors’ Contact Information 179
Index 181
PREFACE
Electric machines are known to be the basic elements of electric

drives of various operating mechanisms used in up-to-date industry and
economy. Induction motors with a squirrel-cage rotor, having
considerable advantages in comparison with other electric machines,
are commonly used in most operating mechanism electric drives
nowadays. Technological processes conditions often imply induction
motor operation at non-rated and asymmetric supply voltage, high
temperature, humidity, switching overvoltage, technological overloads,
etc. The mentioned factors predetermine premature aging of induction
motor units and reduce their service life.
According to statistics data, about 7–10% failures of industrial
induction motors occur due to broken rotor bars. Broken rotor bars are
found more often in an induction motor with a copper and brass rotor
winding than in an induction motor with an aluminum rotor winding.
Besides, broken rotor bars occur in rotors with a welded aluminum
winding more often than in cast rotors.
Broken bars of the rotor squirrel-cage winding cause the distortion
in the magnetic field in the induction motor air gap, increased vibration
of the motor, occurrence of the stator current pulsations in all phases,
increase of losses in the stator and rotor windings, reduction of rotation
viii M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov
frequency under load, decrease of the efficiency. That is why timely

detection of the degree of breakage will allow one to avoid its
development, reduce the restoration time, decrease
the maintenance expenditure, reduce the equipment idle time, increase
the working efficiency of the motors and operating mechanisms.
Such researchers as: Douglas H., Marques С., Thomson W. T.,
Vaskovskii Yu. М., Rogozin G. G., Syvokobylenko V. F., Yatsun М.
А. et al. devoted a great number of papers to the development and
implementation of the methods and systems of the diagnostics of
broken rotor bars.
There exist various methods for diagnostics of induction motor
broken rotor bars. However, an analysis of the conventional diagnostics
methods revealed that most of them require the withdrawal of the
induction motor from the operating process and its disassembling.
There are methods of induction motor broken rotor bars diagnostics in
the operation mode, e.g., the methods of currents spectral analysis, the
analysis of zero-sequence voltage, the analysis of applied magnetic
field parameters. However, these methods do not provide satisfactory
results during the diagnostics in the idle mode and do not take into
consideration the supply mains low quality voltage and the load level
variation influence on the diagnostics results. Moreover, the use of
Fourier transform of current signals does not enable unambiguous
determination of the number and relative position of induction motor
broken rotor bars. Taking the above said into account, development of
the method for the diagnostics of induction motor broken rotor bars is
topical.
The presented monograph contains theoretical and experimental
research that made it possible to solve a topical scientific problem of
improvement of the efficiency of induction motor broken rotor bars
diagnostics by the wavelet-analysis of electromotive force in stator
windings in the mode of motor self-running-out.
The first chapter of the monograph contains an analysis of the
existing methods for the induction motor broken rotor bars diagnostics,
Preface ix
the basic causes of breakages are stated, the advantages and drawbacks
of the known methods of diagnostics are determined.
The second chapter contains research concerning the choice of the
testing condition and diagnostic signal for the broken rotor bars
diagnostics, substantiation of the methods for electromagnetic field
calculation, a comparative analysis of methods for diagnostic signals
processing, and also the investigation of factors influencing the
generation of electromotive force in stator windings. Mathematical
models for the research of different operating modes of an induction
motor with broken rotor bars are presented in the third chapter of the
monograph.
The fourth chapter deals with a method for the induction motor
broken rotor bars diagnostics on the basis of the analysis of
electromotive force in the stator windings with the use of wavelet-
transform. Basing on the developed methods, a method of
decomposition of the phase electromotive force signal into the signals
of the electromotive forces of the active sides of the coil with the use of
the theory of inverse z-transform is proposed.
The fifth chapter of the monograph is devoted to experimental
research of the developed method of the diagnostics of induction motor
broken rotor bars.
The monograph has been written in Kremenchuk Mykhailo
Ostrohradskyi National University. The authors are grateful to
Professors V. I. Milykh, V. Yu. Kucheruk, Associate Professor
О. V. Kachura for reviewing and support of the monograph, Associate
Professors D. G. Mamchur and V. O. Melnykov for the assistance in the
performance of experimental research, and also to a worker of
Kremenchuk Mykhailo Ostrohradskyi National University
K. V. Kovalenko for the assistance in the preparation of the monograph
in the English language.
LIST OF ABBREVIATIONS
IM Induction Motor
EW Exciting Winding
ADC Analog-Digital Converter
DCG Direct Current Generator
BVS Block Of Voltage Sensors
WB Wavelet-Basis
VMP Vector Magnetic Potential
WT Wavelet-Transform
ShCR Short-Circuited Ring
SCR Squirrel-Cage Rotor
FDM Finite Difference Method
FEM Finite Element Method
CWT Continuous Wavelet Transform
CFMM Circuit-Field Mathematical Model
EMF Electromotive Force
EM Electric Machine
EMFl Electromagnetic Field
ED Electric Drive
LIST OF SYMBOLS
A Vector Magnetic Potential

A1 , A2 , A3 Functions Assigning Rotor Breakage
And Stepped Appearance with the Adopted
Value of Teeth on the Rotor
Az The Total Arithmetic Value of VMP in All
the Slots of the Phase
a The Number of Winding Parallel Paths
amax The Maximum Value of the Wavelet Scale
B The Amplitude of Magnetic Induction
of the Fundamental Harmonic of the Field
in the Air Gap
B Relative Magnetic Induction
Bx , B y Magnetic Induction Vector Components
in Cartesian Coordinates
B Magnetic Induction Vector
bbev The Slots Skew in Linear Dimensions
E1 E34 The EMF of Power Supply for the
Investigated IM
E A , EB , EC The EMF of the Stator Of IM
xiv List of Symbols
E AB The EMF Inter-Phase of the Stator Of IM

Ec The EMF of Coil
Eq The Total EMF of the Coil Group
Em EMF Signal Initial Amplitude
Et The EMF of the Winding Turn
Eres The EMF of the Rotor Steel Residual
Magnetization
E Vector of Electric Field
. The EMF of the Winding Phase
E ph
E  z The Z-Image of EMF Signal
Ec1  z  , Ecm  z  , m = 3 The Z-Image of the Coils EMF Signals
E ph  z  The Z-Image of the Winding Phase EMF

Signal
Eq1  z  , Eq 2  z  The Z-Image of the Coil Groups EMF
Signals
Et1  z  , Et 2  z  The Z-Image of EMF Signals of the Two
Active Sides of the Coil
e(t ) Winding EMF
etest1, etest2, etest3 Coils Testing EMFs
etest The Test Signal of the Total EMF of the
Stator Winding Coil Group
f1 The Frequency of the Supply Network
fm The Upper Boundary of Frequencies Band
of Concentrated Fundamental Energy of the
Signal (in Low-Frequency Domain)
fs Discretization Frequency
G The Boundary of the Calculation Area
H the Vector of Magnetic Field
I 2beg Rotor Current at the Moment of IM
Disconnection from the Supply Mains
List of Symbols xv
I adj1 , I adj 2 Currents in Adjacent Bars

I br Current in a Broken Bar
Ib Rotor Bar Current
iA , iB , iC Stator Currents
ia , ib , ic Rotor Currents
J z _ out The Density of Outside Currents, Assigned
in Stator Winding Sections
j Currencies Density
jout Currents Density Caused by Outside EMF

Ki.d . Coefficient of Insulation Durability
K a The Function of the Average Value of the

Wavelet-Expansion Coefficients Sum for
Medium Frequency Area
K * a Created in Relative Units Reduced to the
Maximum Value of the Scale amax for IM
With Broken Rotor Bars
ka The Values of Wavelet-Expansion
Coefficients
L2 Rotor Inductance
LA Phase Inductance
La The Optimum Upper Level of Wavelet-
Decomposition
l The Number of Wavelet-Expansion
Coefficients
l1 The Active Length of the Stator
M The Maximum Value of Mutual Inductance
Me Electromagnetic Torque Equation
M xy Mutual Inductance Between Windings Х
and Y
xvi List of Symbols
 The Multiplication Factor of IM Starting

Ì st
Torque in Relation to the Rated One
m0 Trigonometric Polynomial
Pe1.br. Losses in the Stator Windings of IM with

Broken Bars
Pe1.heal . The Rated Value of Losses of a Healthy IM
Pe 2.br. Losses in the Rotor Of IM with Broken

Bars
Pe2.heal. The Rated Value of Losses of a Healthy IM
Pe1 The Relation of the Losses in the Stator

Windings of IM With Broken Bars to the
Rated Value of the Losses of a Healthy IM
Operating Under Nominal Condition
Pe2 The Relation of the Losses in the Rotor of
IM With Broken Bars to the Rated Value of
the Losses of a Healthy IM Operating
Under Nominal Condition
Prat The Relative Increase of the Value of
Losses in IM Windings
p The Number of IM Poles Pairs
q The Number of Slots Per a Pole And a
Phase
R2 Rotor Resistance
RA , RB , RC , Rs Stator Phases Resistances
Ra , Rb , Rc , Rr Rotor Phases Resistances
Rb.r. Rotor Bar Resistance
Rscr Short-Circuited Ring Resistance
S ph The Total Area of Cross Section of all
Phase Coils Connected in Series
Ss The Sectional Area of the Stator Slot
s IM Slipping
List of Symbols xvii
T0 The Conditional Durability of Insulation

At i.h.  0
t Time
t Discretization Period
tst .br . The Value of Start-Up Time of IM with
Broken Bars
tst .heal . The Value of Start-Up Time of a Healthy
IM
t st The Relation of the Value Of Start-Up
Time of IM With Broken Bars to the Value
of Start-Up Time of a Healthy IM
UA, UB, UC Stator Phases Voltage
Um The Amplitude Value of Stator Phases
Voltage
w The Number of Turns in the Slot
Connected in Series
wc The Number of the Coil Turns
Xb Bar Inductive Reactance
y Winding Pitch in Slots
Z  1,2... Rotor Bars Ordinal Numbers
z    j Arbitrary Complex Variable
Z 1 The Operator of Inverse Z- Transform
Z1 , Z 2 The Number of Teeth of IM Stator and
Rotor
Zb1 Zb34 The Impedance of Rotor Bars
Zb Rotor Bar Impedance in a Complex Form
Z scr1 Z scr 34 The Impedance of Short-Circuited Ring
α The Angle of Phase Zone
bev The Slots Skew in Relative Dimensions
 The Value of the Rotor Rotation Angle in
Relation to the Stator
xviii List of Symbols
 bev A Skew Angle

br .b. The Value of Angles Between Two Broken
Bars
1 The Temperature of Heating of the Stator
Windings of IM with Broken Bars
2 The Temperature of Heating of the Rotor
Bars of IM With Broken Bars
i.h. The Temperature of Insulation Heating
max Maximum Operating Insulation
Temperature Under Nominal Condition
max.all. The Allowable Excess of Temperature
rat The Relative Increase of the Value of

Temperature
 The Angle of Shift of Stator Winding Coils
 The Number of Slots Between the Coils of
Adjacent Phases
i ,  j The Relative Permeances of the i -Th and
j -Th Harmonics
 1 The Relative Permeances of IM Stator
2 The Relative Permeances of IM Rotor
a Permeability
 Motion Speed in Cartesian Coordinates for
Motionless Media
 The Density of Electric Charges
 Pole Pitch
a Rotor Time Constant
i.d . Insulation Durability at Temperature i.h.
s The Time Constant of Decrease of the
Rotor Rotation Frequency
Φ Magnetic Flux
 Scalar Magnetic Potential
List of Symbols xix
 ph Spatial Angle Between The Corresponding

Phase of the Stator and the N-Th Bar of the
Rotor
t  Phi Scaling Function
 Coordinate Origin
1 ,  2 ,  Z Flux Linkages for Rotor Bars in the Self-
Running-Out Mode
 A ,  B , C Stator Phases Flux Linkages
 a , b ,  c Rotor Phases Flux Linkages
 ph The Full Flux Linkage of Stator Winding
Phase
 t  Time Psi-Function, Wavelet-Function
2 The Mechanical Angular Speed Of Rotor
Rotation
 The Current Electric Angular Speed Of IM
0 The Electric Angular Speed of the Field of
Stator
2 Electric Angular Speed
r Rotor Rotation Frequency in the No-Load
Condition
Chapter 1
THE CONTEMPORARY STATE OF

THE PROBLEM OF THE DIAGNOSTICS OF
INDUCTION MOTOR BROKEN ROTOR BARS
1.1. THE ANALYSIS OF SQUIRREL-CAGE INDUCTION

MOTOR FAILURES
Squirrel-cage induction motors represent a most common type of

electric drive in up-to-date technological machines. They are the basic
consumers of all the electric power produced in the world. This wide
application can be explained by a number of advantages, in particular,
by the simplicity of maintenance and reliable operation. However, the
complexity of technological processes is often the cause of IM
operation under hard conditions: at non-rated and asymmetric supply
voltage, high temperature, humidity, commutation overvoltage,
technological overloads, etc. The listed factors and a number of others
predetermine the premature aging of IM units and, consequently, more
frequent failures with further damage for production.
The above mentioned causes of equipment failure can be classified
in the following way: technological ones – about 35%; operation ones
2 M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov
(mainly unsatisfactory protection of electric machines (EM) – 50% and

design ones – 15%.
In most countries of the world the indices of EM failures are as
follows [1, 2]: out of the total number of broken EMs about 80% are
repaired and 20% are substituted by reserve ones. About 40% of the
failures of IM of the power of 5 kW and higher occur due to broken
windings [3–5]. According to various statistic sources [6, 7], there often
appear damages of the stator winding slot parts, rotor bar cage and IM
mechanical part. Rotor broken bars occur in 7% of the cases. The
statistic data of IM failures according to the damage types are shown in
Figure1.1 and in Table 1.1.
Under the conditions of Ukrainian industrial enterprises the park of
electric machines and electromechanical equipment is physically going
out of date year after year. The overwhelming majority of enterprises
try to create their own repair shops, which often results in the reduction
of EM reliability due to the insufficient qualification of the repair
personnel, the absence of the necessary testing and diagnostic
equipment, the violation of restoration and repair technology, etc. This
situation results in the deterioration of the general reliability of all EM
units and elements – magnetic system, stator and rotor windings,
bearings. The breakage of each of these parts causes EM failure.
Figure 1.1. The statistics of induction motor failures (%).

The Contemporary State of the Problem of the Diagnostics … 3
Table 1.1. The statistics of induction motor failures
No. Number Damage No. Number Damage

of failures,% location of failures,% location
1. 2 stator core 7. 17 stator winding
slot parts
2. 5 rotor slot 8. 11 stator end
wedges windings
3. 9 rotor cage 9. 3 winding joint
bars
4. 16 bearings 10. 3 connections
5. 3 fan 11. 8 rotor winding
6. 15 rotor 12. 3 contact rings
interference 13. 6 other damages
with the stator
EM diagnostics is performed to timely reveal damages. The

procedure of carrying out the diagnostics can be started in different
directions: during the motor periodical testing, with the use of
monitoring stationary system, during the additional research performed
according to the results of other methods, e.g., vibrodiagnostics, etc.
Table 1.2. The data of IM failures during 2016 in some production

shops at PJSC «AutoKrAZ», Ukraine
No. PJSC «AutoKrAZ» Total number Number of broken IMs

production shop of IM, pcs. pcs. %
1. Machine assembly shop 985 70 7.1
2. Tool production 425 63 14.8
administration
3. Experimental shop 97 5 5.2
4. Electrotechnical shop 35 3 8.6
5. Total 1542 141 9.1
To confirm the credibility of the statistic data given in Table 1.1 the
statistic data of squirrel-cage IM failures at one of the enterprises of
Poltava region, Ukraine (PJSC «AutoKrAZ») during 2016 were
analyzed. The data are given for four production shops that use IMs in
their technological processes. The data of IM failures are given in Table
1.2, and the distribution according to damage types – in Table 1.3.
The analysis of the given data of IM failures and the distribution
according to the damage types revealed that the number of motors of
some production shops at PJSC “AvtoKrAZ” , that were repaired due to
broken rotor bars, makes on average 8.5%.
Table 1.3. The distribution according to the types of damages
No. PJSC Number of broken IMs

“AvtoKrAZ” With broken With broken With broken
production shop mechanical part stator windings rotor bars
pcs. % pcs. % pcs. %
1. Machine 6 8.6 59 84 5 7.1
assembly shop
2. Tool production 10 15.9 48 76.2 6 9.5
administration
3. Experimental 1 20 3 60 1 20
shop
4. Electrotechnical 1 33.5 2 66.5 – –
shop
5. Total 18 12.8 112 79.4 12 8.5
Paper [8] contains the statistic data according to the results of

troubleshooting performed for 216 series A motors: of the 11-th size
(power up to 100 kW), of the 12-th and 13-th sizes (power of 200–1000
kW). The data are given taking into account the method of SCR
winding performance: a welded copper one, a cast aluminum one and a
welded aluminum one. According to these data, for IMs with a welded
aluminum winding on the rotor the index of bar damaging is 31.9%, for
motors with cast aluminum winding – 8.9%.
The following types of damages are distinguished according to the
character of rotor bar breakage:
 bars break in the slot part;

 bars break at the points of attachment to the short-circuited ring
on one side;
 bars break at the points of attachment to the short-circuited ring
on two sides;
 bars burning-out in the slot.
The distribution of rotor breakages depending on their location and

winding design for each motor size in series A is shown in Figures 1.2–
1.4.
The analysis of the given data revealed that damages most often
occur in IMs with a rotor welded copper winding due to the breaks of
bars at the points of attachment to the short-circuited rings. The cases of
rotor bar breaks in slots are the least common.
copper winding aluminum winding
bars burning-out in the slot 8.9

8,9
bars break beside the short-

circuited ring on two sides 13,5
13.5
circuited ring on one side 51,4
51.4
bar break off the short-

circuited ring
bar break in a slot
0 10 20 30 40 50 60
%
Figure 1.2. The distribution of rotor breakages for series A motors, size 11.
bars burning-out in the slot 2.3

2,3

circuited ring on two sides 16.7
16,7
circuited ring on one side 36.7
36,7
bar break off the short- 31,9
31.9
circuited ring
bar break in a slot 2.3
2,3
0 5 10 15 20 25 30 35 40
%
bars burning-out in the slot

circuited ring on two sides
circuited ring on one side 21.4
21,4
bar break off the short- 20

20
circuited ring
bar break in a slot
0 5 10 15 20 25
%
The analysis of the statistic data confirms the necessity for the
diagnostics of the mentioned IM damages.
1.2. THE RESEARCH OF THE CAUSES OF

THE OCCURRENCE OF INDUCTION MOTOR
BROKEN ROTOR BARS
The paper contains an analysis of the basic causes for the

occurrence of induction motor broken rotor bars.
1.2.1. The Technology of the Manufacture of an IM Rotor

Squirrel Cage
An important factor that predetermines defects consists in the

technology of the manufacture of the electromotor itself. Squirrel cage
casting with aluminum is a complicated technological process during
which problems with the quality of cast rotors may occur [8]. Later, this
factor causes the appearance of shrink holes, cracks on rings, rotor bars
breaks, etc. Determinant conditions for obtaining high-quality squirrel-
cage rotors include the method of cast, the metal melting mode, the
temperature of cores heating. Due to core dense pressing the effective
shrink of aluminum decreases and internal stresses occur in the bar,
which provoke appearance of cracks and bar breaks.
1.2.2. Thermal Overloads
The overheating of the rotor windings (bars and short-circuited

rings) results in melting of a bar or even of a rotor cage. The source of
the overheating may be concentrated in rotor bars (especially during
repeated starts, at motor acceleration and braking) or in the rotor core
with further distribution of the overheat across the bar. These
phenomena result in the delamination of the rotor winding elements. At
supply voltage asymmetry big currents caused by negative phase-
sequence voltage appear in the rotor. Because of the surface effect
inherent in the higher harmonics of these currents they are unevenly
distributed along the sections of the rotor bars. Also, during operation at
low speeds with big sliding due to the surface effect the thermal
gradient along the rotor bar increases, which results in its destruction
[9].
1.2.3. Mechanical Overloads
Analogously to the stator windings, the magnetic forces in the air

gap cause the vibrations of the rotor bars. However, centrifugal forces,
occurring due to rotation, make the bars remain in the slots. At low
rotation speeds the centrifugal forces are insignificant, because of
which the rotor bars vibrate, which results in mechanical wear [10].
Thus, in transient modes (at start, reverse, repeated switch-on) the
rotor cage bars are subjected to the impact of various forces: thermal,
mechanical and electrodynamic ones, which are of a different character
and direction. That is why it is necessary to take into consideration the
action of the torque, thermal deformation, forces caused by torques
from magnetomotive forces higher harmonics, rotor imbalance etc. The
influence of all the above mentioned factors results in the deformation
of the bar, its vibration in the slot and the creation of a different degree
of bar fit against the rotor core teeth. Let us consider the basic physical
processes occurring with broken bars. In case when the rotor bar is
broken or partially broken, current flowing through this bar is
redistributed to the adjacent bars. There the current exceeds the rated
one, correspondingly, heat losses grow. Besides, the distribution of
magnetic flux around the broken bar changes – the flux increases on
one end of the bar and decreases on the other one.
This phenomenon results in the growth of steel losses around the
broken bar. Moreover, the temperature of bars adjacent to the broken
one increases, which may result in their breakage. However, in practice
the situation is less dangerous.
Currents in a broken bar seldom equal to zero – very often a bar is
just partially broken or internal currents flow in it. These currents occur
when the rotor cage is not insulated from the rotor core.
In this case a usual value of current I br comes into the bar, but
along the length of the bar the current flows into adjacent bars ( I adj1 ,
I adj 2 ) through the rotor magnetic circuit (Figure 1.5).
I a d j1
I br
I adj 2
broken bar
Figure 1.5. Currents distribution with a broken rotor bar.
In Figure 1.5: I br – current in a broken bar, I adj1 , I adj 2 – currents in

adjacent bars. The flux of current flowing through the rotor steel in this
case causes the rotor core overheat [11]. Thus, the performed analysis
revealed that most IM failures occur due to damages of the stator,
mechanical part and squirrel-cage rotor bars. Broken rotor bars most
often occur in motors with copper and brass bars laid in slots.
The diagnostics of squirrel-cage rotor broken bars is also
complicated by the fact that direct measurement of the rotor parameters
is impossible. The problem of the development of the new approaches
to the diagnostics of rotor broken bars is topical. That is why timely
determination of the place and degree of damages will allow one to
eliminate their further development, decrease the restoration time,
reduce the maintenance expenditure, avoid the equipment idle time,
improve the efficiency of operation of motors and production
mechanisms.
1.3. THE ANALYSIS OF THE METHODS FOR

THE DIAGNOSTICS OF INDUCTION MOTOR BROKEN
ROTOR BARS AND THEIR CLASSIFICATION
At present a lot of methods are used for the diagnostics of IM

broken rotor bars so, it is necessary to classify these methods.
A number of scientific papers suggest certain classifications of the
methods for of diagnostics of electric machines (EM) technical state. In
particular, paper [7] contains a classification of the methods and the
means of EM technical diagnostics according to different classification
criteria:
 according to the purpose;

 according to the operating condition;
 according to the degree of automation;
 according to the character of the use;
 according to the method of influence on the diagnostics object;
 according to the diagnostics parameters etc.
The given classification allows the description of general features of

the most common methods for the diagnostics of EM technical
condition. However, the analysis of the methods for the diagnostics of
IM rotor broken bars needs specification.
The review and analysis of methods for the diagnostics of rotor
broken bars is also presented in paper [6], the basic advantages and
drawbacks of the considered methods for the diagnostics are
determined, but the authors do not perform their classification.
The application of particular diagnostics methods is determined by
a number of factors: requirements to the diagnostics results, conditions
under which motor diagnostics is performed, the type of technological
process in which IMs take part, their operation mode, software level,
the level of diagnostic equipment automation. Each of the methods
makes it possible to reveal the broken rotor bars at different stages of

the breakage development and can be applied to a certain field of use.
1.3.1. The Criteria of the Assessment of the Efficiency of

