The Diagnostics of Induction Motor Broken Rotor Bars on the Basis of the Electromotive Force Analysis 1536126837, 9781536126839

Table of contents :
Contents
Preface
List of Abbreviations
List of Symbols
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
1.2. The Research of the Causes of the Occurrence of Induction Motor Broken Rotor Bars
1.2.1. The Technology of the Manufacture of an IM Rotor Squirrel Cage
1.2.2. Thermal Overloads
1.2.3. Mechanical Overloads
1.3. The Analysis of the Methods for the Diagnostics of Induction Motor Broken Rotor Bars and Their Classification
1.3.1. The Criteria of the Assessment of the Efficiency of Broken Rotor Bars Diagnostics
Criterion 1
Criterion 2
Criterion 3
Criterion 4
Criterion 5
Method 1
Method 2
Method 3
Method 4
Method 5
Method 6
Method 7
Method 8
Method 9
Method 10
Method 11
Method 12
Method 13
Method 14
Method 15
1.4. Conclusion
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
Static Operating Conditions
Idle Condition
Operation under Load
Short-Circuit Condition
Dynamic Operating Conditions
Start-Up Conditions
Self-Running-Out Condition
2.1.2. The Substantiation of Diagnostic Signals
2.2. The Substantiation of the Method for the Calculation of Induction Motor Electromagnetic Field
2.3. The Substantiation of the Use of the Wavelet-Transform for Diagnostic Signals Processing
2.4. The Choice of the Wavelet-Basis Functions for the Diagnostic Signals Wavelet-Transform
2.5. The Research of Factors Influencing the Generation of Electromotive Force in Induction Motor Stator Windings
2.6. The Generalized Methods of the Analysis of Induction Motor Broken Rotor Bars
2.6.1. The Basic Points of the Diagnostics Method
2.7. Conclusion
Chapter 3
Mathematical Models for the Research of the Method of Induction Motor Broken Rotor Bars Diagnostics
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
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
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
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
3.5. Conclusion
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
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 Decomposition of the Signal of Winding Phase EMF into the Signals of EMF of the Coil Groups
The Decomposition of the Signal of the EMF of the Coil Group into the Signals of Coils EMF
The Decomposition of the Coil EMF Signal into the Signals of the EMF of Two Active Sides of the Coil
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
4.3. Conclusion
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
5.2. The Analysis of the Results of the Experimental Research
5.3. The Assessment of Broken Rotor Bars Influence on Induction Motor Operation
5.4. Conclusion
Conclusion
References
Appendices
Appendix A
Appendix B
Appendix C
Appendix D
Appendix E
Authors’ Contact Information
Index
Blank Page

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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

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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

Copyright © 2018 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected]. NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. 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 Chapter 1

Chapter 2

Chapter 3

Chapter 4

xiii

The Contemporary State of the Problem of the Diagnostics of Induction Motor Broken Rotor Bars

1

Theoretical Foundation for the Research of Induction Motor Broken Rotor Bars in the SelfRunning-Out Condition

25

Mathematical Models for the Research of the Method of Induction Motor Broken Rotor Bars Diagnostics

69

The Method of Induction Motor Broken Rotor Bars Diagnostics with the Use of WaveletTransform

105

vi Chapter 5

Contents 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 wavelettransform. 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 EW ADC DCG BVS WB VMP WT ShCR SCR FDM FEM CWT CFMM EMF EM EMFl ED

Induction Motor Exciting Winding Analog-Digital Converter Direct Current Generator Block Of Voltage Sensors Wavelet-Basis Vector Magnetic Potential Wavelet-Transform Short-Circuited Ring Squirrel-Cage Rotor Finite Difference Method Finite Element Method Continuous Wavelet Transform Circuit-Field Mathematical Model Electromotive Force Electric Machine Electromagnetic Field 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 The Total Arithmetic Value of VMP in All the Slots of the Phase The Number of Winding Parallel Paths The Maximum Value of the Wavelet Scale

Az a

amax

B

B Bx , B y

The Amplitude of Magnetic Induction of the Fundamental Harmonic of the Field in the Air Gap Relative Magnetic Induction

B

Magnetic Induction Vector Components in Cartesian Coordinates Magnetic Induction Vector

bbev

The Slots Skew in Linear Dimensions

E1

E34

E A , EB , EC

The EMF of Power Supply for the Investigated IM 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

E

The EMF of the Rotor Steel Residual Magnetization 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 The Z-Image of the Coil Groups EMF Signals The Z-Image of EMF Signals of the Two Active Sides of the Coil Winding EMF

Eq1  z  , Eq 2  z  Et1  z  , Et 2  z 

e(t ) etest1, etest2, etest3 etest

f1 fm

fs G

H I 2beg

Coils Testing EMFs The Test Signal of the Total EMF of the Stator Winding Coil Group The Frequency of the Supply Network The Upper Boundary of Frequencies Band of Concentrated Fundamental Energy of the Signal (in Low-Frequency Domain) Discretization Frequency The Boundary of the Calculation Area the Vector of Magnetic Field 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

j

The Density of Outside Currents, Assigned in Stator Winding Sections 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 Created in Relative Units Reduced to the Maximum Value of the Scale amax for IM

K * a

L2

With Broken Rotor Bars The Values of Wavelet-Expansion Coefficients Rotor Inductance

LA

Phase Inductance

La

The Optimum Upper Level of WaveletDecomposition The Number of Wavelet-Expansion Coefficients The Active Length of the Stator

ka

l l1

M Me

M xy

The Maximum Value of Mutual Inductance Electromagnetic Torque Equation Mutual Inductance Between Windings Х and Y

xvi Ì

List of Symbols  st

m0

Pe1.br. Pe1.heal .

Pe 2.br. Pe2.heal. Pe1

The Multiplication Factor of IM Starting Torque in Relation to the Rated One Trigonometric Polynomial Losses in the Stator Windings of IM with Broken Bars The Rated Value of Losses of a Healthy IM Losses in the Rotor Of IM with Broken Bars The Rated Value of Losses of a Healthy IM

R2

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 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 The Relative Increase of the Value of Losses in IM Windings The Number of IM Poles Pairs The Number of Slots Per a Pole And a Phase 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 Ss

The Total Area of Cross Section of all Phase Coils Connected in Series The Sectional Area of the Stator Slot

s

IM Slipping

Pe2

Prat p q

List of Symbols

xvii

T0

The Conditional Durability of Insulation At i.h.  0

t t

wc

Time Discretization Period The Value of Start-Up Time of IM with Broken Bars The Value of Start-Up Time of a Healthy IM 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 Stator Phases Voltage The Amplitude Value of Stator Phases Voltage The Number of Turns in the Slot Connected in