Broken Rotor Bars Diagnostics
To analyze the methods for the diagnostics of broken rotor bars a

number of criteria for the assessment of the diagnostics efficiency were
proposed.
Criterion 1
Information value. The criterion determines a possibility to separate
the types of the defects, to locate them.
Criterion 2
The degree of automation of the diagnostics process. This criterion
determines the level of soft- and hardware, the structure and
composition of measuring-diagnostic equipment during experimental
research.
Criterion 3
The expenditure of time for carrying out the diagnostics preparation
operations. According to this criterion, inefficient methods include
those that require the withdrawal of IM from the technological process,
its disassembling and the installation of the required measuring sensors
in the motor gap, etc.
Criterion 4
The expenditure of time for processing of information obtained as a
result of the research and making a decision concerning the existing
damage. The use of a mathematical apparatus with ready software
considerably reduces the time necessary for processing the information

and simplifies the diagnostics process.
When intellectual or expert systems, e.g., neural networks, are used,
the software adaptation is necessary for different diagnostics methods.
Consequently, it determines a necessity for additionally trained
personnel for realization of this method. Accordingly, the following
criterion can be formulated.
Criterion 5
A necessity for specialized personnel for the analysis of the data. In
this case the cost of service personnel labor should be taken into
account.
Thus, correspondence of the methods for IM broken rotor bar
diagnostics to the proposed criteria enables determination of their
feasibility study for the use under different conditions.
It is expedient to mention a number of widespread methods for
induction motor broken bar diagnostics.
Method 1
A method of continuous monitoring [12] of the state of the stator
and rotor windings of IM with SCR according to the data of
measurement of the phase currents and voltages. To assess the IM
technical condition the symmetric components of the stator currents and
voltages, as well as the consumed active power and the angle of slope
of the electromotor mechanical characteristic in the domain of operating
slip are used. This method makes it possible to avoid diagnostics errors
in the presence of pulsations and harmonic components in supply
voltage.
Drawback: the results of the measurement are assessed according to
the complex criterion of the diagnostics, which prevents one from the
localization of the damages and, in its turn, the simplification of IM
repair.
Method 2
This method [13] provides for the measurement of the phase current
and voltage of the stator windings, and a conclusion as to the degree of
bar breakage is made according to the size of pulsation of the third
harmonic of the measured value.
Drawback: the supply mains voltage quality, imbalance and other
damages influence on the diagnostics results.
Method 3
The method is based on the analysis of starting current in the stator
in one of the motor phases [14]. During the diagnostics process every
previous amplitude value of the phase current is compared with the
following one. It is possible to judge about the presence of winding
defects by the obtained difference.
Drawbacks: it can be realized only in the starting mode, the analysis
is difficult because of the influence of IM electromagnetic parameters
during the start, low reliability, especially for low- and medium-power
IMs.
Method 4
A method with the measurement [15] of the instantaneous values of
two phase currents in the constant operation mode under load. The
presence of damage is determined by the appearance of phase portraits
of IM phase currents instantaneous values.
Drawback: the diagnostics is carried out under load, clear criteria
for the determination of broken rotor bars are not stated.
Method 5
The essence of methods based on the analysis of currents spectra,
Motor Current Signature Analysis (MCSA) [16–23] consists in the fact
that the presence of damage causes variation of magnetic field in the
motor air gap and, consequently, weak modulation of the current
consumed by the motor. The damages of IM electric or mechanical part,
including rotor bars, are determined according to the presence of the

typical frequencies of a certain value in the current spectrum, in
particular, according to the presence of two peaks symmetrical with
respect to the mains frequency.
Drawbacks: the supply mains voltage bad quality influence on the
diagnostics results, unsatisfactory results during diagnostics in the idle
mode. Besides, the use of the results of the Fourier transform of current
signals does not allow the unambiguous determination of the degree of
damage and relative position of the broken bars.
Method 6
Methods on the basis of the analysis of current and voltage
envelopes spectra [17, 24] provide for the record of the instantaneous
values of currents and voltages in three stator phases, the separation of
typical frequencies of the electric motor, the comparison of amplitudes
values at typical frequencies with the value of constant component.
Rotor winding defect is determined by the presence of two symmetric
peaks in the current spectrum in relation to supply mains frequency.
This method has the same drawbacks as the previous one.
Method 7
In paper [10] the spectra of IM vibration in the axial direction are
analyzed. The value of the amplitude of signal harmonics for a
damaged rotor is determined, and a conclusion as to the presence of a
damage is made on the basis of the increase of corresponding
components.
Drawback: the necessity for the installation of vibration sensors, the
high cost of vibration complexes.
Method 8
A method based on the analysis of the applied magnetic field
(AMF) [25] consists in the analysis of the variation of the magnetic
induction of IM applied magnetic field representing a combined
magnetic field created by different frequencies of the motor and the

screen magnetic fields. The presence of broken rotor windings causes
the appearance of spatial harmonics in AMF, whose order is lower than
the order of the basic spatial harmonic and which largely determine the
IM AMF level.
Drawback: the research complexity and low accuracy of the
research results.
Method 9
A method based on the thermal action of electric current [9] and
input of such voltage to the rotor rings, at which the current value in
bars exceeds the rated value. A thermal imager is used as a measurer of
rotor bars thermal condition. The state of rotor bars can be judged by
the heating temperature when current flows in the bars: the healthy bars
are heated more than the broken ones.
Drawbacks: the necessity for the input of high values of currents to
the rotor winding, the necessity for motor disassembling.
Method 10
Methods based on the stator current analysis using wavelet
transform [26–33]. The presence of broken rotor winding is determined
according to the corresponding values of typical coefficients of IM
stator current wavelet-spectra.
Drawbacks: analogous to method 5, and also the impossibility of
damages localization.
Method 11
Methods [34, 35] provide for the use of the variation of rotor field
magnetic induction, caused by machine double-sided serration
influence on the fundamental harmonic in the electromotor gap, as a
diagnostics signal. The measurement of magnetic induction is realized
on the basis of hall-effect sensor. According to the analysis of the
variation of magnetic induction values and conductivities the degree of

rotor imbalance is determined.
Drawback: the necessity for the use of a hall-effect sensor, which is
accompanied by the complication of the technical realization of
experimental research due to small sizes of the motor air gap.
Method 12
Methods with the use of neural networks [36–42] make it possible
to use especially educated systems of artificial intellect for IM
diagnostics and forecasting of breakage occurrence. The signal of IM
consumed power in each phase is used as a diagnostic signal; it is
analyzed with the help of an artificial neural network. This method
enables the detection of damage in both IM electrical and mechanical
parts.
Drawback: the set of fuzzy logic rules is formed on the basis of the
performed experimental research, i.e., the assessment of the results is of
an individual character.
Method 13
The method is based on the analysis of electromagnetic torque
spectrum [43]. The method provides for the measurement of the stator
phase currents in IM idle mode, the determination of the
electromagnetic torque and the comparison of a considerable number of
the spectrum harmonics that change for a certain frequencies range.
Drawback: the necessity for taking into account instantaneous
losses in the motor steel for calculations, which cause the complexity of
electromagnetic torque calculation.
Method 14
According to this method [7], an alternating current electromagnet
with a magnetizing winding and a measuring winding is connected to
the tested bar. The diagnostic feature of the bar state consists in the
value of magnetomotive force at constant supply voltage.
Drawback: the necessity for IM withdrawal from the technological

process to install measuring equipment.
Method 15
The method with the use of the wavelet-analysis of IM each phase
start currents [31] makes it possible to reveal broken rotor bars
independently of the level of load on the motor shaft. The analysis of
the signal of IM currents transient process is performed on the basis of
the algorithm that singles out the signal basic component according to
both amplitude and frequency.
Drawback: the impossibility of breakage location, realization only
in start modes, the complexity of the analysis due to the influence of IM
electromagnetic parameters measurement during start-up. The
considered methods of IM broken rotor bars diagnostics can be
classified according to the following signs. Depending on the
conditions according to which the diagnostics is carried out, the
methods of IM broken rotor bars diagnostics can be divided into several
groups:
 The diagnostics of IM state in the process of its operation

without withdrawal of the motor from the technological process
(methods 1–8, 12–13, 15). The advantages of these methods
consist in the possibility for tracing the variation of the motor
parameters in real time.
 The diagnostics of IM state with the withdrawal of the motor
from the technological process and the performance of
diagnostics in the operating mode on special equipment
(method 11). The realization of such methods does not need
motor disassembling and provides for the possibility for
carrying out an extended complex of research. However, the
disadvantages of such methods consist in the necessity to
withdraw IM from the technological cycle.
 IM diagnostics under the condition of completely or partially

disassembled motor (methods 9, 14). These methods allow one
to perform the diagnostics of separate parts of the motor and
reveal damages at the early stages of their development.
However, it is the necessity to disassemble the motor and
withdraw it from the operating process that makes the
disadvantage of these methods.
According to the character of the nature of measurable physical

values, the methods of broken rotor bars diagnostics can be classified in
the following way (Figure 1.6):
1. Methods using the measurement and control of electric values

(methods 1–6, 10, 12, 15), allow performance of the diagnostics
on turned-on equipment and connection directly to the motor
leads. However, the drawback of these methods consists in
supply voltage low quality influence on the diagnostics results,
the unsatisfactory results of diagnostics in the idle mode.
2. Methods with the measurement of electromagnetic values
(methods 8, 11, 14), enable a reliable analysis of IM gap
electromagnetic field that contains information about breakage.
The disadvantage of these methods consists in the complexity
of the installation of sensors in the gap. The analysis of the
external magnetic field signal is more complicated because of
the low interference immunity of the equipment and the
complexity of the research.
3. Methods assessing the mechanical and electromagnetic values
(methods 7, 13), in particular, electromagnetic torque, speed
and vibrations. The advantages of these methods consist in high
reliability and considerable level of their development.
However, a number of drawbacks of these methods should be
mentioned, in particular, when, e.g., vibration is measured, the
research results depend on the point of installation of the
measuring equipment. Besides, there is a necessity for the

installation of sensors in three planes, also, external factors
(supply mains parameters, etc.) influence on the measurement
results should be taken into consideration.
4. Methods based on the measurement of the motor temperature
(method 9), make it possible to assess the energy released at the
motor units. However, the temperature itself is an integral
value, and its analysis is accompanied by the limitation of
diagnostics signs.
electromagnetic torque
speed vibration
mechanical and electromechanical temperature control
physical values used in induction motor diagnostics
electric electromagnetic
magnetic flux and magnetic

current (phase, start, reverse) induction in the gap
magnetomotive force
electromotive force (of stator
phase, turn) magnetic induction of the rotor
field
active consumed power
signal of the external magnetic
field
Figure 1.6. Physical values used in induction motor diagnostics.
modes of IM operation in which breakage diagnostics is performed
steady dynamic
idle mode start-up condition
on-load operation short-circuit condition
short-circuit condition motor self-running-out

with stalled rotor
Figure 1.7. IM operation modes during the diagnostics of broken rotor bars.
According to operating modes in which diagnostics is performed,

the methods for the diagnostics of IM broken rotor bars can be
conditionally divided into two groups (Figure 1.7):
1. Methods for the diagnostics of broken bars under static

operating modes (methods 1, 2, 4–6, 10–10, 13, 14). They are
characterized by the simplicity of realization but during the
diagnostics it is necessary to assign a number of test impacts. It
should be stated that the results of the analysis of certain
damages for the methods of broken bars diagnostics in static
modes of operation should not depend on the level of the motor
load.
2. Methods for the diagnostics of broken bars in dynamic modes of
operation (methods 3, 11, 12, 15). These methods are
characterized by the complexity of analysis, as some IM
parameters change in time in the transient process. However,
the use of these methods is caused by high information value.
According to the way of obtaining data and processing the research

results the methods of diagnostics of IM broken rotor bars can be
divided into two groups (Figure 1.8):
1. Methods for information initial processing (methods 1, 2, 4–7,

9–11, 13, 15). They include such ways of processing the
information-diagnostic signal as:
1.1. The spectral analysis of the signal. The use of the signal
spectra makes it possible to detect the various types of
breakage. In this case the frequency components of the
signal cannot be localized in time, and it prevents one from
carrying out the analysis of non-stationary signals with
complex frequency-time characteristics.
1.2. Wavelet-analysis. This type of information initial processing
proved to be an extremely efficient way of the analysis of
signals with complex spatial-time characteristics. The

relative complexity of mathematical apparatus is typical of
wavelet-transform.
1.3. Approximation. It requires a priori knowledge of the law
according to which the parameter changes; when this law is
simplified, the accuracy of diagnostics deteriorates; there
appears a necessity for additional analysis of the coefficients
or time constants obtained as a result of approximation.
1.4. The determination of local extremums. Only an
inconsiderable part of the information-diagnostic signal is
subject to the analysis.
1.5. Obtaining integral coefficients. During the use of this
method a generated analysis of the signal is performed,
which does not allow the detection of local defects.
methods of information initial processing and making decisions
methods of information initial

methods of making decisions
processing
comparison with the
spectral analysis of the signal threshold value
comparison with standard
wavelet-analysis
models
approximation logic rules
determination of local artificial neural networks

extremums
fuzzy logic
obtaining integral
coefficients statistical analysis
Figure 1.8. Methods for information initial processing and methods for
making decisions.
2. Methods for making decisions (methods 3, 8, 12, 14). They

provide for the analysis of the results of information processing
with making decisions as to the position and the degree of the
defect:
2.1. Comparison with the threshold value of the signal. In this case
comparison with the threshold values causes low information
value.
2.2. Comparison of processes research results obtained on
standard models. In this case idealized standard models are
used; they do not take into account all the real physical
properties of the system.
2.3. Making a decision on the basis of logical rules. When logical
rules are made, it is necessary to take into consideration the
influence of many factors, which causes the complexity of the
analysis.
2.4. Making a decision with the use of artificial neural networks
and fuzzy logic. Neural networks have complex architecture; in
this case there occurs a necessity for instruction in the
networks or making expert rules.
2.5. Statistical analysis. The results of signals processing are
reliable only to a certain level of probability assigned by the
researchers before starting statistical data processing; as a rule,
to obtain results a considerable volume of processing is
required and breakage localization is impossible.
The performed analysis of the diagnostics methods revealed that,

due to the simplicity of realization, the methods of currents spectral
analysis are most common, but they have a number of drawbacks: they
do not take into account supply voltage low quality influence on the
results of diagnostics, do not provide satisfactory results during
diagnostics in the idle mode and do not allow the determination of the
number and relative position of the rotor broken bars. So, there appears
a necessity for the development of a new method of IM broken rotor
bars diagnostics, which would make it possible to determine the number
and relative position of the rotor broken bars.
1.4. CONCLUSION
Thus, the performed analysis of statistical data has revealed that

most IM breakages occur due to broken stator windings and squirrel-
cage rotor bars. The breakage data analysis with the use of different
literary sources has demonstrated that broken bars occur more seldom
in IMs with aluminum winding on the rotor than in IMs with copper
winding on the rotor. Besides, broken bars occur more often in rotors
with welded aluminum winding than in cast rotors. It has been shown
that, unlike the diagnostics of broken stator windings, in the diagnostics
of squirrel-cage rotor broken bars there appear complications related to
the impossibility for direct measurement of rotor parameters.
The performed comparative analysis of the modern methods of IM
broken rotor bars has allowed formulating the criteria for assessment of
the methods efficiency. Basing on the analysis of these criteria it has
been determined that it is possible to consider inefficient the methods
that require the installation of special equipment, the withdrawal of the
diagnostics object from the technological process, the disassembling of
the motor and the installation of additional sensors in its structure.
The proposed criteria of the assessment of the efficiency of
induction motor broken rotor bars diagnostics has shown the necessity
for the development of a diagnostics method that would allow the
determination of the degree of breakage, the number and relative
position of the broken rotor bars without withdrawal of the motor from
the technological process and without its disassembling. The proposed
classification of the existing methods of diagnostics of induction motor
broken rotor bars according to a number of classification signs makes it
possible to determine the corresponding position of the diagnostics
method presented in this book.
Chapter 2
THEORETICAL FOUNDATION FOR

THE RESEARCH OF INDUCTION MOTOR
BROKEN ROTOR BARS IN
THE SELF-RUNNING-OUT CONDITION
2.1. THE SUBSTANTIATION OF THE TESTING CONDITION

AND THE DETERMINATION OF DIAGNOSTIC SIGNALS
FOR THE ASSESSMENT OF IM BROKEN ROTOR BARS
2.1.1. The Substantiation of the Testing Condition
The efficiency of diagnostics depends on the choice of the testing

condition, so, it is one of important stages of the development of
diagnostics method. The first chapter contains an analysis of IM
operating modes (Fig. 1.4), under which broken rotor bars diagnostics
can be performed, and the general drawbacks of the use of these modes
are determined. An analysis of possible IM operating modes was
performed to choose the testing condition.
Static Operating Conditions
Idle Condition
When the diagnostics is performed in the idle condition, the motor,
as a rule, is to be withdrawn from the technological process.
Operation under Load

The diagnostics results are essentially influenced by both the level
of the motor load (its variation in time and vibration) and bad quality of
the supply mains voltage.
Short-Circuit Condition
When research is performed in a short-circuit condition, there
appear limitations. They are caused by the fact that only equivalent
rotor resistance can be determined by the results of IM parameters
identification. That is why the diagnostics results are not very reliable
in such condition.
Dynamic Operating Conditions
Start-Up Conditions
The diagnostics under start-up conditions is limited due to the
complexity of the analysis of the diagnostics results. Besides, the
analysis of diagnostics results is complicated because of the influence
of the efficiency of diagnostics depends on variation of IM
electromagnetic parameters during start-up.
Self-Running-Out Condition
Carrying out diagnostics in self-running-out condition, unlike the
above mentioned conditions of IM operation, has a number of
advantages:
Theoretical Foundation for the Research of Induction Motor … 27
 allows diagnostics performance without withdrawal of the

motor from the technological process and its disassembling;
 eliminates the supply mains voltage bad quality and
technological mechanism operation influence on the diagnostics
results;
 the results do not practically depend on the previous condition
of motor operation. It provides a possibility to carry out the
diagnostics both during scheduled IM repair stops and after
completion of the technological process;
 the stator winding voltage is subject to measurement. The stator
winding itself is the magnetic flux sensor during measurement.
In this case magnetic flux measurement is more informative
than the measurement of current. The diagnostics in self-
running-out condition allows the measurement of magnetic flux
without the additional use of flux sensors or hall-effect sensors;
 the realization of the method does not require additional sources
of testing impacts.
Thus, the performed analysis of IM testing conditions made it

possible to determine that motor self-running-out condition is the most
efficient one for the diagnostics of broken rotor bars.
2.1.2. The Substantiation of Diagnostic Signals
The first chapter of the monograph contains an analysis of basic

physical values used for the diagnostics of broken rotor bars (Figure
1.6). It enabled the determination of the main advantages and
drawbacks. During the research, taking into account IM testing
condition chosen in paragraph 2.1.1, the substantiation of the diagnostic
signal was performed. The research is made under the condition of IM
self-running-out. That is why it is inappropriate to use mechanical,

electromechanical or temperature values as measuring parameters. It
can be explained by the low level or the absence of the mentioned
parameters in IM self-running-out condition.
As stated above, stator EMF contains information about magnetic
field distortion due to the rotor breakage. The shape of EMF signal in
winding conductors reflects the character of magnetic induction
distribution in IM air gap. The presence of slots on the surface of
electric machines stators and rotors is known to cause distortion in
magnetic field in the air gap and appearance of tooth spatial harmonics
in this field [44]. Such harmonics cause additional losses in steel and
short-circuited windings, the distortion of the torque curve, the variation
of induction resistances of differential scattering and the appearance of
noise in the machine.
It is known that magnetic field that rotates with p pairs of poles
induces in the squirrel cage with Z 2 bars a system of currents with a
phase shift by angle  in the adjacent bars. In this case the squirrel cage
creates an infinite series of harmonics rotating directly with ordinal
numbers [45]:
Z2
k  1,  k  0, 1, 2, 3 ... , (2.1)
p
and a series of harmonics, rotating inversely with ordinal numbers:
Z2
k  1,  k  1, 2, 3 ... . (2.2)
p
Thus, relation Z 2 / p determines the number of bars per a pair of

poles. Currents in these bars are shifted in phase like currents in phase
zone of a usual multiphase winding, so these bars are analogous to

phase zones. At sufficiently big relation Z 2 / p the squirrel cage has a
great number of phases, and its magnetizing force contains a small
number of low-order harmonics approaching a sinusoid [46].
To research the IM magnetic field modeling a specific permeance
method was performed. In this case it was assumed that at the initial
moment of time t  0 stator and rotor teeth axes at the origin of
coordinate  coincide and the rotor rotates with electric angular speed:
2  p2 , (2.3)
where 2 – the mechanical angular speed of rotor rotation.

Then stator and rotor relative permeances are determined by
expressions [47]:
  iZ1 
 1  1   i  cos ;
 i  p 
 (2.4)
  1    cos jZ 2      t   ,
 2  j 2 
 j  p 
where  1 ,   2 – the relative permeances of IM stator and rotor,

respectively;
i ,  j – the relative permeances of the i -th and j -th harmonics; i , j
– the numbers of harmonics of the stator and rotor permeances; Z1 , Z 2
– the number of teeth of IM stator and rotor;  – coordinate origin; p
– the number of IM poles pairs; 2 – the electric angular speed of the
rotor.
The general relative specific permeance of IM air gap is of the form
[13]:
 iZ  
  (t )   1  2  1    i cos  1   1 
i  p 
 jZ  
   j cos  2 (  2t )  
j  p 
(2.5)
1    Z 
  i  j cos ( jZ 2  iZ1 )  j 2 2t  
2 i j   p p 
  Z  
 cos ( jZ 2  iZ1 )  j 2 2t   .
 p p  
The first right-hand term of this expression determines the

permeance of the equivalent uniform gap, the second term – stator
permeance harmonics, the third term – rotor permeance harmonics and
the last one – interferential permeance harmonics caused by stator and
rotor mutual influence.
In the course of the research it was found out that the general
relative specific permeance of the gap is shaped as a sinusoid. It does
not reflect unevenness of magnetic field in the motor air gap in any
way. The value of the gap admittance multiplied by the machine
magnetizing force makes it possible to obtain infinite series of the field
harmonics. Some of the harmonics have an important impact on the
machine operation [48]. In this case relative magnetic induction is
determined by expression:
B (t )  b cos  1t    . (2.6)
Thus, when IM tooth design is taken into account, the magnetic

field in the air gap is non-sinusoidal. Figure 2.1 shows IM air gap
magnetic induction taking into consideration the IM rotor tooth design
(in relative units).
B,
r.u.
0.6
0.2
t, s
-0.2 0 0.008 0.016 0.024 0.032
-0.6
Figure 2.1. IM air gap magnetic induction.
The obtained pattern of tooth kink on the IM air gap magnetic

induction curve provides the possibility for the comparison of IM
magnetic field lines with geometrical arrangement of the rotor teeth. Let
us find out in what way IM magnetic field and stator winding
electromotive force are interrelated.
Electromagnetic field is described by Maxwell’s equation system
[49, 50]:
rotH  j ;
B
rotE   ;
t
B  a H ; (2.7)
1
j E  jout ;

divB  0,
where H and E – the vectors of magnetic and electric field,

respectively; j – currencies density; B – magnetic induction vector;
 a – permeability;  – the density of electric charges; jout – currents
density caused by outside EMF.
The solution of Maxwell’s equations is a rather complicated

problem. So, to reduce these equations to the form that is more
convenient for the solution additional functions were introduced: of
vector A and scalar  magnetic potentials.
It is known that IM electromagnetic field is three-dimensional [50].
However, in a number of cases it is sufficient to consider 2D
electromagnetic fields. In such fields all field vectors depend only on
two spatial coordinates. Each of the vectors has only one or two spatial
components. Such tasks include the problems of the analysis of the field
in EM active zone cross section.
The use of vector magnetic potential A in solution of 2D field
problems with such coordinates system orientation when windings
currents are directed along one of its axes is the most efficient. Axes x
and y of the Cartesian coordinate system are located in EM active zone
cross section, and axis z is directed along its longitudinal axis.
In Cartesian coordinates for motionless media (under the condition
that motion speed is   0 ) the scalar equation in relation to the vector
magnetic potential is of the form [50]:
 2 Az  2 Az Az
    J z _ out , (2.8)
x 2
y 2 t
where J z _ out – the density of outside currents, assigned in stator

winding sections. At boundary G of the calculation area equation (2.8)
is supplemented by a first-type homogeneous boundary condition
Az G  0 that reflects field damping outside the area.
Magnetic induction vector components in Cartesian coordinates are
determined by the known values of vector magnetic potential:
Az
Bx  ;
y
(2.9)
A
By   z .
x
In the integral form the expression for the vector magnetic potential
determines its physical sense.
The circulation of the vector magnetic potential along a closed
contour is equal to magnetic flux Φ that pierces this contour:
Adl=Φ. (2.10)
The relation between the vector magnetic potential and the full flux
linkage of the stator winding phase is written by means of equation:
2l1w
 ph 
Ss S
 Az dS , (2.11)
ph
where l1 – the active length of the stator; w – the number of turns in the
slot connected in series; S s – sectional area of the stator slot; Az – the
total arithmetic value of VMP in all the slots of the phase; S ph – the
total area of cross section of all phase coils connected in series.
By the law of electromagnetic induction (according to Maxwell’s
equations) [49–50], the stator winding phase EMF is equal to:
d  ph  t 
e ph (t )   . (2.12)
dt
Thus, EMF in IM stator windings is a derivative caused by the

presence of motor electromagnetic field.
On the grounds of the above given equations it is possible to state

that tooth kink caused by the presence of tooth harmonics will also be
present in the stator winding EMF signal. Thus, this tooth kink provides
for the possibility of the comparison of magnetic field lines with the
geometrical arrangement of the rotor teeth. The presence of broken
rotor bars results in distortion in IM magnetic field. That is why it was
supposed that the stator winding EMF signal contains information about
two components. They are magnetic field unevenness caused by stator
and rotor toothed design and the presence of broken rotor bars.
Damping currents flow in the rotor windings in self-running-out
condition. Rotating electromagnetic field inducing EMF in the stator
windings is created under their action. To confirm this supposition
preliminary experimental research for IM of АIR80V4U2 type
(appendix А) was carried out in the motor self-running-out mode.
During the research a measuring winding was laid into IM stator slots.
It was used to fix EMF instantaneous values. The research was carried
out for a healthy IM and for an IM with broken rotor bars. The results
of the performed research are shown in Figure 2.2. The research results
(Figure 2.2) revealed that EMFs in the measuring winding of an IM
with broken rotor bars contain typical distortions that correspond to
broken rotor bars and are absent in the EMF of the healthy IM. Thus,
stator windings EMF signal in the motor self-running-out mode can be
used for broken rotor bars diagnostics.
E,
V
2.5
0
0.01 0.02 0.03 0.04 0.05 t, s
-2.5
a
Figure 2.2. (Continued)
E,
V
0
0.01 0.02 0.03 0.04 0.05 0.06 t, s
-1
-2 2 broken bars 1 broken bar

b
Figure 2.2. EMF in the measuring winding of a healthy IM (а) and an IM with three
broken rotor bars (b) in the motor self-running-out mode.
2.2. THE SUBSTANTIATION OF THE METHOD FOR

THE CALCULATION OF INDUCTION MOTOR
ELECTROMAGNETIC FIELD
The previously performed research on the choice of the diagnostic

signal suggested a conclusion that stator winding EMF is a derivative. It
is caused by the presence of an electromagnetic field and can be
calculated by dependences (2.7)–(2.12).
It is necessary to research electromagnetic field influence on EMF
in the motor stator windings in detail for the analysis of IM broken rotor
bars influence and the development of a diagnostic method. That is why
it is proposed in the paper to perform stator windings EMF calculation
according to the results of the calculation of IM electromagnetic field.
As a rule, electromagnetic field equations are solved by analytical
or numerical methods [50]. The use of analytic methods makes it
possible to obtain a general solution. It provides for a possibility to have
a complete idea of different parameters influence on EM
electromagnetic field. When numerical methods are used, it is necessary
to carry out calculation for every totality of parameters values. That is
why a general pattern can only be obtained when a big number of
calculations are available.
According to the posed problem, the result of EMF calculation is to

be presented by IM stator windings EMF curve, i.e., a totality of EMF
instantaneous values after IM disconnection from the supply network.
The numerical methods of field calculation allow the solution of this
problem. Besides, EMF calculation in EM by numerical methods makes
it possible to analyze its distribution in the separate elements of
electromagnetic circuit in detail.
Among the numerical methods used for field calculations the most
common ones include: the finite difference method (FDM), the finite
element method (FEM), the integral equation method (IEM) and the
boundary integral approach (BIA).
The finite difference method [51] uses the substitution of the field
differential equations by the finite-difference equations. In this case the
field calculation area is realized by a rectangular finite differences grid
with a uniform superposition step. The smaller the value of finite
differences step is the more accurately the continuous distribution of the
field function in the calculation area is approximated with the help of its
discrete model.
The basic drawback of this method consists in the complexity of the
exact description of the boundaries and the optimum superposition of
the finite-difference grid on the calculation area.
According to the problems of electromechanics, FEM allows the
calculations of electrical, magnetic, temperature and other fields. The
basic FEM idea consists in the following: any continuous function e.g.,
vector or scalar magnetic potential, induction, temperature, etc. can be
approximated by a discrete model. Such a model is created on a
multitude of piece-wise-continuous functions. The solution of the field
equations in FEM is determined on the condition of minimum power
functional or orthogonality of field equations discrepancy and
interpolational functions of finite elements [51, 52].
At present FEM has sufficient theoretical justification and is a
generalized method of numerical solution of differential equations [35,
50–52].
The integral methods of EM field calculation (the integral equation

method and the boundary integral approach) are based on the transform
of Maxwell’s equations to the integral equations formulated in relation
to the field secondary sources [50]. Their application is most efficient in
the case when the amount of areas occupied by field sources and
ferromagnetic materials is insignificant in comparison with the whole
volume of the calculation area.
In some cases the application of the integral equation method and
the boundary integral approach is insufficiently efficient. It refers to
various EMs whose calculation area is mainly filled with ferromagnetic
magnetic circuits. It is for this reason that the mentioned methods are
not practically used in EM field analysis.
At present FDM and FEM are most commonly used for EM
calculation and design in practice. FEM is more popular and has a
number of advantages:
 a possibility for accurate description of curved boundaries of

the areas;
 the simplicity of the variation of the discretization of the area at
its different sections for improved accuracy of calculations at
the smallest number of the calculation grid knots;
 a possibility for assigning the second type boundary conditions
as well as mixed boundary conditions at the boundaries of any
length;
 a possibility for overlapping the boundary conditions with
breaking surface load.
When FDM or FED is used, there appear a great number of knots at

at the boundaries of any length as well calculation. It is due to a
complex boundary of tooth area, a large size of the calculation area and
presence of media with different permeance. The difficulty of grid
laying consists in the fact that EM field calculation area has a small air
gap that requires a high degree of discretization. At the same time, it is

possible to increase the grid step at peripheral sections where the field
variation gradient is small. So, Figure 2.3 shows an element of the grid
laid on the field calculation area of a part of IM (with elements of the
stator, rotor and an air gap).
FEM takes into account the complex configuration of the
calculation area. Moreover, FEM makes it possible to take into
consideration the structural materials physical characteristics
nonlinearities, the uniformity of calculation procedures, etc.
Figure 2.3. An example of the finite elements grid of a part of EM.
It is generally known that all physical fields, including

electromagnetic field, are three-dimensional. However, in many cases it
is impossible to obtain the exact analytic solution for three-dimensional
fields, and finding numerical solutions is often connected with the
excessive amount of calculations.
An approximate solution with sufficient accuracy can be found by
reducing a spatial problem to a plane one, i.e. without taking into
account the field variation along one of the coordinates. As a result of
this approach, it is possible to find an analytical solution for many
problems, including IM EMF calculation, and essentially decrease the
laboriousness of calculation and time expenditure in numerical
calculation.
Thus, the performed analysis of field calculation methods made it

possible to determine that it is expedient to calculate the
electromagnetic field signal by means of numerical methods. The
analysis of most common numerical methods revealed that FEM has a
number of advantages in comparison with other numerical methods. It
enabled making a conclusion as to the expediency of its use in IM EMF
calculation for the determination of EMF in the stator windings.
It is shown that the motor electromagnetic field calculation can be
performed with the use of a plane-parallel (2D) model in IM cross
section.
Thus, to calculate EMF in IM stator windings it is necessary to
calculate the electromagnetic field in IM cross section in a self-running-
out mode using FEM.
2.3. THE SUBSTANTIATION OF THE USE OF

THE WAVELET-TRANSFORM FOR DIAGNOSTIC
SIGNALS PROCESSING
A significant number of methods of IM broken bars diagnostics are

based on the use of the spectral analysis as a method for diagnostic
signals processing [53–57]. The spectral analysis methods are based on
the processing of signals of the such electric values as current, voltage,
consumed instantaneous power [58], vibrations, etc.
The methods for spectral analysis of the stator phase current signals
are most commonly used [16, 58, 58]. It is a method for diagnostics of
the alternating current motor and mechanical devices connected to it, at
which:
 during an assigned time period the values of currents consumed

by the electromotor are recorded by current sensors;
 the obtained signal is transformed from the analog into the

digital form;
 the obtained signal is decomposed into a spectrum by means of
Fourier transform;
 a spectral analysis of the obtained signal is carried out and the
values of amplitudes at typical frequencies are compared with
the signal level at the frequency of the supply network.
IM broken rotor bars (bars breakage, slackening of bars fastening

against contact rings, undetected casting flaws) are determined by the
presence of two current spectrum side components symmetrical about
the frequency of the supply network.
The frequency of the side components in the spectrum is calculated
according to expression [11]:
f p  f1 (1 2ks), (2.13)
where f1 – the frequency of the supply network (50 Hz); k  1,2,3,... –

harmonics number; s – IM slipping.
IM slipping is determined by expression:
0  
s . (2.14)
0
I,A
1
0.1
0.01
10 3
10 4
0 50 100 150 f, Hz
Figure 2.4. The current spectrum of a healthy IM.

I, A Broken bars (44.5 Hz and 55.5 Hz)

1
0.1
0.01
10 3
10 4
0 50 100 150 f, Hz
Figure 2.5. The current spectrum of an IM with broken rotor bars.
drawbacks of Fourier transform
low accuracy of determination of local features of signals

or instantaneous variations of signals frequency components
impossibility to detect breakage at frequencies that are not

multiple of the first harmonic frequency
impossibility of the analysis of non-stationary signals

with complex frequency-time characteristics
impossibility to localize the point and the degree

of the damage
Figure 2.6. The drawbacks of Fourier transform at diagnostics of IM broken rotor bars.
Figures 2.4–2.5 contains current spectra for the researched healthy

IM and an IM with broken rotor bars [11].
The analysis of the obtained results revealed that the broken IM
current spectrum contains side harmonics. These harmonics are located
in relation to the fundamental one and point to the presence of broken
rotor bars. Thus, the methods of currents spectral analysis enable
determination of IM broken rotor bars. However, due to the use of
Fourier transform as a software, the method of spectral analysis has a
number of drawbacks (Figure 2.6).
So, according to the results of the preliminary experimental

research in IM self-running-out mode (Figure 2.2, b), it is determined
that broken rotor bars occur in the presence of distortions in EMF
signal. That is why one can formulate a hypothesis that the sign of the
presence of broken bars consists in stator winding EMF signal peculiar
features manifested in the distortion of its form. It is known that in the
self-running-out mode the rotor rotation frequency decreases, i.e., the
signal damping time constant changes. Probably, the periodicity of
repetition of breakage signs (signal form distortion) also changes with
the change of the signal period. That is why the use of Fourier
transform for the analysis of the damping signals is only possible at the
time sections at which the signal period and information signs of
damages do not change. It is due to this fact (the impossibility of
damages detection because of their shift on EMF signal) that the use of
Fourier transform for diagnostic signals processing is inefficient. So, to
improve diagnostics efficiency there arises a necessity for the use of
another method for diagnostic signals processing.
Wavelet-transform (WT) is one of the modern efficient methods for
signal processing [59–61]. This method is a generalized form of
spectral analysis of signals in both frequency and time domains.
Wavelet-transform allows the analysis and processing of the signals and
functions that are non-stationary in time and non-uniform in space [59].
Wavelet-transform is divided into a discrete wavelet-transform
(DWT) used for the transformation and encoding the signals and a
continuous wavelet-transform (CWT) used for the signals analysis [59].
WT basic functions may include various functions with a compact
support – pulse-modulated sinusoids, functions with level leaps, etc.
They provide for good mapping and the analysis of signals with local
features, including leaps, breakages and values differences with high
peaks [59].
The most common and widely used basic wavelet functions are
shown in Figure 2.7.
In a general case WT is based on the use of two continuous,

mutually dependent functions integrated by an independent variable
[59]:
 a wavelet-function   t  – a time psi-function with zero value

of the integral and a frequency Fourier-image    . The signal
local features are marked by this function. The functions that
are well localized in the time and frequency domain are usually
used;
 a scaling function   t  – a phi scaling function with a single
integral value, on its basis a signal rough approximation is
performed. Phi-functions are inherent not in all but, as a rule,
only in orthogonal wavelets. They are necessary for the
transformation of acentric and rather long signals at separate
analysis of low-frequency and high-frequency components.
basic functions of wavelets
pre-wavelets (Gauss wavelets, Morlet wavelets, Mexican hat wavelet)
regular and discrete Meyer wavelets
orthogonal wavelets with a compact support
biorthogonal wavelets
complex wavelets (complex Gauss wavelets, complex Morlet wavelets,

complex Shennon wavelets, complex frequency
B-spline wavelets)
Figure 2.7. The basic functions of wavelets.

WT carries a huge amount of information about the signal, but, on

the other hand, it is extremely excessive, as each point on the phase
plane influences its result. For the accurate restoration of the signal is it
sufficient to know its wavelet-transform on a certain rather rare grid in
a phase plane. Thus, all the information about the signal is contained in
this set of values that is quite small.
The main idea of the wavelet-transform consists in wavelet scaling
by a certain permanent number of times and its shift in time by a fixed
distance depending on the scale [61, 62]. In this case all the shifts of
one scale are to be orthogonal in pairs – such wavelets are called
orthogonal ones. At such transform a signal convolution with a certain
function (so-called scaling function) and with a wavelet connected with
this scaling function is performed. It results in a “smoothed” version of
the initial signal and a set of “details” that differs the smoothed signal
from the initial one. Using this transform successively it is possible to
obtain the result of the required degree of specification and a set of
details on different scales.
Wavelets differ in the purpose and from the point of the
decomposition-reconstruction of signals [62]. For the high-quality
analysis of signals and local features in the signals non-orthogonal
wavelets can be used; though they do not provide signals
reconstruction, but allow the assessment of signals information content
and the dynamics of the change of this information.
Generalizing the above said, one can single out the possibilities and
advantages of WT use for the diagnostics of the rotor broken bars
(Figure 2.8).
As stated above, at the wavelet-transform of the signal there occurs
wavelet scaling by a certain permanent number of times and its shift in
time by a fixed distance depending on the scale. To detect the
information signs of broken rotor bars it is necessary to use
corresponding wavelet-basis that repeats EMF signal properties shifted
in time.
possibilities and advantages of wavelet-transform
analysis in the frequency and time domains
analysis and processing of signals non-stationary in time

or non-uniform in space
possibility of choice of different basic functions for the analysis

of signals certain properties
possibility of singling out and analysis of signals local features
Figure 2.8. Possibilities and advantages of signal wavelet-transform at diagnostics of

IM broken rotor bars.
Thus, the performed comparative assessment of the diagnostic

signal processing methods for the problems of the broken rotor bars
diagnostics made it possible to determine the drawbacks of Fourier
transform. The analysis of WT characteristic features enabled us to
determine that it is expedient to use the wavelet-transform as a method
of diagnostic signals processing for the rotor broken bars diagnostics on
the basis of the analysis of IM stator windings EMF [63–64].
2.4. THE CHOICE OF THE WAVELET-BASIS FUNCTIONS

FOR THE DIAGNOSTIC SIGNALS WAVELET-TRANSFORM
The choice of the wavelet function is an important problem in the

performance of diagnostics with the use of wavelet-transform (WT).
The efficiency of the diagnostic signal analysis depends on the results
of this choice [61].
As a rule, the choice of the wavelet is determined by the
information that is to be obtained from the signal analysis. Taking into
account the characteristic features of different wavelets in the time and

frequency domains it is possible to find certain properties in the
analyzed signals that are not seen in the signals.
To determine the possibilities of WT at the analysis of EMF signal
in the motor stator windings it is proposed to carry out a number of
preliminary researches for the following test signals:
 an ideal signal – a sinusoid modulated by periodical high-

frequency oscillations;
 a signal approximate to the expected signal of EMF in stator
windings of the IM with broken rotor bars (without taking into
consideration the damping character of EMF).
In a general case, as a rule, the choice of the wavelet can be

determined by the following basic factors:
 the purpose of the analysis;

 the type of the signal;
 the characteristic features of the signal structure;
 the signal-interference conditions.
The analysis of these factors and the choice of the wavelet-function

are described below.
 The purpose of the analysis: The purpose of the analysis in the

research of the mentioned signals consists in detecting the
information signs caused by the presence of broken rotor bars.
The wavelet-analysis of the signals can be performed without
their further reconstruction, so, the use of any wavelets is
admissible (both orthogonal and non-orthogonal).
 The type of the signal: The researched signals are continuous

and are of an oscillating character. So, it is expedient to carry
out a continuous wavelet-transform to detect the breakage.
 The characteristic features of the signal structure: The analyzed
signals are sinusoids with the addition of periodic high-
frequency oscillations. It is found out from literature [59, 61]
that to analyze oscillating signals having the shape of a sinusoid
it is possible to use orthogonal wavelets with compact
supporters.
 Signal-interference conditions: The researched signals have a
low level of obstacles, which provides WT performance without
additional operations as to their elimination.
To single out EMF information signs in the stator windings at the

use of WT at first it is necessary to choose a general class of wavelets,
then form a multitude of wavelet-bases (WB) from the set of bases with
ordinal numbers for particular wavelet families, determine the optimum
level of decomposition and choose the most optimum basis of wavelet.
The process of determination of the optimum wavelet-basis consists
of the following stages:
The choice of the type of frequency-time transform: A continuous
wavelet-transform is chosen; it is characterized by excellent frequency-
time localization, the accessibility of different exciting WB.
On the basis of the required characteristics of wavelet-transform a
general WB class is chosen: Orthogonal wavelets with a compact
supporter are chosen; the presence of a number of zero moments
according to the number of the basis ordinal index is typical of them.
Also, a rapid calculation algorithm is well realized.
The choice of WB family with a set of ordinal indices: Daubechies,
Symlet and Coiflets families are chosen as they are deigned on the basis
of the necessary requirements of the wavelet-analysis, orthogonality,

compactness of the supporter; in this case symmetry and smoothness
grow with the increase of the wavelet ordinal index.
The determination of the optimum level of decomposition: It is
determined on the basis of maximum frequency and the frequency of
initial signal discretization. The optimum limit level for the analyzed
non-stationary signals is calculated by expression [59]:
  f 
La   log 2  s    1    log 2  2 f m t    1 , (2.15)
 
  fm  
where La – the optimum upper level of wavelet-decomposition; f m –

the upper boundary of frequencies band of concentrated fundamental
energy of the signal (in low-frequency domain); f s – discretization
frequency; t – discretization period. i.e. further decomposition of
the analyzed signal to the levels that exceed the found threshold La will
not be efficient.
The determination of optimum WB: Orthogonal wavelets with a
compact support can be used for the analysis of oscillation sinusoidal
signals. These wavelets include: Daubechies wavelets (dbN), Symlet
wavelets (symN) and Coiflets wavelets (coifN) 166], where N –
numerical index (1,2,...) (Figure 2.9).
Figure 2.9 contains the wavelets of one order ( N  4 ). At N  2
Daubechies and Symlet wavelets are of the same type and only differ in
the psi-function sign. Daubechies wavelets are asymmetrical. Wavelets
approaching symmetry and obtained from Daubechies wavelets [65] are
called Symlet wavelets.
Such wavelets are used for a continuous and discrete, as a rule,
rapid wavelet-transform and reconstruction of the signal. They have phi
and psi-functions, as well as the functions of scaling filter, which is the

basic one for the calculation of signals decomposition and
reconstruction filters calculation.
scaling function phi φ ψ

φ ψ
wavelet function psi
1.5
wavelet function psi 1

0.5
0.5
0
1 2 3 4 5 6 j 0
1 2 3 4 5 6 7 j
-0.5 -0.5
scaling function phi
-1
a b
φ ψ
1
wavelet function psi
0.5
0
5 10 15 20 j
scaling
-0.5 function phi
Figure 2.9. Daubechies wavelets (а), Symlet wavelets (b) and Coiflets wavelets (c)
of the fourth order.
Let us perform a wavelet-transform for an ideal signal – a sinusoid

modulated by periodic high-frequency oscillations with the use of all
three mentioned WBs as a basic function. On the basis of the obtained
wavelet-spectra (Figure 2.10) it is possible to state that at low scales of
wavelet high-frequency oscillations of the signal can be seen and at big

scales – a low-frequency component of the signal is mapped.
a
Analyzed Signal (length = 1001)
1
-1
100 200 300 400 500 600 700 800 900 1000
Ca,b Coefficients - Coloration mode : init + by scale + abs
127
120
113
106
99
92
85
78
71
64
57
50
43
36
29
22
15
8
1
Scale of colors from MIN to MAX
b
Analyzed Signal (length = 1001)
1
-1
100 200 300 400 500 600 700 800 900 1000
Ca,b Coefficients - Coloration mode : init + by scale + abs
127
120
113
106
99
92
85
78
71
64
57
50
43
36
29
22
15
8
1
Scale of colors from MIN to MAX
Figure 2.10. Sinusoid signal with a high-frequency component and its wavelet-
spectrum with the use of Daubechies bases (а), Symlet bases (b) and Coiflets bases (c).
In this case, the use of each of the wavelets as the basis allows
determination of a high-frequency component caused by the presence of
tooth harmonics. So, it can be stated that one of the mentioned bases
can be used for the analysis of sinusoidal signals containing a high-
frequency component.
The paper contains an analysis of the testing signal approximate to
the anticipated signal of EMF in IM stator with broken rotor bars. Let
us simulate testing signals of the stator windings coils EMF, taking into
consideration the fact that only the motor rotor has a stepped
appearance.
We assume that the stator winding coil group consists of three coils
whose EMF vectors are shifted by angle θ (Figure 2.11).
Let us assume that coils testing EMFs are calculated by the
following expressions:
etest1  t   A1  t  cos  2  t   ;
etest 2  t   A2  t  cos  2  t     ; (2.16)
etest 3  t   A3  t  cos  2  t   2  ,
where A1 (t ) , A2 (t ) , A3 (t ) – functions assigning the rotor breakage and

a stepped appearance with the adopted value of teeth on the rotor
N  34 (the number of teeth is chosen according to the number of rotor
bars of the analyzed IM) (t )   (t )dt – the value of the rotor rotation
angle in relation to the stator;  – the angle of shift of stator winding
coils. In this case functions A1(t), A2(t), A3(t) are analytically written
down in such a way that they take into account the availability of some
excitations in the signal. These excitations are caused by broken rotor
bars and are repeated in a period of rotor rotation (Figure 2.12).
The shape of the coil EMF test signal may influence the choice of
the wavelet function. So, research of EMF test signals was carried out
in the presence of excitation imitating:
 one broken bar;

 two adjacent broken bars;
 two broken bars shifted in relation to each other by a certain
shift angle.
E test 1
E test 2

E test 3

Figure 2.11. The vectors of the coils testing EMF.

A1, A2, A3,
r. u.
1.8
1.2
A1 ( t )
0.6 A2 ( t )
A3 ( t )
t, s
0 0.008 0.016 0.024 0.032
Figure 2.12. Functions assigning the rotor breakage and a stepped appearance.
The results of research of test signals in the presence of one broken

bar are shown in Figures 2.13–2.16, with two broken bars – in
appendix B.
Test signal of the total EMF of the stator winding coil group is
calculated with the use of expression:
etest  t   etest1  t   etest 2  t   etest 3  t  .

(2.17)
The test signals approximate to the anticipated stator winding coils

EMF are shown in Figure 2.13. It is assumed that the shift angle
between coils  equals 10 degrees.
To choose the wavelet fundamental function the wavelet-transform

of the coils EMF test signals is performed with the use of Daubechies,
Symlet and Coiflets wavelet-bases. The results of the research with the
imitation of one broken bar are given in Figures 2.14–2.16. The results
of wavelet-transform of coils EMF test signals with the imitation of two
broken bars (located at a different angle in relation to each other) are
given in appendix B. The analysis of the obtained wavelet-spectra with
the use of different wavelet-bases revealed that the application of
Coiflets wavelet, due to the properties of the basis itself, is considerably
excessive, which ambiguously influences the reliability of the
diagnostics results. This basis is asymmetric.
The use of Daubechies and Symlet wavelets allows the
determination of excitation in coils EMF test signals that appear on
wavelet-spectrum in the form of typical sections with wavelet-
coefficients. It is seen in Table 2.1. that the obtained wavelet-spectra
with Daubechies and Symlet bases are identical. It can be explained by
the fact that at N  2 Daubechies and Symlet wavelets are of the same
type and differ only in the sign of the psi-function. Daubechies and
Coiflets wavelets are asymmetric and insufficiently periodic. As
mentioned above, Symlet wavelets are to some extent approximate to
symmetric ones (Figure 2.9, b). So, it is possible to come to the
conclusion that it is expedient to use Symlet wavelet-basis when there
are superimposed excitations in the test signals of windings EMF.
etest1, etest2, etest3, e te s t 1 ( t )

r. u.
e te s t 2 ( t )
1.8
e te st 3 ( t )
0.6
t, s
0 0.008 0.016 0.024 0.032
-0.6
-1.8
Figure 2.13. The test signals of stator winding coils EMF, approximate to the
anticipated EMF signals.
Figure 2.14. The test signal of coil EMF etest1 with superimposed excitation that
corresponds to one broken bar and its wavelet-spectrum with the use: of Daubechies
wavelet (a), of Symlet wavelet (b), of Coiflets wavelet (c).
Symlet wavelet can be presented in the form [65]:
ˆ     ei / 2 m0   / 2   ˆ   / 2  ,
 (2.18)
where  K , ̂    – scaling function:

ˆ      2 
1/ 2

 m0 2 j ;
j 1
 (2.19)
m0    – trigonometric polynomial:
N
 1  ei 
m0      L  ,
 2  (2.20)
 
where N – wavelet order; L    – filter with a finite impulse response

(FIR-filter), assigned in the form:
 sin 2 ξ 
L ξ   P  ; (2.21)
 2 
 
P  y  – polynomial of the type:
N 1  N 1 k  k
P y   y . (2.22)
k 0  k 
Thus, to analyze the stator winding EMF signals it is expedient to

use Symlet wavelet as WB. The wavelet-spectrum of test signal of the
stator winding coil group EMF is obtained with the use of Symlet
wavelet (Figure 2.17).
Figure 2.17. The test signal of stator winding coil group EMF etestΣ and its
wavelet-spectrum.
Unlike EMF signals in measuring windings (Figure 2.2), the

characteristic signs of damages in the form of particular sections
superimpose in wavelet-spectra during the analysis of EMF signals of
the coil group. Because of this there appears a necessity for the analysis
of factors influencing the generation of electromotive force in IM stator
winding elements.
2.5. THE RESEARCH OF FACTORS INFLUENCING

THE GENERATION OF ELECTROMOTIVE FORCE IN
INDUCTION MOTOR STATOR WINDINGS
The performed analysis of different types of IM stator windings

revealed that electric machines with distributed windings are more
complex in manufacture than machines with concentrated windings
[45]. Nevertheless, distributed windings are mainly used in alternating
current machines. It can be explained by the fact that the useful
transform of energy in most alternating current machines is performed
by the first harmonic of EMF, current and induction; higher harmonics
cause additional power losses. Concentrated windings do not provide
the change of magnetomotive force (MMF) (and, correspondingly,

induction) in space and EMF in time that would be approximate to
sinusoidal law.
For a stator winding distributed in space the number of slots per a
pole and a phase q  1 , i.e., the number of turns necessary under each
pair of poles is not concentrated in one winding but is distributed across
several ( q  1 ) smaller windings connected in series, located in q
adjacent slots. Such element made of q coils is called a coil group. In
this case all q coils are connected in series so that the beginning of
every coil is connected to the end of the previous one.
A concentrated winding is an EM winding created by q coils
whose sides are located in the same slots (Figure 2.18, а).
MMF MMF
a b
Figure 2.18. The simplified MMF curves of concentrated (а) and distributed
(b) windings.
The magnetomotive force (MMF) curve of such concentrated

winding is approximate to a rectangular and, apart from the first
harmonic, it contains a spectrum of higher-order harmonics.
If these coils are located, one at a time, in q adjacent slots, MMF
curve (Figure 2.18, b) will be of the shape of a stepped trapezoid. The
higher harmonics of such MMF are much less marked than in a
rectangular curve. Nevertheless, the total EMF of the distributed

winding will be less than of the concentrated one. The axes of
distributed coils in q adjacent slots are shifted in relation to each other
by electrical angle Z  2p / Z radian. EMF vectors are shifted in
relation to each other by the same angle, so, the total EMF will be equal
not to an algebraic but to a geometrical sum of the EMF of all the coils
making the group, i.e., Eq   Ec . Relation Eq of the distributed
winding to the calculated EMF equal to the product of the number of
coils and the EMF of each of them qEc , is called the distribution
coefficient.
Thus, the value of EMF in the stator distribution winding will differ
from the value of EMF in the concentrated winding by k p times. In
this case winding distribution in slots causes decrease of EMF
amplitude.
In modeling design features of the analyzed IM, namely, parameters
of IM stator winding, were taken into account:
 the winding type – single-layer lap winding;

 the number of poles 2p=4;
 the number of stator slots Z1=36;
 the number of slots per a pole and a phase q=3;
 the number of winding parallel paths a=1;
 the winding pitch in slots y=9;
 the number of slots between the coils of adjacent phases
of the winding λ=6.
Every phase of the stator winding consists of two coil groups, each
of which, in its turn, contains three coils. A winding phase coil is
formed by a group of turns connected in series and put into the same
slots.
Let us consider the generation of the EMF of IM stator winding

phase.
The coil EMF is determined by expression [45]:
Ec  wc Et , (2.23)
where wc – the number of the coil turns; Et – the EMF of the winding
turn.
Ec 2
Ec1 Ec 3
 
/2 /2
  q
q 3
0
a b
Ec 3
 Eq
Ec 2
 
0

Ec1
Figure 2.19. Coil group in the magnetic field (а), coils EMF vectors (b) and vector
diagram for determination of EMF of the coil group (c) of IM stator winding.
A coil group contains a number of coils with the same number of

turns located in the adjacent slots. The coils are connected in series and
belong to one phase of the winding (Figure 2.19, а). The coils in the
coil group are shifted by electric angle  , correspondingly, the EMFs of
these coils are shifted by the same angle (Figure 2.19, b). EMF Eq of
the coil group is equal to the geometric sum of the EMF of separate
coils of this group (Figure 2.19, c).
The designations in Figure 2.19: B – the amplitude of magnetic
induction of the fundamental harmonic of the field in the air gap;  –
pole pitch; α – the angle of phase zone.
In the general case the EMF of the stator winding phase is equal to
the geometric sum of the EMF of all coil groups forming this phase.
For the analyzed IM the EMF of the winding phase is equal to the sum
of EMFs of six coils:
. 6 .
E ph   Eci , (2.24)
i 1
.
where E ci – the EMF of the i-th coil; i – coil number.
Coils EMFs are shifted in relation to each other. So, information
signs available in the signals of the EMF of every separate coil (caused
by the presence of broken rotor bars) are mutually superimposed when
EMFs are summed up.
Another factor influencing the amplitude of EMF signal is rotor
slots skewing. In a small machine in which increase of q is
complicated skewed slots are made to eliminate stepped harmonics.
Stator or rotor slots are located not parallel to the machine axis but
under a certain angle  bev to it, which is called a skew angle.
The slots skew is assessed in linear bbev or relative bev dimensions
(Figure 2.20). These dimensions demonstrate by how many millimeters
or by what part of the step pitch along the air gap circular arc the
direction of the slot axis is changed in comparison with its position at

non-skewed angles.
stator slots
bbev
rotor slots
Figure 2.20. The skew of rotor slots in relation to stator slots.
The central angle determined by an arc equal to bbev , is called a

skew angle:
 cr  bbev  /  . (2.25)
The slots skew decreases the value of the EMF induced in the
winding turns.
Thus, the skew of rotor slots, as well as distribution of stator
winding by the slots, causes the decrease of EMF amplitude in the
winding.
2.6. THE GENERALIZED METHODS OF THE ANALYSIS

OF INDUCTION MOTOR BROKEN ROTOR BARS
The performed research enabled making a conclusion about the

expediency of carrying out diagnostics with the use of wavelet-analysis
of EMF signal in the stator windings in IM self-running-out mode.
Figure 2.21. The position of the proposed method for IM broken bars diagnostics in the general classification.
The position of the proposed method of IM broken rotor bars

diagnostics can be presented in the general classification in the form
(Figure 2.21).
According to Figure 2.21, the basic points of the method are
determined. These points differ it from other methods and demonstrate
basic advantages over the known methods of broken rotor bars
diagnostics.
2.6.1. The Basic Points of the Diagnostics Method
The method of broken rotor bars diagnostics by the analysis of EMF

in IM stator windings provides for diagnostics without the removal of
the motor from the technological process. It is especially useful under
the conditions of technological processes whose long-term stop is
impossible and may result in considerable equipment outage or
emergencies.
The use of the method does not require the availability of input test
impacts, which simplifies diagnostics performance and does not require
the installation of additional sources of input action.
The method provides for the performance of diagnostics in the self-
running-out mode. Consequently, there is no need to take into account
the level of load, which eliminates the necessity for installation of load
devices. Besides, the performance of diagnostics in IM self-running-out
mode eliminates the influence of such disturbing factors as low quality
of the supply mains and operation of the technological mechanism.
The EMF induced in the stator windings by damping currents in IM
self-running-out mode is proposed as a diagnostic signal. In this case
the installation of additional measuring coils or hall-effect sensors is not
required. During the diagnostics it is only necessary to connect voltage
sensors to measure EMF in IM stator windings.
To process diagnostic signals the method provides for the use of

wavelet-transform. It enables the improvement of diagnostics efficiency
due to the analysis of signals both in the frequency and the time
domain. This condition is achieved by the determination of relative
location of broken rotor bars.
2.7. CONCLUSION
The performed analysis of IM test conditions made it possible to

substantiate the use of the motor self-running-out mode for the
diagnostics of the rotor broken bars. Self-running-out condition allows
carrying out diagnostics without withdrawal of the motor from the
technological process and its disassembling, eliminates the supply
mains low quality influence on the diagnostics results and does not
depend on the previous condition of the motor operation.
It is shown that EMF signal in the stator windings is a derivative of
IM electromagnetic field signals and contains a tooth kink. It enables
the comparison of IM electromagnetic field lines with geometric
location of the rotor teeth. It is experimentally proved that EMF signal
in the stator additional measuring winding contains information signs of
the rotor broken bars in the form of distortions of the signal shape. It is
found out that the EMF induced in IM stator windings by the rotor
damping currents can be used in the self-running-out mode as a
diagnostic signal for the determination of IM rotor broken bars.
The performed analysis of the methods for electromagnetic field
calculation allowed the substantiation of the use of numerical methods.
It is found out that to determine EMF signals in IM stator windings in
the self-running-out condition it is necessary to calculate
electromagnetic field in IM cross-section using the finite element
method.
A comparative analysis of the methods for processing the

diagnostics signals made it possible to determine the disadvantages of
Fourier transform for the diagnostics of IM rotor broken bars. It is
proved that that the use of spectral analysis methods does not allow the
unambiguous determination of the number and relative position of the
rotor broken bars. It is found out that it is expedient to use wavelet-
transform as a method for processing the EMF signal in IM stator
windings.
The performed analysis of wavelet-bases, taking into account their
typical features in the time and frequency domains made it possible to
find out that it is expedient to use orthogonal wavelets with a compact
support for the analysis of EMF signals in IM stator windings.
The analysis of design features of various IM windings revealed,
that the generation of EMF in windings can be influenced by such
factors as: the number of the motor poles pairs, the circuit of coil
groups connection in the winding phase and the type of the stator
winding. These factors contribute to the mutual superimposition of
information signs of the rotor broken bars in EMF signal.
Chapter 3
MATHEMATICAL MODELS FOR

THE RESEARCH OF THE METHOD
OF INDUCTION MOTOR BROKEN ROTOR
BARS DIAGNOSTICS
A number of mathematical models are used for the research of the

method of IM broken rotor bars diagnostics on the basis of the stator
winding EMF analysis: IM circuit mathematical models and a
mathematical model using the finite element method. Relation between
mathematical models, input and output data are shown in Figure 3.1 in
the form of a block diagram.
Published data IM IM circuit mathematical models

Number of the rotor For determination
bars of distribution of currents in
IM circuit mathematical model the rotor bars at the moment
Value of in a three-phase coordinates systems of IM disconnection from
electromagnetic with presentation of the rotor the supply mains taking into
parameters of IM as a system of short-circuited bars account the mode of motors
stator and rotor operation
windings
Rotor geometric
parameters Currents in the rotor bars
For specification
Number of the rotor of distribution of currents
bars in the rotor bars at the
Circuit mathematical model on the basis moment of IM disconnection
Resistance of the of IM rotor equivalent circuit from the supply mains taking
rotor bars and into account the broken bars
ShCR relative position
Specified values of currents

IM geometric
in the rotor bars
parameters
For determination
Magnetic and of instantaneous values
electric properties Mathematical model of a two- of the vector magnetic
of IM materials dimensional electromagnetic field in IM potential in the stator
cross section with the use of FEM windings by the results
of calculation of magnetic
Parameters of model field in IM cross section
calculation (number
of rotor rotation Instantaneous values of vector
periods, angle of magnetic potential in IM stator
rotor rotation) windings
Calculation of electromotive force

in IM stator windings
Figure 3.1. Relation between mathematical models during the research

of the method for IM broken rotor bars diagnostics.
Mathematical Models for the Research of the Method … 71
3.1. THE CREATION OF AN INDUCTION MOTOR

MATHEMATICAL MODEL IN A THREE-PHASE
COORDINATE SYSTEM FOR THE DETERMINATION
OF CURRENTS IN THE ROTOR BARS AT THE MOMENT
OF MOTOR DISCONNECTION FROM THE SUPPLY MAINS
To calculate the electromagnetic field of IM in self-running-out

mode it is necessary to know the initial values of currents in the rotor
bars at the moment of motor disconnection from the supply mains.
Currents values can be obtained with the use of IM mathematical model
in a three-phase coordinate system.
When stator windings EMF is calculated in the self-running-out
mode it is necessary to take into account current attenuation in the rotor
bars. Rotor current at the moment of IM disconnection from the supply
mains is determined by expression:
I 2beg  I 2(t 0)  k2 I1(t 0) , (3.1)
where I 2(t 0) – the rotor current of the previous steady condition at the
moment of IM disconnection from the supply mains; k2 – rotor
coupling coefficient; I1(t 0) – the stator current of the previous steady
condition at the moment of IM disconnection from the supply mains.
At the following time moments the currents in the rotor bars change
according to exponential law with time constant:
  L2 / R2 , (3.2)
where L2 – rotor inductance; R2 – rotor resistance.

It is known that, in case when one or several rotor bars are broken
in IM, the currents in the rotor bars redistribute. Besides, magnetic flux
distribution around the broken bar changes – the flux increases at one
end of the bar and decreases at the other end.
To determine the initial values of the current in the rotor bars at the
moment of motor disconnection from the supply mains an IM
mathematical model in a three-phase coordinate system is improved.
The model is based on the known IM mathematical model in a
three-phase coordinate system [66–67]. The improvement of the known
model consists in the fact that the rotor is simulated in the form of a
system of short-circuited bars. Besides, IM electromagnetic part is
presented as a system of magneto-connected windings located on the
stator and rotor.
The system of equations of electric balance of the stator circuit is of
the form:
 d  A (t )
u A (t )  i A (t ) RA  dt ;

 d  B (t )
u B (t )  iB (t ) RB  ; (3.3)
 dt
 d C (t )
uC (t )  iC (t ) RC  dt ,

where u A (t ), uB (t ), uC (t ) – stator voltages; iA (t ), iB (t ), iC (t ) – stator

currents;  A (t ),  B (t ), C (t ) – stator phases flux linkages;
RA  RB  RC  Rs – stator phases resistances.
The system of equations of electric balance of the rotor circuit:
 d a t 
0  ia  t  Ra  ;
 dt
 d b  t 
0  ib  t  Rb  ; (3.4)
 dt
 d c  t 
0  ic  t  Rc  ,
 dt
where ia  t  , ib  t  , ic  t  – rotor currents; a  t  , b  t  , c  t  – rotor
phases flux linkages; Ra  Rb  Rc  Rr – rotor phases resistances.
Taking into account the number of bars of the analyzed IM rotor the
system of equations of electric balance of the rotor circuit takes the
form:
 d 1 (t )
0  i1 (t ) R1  dt ;

0  i (t ) R  d  2 (t ) ;
 2 2
dt

0  i (t ) R  d  3 (t )
3 3 ;
 dt (3.5)
...

0  i (t ) R  d  Z 1 (t ) ;
 Z 1 Z 1
dt

0  i (t ) R  d  Z ( t )
,
 Z Z
dt
where Z  1,2... – rotor bars ordinal numbers.

The flux linkage of every phase of IM stator and rotor depends on

the value of the winding own inductance and mutual inductance with all
the other windings.
For stator phase A:
 A  LAiA  M ABiB  M AC iC  M A1i1  M A2i2  M A3i3 

(3.6)
...  M A Z 1iZ 1  M AZ iZ ,
where LA – phase inductance; M xy – mutual inductance between

windings х and y.
Mutual inductances between windings:
M AB  M AC  M BC  M S (3.7)
M12  M 23  M 34  ...  M  Z 1 Z 

 M13  M14  M15  ...  M 1 Z 1  M1Z 
(3.8)
 M 24  M 25  M 26  ...  M 2 Z 1  M 2 Z 
 ...  M 30 Z 1  M 31Z  M r .
The relative spatial position of rotor and stator windings changes,

which results in the change of the value of mutual inductance between
the windings.
The maximum value of mutual inductance corresponds to the
coincidence of the axes of two phases, and when the axes are located
perpendicularly to each other, mutual inductance is equal to zero. That
is why mutual inductance between the stator and rotor windings will
change according to the harmonic law for stator phase A:
M A1  M cos ;
 2 
M A2  M cos    ;
 Z 
 2 
M A3  M cos    2  ;
 Z 
(3.9)
...
 2 
M A Z 1  M cos     Z  2   ;
 Z 
 2 
M AZ  M cos     Z  1  ,
 Z 
where M – the maximum value of mutual inductance;  – rotor

rotation angle.
For stator phase B:
 2 
M B1  M cos    ;
 3 
 2 2 
M B 2  M cos     ;
 3 Z 
 2 2 
M B 3  M cos     2 ;
 3 Z  (3.10)
...
 2 2 
M B Z 1  M cos      Z  2  ;
 3 Z 
 2 2 
M BZ  M cos      Z  1  .
 3 Z 
For stator phase C:

 2 
M C1  M cos    ;
 3 
 2 2 
M C 2  M cos     ;
 3 Z 
 2 2 
M C 3  M cos     2 ;
 3 Z  (3.11)
...
 2 2 
M C  Z 1  M cos      Z  2 ;
 3 Z 
 2 2 
M CZ  M cos      Z  1  .
 3 Z 
Taking into account the written expressions, the equations of flux

linkage for stator phase A take the form:
 2 
 A  LAi A  M s iB  M s iC  Mi1 cos   Mi2 cos    
 Z 
 2   2 
 Mi3 cos    2   ...  MiZ 1 cos     Z  2    (3.12)
 Z   Z 
 2 
 MiZ cos     Z  1  .
 Z 
Taking into consideration iA  iB  iC  0 :
 2 
 A   LA  M s  i A  Mi1 cos   Mi2 cos    
 Z 
 2   2 
 Mi3 cos    2   ...  MiZ 1 cos     Z  2    (3.13)
 Z   Z 
 2 
 MiZ cos     Z  1  .
 Z 
Flux linkages for phases B and C:
 2   2 2 
 B   LB  M s  iB  Mi1 cos      Mi2 cos     
 3   3 Z 
 2 2   2 2 
 Mi3 cos     2   ...  MiZ 1 cos      Z  2  
 3 Z   3 Z 
 2 2 
 MiZ cos      Z  1  .
 3 Z 
 2   2 2 
 C   LC  M s  iC  Mi1 cos      Mi2 cos     
 3   3 Z 
 2 2   2 2 
 Mi3 cos     2   ...  MiZ 1 cos      Z  2  
 3 Z   3 Z 
 2 2 
 MiZ cos      Z  1  .
 3 Z 
Then the equation of derivative from flux linkage of stator phase A:
d A
  dtA  M dt1 cos   M i1 sin  
di di
 LA  M s
dt
di2  2   2 
M cos      M i2 sin    
dt  Z   Z 
di  2   2 
 M 3 cos    2   M i3 sin    2  
dt  Z   Z 
diZ  1  2 
...  M cos     Z  2   
dt  Z 
 2 
 M iZ  1 sin     Z  2   
 Z 
di  2   2 
 M Z cos     Z  1   M iZ sin     Z  1  ,
dt  Z   Z 
where  is determined by the differentiation of equation     t  dt .

Analogously, the equations of derivatives from flux linkages of

stator phases B and C:
dB di di  2   2 
  LB  M s  B  M 1 cos      M i1 sin    
dt dt dt  3   3 
di  2 2   2 2 
 M 2 cos       M i2 sin     
dt  3 Z   3 Z 
di3  2 2   2 2 
M cos     2   M i3 sin    2 
dt  3 Z   3 Z 
diZ 1  2 2 
...  M cos      Z  2  
dt  3 Z 
 2 2 
 M iZ 1 sin      Z  2  
 3 Z 
di  2 2   2 2 
 M Z cos      Z  1   M iZ sin      Z  1  .
dt  3 Z   3 Z 
d C di di  2   2 
  LC  M s  C  M 1 cos      M i1 sin    
dt dt dt  3   3 
di  2 2   2 2 
 M 2 cos       M i2 sin     
dt  3 Z   3 Z 
di  2 2   2 2 
 M 3 cos     2   M i3 sin    2 
dt  3 Z   3 Z 
di  2 2 
...  M Z 1 cos      Z  2  
dt  3 Z 
 2 2 
 M iZ 1 sin      Z  2  
 3 Z 
di  2 2 
 M Z cos      Z  1  
dt  3 Z 
 2 2 
 M iZ sin      Z  1  .
 3 Z 
The basic difference of IM mathematical model from the known

one [66–67] consists in presentation of the rotor in the form of a system
of short-circuited bars. This fact was taken into account during the
creation of the complete equations of electric balance for rotor circuits.
That is why a system of equations of the rotor circuit will consist of Z
equations; their number corresponds to the number of IM rotor bars.
Equation of flux linkage of the rotor bars:
 2   2 
1   L1  M r  i1  Mi2 cos      Mi3 cos    2  
 Z   Z 
 2   2 
...  MiZ 1 cos     Z  2    MiZ cos     Z  1  
 Z   Z 
 2   2 
 Mi A cos   MiB cos      MiC cos    .
 3   3 
 2   2 
 2   L2  M r  i2  Mi1 cos      Mi3 cos    
 Z   Z 
 2   2 
 Mi4 cos    2   ...  MiZ 1 cos     Z  3  
 Z   Z 
 2   2 
 MiZ cos     Z  2    Mi A cos    
 Z   Z 
 2 2   2 2 
 MiB cos       MiC cos     .
 3 Z   3 Z 
…
 2   2 
 Z   LZ  M r  iZ  Mi1 cos      Mi2 cos    2  
 Z   Z 
 2   2 
...  MiZ  2 cos     Z  2    MiZ 1 cos     Z  1  
 Z   Z 
 2   2 2   2 2 
 Mi A cos      MiB cos       MiC cos     .
 Z   3 Z   3 Z 
The value of shift angles between the rotor bars can be written
down in the following general form:
   N  1 K   ph , (3.14)
where N – the ordinal number of the rotor bar; K  2 / Z – a spatial

angle between the rotor bars;  ph – a spatial angle between the
corresponding phase of the stator and the n-th bar of the rotor (for stator
phase A:  ph  0 , for stator phase B:  ph  2 / 3 , for stator phase
C:  ph  4 / 3 ).
Then the equation of derivative from the flux linkage of one bar of
the rotor:
d 1 di di  2   2 
  L1  M r  1  M 2 cos      M i2 sin    
dt dt dt  Z   Z 
di  2   2 
 M 3 cos    2   M i3 sin    2  
dt  Z   Z 
diZ 1  2   2 
...  M cos     Z  2    M iZ 1 sin     Z  2   
dt  Z   Z 
diZ  2   2 
M cos     Z  1   M iZ sin     Z  1  
dt  Z   Z 
di di  2 
 M S A cos   M S i A sin   M S B cos    
dt dt  3 
 2  diC  2 
 M S iB sin      MS cos    
 3  dt  3 
 2 
 M S iC sin    .
 3 
The equations of electrical balance for all the rotor bars are obtained
analogously.
Stator phases voltage:
UA=Umcosγ;
UB=Umcos(γ+2π/3); (3.15)
UC=Umcos(γ–2π/3);
where U m  2U n – the amplitude value of stator phases voltage.

The mechanical part of the mathematical model is represented by an
equation for electromagnetic torque and an equation of rotor motion
[68].
Electromagnetic torque equation:
2p
Me   C   B  iA    A  C  iB    B   A  iC  . (3.16)
3 3
Rotor motion equation:
d 1
 p  Me  Mc  . (3.17)
dt J
Taking into account the above said, the block diagram of the
mathematical model will be of the form (Fig. 3.2). The block diagrams
of separate blocks of IM mathematical model are given in appendix B.
UA,UB,UC IA,IB,IC Ir1

M(t)
IA,IB,IC
Ir2
Ir1,Ir2...ІrZ STATOR ΨA,ΨB,ΨC ROTOR MECHANICAL ω(t)
... PART
ІrZ ΨA,ΨB,ΨC
γ(t) γ(t)
Figure 3.2. The block diagram of IM mathematical model for the determination of
currents in rotor bars at the moment of motor disconnection from the supply network.
The above given expressions are true for IM operating mode. In the
motor self-running-out mode the flux linkages for every phase of the
stator will depend only on the value of currents in the bars of the rotor
that continues rotating:
 2   2 
 A  Mi1 cos   Mi2 cos      Mi3 cos    2  
 Z   Z 
 2   2 
...  MiZ 1 cos     Z  2    MiZ cos     Z  1  .
 Z   Z 
 2   2 2 
 B  Mi1 cos      Mi2 cos     
 3   3 Z 
 2 2 
 Mi3 cos     2   ... 
 3 Z 
(3.18)
 2 2 
 MiZ 1 cos      Z  2  
 3 Z 
 2 2 
 MiZ cos      Z  1  .
 3 Z 
 2   2 2 
 C  Mi1 cos      Mi2 cos     
 3   3 Z 
 2 2   2 2 
 Mi3 cos     2   ...  MiZ 1 cos      Z  2  
 3 Z   3 Z 
 2 2 
 MiZ cos      Z  1  .
 3 Z 
The flux linkages for rotor bars in the self-running-out mode:

 2   2 
1   Lr  M r  i1  Mi2 cos      Mi3 cos    2  
 Z   Z 
 2   2 
...  MiZ 1 cos     Z  2    MiZ cos     Z  1  .
 Z   Z 
 2   2 
 2   Lr  M r  i2  Mi1 cos      Mi3 cos    
 Z   Z 
 2   2 
 Mi4 cos    2   ...  MiZ 1 cos     Z  3  
 Z   Z 
 2 
 MiZ cos     Z  2   .
 Z 
…
 2   2 
 Z   Lr  M r  iZ  Mi1 cos      Mi2 cos    2  
 Z   Z 
 2   2 
...  MiZ 2 cos     Z  2    MiZ 1 cos     Z  1  .
 Z   Z 
The mathematical model is rather extensional and contains a lot of
cross links, but it has a number of advantages:
 a possibility for the research of operating conditions of IM with

any number of the broken rotor bars;
 a possibility for the research of operating conditions of IM
taking into account the location of the rotor bar breakage;
 a possibility for the analysis and research of any conditions of
motor operation, such as a no-load operation, a load mode, a
self-running-out mode, etc.;
 the broken rotor bars are simulated by equating the current
flowing through the broken bar to zero.
According to the results of modeling for IM of АIR80V4U2 type

the current transient processes in several IM rotor bars are given:
1. At no-load start and the motor self-running-out mode (Figure

3.3).
2. At no-load start, subsequent load-on ( t  0.3 s) and the motor
self-running-out mode ( t  0.3 s) (Figure 3.4).
The analysis of modeling results revealed that at the moment of IM

disconnection from the supply mains both in no-load mode and under
load there occur a considerable surge of amplitudes of rotor bars
currents. Later, when IM is disconnected, rotor bar currents change
according to the exponential law.
Ir, A
1.035
0.623
Ir, A 0.212
0.473 0.501 0.53 0.559 0.588 t, s

20
0 0.15 0.3 0.45 t, s
 20
Figure 3.3. The transient processes of currents in IM rotor bars at no-load start and
the motor self-running-out mode ( t  0.5 s).
Ir, A
 0.391 0.476 0.503 0.53 0.557 0.584 t, s
 1.608
Ir, A
 2.824
20
0 0.15 0.3 0.45 t, s
 20
Figure 3.4. The transient processes of currents in IM rotor bars at no-load start,
subsequent load-on ( t  0.3 s) and the motor self-running-out mode ( t  0.5 s).
Ir,A
0 10 20 30 Z
-1
Figure 3.5. The distribution of currents in the rotor bars of a healthy IM at

the moment of motor disconnection from the supply mains.
The distribution of currents in the rotor bars of a healthy IM at the

moment of motor disconnection from the supply mains (when
previously IM operated under no-load mode) is given in Figure 3.5.
Figure 3.6 shows the distribution of currents in the rotor bars of IM
with one broken bar at the moment of motor disconnection from the
supply mains. Thus, an IM mathematical model in a three-phase
coordinate system with representation of a rotor as a system of short-
circuited bars allows obtaining distribution of the instantaneous values
of currents in the rotor bars at the moment of motor disconnection from
the supply mains. In this case it is possible to obtain current distribution
for both the previous IM operation in a no-load mode and for operation
under load. However, the mathematical model does not enable separate
consideration of the bar and a part of the short-circuited ring as winding
elements. The mathematical model does not take into account the
number of the stator poles pairs either.
Ir,A
0 10 20 30 Z
-1
Figure 3.6. The distribution of currents in the rotor bars of an IM with one broken
bar at the moment of motor disconnection from the supply mains.
3.2. WORKING OUT A CIRCUIT MODEL OF AN INDUCTION

MOTOR ROTOR FOR THE SPECIFICATION OF CURRENTS
IN THE ROTOR BARS AT THE MOMENT OF MOTOR
DISCONNECTION FROM THE SUPPLY MAINS
A circuit model of IM rotor is developed for the specification of the

initial currents in the rotor bars at the moment of motor disconnection
from the supply mains. Apart from the resistances of the bars, the
resistances of short-circuited rings are taken into consideration in the
mathematical model. The number of rotor electric circuits in the model
corresponds to the number of bars [49].
The research was carried out for two IMs: АIR80V4U2 type, 1.5
kW and 4АN200L2UЗ type, 75 kW (the motor rated data are given in
appendix А). The results of calculation for IM of the power of 1.5 kW
are given below. IM rotor equivalent circuit is given in Figure 3.7.
E1 Zb1
Zscr1 Zscr1
E2 Zb2
Zscr2 Zscr2
E3 Zb3
Zscr3 Zscr3
E34
Zb34
Zscr34 Zscr34
Figure 3.7. IM rotor equivalent circuit.

In Figure 3.7: E1 E34 – the EMF of power supply for the

investigated IM; Zb1 Zb34 – the impedance of rotor bars;
Z scr1 Z scr 34 – the impedance of short-circuited ring.
IM rotor equivalent circuit parameters are calculated taking into
account the geometric parameters of the rotor and transformation
coefficient:
4
 bar resistance Rb.r.  1.985 10 Ohm;
4
 bar inductive reactance X b  2.158 10 Ohm;
5
 short-circuited ring resistance Rscr  1.472 10 Ohm.
Rotor bar EMF changes by the sinusoid law and is written down in
a complex form as:
E  I b Zb , (3.19)
where I b – rotor bar current.

Rotor bar impedance in a complex form:
Zb  Rb.r.  jX b . (3.20)
In accordance with the given IM rotor equivalent circuit the broken

rotor bars are simulated as a complete loss of electric contact with the
rotor short-circuited ring. The location of the broken bars in the model
corresponds to the location of the artificially introduced breakages of
the rotor at the experimental sample of the analyzed IM (АIR80V4U2).
Artificially introduced rotor breakages are obtained by means of holes
drilled on the rotor of the experimental IM (in Figure 3.8 they are
designated by numbers 1, 2 and 14). It enabled the disturbance of
electric coupling between the bars and imitation of a breakage. A circuit
model can be used for modeling rotors for IMs of different power with
different number of broken bars. The distribution of currents in the
rotor bars at the moment of motor disconnection from the supply mains
for a healthy IM and an IM with one, two and three broken bars is
shown in Figures 3.9.–3.12.
1 2
14
Figure 3.8. The location of broken bars on the IM rotor.
Ib, A
15
10
5
0
0 5 10 15 20 25 30 35 Z2
-5
-10
-15
Figure 3.9. The distribution of currents I b in the rotor bars for a healthy IM at
the moment of motor disconnection from the supply mains.
Ib, A
15
10
5
0
5 10 15 20 25 30 Z2
-5
one broken bar
-10
-15
Figure 3.10. The distribution of currents I b in the rotor bars for an IM with one
broken bar at the moment of motor disconnection from the supply mains.
Ib, A
15
10
5
0
0 5 10 15 20 25 30 Z2
-5
-10 two broken bars
-15
Figure 3.11. The distribution of currents I b in the rotor bars for an IM with two
broken bars at the moment of motor disconnection from the supply mains.
Figure 3.12. The distribution of currents I b in the rotor bars for an IM with three
broken bars at the moment of motor disconnection from the supply mains.
As the obtained results show, the distribution of instantaneous

currents of a healthy rotor bars at the moment of disconnection from the
supply mains is of a sinusoid form. If there is a breakage, the current in
the broken bar is equal to zero and the currents in the other bars of the
rotor are redistributed. It meets the first Kirchhoff’s law. The obtained
values of currents in the rotor bars at the moment of motor
disconnection from the supply mains are used for the calculation of IM
electromagnetic field. Thus, the developed circuit mathematical model
makes it possible to obtain the value of currents in IM rotor bars at the
moment of motor disconnection from the supply mains with different

number of broken bars and taking into account their geometric location.
3.3. A MATHEMATICAL MODEL WITH THE USE OF

THE FINAL ELEMENT METHOD FOR THE CALCULATION
OF INDUCTION MOTOR ELECTROMAGNETIC FIELD IN
THE SELF-RUNNING-OUT MODE
It is known that electric machines mathematical modeling with the

use of the classical field theory allows the research of their
characteristics on the basis of the analysis of simplified circuit models.
The analysis of the steady modes of IM operation is performed on the
basis of the results of calculation of their electromagnetic field. The
analysis of the transient and dynamic modes is performed with the help
of circuit-field mathematical models (CFMM). They are based on the
general solution of differential equations of windings electric circuits
and the equations of non-stationary electromagnetic field in IM active
zone [50].
Most often, the electromagnetic field within the limits of the motor
steel active package is plane-parallel, i.e., in any cross-section of the
machine the magnetic field pattern is constant. The use of FEM in IM
calculation makes it possible to rather accurately take into account the
geometry of the motor in its cross-section taking into consideration the
non-linearity of magnetizing curve and other features such as, for
example, possible defects of IM magnetic and turns system.
A mathematical model of a two-dimension electromagnetic field in
IM cross-section, based on FEM, is proposed in the monograph for the
assessment of the broken rotor bars influence. The equations for the
field are given in Chapter 2 (expressions 2.7–2.8).
The calculation of the electromagnetic field in IM cross-section was
performed with the use of Femm software package for the calculation
of two-dimension fields on the basis of FEM [69]. The software

package enables the solution of both linear and non-linear problems and
its specific feature consists in the simplicity of the use and rather high
operation speed.
An IM model taking into account the motor geometry, the magnetic
and electrical properties of its active materials is developed for the
numerical calculation of electromagnetic field. The geometric
parameters of the analyzed IMs are given in appendix A.
The design of IM made of real materials is characterized by a
number of features: geometry deviation from symmetry, the
heterogeneity of properties (the deviation of magnetic and electrical
properties from the set values) etc. That is why when a model is
created, basic assumptions determining the degree of idealization of the
properties of design physical and geometric characteristics are taken
into account.
The following assumptions were made during the creation of the
model:
 the electromagnetic field within IM active space is plane-

parallel;
 the dependence between electromagnetic field induction and
strength is linear;
 the IM magnetic circuit resistance is infinite, i.e., eddy currents
in steel are absent;
 the rotor “squirrel cage” joint rings have zero resistance;
 the surface effect in two-dimension electromagnetic field in the
cross-section is taken into account by the introduction of
boundary conditions;
 the rotor slots slants are not taken into account in the model;
 a broken bar is considered as a complete disturbance of the
electric coupling with the rotor short-circuited ring.
Magnetic calculation by the finite element method consists of the

following stages:
1. The development of the model geometry.

2. The choice of elements types, the introduction of materials
properties, the assigning of materials and elements properties to
geometric areas.
3. The assumption of the initial values of currents in the rotor bars.
4. The assumption of the boundary conditions. For the model
external boundaries the Dirichlet condition is used. In this case
the field vector magnetic potential is assumed equal to zero
(A=0). Zero Dirichlet condition determines the behavior of the
normal component of magnetic induction at the model
boundary.
5. Decomposing the model areas into a grid of finite elements.
6. The numerical solution of the equation system.
The calculation of IM electromagnetic field was performed for two

complete rotor revolutions. During one rotor revolution 360 modeling
steps are used, each step corresponds to the rotor revolution by one
electrical degree. Correspondingly, a time step of the algorithm of IM
5
electromagnetic field calculation is equal to t  5.5 10 s. In this case
the reduction of the frequency of rotor rotation in the self-running-out
mode is taken into account [63].
The calculation of electromagnetic field in IM cross-section is
performed in a package mode with the use of LUA programming
language. With this purpose in view a program with the help of LUA-
script was worked out (appendix D). The set of LUA-script commands
can be changed in accordance with the tasks posed during the
calculation. The sequence of operations in calculation of
electromagnetic field in IM cross section in the motor self-running-out
mode is given in Figure 3.13.
assignment of initial values of currents in the rotor bars
assignment of parameters for model calculation: calculation time,

the value of the rotor rotation angle, rotor rotation pitch and period
beginning of the cycle for calculation of the motor self-running-out mode
opening of a file with IM geometry and memorization of the model

temporary file
rotor revolution by the chosen value of rotation angle
calculation of currents in the rotor bars by exponential law

and step-by-step assigning their values
numerical calculation of the system of electromagnetic field equations
mapping and memorization of distribution of magnetic flux lines

and magnetic induction density in IM cross-section
measurement of the values of magnetic potential at every step

of the rotor rotation
recording of the measured values into the text file
end of the cycle for calculation of the motor self-running-out mode
Figure 3.13. The sequence of operations in the calculation of electromagnetic field in

IM cross-section in the self-running-out mode with the use of the finite element
method.
3.4. THE ANALYSIS OF THE PROCESS OF GENERATION

OF ELECTROMOTIVE FORCE IN THE STATOR WINDINGS
UNDER THE INFLUENCE OF ELECTROMAGNETIC FIELD IN
THE AIR GAP
According to the sequence of operations (Figure 3.13) the

calculation of electromagnetic field in IM cross-section under the self-
running-out condition is carried out. As a result of electromagnetic field
calculation, the instantaneous values of vector magnetic potential and
magnetic induction at every step of the rotor rotation are obtained.
Figure 3.14 contains the calculation area of the cross-section of a
healthy IM type АIR80V4U2 with a grid of triangular finite elements.
The parameters of the created model: 30274 nodes, 60087 elements.
When IM electromagnetic field is calculated in the cross-section, the
small dimensions of the motor air gap and the area around the rotor
slots are taken into account. Taking this into consideration, the finite
elements grid is compacted in the area of the air gap and the rotor slots.
It allows the improvement of the accuracy of electromagnetic field
calculation. The calculation of electromagnetic field in IM cross-section
after the discretization of the operation area by a finite elements grid
resulted in obtaining the distribution of magnetic flux lines (Figure
3.15) at the moment of IM disconnection from the supply mains for a
healthy motor (а) and with three broken rotor bars (b). In Figure 3.15
(b) the broken bars are marked black.
The results (Figure 3.15) revealed that in the healthy motor a
symmetrical distribution of electromagnetic field lines can be seen. In
the presence of broken rotor bars the motor electromagnetic field
becomes asymmetric. The following step consists in the analysis of the
distribution of IM electromagnetic field in dynamic mode, namely, in
the motor self-running-out mode. The final result of the calculation of
IM electromagnetic field consists in the determination of the values of
vector magnetic potential. Formula (2.12) is used for the calculation of
EMF signals in the stator windings (for two complete rotor revolutions)
in the motor self-running-out mode. These signals are used for the
assessment of electromagnetic field distortion. The EMF signals of one
active side of the coil, the coil, the coil group and the winding phase of
the stator of a healthy IM, type АIR80V4U2, obtained as a result of the
calculation, are shown in Figures 3.16–3.19.
Figure 3.14. The calculation area of a healthy IM cross-section with a grid of

finite elements.
а b
Figure 3.15. The distribution of magnetic flux lines at the initial moment of the
motor disconnection from the supply mains for a healthy IM (а) and for an IM
with three broken rotor bars (b).
Е, V
20
10
0 0.01 0.02 0.03 0.04 t, s
10
Figure 3.16. The EMF signal of one active side of the coil of a healthy IM
stator winding.
Е, V
40
20
0 0.01 0.02 0.03 0.04 t, s
-20
Figure 3.17. The EMF signal of the coil of a healthy IM stator winding.
Е, V
100
0 0.01 0.02 0.03 0.04 t, s
100
Figure 3.18. The EMF signal of the coil group of a healthy IM stator winding.
Е, V
200
100
0 0.01 0.02 0.03 0.04 t, s
100
Figure 3.19. The EMF signal of a healthy IM stator winding phase.
Е, V one broken rotor bar

20
10
0 0.01 0.02 0.03 0.04 t, s
10
20
Figure 3.20. The EMF signal of one active side of the coil of the stator winding of IM
type АIR80V4U2 with one broken rotor bar.

40
20
0 0.01 0.02 0.03 0.04 t, s
 20
one broken rotor bar

 40
Figure 3.21. The EMF signal of the coil of the stator winding of IM type АIR80V4U2
with one broken rotor bar.

100
0 0.01 0.02 0.03 0.04 t, s
100 one broken rotor bar
Figure 3.22. The EMF signal of the coil group of the stator winding of IM type
АIR80V4U2 with one broken rotor bar.
Е, V
200
100
0 0.01 0.02 0.03 0.04 t, s
 100
 200
Figure 3.23. The EMF signal of the stator winding phase of IM type АIR80V4U2
Then modeling for IM with one broken rotor bar was performed.
The obtained EMF signals of one active side of the coil, the coil, the
coil group and the stator winding phase of IM, type АIR80V4U2 with
one broken rotor bar are shown in Figure 3.20–3.23.
The results of the analysis revealed that the EMF signal of one
active side of the coil contains both tooth kink and signal shape
distortions caused by the presence of broken rotor bars. A visual
analysis showed that in the EMF signal of the coil (Figure 3.20) the
information signs that manifest in the signal shape distortion, become
less noticeable and in the EMF signals of the coil group and the
winding phase (Figure 3.22–3.23) - they are practically absent.
Modeling of IM with two adjacent broken bars was performed in an
analogous way. The results of modeling are shown in Figures 3.24–
3.27.
The analysis of the results revealed that with the growth of the
number of broken bars (adjacent), information signs that are available
in the EMF signal of one active side of the coil intensify and manifest
in greater distortion of EMF signal shape.
two broken rotor bars

Е, V
10
0 0.01 0.02 0.03 0.04 t, s
10
Figure 3.24. The EMF signal of one active side of the coil of the stator winding of IM
type АIR80V4U2 with two broken rotor bars.
Е, V two broken rotor bars
20
0 0.01 0.02 0.03 0.04 t, s
20
Figure 3.25. The EMF signal of the coil of the stator winding of IM type АIR80V4U2
with two broken rotor bars.
Е, V
50
0 0.01 0.02 0.03 0.04 t, s
50
Figure 3.26. The EMF signal of the coil group of the stator winding of IM type
АIR80V4U2 with two broken rotor bars.
Е, V
200
100
0 0.01 0.02 0.03 0.04 t, s
100
Figure 3.27. The EMF signal of the stator winding phase of IM type АIR80V4U2
To confirm the universal character of the developed mathematical

model with the use of FEM the research for two IMs was carried out: a
two-pole IM of the power of 1.5 kW and a single-pole IM of the power
of 75 kW.

Е, V
20
10
0 0.01 0.02 0.03 0.04 t, s

10
20
Figure 3.28. The EMF signal of one active side of the coil of the stator winding
of IM type 4АN200L2U3 with one broken rotor bar.

Е, V
40
20
0 0.01 0.02 0.03 0.04 t, s

20
40
Figure 3.29 – The EMF signal of the coil of the stator winding of IM type
4АN200L2U3 with one broken rotor bar.
Е, V
200
100
0 0.01 0.02 0.03 0.04 t, s

100
200
Figure 3.30. The EMF signal of the stator winding phase of IM type 4АN200L2U3
The results of modeling for a single-pole IM, type 4АN200L2U3,

with one broken rotor bar are given in Figures 3.28–3.30. The results of
modeling for the mentioned motors with different number of broken
rotor bars and their relative position are given in appendix E. The
analysis of the results of modeling IM, type 4АN200L2UЗ, revealed

that the EMF signal of one active side of the coil contains broken bars
information signs that also manifest in the signal shape distortion. It
should be mentioned that the number of rotor teeth for the said IM is
Z2=46. The modeling results are given only for an IM with one broken
bar. In EMF signal of the winding phase the breakage information signs
that manifest in the form of the signal shape distortion are absent
because of the mutual superposition of the information signs.
For the preliminary assessment of EMF signal in the stator
windings for a healthy IM and an IM with three broken rotor bars its
approximation is performed:

e  t   Eme
t / a
 
 Eres sin  pte
t /  s
 0 ,  (3.21)
where Em – EMF signal initial amplitude; a – rotor time constant;

Eres – the EMF of the rotor steel residual magnetization; r – rotor
rotation frequency in the no-load condition;  s – the time constant of
the decrease of the rotor rotation frequency.
Е, V Е, V
1 2
200 150
2
150 1
100
100
50 50
0 0.005 0.01 t, s 0 0.005 0.01 t, s

a b
Figure 3.31. The fragments of the initial (1) and approximated (2) signals of EMF
in the stator windings of a healthy IM (а) and an IM with three broken rotor bars (b).
Е, V 2
40
1
20
0 0.02 0.04 t, s
-20
Figure 3.32. The differences of the approximated and calculated EMF signals
in the stator windings: 1 – for a healthy IM, 2 – for an IM with three broken rotor
bars.
The comparison of half-periods of the initial and approximated

according to (3.18) the EMF signals of one active side of the coil of the
healthy IM and an IM with three broken rotor bars is given in Figure
3.31. The comparison results revealed that in the healthy IM the EMF
signal is shaped as a regular sinusoid with a high-frequency component
caused by the motor teeth design. In the presence of breakages the
shape of EMF signal deviates from the sinusoidal form. The dynamic
models of a healthy IM and an IM with different number of broken
rotor bars are also obtained due to the results of electromagnetic field
calculation. These models make it possible to visually demonstrate the
distribution of electromagnetic field lines of IM with broken rotor bars
in the motor condition at every step of the rotor rotation. Thus, the
proposed method for the calculation of an induction motor
electromagnetic field with the use of a circuit model and a model based
on MEF makes it possible to assess the broken rotor bars influence on
EMF signals in the stator windings in the motor self-running-out mode.
3.5. CONCLUSION
The monograph contains an IM mathematical model in a three-

phase coordinate system with the representation of the rotor as a system
of short-circuited bars. The model allows the research of operation

modes of IM with any number of broken rotor bars taking into account
the location of the breakage and providing the possibility for the
research of different modes of motor operation. The mathematical
model enables obtaining the distribution of instantaneous values of
current in the rotor bars in the motor self-running-out mode
independently of the previous mode of IM operation condition.
An IM rotor circuit mathematical model in which the number of
rotor electrical circuit corresponds to the number of the bars is worked
out. The model enables the specification of the initial values of currents
in the rotor bars at the moment of the motor disconnection from the
supply mains. The number of broken bars and the geometric location of
breakages is also taken into account in the developed model.
An IM mathematical model with the use of the finite element
method makes it possible to perform calculation of electromagnetic
field in IM cross-section in self-running-out mode. The mathematical
model is used for the research of information signs of broken rotor bars
in EMF signals in IM stator windings.
A software module with the use of LUA programming language is
worked out for the calculation of electromagnetic field in the cross-
section of IM with broken bars. The use of the software module allows
obtaining the instantaneous values of EMF in the stator winding
elements in the motor self-running-out mode under the automated
condition for the IMs of various powers with an arbitrary pitch of rotor
rotation.
Chapter 4
THE METHOD OF INDUCTION MOTOR

BROKEN ROTOR BARS DIAGNOSTICS WITH
THE USE OF WAVELET-TRANSFORM
4.1. THE ANALYSIS OF ELECTROMOTIVE FORCE SIGNALS

IN INDUCTION MOTOR STATOR WINDINGS BY MEANS
OF WAVELET-TRANSFORM
The performed research (p. 2.4) made it possible to find out that
orthogonal wavelets with a compact support can be used for the
analysis of sinusoid-shape wave signals. To reveal the local features of
EMF signal in the stator winding taking into account wavelets
properties a wavelet analysis with the use of the Symlet wavelet was
carried out.
As stated above, the EMF signal of the stator winding phase does
not contain explicit signs typical of broken rotor bars, unlike EMF
signals of one active side of the coil. To confirm this fact an analysis of
the obtained signals for an IM with one broken rotor bar was performed
with the use of continuous WT (Figures 4.1–4.4).
Figure 4.1. The EMF signal of one active side of the coil of the stator winding of an IM
with one broken rotor bar and its wavelet-spectrum.
Figure 4.2. The EMF signal of the stator winding coil of an IM with one broken rotor
bar and its wavelet-spectrum.
Figure 4.3. The EMF signal of the stator winding coil group of an IM with one broken
rotor bar and its wavelet-spectrum.
The Method of Induction Motor Broken Rotor Bars … 107
Figure 4.4. The EMF signal of the stator winding phase of an IM with one broken rotor
bar and its wavelet-spectrum.
with two broken rotor bars and its wavelet-spectrum.
Figure 4.6. The EMF signal of the stator winding coil of an IM with two broken rotor
bars and its wavelet-spectrum.
Figure 4.7. The EMF signal of the stator winding coil group of an IM with two broken
rotor bars and its wavelet-spectrum.
Figure 4.8. The EMF signal of the stator winding phase of an IM with two broken rotor
bars and its wavelet-spectrum.
with three broken rotor bars and its wavelet-spectrum.
Figure 4.10. The EMF signal of the stator winding coil of an IM with three broken
Figure 4.11. The EMF signal of the stator winding coil group of an IM with three
broken rotor bars and its wavelet-spectrum.
Figure 4.12. The EMF signal of the stator winding phase of an IM with three broken
The analysis of the results revealed that the wavelet-spectrum of

EMF signal of one active side of the coil (Figure 4.1) in the area of high
frequencies contains tooth harmonics and their number corresponds to
the number of the real IM rotor bars. The analysis of wavelet-spectrum
in Figure 4.1 also shows that typical sections marked with a dotted line
correspond to the location of the broken bar.
The results of continuous wavelet-transform for EMF signals in the
stator winding elements of an IM with two or three broken rotor bars
are shown in Figures 4.5–4.12. In this case two broken bars are adjacent
and the third one is at a distance of a polar division from them.
The obtained results demonstrated that the presence of typical
sections on the wavelet-spectra enables the determination of the relative
position of breakages independently of the number of the broken bars
[63].
Thus, the wavelet-transform of EMF signals in the stator winding
elements allows the determination of the number and relative position
of the broken bars.
For the quantitative assessment of the influence of broken rotor bars
it is proposed to use the analysis of the values of wavelet-expansion on
spectra with typical sections corresponding to the location of broken
bars. With this purpose in view, the low-frequency area and the high-
frequency spectrum part at which tooth frequencies occur are
conditionally cut off during the analysis.
Typical sections with surges of wavelet-coefficients corresponding
to the broken bars are in the area of medium frequencies. The lower
boundary of this area is limited by the section at which high-frequency
components caused by the tooth kink begin to appear.
So, it is proposed to use the function of the average value of
wavelet-expansion coefficients sum for medium frequency area:
k a
K a  a
, (4.1)
l
where k a – the values of wavelet-expansion coefficients; a and A –

the initial and final values of wavelet-spectrum scales, respectively,
A  a  (5..10) ; l – the number of wavelet-expansion coefficients.
Expression (4.1) was used as the basis for the creation of wavelet-
expansion coefficients K  a from wavelet shift b for a healthy IM and
an IM with one and three broken rotor bars, respectively (Fig. 4.13).
Ka
200
100
0 100 200 300 400 500 b
-100
-200
healthy IM
three broken rotor bars
Figure 4.13. The functions of the average values of wavelet-expansion coefficients

sums for the medium-frequency area.
K a
Figure 4.14 contains function K * a  100 % created in relative
amax
units reduced to the maximum value of the scale amax  64 for an IM
with broken rotor bars. Research demonstrated that the value of
coefficient K  a in the presence of several broken bars grows
approximately proportionally to their number.
K

19.2 % 1 – one broken rotor bar
 a , 2 – two broken rotor bars
%
2 2
13.7 8.9 %
1
1
4
0 0.01 0.02 0.03 t, s
-5.8
Figure 4.14. The function of the average values of wavelet-expansion coefficients

sums for the medium-frequency area.
According to (4.1) the surfaces of the coefficients of wavelet-

expansion of EMF signals in the stator windings are obtained (Figure
4.15); they reflect both the variation of the signal frequency-time
characteristics and the amplitude values of wavelet-coefficients in 3D
space.
a b
Figure 4.15. The surfaces of the coefficients of the wavelet-expansion of EMF signals
in the stator windings: (а) – for a healthy IM, (b) – for an IM with three broken
rotor bars.
The use of the 3D wavelet-spectra of signals enables the visual

assessment of the impact of the amplitude of wavelet-coefficients
surges corresponding to the location of broken rotor bars.
4.2. THE METHOD FOR DECOMPOSITION OF THE SIGNAL

OF THE ELECTROMOTIVE FORCE OF THE STATOR
WINDING PHASE
4.2.1. The Method for Decomposition of the Signals

of Electromotive Forces in the Stator Windings
The proposed method for the diagnostics of IM broken rotor bars

makes it possible to determine the relative position of broken bars.
However, difficulties occur during the analysis of the signals of the
EMF of the coil, the coil group and the winding phase. So, in the
wavelet-spectrum of the coil EMF (Figure 4.2) one can see a
duplication of sections corresponding to the location of broken bars. It
can be explained by summing up of the signals of EMF of the coil two
active sides shifted in space by the angle equal to π/2 (for an IM with
two pairs of poles). So, in the wavelet-spectrum the sections with
wavelet-coefficients characterizing the damage are also shifted by this
space angle. The analysis of the wavelet-spectra of the signals of the
EMF of the coil group (Figure 3.23) and the winding phase on the
whole (Figure 3.24) revealed that there occurs “smearing” of typical
sections on the wavelet-spectra due to summing up the EMF signals. It
complicates the reliable positioning of broken rotor bars. So, it was
proposed to perform the analysis of the signal of the EMF of one active
side of the stator winding coil after its singling out of the total signal of
the EMF of the winding phase. It is enabled by the decomposition of
the winding elements corresponding signals into their components.
During the decomposition the type of IM stator windings
connection was taken into account. So, with a “star” connection of the
stator windings (Figure 4.16) when voltage sensor is connected to
terminals of two phases, inter-phase EMF is fixed. For example, with
connection to phases А and В the EMF inter-phase value equal to
EAB  EA  EB  U AB is measured. When stator windings are connected

according to a “star” scheme, the equation of electromotive forces
EA  EB  EC  0 is true.
If the stator windings are connected according to a “triangle”
scheme with the connection of a voltage sensor to the terminals of two
phases, linear EMF is fixed. I.e. during connection to phases A and B
the linear value of EMF E A  U AB is measured. When stator windings
are connected according to a “triangle” scheme, it is necessary to take
into account that EAB  EBC  ECA  0 .
EA
EC E AB
ECA  EB
120
EA
120
EB
E BC
 EC
Figure 4.16. The vector diagram of EMF with the stator windings connection
according to a “star” scheme.
One of signal processing methods allowing signals decomposition

consists in z-transform [70]. It should be mentioned that in this case the
signals are to be presented in a discrete form.
Z-transform represents the decomposition of functions into series of
power polynomials according to z. The sense of value z in z-polynomial
consists in the fact that it is an operator of a unit delay by the function
coordinates.
It is possible to imagine a continuous signal of the winding EMF

e(t ) in the form of a sequence of numbers e  k  as a result of its
discretization in equal periods of time kT , where k  0,1,... , T – the
period of discretization.
Polynomial according to z corresponds to this function; values e  k 
are its successive coefficients:
K
e  k   e  k t   TZ e  k t    ek z k  E z , (4.2)
k 0
where z    j – an arbitrary complex variable; E  z  – the z-image

of EMF signal e  k  .
Expression (4.2) is an analytical record of direct z-transform.
One of the properties of z-transform consists in delay by n strokes:
e  k   e  k  n  . Taking it into consideration,:
K K K
E  z   e  k  zk   e  k  n zk z n  e  k  n  z k n  z n E  z  .
k 0 k 0 k 0
Consequently, the multiplication of the signal z-image by multiplier

z n provides for the signal shift by n strokes of discretization.
During the decomposition of the signals of the stator winding
elements EMF the inverse z- transform was used, its analytical
expression is of the form:
e  k   Z 1  E  z  , (4.3)
where Z 1 – the operator of inverse z- transform.

The use of signal discretization delay by n strokes makes it
possible to take into account the angles of EMF signal shift in relation
to each other in the stator winding elements.
The decomposition of EMF signals in the stator winding elements

was carried out step-by-step. The EMF signal of the stator winding
phase is first divided into EMF signals of the coil group of this winding.
After that the EMF signal of one of the coil groups at the known angles
of shift between the coils in the stator slots is divided into signals of the
coils EMF, which, in its turn, is divided into the signals of the EMF of
two active sides of the coil.
E, V phase EMF
1st stage
100
0 0.01 0.02 0.03 0.04 t, s

Division of the phase -100
EMF signal into
signals of EMF of the
E, V
EMF of the coil group 1 EMF of the coil group 2
coil groups 100
broken bars E, V
100
0 0.01 0.02 0.03 0.04 t, s 0 0.01 0.02 0.03 0.04 t, s
nd
2 stage -100 -100
EMF of the coil 1 EMF of the coil 2 EMF of the coil 3

E, V broken bars E, V E, V
Division of the coil
20 20 20
group EMF signal into
0.04 t, s 0 0.01 0.02 0.03 0.04 t, s 0 0.01 0.02 0.03 0.04 t, s
signals of the coils EMF 0 0.01 0.02 0.03
-20 -20 -20
3rd stage
EMF of the coil active sides 1 E, EMF of the coil active sides 2
E,
V V
broken bars
Division of the coil 10
10
EMF signal into
0 0 0.01 0.02 0.03 0.04 t, s
0.01 0.02 0.03 0.04 t, s
signals of EMF of the -10
-10
coils active sides
Figure 4.17. The block diagram of the decomposition of the signal of the EMF of IM
stator winding phase.
A block diagram of the decomposition of the EMF signal of the

winding of the stator of the analyzed IM, type AIR80V4U2, taking into
account the specific features of the design (the winding phase consists
of two coil groups, each of which, in its turn, contains three coils) is
To simplify the decomposition the following definitions of the
signals are introduced:
 an initial signal – the signal of EMF of one of the elements of

the stator winding, calculated according to the results of the
simulation of electromagnetic field in IM cross section;
 a detached signal – the signal of EMF of one of the elements of
the stator winding, obtained by the decomposition of the total
signal into signals of its components.
The Decomposition of the Signal of Winding Phase EMF into the

Signals of EMF of the Coil Groups
For a multi-polar IM the signal of the EMF of the winding phase is
divided into the signals of the coil groups EMF. The IMs with p  1
coils are not united into coil groups, so, this stage of decomposition is
not performed for them.
The block diagram of the decomposition of the phase EMF signal is
given in Figure 4.18. For the analyzed IM ( p  2 ) the winding phase
EMF signal E ph  z  is divided into coil groups EMF signals Eq1  z 
and Eq 2  z  . During the decomposition the space angles of the position
of the winding elements in the stator slots are taken into consideration.
So, for the analyzed IM the coil groups are located in the slots at angle
 (in Figure 4.18 this angle corresponds to the number of
increments k).
E ph  z 
k
z
Eq2  z 
E q1  z  Eq2  z 
Figure 4.18. The block diagram of the decomposition of the signal

of the winding phase EMF.
The comparison of two signals of the EMF of the stator winding

coil group (the initial and the detached ones) for an IM with one broken
rotor bar is given in Figure 4.19.
For the analysis of the information signs of breakages the wavelet-
transform of the initial and detached signals of the coil group EMF was
performed (Figures 4.20–4.21).
Figure 4.19. The comparison of the initial and detached signals of the EMF of
the stator winding coil group of an IM with one broken rotor bar.
Figure 4.20. The Initial signal of the EMF of the coil group
and its wavelet-spectrum.
Figure 4.21. The detached signal of the EMF of the coil group and its wavelet-
spectrum.
The Decomposition of the Signal of the EMF of the Coil Group into
the Signals of Coils EMF
One of the signals of the EMF of the coil group Eq  z  obtained as
a result of the decomposition is divided into the signals of coils EMF.
For the analyzed IM the decomposition of the signal of the EMF of the
coil group Eq1  z  into the signals of the EMF of the coils Ecm  z  was
performed, where m – the number of coils forming a coil group (for
the analyzed IM m  3 ), k – the number of increments corresponding
to the angle of shift between the coils (for the analyzed IM at this stage
of decomposition k  2 ) (Fig. 4.22).
Z1
E q1  z 
Ec2  z  k k
z z
E c1  z  Ec2  z  E cm  z 
E cm  z 
Figure 4.22. The block diagram of the decomposition of the signal of the EMF of
the coil group.
The comparison of the initial and detached signals of the EMF of

the stator winding coil of an IM with one broken rotor bar is shown in
Figure 4.23.
20
E, V 10
initial signal detached signal
40
0.04 0.042 0.044
20
0
0.02 0.04 0.06 t, s
-20
-40
Figure 4.23. The comparison of the initial and detached signals of the EMF of the
stator winding coil of IM with one broken rotor bar.
Figure 4.24. The initial signal of coil EMF and its wavelet-transform.
Figure 4.25. The detached signal of coil EMF and its wavelet-transform.
The analysis of the obtained signals of the coil EMF revealed that
the distortion of the shape of the detached EMF signal certifies the
presence of broken rotor bars, which is confirmed by corresponding
research in Chapter 3.
To reveal the information signs of the breakage of the initial and
detached signals of EMF of the coil a continuous wavelet-transform is
performed. The results of the CWT are shown in Figures 4.24–4.25.
The obtained results demonstrated that the wavelet-spectrum of the
detached signal of the IM stator winding coil EMF contains typical
sections corresponding to the location of broken rotor bars. At the same
time the “duplication” of the said sections can be seen on the wavelet-
spectrum of the detached signal of the coil EMF (Figure 4.25), which is
also typical of the coil EMF signal obtained according to the results of
calculation of IM electromagnetic field (Figure 4.2). So, the obtained
results of the wavelet-transform (Figure 4.25) coincide with the results
of the wavelet-expansion for the initial signal of EMF of the stator
winding coil (Figure 4.2).
Thus, at the stage of singling out the signals of the coil EMF it is
possible to come to the conclusion that the use of the method for
decomposition of the winding phase EMF signal into the signals of the
EMF of its elements allows the determination of the information signs
of broken rotor bars.
The Decomposition of the Coil EMF Signal into the Signals of the
EMF of Two Active Sides of the Coil
The final stage of the decomposition of the signal of EMF of the
stator winding phase consists in singling out the signals of EMF of two
active sides of the coil Et1  z  and Et 2  z  from the signal of EMF of
coil Ec1  z  (Figure 4.26).
E c1  z 
k
z
Et 2  z 
E t1  z  Et 2  z 
Figure 4.26. The block diagram of the decomposition of the coil EMF signal.
The results of the comparison of the initial and detached signals of

the EMF of one active side of the coil are given in Figure 4.27, and the
results of their wavelet-transform – in Figures 4.28–4.29.
E, V detached signal
20 initial signal
10
0.02 0.04 0.06 t, s

-10
-20
Figure 4.27. The comparison of the initial and detached signals of the EMF of one
active side of the coil of the stator winding of an IM with one broken rotor bar.
Figure 4.28. The initial signal of the EMF of one active side of the coil and its wavelet-
transform.
Figure 4.29. The detached signal of the EMF of one active side of the coil and its
wavelet-transform.
The analysis of the results of the decomposition and wavelet-

transform of the signals revealed that the information signs of the initial
signal of the EMF of one active side of the coil are also present in the
detached EMF signal.
In this case the high-frequency component is not reproduced
completely, but breakage information signs manifested in distortion of
EMF signal shape remain. Besides, as obvious from the obtained
wavelet-spectrum (Figure 4.29), the EMF signal distortion,
corresponding to the location of the broken bar, is reflected on the
wavelet-spectrum in the form of typical sections with wavelet-
coefficients.
To confirm the efficiency of the method of the decomposition of the
EMF signal of the stator winding phase into the signals of the EMF of
the active sides of the coil the research of an IM with two broken bars
was carried out. In this case it is assumed that the broken bars are
shifted in relation to each other. The scheme of the experiments for IMs
with different relative position of the broken rotor bars is shown in
Table 4.1. The values of the angles between two broken bars are chosen
arbitrarily ( br.b.  84.7 or br.b.  169.4 ) to research the possibilities
of the proposed method of decomposition.
The results of the research for step-by-step singling out the signals
of the EMF of the stator winding elements at the relative position of the
broken rotor bars at a distance of the space angle of br.b.  84.7 are
given in Figures 4.30–4.34.
Table 4.1. The scheme of experiments for IMs with different

relative position of the broken rotor bars
No. Experiment Relative position of the broken bars, degrees

1. one broken bar no
2. two broken bars br.b.  31.8
3. br.b.  84.7
4. br.b.  169.4
Е, V initial signal
detached signal
100
0
0.02 0.03 0.04 0.05 t, s
-100
stator winding coil group of IM with two broken rotor bars ( br .b.  84.7 ).
Е, V detached signal
50 initial signal
25
0.04 0.08 t, s
-25
-50
stator winding coil of IM with two broken rotor bars ( br .b.  84.7 ).
Figure 4.32. The detached signal of the EMF of the stator winding coil of an IM
with two broken rotor bars ( br .b.  84.7 ) and its wavelet-spectrum.
Е, V
initial signal detached signal
20
10
0.02 0.04 0.06 0.08 t, s

-10
-20
Figure 4.33. The comparison of the initial and detached signals of the EMF
of one active side of the stator winding coil of an IM with two broken rotor
bars ( br .b.  84.7 ).
Figure 4.34. The detached signal of the EMF of one active side of the stator
winding coil of an IM with two broken rotor bars ( br .b.  84.7 ) and its wavelet-
spectrum.
As the results of the decomposition and wavelet-transform of

signals demonstrate, the detached signal of the EMF of one active side
of the coil corresponds to the analogous EMF signal obtained as a result
of the calculation of electromagnetic field (Figure 4.1). The wavelet-
analysis of the detached signal of the EMF of one active side of the coil
(Figure 4.34) revealed that typical sections with wavelet-coefficients
are shifted at the wavelet-spectrum by the angle of br.b.  84.7 , by
which value the broken bars are also shifted in space.
Figure 4.35. The detached signal of the EMF of one active side of the coil of the
stator winding with two broken rotor bars ( br .b.  31.8 ) and its wavelet-spectrum.
Figure 4.36. The detached signal of the EMF of one active side of the coil of
the stator winding with two broken rotor bars ( br .b.  169.4 ) and its wavelet-
spectrum.
The results of the wavelet-expansion for the experiments with

relative position broken rotor bars at the distance of angles br.b.  31.8
or br.b.  169.4 (Figures 4.35–4.36) are obtained in an analogous way.
So, the use of the method for the decomposition of the signal of the
EMF of the winding phase into signals of the EMF of active sides of the
coils allows singling out the broken bars information signs that are
impossible to be detected in phase EMF signal.
Thus, the use of the wavelet-analysis of the signal of the
electromotive force of one active side of the coil of the IM stator
winding, obtained by singling out from the total signal of electromotive
force of the winding phase, makes it possible to improve the reliability
of IM broken rotor bars diagnostics.
4.2.2. The Method for the Decomposition of the Coefficients

of the Wavelet-Expansion of the Signals of the Electromotive
Forces in the Stator Winding
Paragraph 4.2.1 contains the description of a proposed method for

the decomposition with the use of the theory of inverse z-transform for
singling out the signals of the EMF of the active sides of the coils from
the signal of the EMF of the winding phase. Broken bars information
signs manifested on the wavelet-spectra in the form of typical sections
with wavelet-coefficients are singled out according to the results of the
wavelet-spectra of the obtained signals. It improves the reliability of the
broken rotor bars diagnostics. The procedure of singling out the
breakage information signs can be simplified by means of the
decomposition of the diagnostics coefficient. It represents a sum of
coefficients of the wavelet-expansion of the signal of the EMF of the
stator winding phase for the medium frequency area (Figure 4.13).
The decomposition of the wavelet-expansion coefficients was
performed on the basis of the algorithm also used in the decomposition
of the signal of the winding phase EMF (paragraph 4.2.1). In this case
the function of the average value of the sum of wavelet-expansion
coefficients for the medium frequency area is used as an initial signal
(Figure 4.13). The signals used as a result of decomposition are shown
in Fiure 4.37.
Ka
200
100
0.016 0.024 0.032 t, s

-100
-200
Figure 4.37. The signal of function K a of the coil.
The analysis of the obtained results revealed that the signal of

function K  a contains breakage information signs, namely, signal
surges, “duplicated” due to summing-up the signals of the active sides
of the coil. The decomposition of the obtained coil signal into the
signals of the active sides of the coil makes it possible to divide
breakage information signs. The comparison of the initial and detached
signals of function K  a of the active side of the coil is given in Figure
4.38.
Ka initial signal

detached signal
100
0.016 0.024 0.032 t, s
-100
Figure 4.38. Comparison of the initial and detached signals of function

K  a of EMF of the active side of the coil.
As a result of the decomposition the difference of the coefficient of

the wavelet-expansion of the EMF of the coil and one active side of the
coil is obtained (curves 2 in Figs. 4.40 and 4.41 respectively).
Ka
10 detached signal initial signal
of the coil group of the phase
5
-10
0
0.028 0.036 0.044 0.052 t, s
-5
Figure 4.39. The functions of the average values of the sums of the wavelet-expansion
coefficients for the area of the medium frequencies of the signals of the EMF of the
phase and the coil group.
Ka
6
2
4
1
2
0 t, s
0.028 0.036 0.044 0.052
-2
-4
-6
1 – initial signal of the K  a coil

2 – detached signal of the K  a coil
coefficients for the area of the medium frequencies of the coil EMF signal.
The analysis of the obtained results (Figures 4.40–4.41) showed that

the use of the decomposition of function K  a allows the determination
of the value of the amplitude of the surge reflecting the degree of rotor
breakage (analogously to Figure 4.14).
Ka
6
4
2
2
1
0
0.028 0.036 0.044 0.052 t, s
-2
-4
-6
1 – initial signal of one active side of the coil K  a

2 – initial signal of one active side of the coil K  a
coefficients for the area of the medium frequencies of the signal
of the EMF of one active side of the coil.
Thus, the use of the decomposition of the function of the average

values of the sums of the wavelet-expansion coefficients for the area of
the medium frequencies makes it possible to divide breakage
information signs and improve the reliability of broken rotor bars
diagnostics.
4.3. CONCLUSION
The analysis of the stator windings EMF signals wavelet-spectra

obtained as a result of electromagnetic field calculation has been carried
out for different number of broken rotor bars. The analysis has revealed
that typical sections on the wavelet-spectra correspond to the location
of broken bars. It has been demonstrated that the area of frequencies at

which broken bars occur are located on the wavelet-spectrum between
the area of rotation frequency and the area of tooth frequencies.
It has been shown that the use of the function of the average values
of wavelet-expansion coefficient sums for the area of the medium
frequencies at which stator breakages manifest enables the quantitative
assessment of damage influence.
The proposed method for the diagnostics of IM broken rotor bars on
the basis of the wavelet-analysis of the EMF signal in the stator
windings under the motor self-running-out condition makes it possible
to determine the relative position of broken rotor bars.
A method for the decomposition of EMF signals in IM stator
windings with the use of the theory of inverse z-transform has been
developed. It enables the improvement of the reliability of the
diagnostics of IM broken rotor bars due to singling out the information
signs available in the signal of EMF of one active side of the coil.
The use of the decomposition of the function of the average values
of the sums of the wavelet-expansion coefficients for the area of
medium frequencies makes it possible to simplify the procedure of
singling out the information signs of broken rotor bars.
Chapter 5
THE EXPERIMENTAL VERIFICATION OF

THE METHOD FOR THE DIAGNOSTICS OF
INDUCTION MOTOR BROKEN ROTOR BARS
5.1. THE DESCRIPTION OF THE EXPERIMENT

AND MEASURING AND DIAGNOSTICS EQUIPMENT
A laboratory stand with a computer-aided measuring system was

used for the experimental verification of the devised method. The
measuring stand contains a computer, which allows carrying out
complicated calculation with high accuracy and operation speed and
providing for the automation of the tests: automated measuring and
mathematic processing of the measured signals.
The computer-aided laboratory stand enables testing both under no-
load and under load conditions.
The power circuit of the complex consists of a switch, a block of
voltage sensors for measuring voltage in every phase of the motor stator
winding.
The experimental research of the developed method for the

diagnostics of IM broken rotor bars is carried out according to the
following algorithm (Figure 5.1).
IM of АIR80V4U2 type, a block of voltage sensors for the
measurement of the instantaneous values of voltages in IM stator
winding phases, an analog-digital converter and a personal computer
for processing the experimental data were used for the experimental
research [63, 71]. The appearance of the measuring complex is shown
in Figure 5.2. The functional arrangement of the laboratory stand is
Beginning
IM disconnection from the supply mains
Measurement and digital-analog conversion of

instantaneous values of voltage in IM stator
windings in the motor self-running-out mode
Wavelet-transform of voltage signals in the stator

windings
Calculation of diagnostic coefficients
Determination of the number and relative position

of IM broken rotor bars
End
Figure 5.1. The algorithm of the performance of the diagnostics of IM broken

rotor bars.
The Experimental Verification of the Method … 135
Figure 5.2. The appearance of the measuring complex.
The block of voltage sensors is based on amplifiers with galvanic

decoupling HCPL 7800A; its specifications are given in Table 5.1.
An external USB module ADA-1406, a device for the collection of
analog and digital data, is used as an analog-digital converter.
ADA-1406 is a multifunctional measuring module connected to a PC
through a USB-interface.
Table 5.1. The basic technical characteristics of HCPL 7800A
Characteristic Value
Pass band 85 kHz
Input voltage ±200 mV (±300 mV max)
Input voltage shift, max. 0.9 mV
Nonlinearity, max 0.3%
Measurement accuracy 1%
Initial voltage:
- minimum 1.18 V
- medium 2.39 V
- maximum 3.61 V
Thermal drift 4.6 mcV/°С
Supply voltage +5V
Input resistance 530 kOhm
~ 380 V
QF1
Ua Ub Uc
PC ADC BVS
A В C
DCG
IM
RDCG
EW
UEW
Figure 5.3. The functional arrangement of the laboratory stand.
In Figure 5.3: PC – a personal computer; ADC – an analog-digital

converter; DCG – a direct current generator; BVS – a block of voltage
sensors; EW – an exciting winding.
A multichannel ADC 14-digit module provides for the operation
with 8 differential channels or 16 channels with a common ground.
The circuit diagram of the voltage sensor is shown in Figure 5.4.
and measuring module board – in Figure 5.5.
V1
DA 1
DC/DC 5/5
XS 4
1 2 4 6
C1 № Circuit
1 U1out
R5
2 U2out
+ 3 U3out
C4
C2 C3 4 U4out
C5
5 +12V
DA 3
6 -12V
DA 2 1 8
R1 7 GND
1 8 2 7
R3 UD140 8 +5V
2 7 3 U1408А 6
XS 2 HCPL
3 7800 6 4 5
№ Circuit R2 4 5
R4
1 U1.1in
C6
2 U1.2in
R6
3 U2.1in
4 U2.2in
5 U3.1in
6 U3.2in V2
7 U4.1in
DA 4
8 U4.2in
1
DC/DC
V3 5/± 12 V 4
V4
Figure 5.4. The circuit diagram of the voltage sensor.
Figure 5.5. A four-channel voltage measuring module.
5.2. THE ANALYSIS OF THE RESULTS OF

THE EXPERIMENTAL RESEARCH
Three equal IMs of АIR80V4U2 type were used for the

experimental verification of the method for the diagnostics of the rotor
broken bars (published data of the analyzed IM are given in appendix

А). Accordingly, three equal rotors that can substitute each other were
used for the research of broken rotor bars. Broken bars were simulated
by drilling holes on one of the rotors at the points of bars connection to
short-circuited rings. It provided for the breakage of the electric
coupling between the bars and the short-circuited rings and, thus, the
creation of a situation corresponding to broken rotor bars. A scheme of
position of artificially broken bars on the rotor is shown in Figure 5.6.
During the experimental research the IM was disconnected from the
mains when the motor operated before the disconnection both in no-
load mode and under load. The measurement of the instantaneous
values of EMF in the stator phases of a healthy IM and an IM with a
different number of broken rotor bars was performed with the use of a
block of sensors. The oscillogram of the stator phase voltages of a
healthy IM during the previous operation under load and under motor
self-running-out condition is shown in Figure 5.7.
Figure 5.6. The position of broken bars on the rotor.

Figure 5.7. The oscillogram of the stator phase voltages of a healthy IM.
When the IM is disconnected from the mains there appears a short-

time electric arc. The duration of the electric arc burning is mainly
determined by power accumulated in the windings. For the analyzed
IMs the duration of arc burning does not exceed ten milliseconds
(Figure 5.8), which is much less than the period of signal attenuation;
i.e., due to the short time of the commutation processes at IM
disconnection from the mains, they are excluded from the analysis and
do not influence the results of the diagnostics.
U, V
I, A
500 IM disconnection
Ua Ub Uc
Ib
Ia
10.262 10.265 10.267 t, s
Ic
-500
Figure 5.8. Voltage and current during IM disconnection from the supply mains.
E, V
100
0
0 0.05 0.01 0.015 t, s
-100
Figure 5.9. The experimental signal of the EMF of the stator winding phase
of IM with broken rotor bars.
Figure 5.9 contains a part of the experimental signal of the EMF of

the stator winding phase of IM with broken rotor bars in the self-
running-out mode.
The visual analysis of the experimental signal of the EMF of IM
stator winding phase shows, that the information signs of broken bars
are absent in the signal. The analysis of the wavelet-spectrum of the
experimental signal of the stator winding phase EMF (Figure 5.10)
revealed that typical sections corresponding to the location of the rotor
broken bars are absent in it. It confirms the correctness of conclusions
formulated as a result of modeling as to the mutual superposition of the
information signs of broken bars in EMF signal.
Figure 5.10. The experimental signal of the EMF of the stator winding phase
of IM with rotor broken bars and its wavelet-spectrum.
So, according to the proposed method of the decomposition of the

signal of the EMF of the coils active sides (chapter 4), the signal of the
EMF of one active side of the stator winding coil was singled out from
the experimental signal of the phase EMF.
The obtained signal of the EMF of one active side of the stator
winding coil and its wavelet-spectrum are shown in Figures 5.11–5.12.
E,
V
broken bars
7.5
0 0.005 0.01 0.015 t, s
-7.5
-15
Figure 5.11. The EMF signal of one active side of the coil, singled out
from the experimental signal of the EMF of the stator winding phase.
Figure 5.12. The EMF signal of one active side of the coil, singled out from the
experimental signal of the EMF of the stator winding phase, and its wavelet-spectrum.
The obtained results demonstrated that broken rotor bars are

determined by the analysis of the signal of the EMF of one active side
of the coil, singled out from the signal of the EMF of the stator winding
phase.
Thus, the results of the experimental research coincide with the
results of modeling. Some deviations of the singled-out signal of the
EMF of one active side of the coil, obtained on the basis of the
experimental research, from the results of modeling are explained by
the repeated transformations of discrete signals.
5.3. THE ASSESSMENT OF BROKEN ROTOR BARS

INFLUENCE ON INDUCTION MOTOR OPERATION
As stated above, broken bars in the short-circuited winding of the

rotor cause increased losses in the stator and rotor windings, more
vibrations of the motor, the reduction of rotation frequency under load,
the occurrence of stator current pulsations in all phases.
It is proposed to use the following indices for the assessment of the
broken rotor bars influence on IM operation:
 the relation of the losses in the stator windings of IM with

broken bars Pe1.br. to the rated value of losses of a healthy IM
Pe1.heal . operating under the nominal condition:
Pe1.br .
Pe1  , r.u.;
Pe1.heal .
 the relation of the losses in the rotor of IM with broken bars
Pe 2.br. to the rated value of the losses of a healthy IM Pe 2.heal.
operating under the nominal condition: Pe1.br. , r.u.;
Pe2 
Pe1.heal .
 the temperature of heating of the stator windings of IM with

broken bars: 1 , С;
 the temperature of heating of the rotor bars of IM with broken
bars:  2 , С;
 the relation of the value of the start-up time of IM with broken
bars tst .br. to the value of the start-up time of a healthy IM
t
tst .heal . : tst  st.br. t , r.u.;
st .heal .
 the multiplication factor of IM starting torque in relation to the


rated one: Ì st , r.u.
As a result of the simulation with the use of a mathematical model

described in paragraph 3.1., for the analyzed IM of АIR80V4U2 type
the values of the proposed indices for a different number of broken
rotor bars are obtained (Table 5.2).
Table 5.2. The proposed indices of the operation of IM

with broken bars
Pe1 , r.u. Pe2 , r.u. t st , r.u. 

No. Number of broken bars
Ì st , r.u.
1. 3% (one broken bar) 1.07 1.01 1.08 1
2. 6% (two broken bars) 1.21 1.23 1.7 0.99
3. 9% (three broken bars) 1.4 1.51 2.4 0.95
4. 12% (four broken bars)    
6.0 6.2 0.66

– the mathematical modeling of IM with 12% broken bars demonstrates non-
operability of the motor.
So, the analysis of the obtained results revealed, that the number of
broken bars about 10% is critical for IM. The analysis of IM starting

torque multiplication factor Ì st showed that the value of this index
decreases with the increase of the number of broken bars. Besides,
during the operation of IM with broken bars the starting-up time

significantly grows, which causes essential power losses in the start-up
condition.
It is known that during engineering calculations it is possible to
assume that the growth of losses in IM windings increases the
insulation temperature proportionally to the value of these losses [45]:
Prat  rat , (5.1)
where Prat – the relative increase of the value of losses in IM

windings; rat – the relative increase of the value of temperature.
So, the obtained values of indices Pe1 and Pe2 can be used for the
determination of the relative increase of the temperature of the stator
and rotor windings in the presence of broken bars.
The maximum allowable temperature of any part of EM is
determined as a sum of the allowable excess of temperature and the
maximum allowable ambient temperature 40 С (accepted for general-
purpose EM) [45]:
1  max  max.all.  40 , (5.2)
where max – the maximum operating insulation temperature under the

nominal condition; max.all. – the allowable excess of temperature.
Taking into account the class of heat resistance of the analyzed IM
winding insulation (insulation class F), using state standard GOST
8865-93, the value of the maximum operating temperature of insulation
is found at IM rated load: max  105 С. The allowable excess of
temperature for the windings of alternating current EM of the power up
to 5000 kW is max.all.  100 С [45].
So, the maximum allowable temperature of the stator windings

heating is: 1  205 С. The maximum allowable temperature of the
rotor bars heating is determined in an analogous way [45]: 2  215 С.
IM stator winding heating temperature 1 , С and rotor bars
heating temperature  2 , С are determined according to (5.2). The
calculated values of temperature are given in Table 5.3.
It is seen in Table 5.3 that with 9% of broken rotor bars the values
of IM stator and rotor windings temperature approach the maximum
allowable temperature of heating.
If 12% of the rotor bars are broken, the stator windings heating
temperature considerably exceeds the maximum allowable temperature,
and the rotor heating temperature almost exceeds the melting
temperature of the material (for the analyzed IM whose rotor cage bars
are made of aluminum the melting temperature is 658 С).
Table 5.3. The temperature of the stator and rotor windings

of IM with broken rotor bars

1 , С 2 , С
1. 3% (one broken bar) 152.4 106
2. 6% (two broken bars) 167 209.2
3. 9% (three broken bars) 187 198.6
4. 12% (four broken bars)  
670 691

– the mathematical modeling of IM with 12% broken bars demonstrates the non-
operability of the motor.
Thus, in the presence of three broken rotor bars (9%) the value of
losses in the stator and rotor windings grows almost by 1.5 times; IM
load-carrying capacity reduces to the value of 0.95; the motor
acceleration time is twice as much; the temperature of heating of the
stator windings and the rotor bars reaches the values approaching the
maximum allowable ones. So, for the analyzed IM the maximum

number of broken rotor bars is not to exceed three.
It should be taken into consideration that, due to broken rotor bars,
IM stator winding heating increases and insulation overheats, which
essentially reduces its durability. It limits the operation time of IM with
broken rotor bars.
To determine the durability of IM insulation “a rule of eight
degrees” is used, according to which the increase of temperature by
every eight degrees above the maximum allowable one makes the
insulation durability half as long. This rule is analytically written down
in the following way:
i.d .  T0  e Ki.d . i.h. , (5.3)
where i.d . – the insulation durability at temperature i.h. ; i.h. – the

temperature of insulation heating; T0 – the conditional durability
of insulation at i.h.  0 ( T0  6.225 104 years at i.d .  7 years and

i.h. =105 С); Ki.d . – the coefficient of insulation durability.
Modeling resulted in obtaining a relation of the losses in the stator
and rotor windings of IM with broken rotor bars to the rated value of
losses P rat (Table 5.4).
Table 5.4. The relation of losses in the stator and rotor windings
to the rated value of losses

Pe1  Pe2
, r.u.
P rat
1. no 1
2. 3% (one broken bar) 1.04
3. 6% (two broken bars) 1.22
4. 9% (three broken bars) 1.44
Thus, with 3% of broken rotor bars the value of heat losses grows
by 4%, which causes the increase of windings temperature also by 4%.
So, taking into account the growth of windings temperature, the
insulation durability is:
 for 3% of broken rotor bars:

i.d .  6.225 104  e0.07281251.04
 
 4.8 years;

i.d .  6.225 104  e0.07281251.22
 
 0.94 years;

i.d .  6.225 104  e0.07281251.44
 
 0.13 years.
So, the full-load operation of IM with 3% of broken rotor bars

reduces the durability of insulation of the stator windings by 1.5 times,
with 6% of broken rotor bars – by 7.5 times, and with 9% – by 53
times.
Thus, for IM in a general case the maximum allowable number of
broken bars is the number that does not exceed 10% of the total number
of the rotor bars. This degree of rotor breakage causes considerable
losses in windings, essential decrease of the duration of winding
insulation and considerable overheat of the stator and rotor windings.
So, when diagnostics is completed, if such a degree of breakage is
revealed, there arises a necessity for the withdrawal of the broken IM
from the technological process and its transfer to the repair shop. If the
number of broken rotor bars is within 10%, the calculated predictable
service life of the broken IM is determined, and the maintenance
personnel takes a decision as to the withdrawal of the IM from the
technological process.
5.4. CONCLUSION
The developed hard- and know are enable the efficient realization
of the proposed methods for the diagnostics of induction motor broken
rotor bars.
The experimental research of the method for the diagnostics of IM
broken rotor bars at the devised laboratory computer-aided stand
confirmed the efficiency of the proposed diagnostics method.
It is demonstrated that, when there are about 10% of broken rotor
bars, the stator and rotor windings are greatly overheated, which is a
reason for the withdrawal of the motor from the technological process
and the necessity for its repair.
CONCLUSION
A method for the diagnostics of induction motor broken rotor bars

by means of the analysis of the signal of the electromotive force in the
stator windings under motor self-running-out condition with the use of
wavelet-transform and the decomposition of signals on the basis of the
theory of inverse z-transform is proposed in the monograph.
The performed analysis of the contemporary methods for the
diagnostics of IM broken rotor bars and the proposed classification
proved the necessity for working out a diagnostics method that would
allow the determination of broken rotor bars without withdrawal of the
motor from the technological process, its disassembling and
introduction of additional sensors into its structure.
The proposed criteria of the assessment of the efficiency of
diagnostics methods made it possible to substantiate the use of the
signal of electromotive force induced in IM stator windings by rotor
decaying currents in the motor self-running-out condition for the
diagnostics of induction motor broken rotor bars.
The developed circuit mathematical IM models enable obtaining

distribution of the instantaneous values of currents in the rotor bars
under the motor self-running-out condition depending on the previous
condition of IM operation. These models are applied to motors with
different number of broken bars, taking into account the geometric
position of the breakage.
The proposed methods for the calculation of electromagnetic field
in IM cross section with the use of circuit models and a model based on
FEM allow the assessment of broken rotor bars influence on the signals
of EMF in the stator windings under the motor self-running-out
condition.
The developed software module with the use of LUA programming
language enables the calculation of electromagnetic field in the motor
cross section and the determination of the instantaneous values of EMF
in the stator winding elements under the self-running-out condition of
the motor in the automatic mode. The module is applicable to the IMs
of different powers with an unconditioned pitch of the rotor rotation for
the research of information signs.
The analysis of the structural features of various IM types and the
results of modeling is performed. The analysis revealed that the
following structural factors may influence the formation of the signal of
EMF in the stator windings: the number of motor ports, the circuit of
the connection of coil groups in the winding phase and the type of the
stator winding. These factors cause the mutual superposition of the
information signs of breakage in EMF signal.
A method for the decomposition of the signal of the EMF of IM
stator winding phase into the signals of the EMF of active sides of coils
with the use of information about the number of stator slots, winding
type, the number of ports is developed on the basis of the theory of
inverse z-transform. The method makes it possible to improve the
Conclusion 151
reliability of the diagnostics of IM broken rotor bars due to singling out

the information signs available in the signal of EMF of one active side
of the coil.
The proposed function of the average value of the sum of the
coefficients of wavelet-transform for the area of the medium
frequencies of EMF signals enables the significant simplification of the
procedure of singling out the signal of the EMF of one side of the coil
from the signal of the EMF of the phase.
The analysis of the theoretic and experimental research revealed
that the proposed method for the diagnostics of IM broken rotor bars in
the motor self-running out condition with the use of wavelet-analysis of
the signal of EMF in the stator windings makes it possible to determine
the number and the relative position of broken rotor bars.
The threshold value of the number of broken rotor bars is determined
by the results of the research of variable loss in an induction motor and
its predicted operation time. This value is about 10% of all the bars; at
this value it is recommended to produce a warning signal to inform the
personnel about the necessity for the repair of the motor.
REFERENCES
[1] Aderiano M. da Silva, Povinelli J. Richard, Demerdash A. O.

Nabeela. Induction Machine Broken Bar and Stator Short-Circuit
Fault Diagnostics Based on Three-Phase Stator Current
Envelopes. IEEE Transactions On Industrial Electronics. – Vol.
55, No. 3. – P. 1310–1318.
[2] Amine Y., Henao Humberto, Capolino Gérard-André. Broken
Rotor Bars Fault Detection in Squirrel Cage Induction Machines.
IEEE. – 2005. –P. 741–747.
[3] Zagirnyak M., Kalinov A., Melnykov V. Sensorless vector-control
system with the correction of stator windings asymmetry in
induction motor. Przeglad Elektrotechniczny. – 2013. – Vol. 89,
No. 12. – P. 340–343.
[4] Zagirnyak М., Chumachova A., Kalinov A. Correction of
operating condition of a variable-frequency electric drive with a
non-linear and asymmetric induction motor. Proceedings of
International IEEE Conference EUROCON. – 2013. –
P. 1033–1037.
[5] Zagirnyak M., Mamchur D., Kalinov A. Comparison of induction
motor diagnostic methods based on spectra analysis of current and
154 References
instantaneous power signals. Przeglad Elektrotechniczny. – 2012.

– Vol. 12b. – P. 221–224.
[6] Panadero Rubén P., Llinares J. P., Alarcon V. C., Pineda S. M.
Review Diagnosis Methods of Induction Electrical Machines
based on Steady State Current. Department of Electrical
Engineering, Polytechnic University of Valencia Campus of Vera.
– Valencia (Espaсa). – 2011. – P. 45–53.
[7] Kucheruk V. Yu. Elements of the theory of building technical
diagnostics systems of electric motors. Monograph. – Vinnytsya:
Universam–Vinnytsya, 2003. – 195 p.
[8] Shyrnin I. G., Tkachuk A. N. Short-circuited windings of rotors of
engines of underground machines. Proceedings of the Lugansk
Branch of the International Academy of Informatics: Scientific
Journal. – 2000.– Issue 2 (9). – P. 97–104.
[9] Ying Xie. Performance Evaluation and Thermal Fields Analysis
of Induction Motor With Broken Rotor Bars Located at Different
Relative Positions. College of Electrical and Electronic
Engineering, Harbin University of Science and Technology,
China, IEEE Transactions on Magnetics. – 2010. – Vol. 46, No. 5.
– P. 1243–1250.
[10] Oviedo S. J., Quiroga J. E., Borrás C. Experimental Evaluation of
Motor Current Signature and Vibration Analysis for Rotor Broken
Bars Detection in an Induction Motor. Proceedings of the 2011
International Conference on Power Engineering, Energy and
Electrical Drives. – May 2011. – P. 125–131.
[11] Zagirnyak M. V., Mamchur D. G., Kalinov A. P., Chumachova A.
V. Diagnostics of induction motors based on analysis of the power
consumption signal. Monograph. – Kremenchuk: PP Shcher-
batykh O. V., 2013. – 208 p.
[12] Nemec M., Drobnič K., Nedeljković D., Fišer R., Ambrožič V.
Detection of Broken Bars in Induction Motor Through the
References 155
Analysis of Supply Voltage Modulation. IEEE Transactions On

Industrial Electronics. – 2010. – Vol. 57, No. 8. – P. 2879–2888.
[13] Gashimov M. A., Gadzhiev G. A., Mirzoev S. M. Diagnostics of
eccentricity and breakage of rotor rods in asynchronous motors
without their disconnection. Elektrotekhnika. – 1998. – No. 10. –
P. 46–51.
[14] Pineda-Sanchez M., Riera-Guasp M. J., Antonino-Daviu A.,
Roger-Folch J., Perez-Cruz J., Puche-Panadero R. Instantaneous
Frequency of the Left Sideband Harmonic During the Start-Up
Transient: A New Method for Diagnosis of Broken Bars. IEEE
Transactions On Industrial Electronics. – 2009. – Vol. 56, No. 11.
– PP. 4557–4570.
[15] Caner Aküner, Temiz Ismail. Symmetrically broken rotor bars
effect on the stator current of squirrel-cage induction motor.
Przegląd Elektrotechniczny (Electrical Review). – 2011. – No. 3.
– P. 313–314.
[16] Calis H., Unsworth P. J. Fault diagnosis in induction motors by
motor current signal analysis. Proceedings SDEMPED. – 1999. –
P. 237–241.
[17] Mehala N., Dahiya R. Condition monitoring methods, failure
identification and analysis for Induction machines. International
Journal of Circuits, Systems and Signal Processing. – 2009. –
Issue 1, Vol. 3. –P. 10–17.
[18] Mehala N., Dahiya R. Motor current signature analysis and its
applications in induction motor fault diagnosis. International
journal of systems applications, engineering & development. –
2007. – No. 1. – Р. 29–35.
[19] Thomson William T. On-Line Motor Current Signature Analysis
Prevents Premature Failure of large Induction Motor Drives.
IEEE. – 2009. – Vol. 24. – P. 30–35.
[20] Thomson William T., Gilmore Ronald J. Motor Current Signature
Analysis To Detect Faults In Induction Motor Drives.
156 References
Proceedings Of The Thirty-Second Turbomachinery Symposium. –

2003. – P. 145–156.
[21] Vaimann T., Ants K. Detection of broken rotor bars in three-phase
squirrel-cage induction motor using fast Fourier transform//10th
International Symposium “Topical Problems in the Field of
Electrical and Power Engineering” Pärnu, Estonia. – 2011. – P.
52–56.
[22] Zagirnyak M., Mamchur D., Kalinov A. A comparison of
informative value of motor current and power spectra for the tasks
of induction motor diagnostics. Proceedings of 2014 IEEE 16th
International Power Electronics and Motion Control Conference
and Exposition (PEMC). – Antalya, Turkey, 2014. – P. 541–546.
[23] Zagirnyak M., Mamchur D., Kalinov A. Induction motor
diagnostic system based on spectra analysis of current and
instantaneous power signals. Proceedings of SOUTHEASTCON. –
Lexington, USA, 2014. – P. 1–7.
[24] Kral C., Haumer A., Grabner C. Modeling and Simulation of
Broken Rotor Bars in Squirrel Cage Induction Machines.
Proceedings of the World Congress on Engineering. – WCE 2009,
London. – Vol 1. – P. 1–6.
[25] Ceban A., Pusca R. and Romary R. Study of Rotor Faults in
Induction Motors Using External Magnetic Field Analysis. IEEE
Transactions on Industrial Electronics. – 2012. – Vol. 59, No. 5. –
P. 2082–2093.
[26] Ordaz-Moreno Alejandro, Romero-Troncoso Rene de Jesus, Vite-
Frias Jose Alberto, Rivera-Gillen Jesus Rooney, Garcia-Perez
Arturo. Automatic Online Diagnosis Algorithm for Broken Bar
Detection on Induction Motors Based on Discrete Wavelet
Transform for FPGA Implementation. IEEE Transactions on
Industrial Electronics. – 2008. – Vol. 55, No. 5. –P. 2193–2202.
[27] Jose Antonino-Daviu A., Riera-Guasp M., Folch José Roger,
Palomares M. Pilar M. Validation of a New Method for the
References 157
Diagnosis of Rotor Bar Failures via Wavelet Transform in Industrial

Induction Machines. IEEE Transactions On Industry Applications. –
2006. – Vol. 42, No. 4. – P. 990–996.
[28] Kechida R. Menacer A. DWT Wavelet Transform for the Rotor
Bars Faults Detection in Induction Motor. Electric Power and
Energy Conversion Systems (EPECS). – 2nd International
Conference. – 2011. – P. 1–5.
[29] Keskes H., Braham A., Lachiri Z. Broken Rotor Bar Diagnosis in
Induction Machines through Stationary Wavelet Packet Transform
under Lower Sampling Rate. First International Conference on
Renewable Energies and Vehicular Technology. – 2012. – P. 452–
459.
[30] Khadim Main S., Giri V. K. Broken Rotor Bar Fault Detection in
Induction Motors Using Wavelet Transform. International
Conference on Computing, Electronics and Electrical
Technologies. – 2012. – P. 1–6.
[31] Mehala N., Dahiya R. Rotor Faults Detection in Induction Motor
by Wavelet Analysis. International Journal of Engineering
Science and Technology. – 2009. – Vol. 1(3). – P. 90–99.
[32] Riera-Guasp M., Jose A. Antonino-Daviu, Pineda-Sanchez M.,
Puche-Panadero R., Perez-Cruz J. A General Approach for the
Transient Detection of Slip-Dependent Fault Components Based
on the Discrete Wavelet Transform. IEEE Transactions on
Industrial Electronics. – 2008. – Vol. 55, No. 12. – P. 4167–4180.
[33] Wei Y., Shi B., Cui G., Yin J. Broken Rotor Bar Detection in
Induction Motors via Wavelet Ridge. International Conference on
Measuring Technology and Mechatronics Automation. – 2009. –
P. 625–628.
[34] Faiz J., Ebrahimi B. M. Mixed fault diagnosis in three-phase
squirrel-cage induction motor using analysis of air-gap magnetic
field. Progress In Electromagnetics Research. – 2006. – P. 239–
245.
158 References
[35] Zouzou Salah E., Khelif Samia, Halem Noura, Sahraoui M.

Analysis of Induction Motor with broken rotor bars Using Finite
Element Method. Electrical Engineering Laboratory of Biskra. –
2011. – P. 1–5.
[36] Matic D., Kulic F., Alarcon V. C., Puche-Panadero R. Artificial
neural networks broken rotor bars induction motor fault detection.
10th Symposium on Neural Network Applications in Electrical
Engineering. – 2010. doi: 10.1109/NEUREL.2010.5644051.
[37] Altug S., Yuen C. Mo, Joel Trussell H. Fuzzy Inference Systems
Implementedon Neural Architectures for Motor Fault Detection
and Diagnosis. IEEE Transactions On Industrial Electronics. –
Vol. 4, No. 6. – 1999. –P. 1132–1136.
[38] Aroui T., Koubaa Y., Toumi A. Application of Feedforward
Neural Network for Induction Machine Rotor Faults Diagnostics
using Stator Current. Research Unity of Industrial Process
Control. – Vol. 3, No. 4. – 2007.–P. 213–226.
[39] Bayir R., Bay O. F. Kohonen Network based fault diagnosis and
condition monitoring of serial wound starter motor. IJSIT Lecture
Note of International Conferense on Intelligent Knowledge
Systems. – Vol. 1, No. 1. – 2004. – P. 130–136.
[40] Cupertino F., Giordano V., Mininno E., Salvatore L. Application
of Supervised and Unsupervised Neural Networks for Broken
Rotor Bar Detection in Induction Motors. Polytechnic University
of Bari V. Orabona, Bari, Italy. – 2001. –P. 1895–1901.
[41] Mazur D. Detection of Broken Rotor Bars in Induction Motors
Using Unscented Kalman Filters. Proceedings of the 2nd
International Congress on Computer Applications and
Computational Science. – 2011. –P. 503–511.
[42] Dias C. G., Chabu I. E., Bussab M. A. Hall Effect Sensor and
Artificial Neural Networks Applied on Diagnosis of Broken Rotor
Bars in Large Induction Motors. IEEE International Conference
References 159
on Computational Intelligence for Measurement Systems and

Applications La Coruna. – 2006. –P. 34–39.
[43] Seyed Abbas T., Malekpour M. A Novel Technique for Rotor Bar
Failure Detection in Single-Cage Induction Motor Using FEM and
MATLAB/SIMULINK. Mathematical Problems in Engineering.
Hindawi Publishing Corporation. – 2011. – Vol. 3. – P. 14–22.
[44] Zagirnyak M. V. Electromagnetic calculations. Textbook. –
Khar'kov: Typohrafyya Madryd, 2015. – 320 p.
[45] Ivanov-Smolensky A. Electrical Machines. – Мoscow: MIR
Publishers, 1983. – 280 p.
[46] Zagirnyak М. V., Prus V. V., Nevzlin B. I. Functional
interrelation of the parameters of electric machines, devices and
transformers with a generalized linear dimension. Monograph. –
Khar'kov: Izdatel'stvo «Tochka», 2014. – 188 p.
[47] Zagirnyak M. V., Almashakbeh Atef S., Qawaqzeh M. Z.
Functional interrelation of the parameters of electric machines,
devices and transformers. – Nova Science Publishers, 2017. –
201 p.
[48] Cupertino F., de Vanna E., Salvatore L. and Stasi S. Analysis
Techniques for Detection of IM Broken Rotor Bars After Supply
Disconnection. IEEE Transactions On Industry Applications. –
2004. – Vol. 40, No. 2. – P. 526–533.
[49] Vaskovskyi J. M., Kovalenko M. A. Diagnostics of latent defects
of the short-circuited rotor winding of asynchronous motor by an
induction method. Technical Electrodynamics. – 2013. – No. 2. –
P. 69–74.
[50] Vaskovskyi J. M. Field analysis of electrical machines. Textbook.
– K.: NTUU “KPI”, 2007. – 192 p.
[51] Boule O. B. Methods for calculating the magnetic systems of
electrical apparatus, magnetic circuit, field and program FEMM.
Textbook. – M.: Izdatel’skyy tsentr «Akademiya», 2005. – 336 p.
160 References
[52] Singiresu S. Rao. The Finite Element Method in Engineering. –

Butterworth-Heinemann, 2010. – 726 p.
[53] Al-Mashakbeh Atef S., Mamchur D., Kalinov A., Zagirnyak M. A
diagnostic of induction motors supplied using frequency converter
basing on current and power signal analysis. Przegląd
Elektrotechniczny. –2016. – No 12. – P. 5–8.
[54] Zagirnyak М., Mamchur D., Kalinov A., Al-Mashakben Atef S.
Induction motors faulth detection based on instantaneous power
spectrum analysis with elimination of the supply mains influence.
ACEEE International Journal on Electrical and Power
Engineering. – 2013. – Vol. 4, No. 3. – P. 7–17.
[55] Qawaqzeh M. Z., Kalinov A., Loyous V., Zagirnyak M.
Experimental research of the loading system for an induction
motor with the use of a double-fed machine. Przegląd
Elektrotechniczny. – 2017. – No. 1. – P. 173–176.
[56] Zagirnyak M. V., Rodkin D. I., Romashykhin Iu. V., Chornyi O.
P. Energy method identification of induction motors. –
Kremenchug: ChP. Shcherbatyh A. V., 2013. – 164 p.
[57] Arezki M., Naît-Saïd Mohamed-Saïd, Benakcha A Hamid, Drid
Saïd. Stator Current Analysis Of Incipient Fault Into
Asynchronous Motor Rotor Bars Using Fourier Fast Transform.
Journal of Electrical Engineering. – Vol. 55, No. 5–6. – 2004. –
P. 122–130.
[58] Benbouzid Mohamed El Hachemi., Vieira Michelle, Theys
Céline. Induction Motors’ Faults Detection and Localization
Using Stator Current Advanced Signal Processing Techniques.
IEEE Transactions On Power Electronics. –1998. – Vol. 14. – P.
4–11.
[59] Mallat S. A Wavelet Tour of Signal Processing: The Sparse Way.
– Academic Press, 2008. – 832 p.
[60] Zagirnyak M., Romashykhina Zh., Kalinov A. Diagnostic signs of
induction motor broken rotor bars in electromotive force signal.
References 161
Proceedings of 17th International Conference Computational

Problems of Electrical Engineering. – 2016. – P. 1–4. doi:
10.1109/CPEE.2016.7738722.
[61] Strang G., Nguyen T. Wavelets and Filter Banks. – SIAM, 1996. –
490 p.
[62] Chui Charles K. An Introduction to Wavelets. – Elsevier, 2016. –
278 p.
[63] Zagirnyak M., Romashihina Zh. , Kalinov A. Diagnostic of
broken rotor bars in induction motor on the basis of its magnetic
field analysis. Acta Technica Jaurinensis. – 2013. – Vol. 6, No. 1.
– P. 115–125.
[64] Zagirnyak M., Romashykhina Zh. , Kalinov A. Analysis of the
induction motor magnetic field for diagnostics of rotor bar
damages. 4th Symposium on Applied Electromagnetics. – 2012. –
Р. 87–88.
[65] Daubechies I. Ten Lectures on Wavelets. – Society for Industrial
and Applied Mathematics, 1992. – 254 р.
[66] Cabanas F., Glez F. Pedrayes, González M. Ruiz, Melero M. G.,
Orcajo G. A., Cano J. M., Rojas C. H. A new On-Line Method for
the Early Detection of Broken Rotor Bars in Asynchronous
Motors Working under Arbitrary Load Conditions. IEEE. – 2005.
– P. 662–669.
[67] Dede E. M., Lee J., Nomura T. Multiphysics Simulation:
Electromechanical System Applications and Optimization. –
Springer, 2014. – 212 p.
[68] Tyukov V. A., Pastuhov V. V., Korneev K. V. Three-drum model
for determining the diameter of the rod of a short-circuited rotor
of an induction motor. Izvestiya Tomskogo politehnicheskogo
universiteta. – 2011. – Vol. 319, No. 4. – P. 99–102.
[69] http://www.femm.info/wiki/HomePage.
162 References
[70] Zagirnyak M, Kalinov A., Romashykhina Zh. Decomposition of

electromotive force signal of stator winding in induction motor at
diagnostics of the rotor broken bars. Scientific Bulletin of National
Mining University. – 2016. – No. 4 (154). – P. 54–61.
[71] Zagirnyak M., Romashykhina Zh., Kalinov A. The diagnostics of
induction motors rotor bar breaks based on the analysis of
electromotiveforce in the stator windings. Electrical engineering
& Electromechanics. – 2014, No. 6. – P. 34–42.
APPENDICES
APPENDIX A
The Catalog of the Data and Geometric Parameters of

the Investigated Induction Motors
Table А1. The published data of АIR80V4U2 IM
Parameter Value
Rated power, kW 1.5
Rated rotational speed, rev/min 1395
Efficiency,% 77
Power coefficient, r.u. 0.81
Rated voltage, V 220/380
Rated current, А 6.3/3.6
Winding connection Δ/Υ
Stator windings resistive impedance, Ohm 5
Rotor windings reduced resistive impedance, Ohm 3.69
Stator winding inductive reactance, Ohm 4.35
Rotor windings reduced inductive reactance, Ohm 4.01
164 Appendices
Table A2. The geometric parameters of АIR80V4U2 IM
Geometric parameter Value

Length of the stator core, mm 104
Number of the stator slots 36
Number of the rotor slots 34
External diameter of the stator core, mm 132.6
Internal diameter of the stator core, mm 90.5
Height of the motor, mm 184
Length of the shaft, mm 370
Diameter of the shaft, mm 24.9
Diameter of the rotor, mm 85.7
Height of the rotation axis, mm 80
Length of the rotor, mm 124.6
Length of the rotor bar, mm 106
Diameter of the bearing, mm 52.1
Height of the stator slot, mm 20
Upper width of the stator slot, mm 3
Lower width of the rotor bar, mm 7
Upper width of the rotor bar, mm 4.5
Height of the rotor slot, mm 1.2+2.5=3.7
Diameter of the rotor ring, mm 82.2
Height of the rotor ring, mm 2.5
Width of the rotor ring, mm 9.6
Table А3. The published data of 4АN200L2U3 IM
Parameter Value
Rated power, kW 75
Rated rotational speed, rev/min 2940
Efficiency,% 92
Power coefficient, r.u. 0.9
Rated voltage, V 220/380
Rated current, А 79.5/54.5
Winding connection Δ/Υ
Parameter Value
Stator windings resistive impedance, Ohm 0.14
Rotor windings reduced resistive impedance, Ohm 0.075
Stator winding inductive reactance, Ohm 0.585
Rotor windings reduced inductive reactance, Ohm 0.643
Appendices 165
Table A4. The geometric parameters of 4АN200L2U3 IM
Geometric parameter Value

Length of the stator core, mm 170
Number of the stator slots 36
Number of the rotor slots 46
External diameter of the stator core, mm 349
Internal diameter of the stator core, mm 200
Diameter of the shaft, mm 80
Diameter of the rotor, mm 198
Height of the rotation axis, mm 200
Length of the rotor, mm 170
Length of the rotor bar, mm 170
Diameter of the bearing, mm 60
Height of the stator slot, mm 26
Upper width of the stator slot, mm 4
Lower width of the rotor bar, mm 3
Upper width of the rotor bar, mm 7.5
Height of the rotor slot, mm 30
Diameter of the rotor ring, mm 145.75
Height of the rotor ring, mm 33.655
Width of the rotor ring, mm 52.25
166 Appendices
APPENDIX B
The Test Signals of the Electromotive Forces of Coils with Different

Position of Alias Disturbance, which Imitate two Broken Bars and
their Wavelet-Spectra with the Use of Different Wavelet-Basis
Figure B1. The test signal of the EMF of coil etest1 with alias disturbance, which
correspond to two broken bars and their corresponding wavelet-spectra with the use: of
Daubechies wavelet (a), of Symlet wavelet (b), of Coiflets wavelet (c).
Appendices 167
168 Appendices
Appendices 169
Figure B4. The test signals of the EMF of coils with alias disturbance, which
correspond to two broken bars relatively located at the angle of 95o and their
corresponding wavelet-spectra with the use: of Daubechies wavelet (a), of Symlet
wavelet (b), of Coiflets wavelet (c).
170 Appendices
APPENDIX C
The Block Diagrams for Simulating an Induction Motor Circuit

Mathematical Model with the Presentation of the Rotor in
the Form of a System of Short-Circuited Bars
y
1
[y]
cos K 34
From 1 1 8
1
Gain 8
K 34 cos 2
[y]
2
From 18
cos
cos
K 34 2
3
3
Gain 3 K 34 9
cos
4 Gain 10
cos
[y] 4
From 2 [y]
From 14
K 34
3
Gain 1
cos
10
K 34
[y]
Gain 9
From 4
cos
[y]
Figure C1. A block diagram for simulating the shift angles between the stator From 15
K 34
phase A winding and the rotor
4
bars.
Gain 4
K 34 11
[y]
Gain 7
From 10
cos
5
[y]
5
K 34 5 From 17
cos 6
Gain 6 6
[y]
From 6 cos
12
K 34
Gain 12
cos cos
6 [y]
K 34 7
7 From 22
Gain 5
Figure C2. A block diagram

[y]
of IM mechanical part. cos
8
8 K 34
13
From 7
Gain 11
[y]
K 34 7
From 24
Gain 2
[y]
From 9
Appendices 171
2
34 PsiA
UA
dPsiA /dt
PsiA
1 IA
1/Ls
s
1
Integrator 1 Gain 3 IA
Gain 2
Rs
Gain 1
I1
k1 1
I2
Product 4
2
Gain 4
Product 1 I3
Lm 3
Product 2
I4
4
Gain 5 Product 3 I5
5
1/11
Product 5 I6 cosy
6
1
Product 6 I7
7 2
3
I8
Product 8
8 4
5
I9
Product 7
9 6
I10 7
Product 9 10
8
9
Product 10 I11
11 10
I12 11
Product 11 12
12
I13 13
Product 12 13
14
Product 13 I14 15
14
16
y
Product 14 17 y 35
I15
18
15
Product 15 19
I16 20
Product 16 16
21
I17 22
17
Product 17 23
I18
18 24
Product 18
I19 25
19
26
Product 19
27
I20
Product 20 20 28
I21 29
21
Product 21 30
I22
22 31
Product 22 32
I23 33
Product 23 23
I24
24
Product 24
I25
25
Product 25
I26
26
Product 26 I27
27
I28
Product 27
28
I29
Product 28
29
I30
Product 29
30
I31
Product 30
31
I32
Product 31 32
I33
Product 32 33
Product 33
Figure C3. A block diagram of IM stator phase A.

172 Appendices
APPENDIX D
Instruction Sequence in the Calculation of the Electromagnetic

Field in Induction Motor Cross Section with the Use of LUA
Programming Language
I1n=-6.176643337
I2n=-1.048275981
I3n=4.334702638
I4n=9.252529795
I5n=0
I6n=15.01380419
I7n=15.08149291
I8n=13.23742675
I9n=9.726220843
I10n=5.010318713
I11n=-0.2884278
I12n=-5.470197361
I13n=-9.850188358
I14n=-12.85026538
I15n=-14.07665185
I16n=-13.37301515
I17n=-10.8416652
I18n=-6.829928836
I19n=-1.883488046
I20n=3.327053568
I21n=8.096487157
I22n=11.780023
I23n=13.88020768
I24n=14.11400968
I25n=12.45102255
I26n=9.117630245
Appendices 173
I27n=4.566569145
I28n=-0.584012643
I29n=-5.633788235
I30n=-9.894566936
I31n=-12.78299851
I32n=-13.89921019
I33n=-13.08084526
I34n=-10.42532979
T=0.159
tn=0
tk=(0.041*2)
k=(360*2) the number of models for calculation is assigned
alfa=0
n=1
h=tk/k
for t=tn,tk,h do
open("34.FEM")
mi_saveas("temp.fem")
mi_selectgroup(37)
mi_move_rotate(0,0,alfa)
I1=I1n*exp(-t/T)
I2=I2n*exp(-t/T)
I3=I3n*exp(-t/T)
…
I34=I34n*exp(-t/T)
mi_addcircprop("1",I1,1) the values of current in the rotor bars
are assigned
mi_addcircprop("2",I2,1)
…
alfa=alfa+1
174 Appendices
mi_analyze(1)
mi_loadsolution()
mo_groupselectblock(1) stator slots are chosen
A1=mo_blockintegral(1) the values of vector magnetic
potential in the chosen elements of
the stator winding are calculated
handle=openfile("AD1.txt","a");
write(handle,A1,"\n");
closefile(handle);
mo_clearblock()
mo_groupselectblock(2)
A2=mo_blockintegral(1)
closefile(handle);
mo_clearblock()
closefile(handle);
mo_clearblock()
…
handle=openfile("AD30...txt","a");
closefile(handle);
mo_clearblock()
n=n+1
end
Appendices 175
APPENDIX E
The Electromotive Force Signals in the Elements of Stator Winding

for the Analyzed Induction Motor
broken bars
Е, V
10
0 0.02 0.04 0.06 0.08 t,s
10
Figure E1. The signal of the EMF of one active side of the coil of the stator winding of
АIR80V4U2 IM with three broken rotor bars.
broken bars
Е, V
20
0 0.02 0.04 0.06 0.08 t,s
20
Figure E2. The signal of the EMF of the coil of the stator winding of АIR80V4U2 IM
with three broken rotor bars.
176 Appendices
Е, V
50
0 0.02 0.04 0.06 0.08 t,s
50
Figure E3. The signal of the EMF of the coil group of the stator winding of
АIR80V4U2 IM with three broken rotor bars.
Е, V
200
100
0 0.02 0.04 0.06 0.08 t, s
100
Figure E4. The signal of the EMF of the phase of the stator winding of АIR80V4U2
IM with three broken rotor bars.
Е, V broken bars
20
10
0 0.02 0.04 0.06 0.08 t, s

10
20
Figure E5. The signal of the EMF of one active side of the coil of the stator winding of
4АN200L2U3 IM with two broken rotor bars.
Appendices 177
Е, V broken bars
40
20
0 0.02 0.04 0.06 0.08 t, s

20
40
Figure E6. The signal of the EMF of the coil of the stator winding of 4АN200L2U3 IM
Е, V
200
100
0 0.02 0.04 0.06 0.08 t,s

100
200
Figure E7. The signal of the EMF of the phase of the stator winding of 4АN200L2U3
IM with two broken rotor bars.
Е, V broken bars
20
10
0 0.02 0.04 0.06 0.08 t, s

10
20
Figure E8. The signal of the EMF of one active side of the coil of the stator winding
of 4АN200L2U3 IM with three broken rotor bars.
178 Appendices
Е, V broken bars
40
20
0 0.02 0.04 0.06 0.08 t, s
20
Figure E9. The signal of the EMF of the coil of the stator winding of 4АN200L2U3 IM
with three broken rotor bars.
Е,V
200
100
0 0.02 0.04 0.06 0.08 t, s

100
200
Figure E10. The signal of the EMF of the phase of the stator winding of 4АN200L2U3
IM with three broken rotor bars.
AUTHORS’ CONTACT INFORMATION
Mykhaylo Zagirnyak
Rector, D. Sc. (Eng.), professor
20, Pershotravneva ul, Kremenchuk, Ukraine
Email: mzagirn@kdu.edu.ua;mzagirn@gmail.com
Zhanna Romashykhina
Senior Lecturer, PhD (Eng.)
Email: romashykhina.zhanna@gmail.com
Andrii Kalinov
Associate Professor, PhD. (Eng.).
Email: andrii.kalinov@gmail.com
INDEX
A D
air gap, vii, xiii, 8, 13, 16, 28, 29, 30, 31, Daubechie wavelets, 48, 49
38, 62, 94 decomposition of the coil EMF signal, 121,
angle of shift of stator winding coils, xviii, 122
51 decomposition of the signal of winding
phase EMF, 117
diagnostic signal, ix, 16, 20, 21, 25, 27, 35,
B
39, 42, 45, 65, 66
disconnection from the supply main, xiv,
bar inductive reactance, 87
71, 72, 84, 85, 86, 88, 89, 94, 95, 104,
block of voltage sensors, 133, 134, 135,
139
136
discretization frequency, xiv, 48
discretization period, 48
C distribution of currents, 85, 88, 89
circuit model of IM rotor, 86

circuit-field mathematical model, xi, 90 E
coefficient of insulation durability, 146
electric machine, vii, xi, 2, 10, 28, 58, 90,
Coiflet wavelets, 48, 49, 53
159
continuous wavelet transform, xi
electromagnetic field, ix, xi, 18, 31, 32, 33,
cross-section of IM, 104
34, 35, 38, 39, 66, 71, 89, 90, 91, 92, 93,
current in a broken bar, 9
94, 103, 104, 117, 121, 126, 130, 150,
current spectrum, 14, 40, 41
172
182 Index
electromotive force, viii, ix, xi, 31, 58, 94, magnetic induction, xiii, 14, 15, 28, 30, 31,
105, 113, 114, 127, 149, 160, 162, 166, 32, 62, 92, 94
175 mathematical model, ix, 69, 70, 71, 72, 79,
electromotive force of the winding phase, 81, 83, 85, 86, 89, 90, 100, 103, 104,
127 143, 145, 170
experimental signal, 140, 141 mutual inductance, xv, 74, 75
F N
finite difference method, xi, 36 nominal condition, xvi, xviii, 142, 144
finite element method, xi, 36, 66, 69, 92, number of IM poles pairs, xvi, 29
93, 104, 158, 160 number of poles, 60
finite elements grid, 38, 94 number of slots per a pole and a phase, 59,
flux linkage, xix, 33, 72, 73, 74, 76, 77, 78, 60
79, 80, 82 number of stator slots, 60, 150
Fourier transform, viii, 14, 40, 41, 42, 45, number of the coil turns, xvii, 61
67, 156 number of turns in the slot, xvii, 33
number of wavelet-expansion coefficients,
xv, 111
H
high-frequency components, 43, 110 O
orthogonal wavelet, 43, 44, 47, 48, 67, 105

I
impedance of rotor bars, xvii, 87 P

impedance of short-circuited ring, xvii, 87
inverse z-transform, ix, 127, 131, 149, 150 permeability, 31
phi scaling function, xix, 43
L
R
losses in IM windings, xvi, 144
losses in the rotor of IM, xvi, 142 relative permeances of IM stator, xviii
losses in the stator windings of IM, xvi, 142 rotor bar current, 84, 87
rotor current, 71, 73
rotor rotation angle, xvii, 51, 75
M
rotor time constant, 102
magnetic field, vii, viii, xiv, 13, 14, 18, 28,
29, 30, 31, 34, 61, 90, 156, 157, 161 S
magnetic flux, 8, 27, 33, 72, 94, 95
scalar magnetic potential, xviii, 36
Index 183
self-running-out mode, xix, 34, 35, 39, 42, temperature of heating of the stator
63, 65, 66, 71, 82, 83, 84, 92, 93, 94, windings, xviii, 143, 145
103, 104, 140 temperature of insulation heating, xviii, 146
short-circuited ring, xi, 5, 7, 85, 86, 87, 91, three-phase coordinate system, 71, 72, 85,
138 103
skew angle, xviii, 62, 63 time psi-function, 43
spectral analysis, viii, 20, 22, 39, 40, 41, 42,
67
V
squirrel-cage rotor, vii, xi, 7, 9, 23
stator current, vii, 12, 15, 71, 72, 142, 153,
vector magnetic potential, xi, xiii, 32, 33,
155, 158, 160
92, 94, 174
stator phase voltage, 138, 139
stator slot, xvi, 33, 34, 63, 116, 117, 164,
165, 174 W
Symlet wavelets, 48, 49, 53
wavelet order, 57
wavelet-basis, xi, 44, 45, 47, 53, 166
T winding pitch in slots, 60
winding type, 60, 150
temperature of heating of the rotor bars,
xviii, 143

(Energy Science, Engineering and Technology) Mykhaylo v. Zagirnyak, Zhanna IV. Romashykhina, Andrii P. Kalinov - The Diagnostics of Induction Motor Broken Rotor Bars On The Basis of The Electromotive

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ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY

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NOTICE TO THE READER

Library of Congress Cataloging-in-Publication Data

Chapter 5 The Experimental Verification of the Method

Electric machines are known to be the basic elements of electric

frequency under load, decrease of the efficiency. That is why timely

A Vector Magnetic Potential

E AB The EMF Inter-Phase of the Stator Of IM

Ec1  z  , Ecm  z  , m = 3 The Z-Image of the Coils EMF Signals

E ph  z  The Z-Image of the Winding Phase EMF

I adj1 , I adj 2 Currents in Adjacent Bars

jout Currents Density Caused by Outside EMF

K a The Function of the Average Value of the

 The Multiplication Factor of IM Starting

Pe1.br. Losses in the Stator Windings of IM with

Pe 2.br. Losses in the Rotor Of IM with Broken

Pe1 The Relation of the Losses in the Stator

T0 The Conditional Durability of Insulation

 bev A Skew Angle

rat The Relative Increase of the Value of

 ph Spatial Angle Between The Corresponding

THE CONTEMPORARY STATE OF

1.1. THE ANALYSIS OF SQUIRREL-CAGE INDUCTION

Squirrel-cage induction motors represent a most common type of

(mainly unsatisfactory protection of electric machines (EM) – 50% and

Figure 1.1. The statistics of induction motor failures (%).

Table 1.1. The statistics of induction motor failures

No. Number Damage No. Number Damage

EM diagnostics is performed to timely reveal damages. The

Table 1.2. The data of IM failures during 2016 in some production

No. PJSC «AutoKrAZ» Total number Number of broken IMs

Table 1.3. The distribution according to the types of damages

No. PJSC Number of broken IMs

Paper [8] contains the statistic data according to the results of

 bars break in the slot part;

The distribution of rotor breakages depending on their location and

copper winding aluminum winding

bars burning-out in the slot 8.9

bars break beside the short-

bar break off the short-

copper winding aluminum winding

bars burning-out in the slot 2.3

bars break beside the short-

copper winding aluminum winding

bars burning-out in the slot

bar break off the short- 20

1.2. THE RESEARCH OF THE CAUSES OF

The paper contains an analysis of the basic causes for the

1.2.1. The Technology of the Manufacture of an IM Rotor

An important factor that predetermines defects consists in the

1.2.2. Thermal Overloads

The overheating of the rotor windings (bars and short-circuited

1.2.3. Mechanical Overloads

Analogously to the stator windings, the magnetic forces in the air

Figure 1.5. Currents distribution with a broken rotor bar.

In Figure 1.5: I br – current in a broken bar, I adj1 , I adj 2 – currents in

1.3. THE ANALYSIS OF THE METHODS FOR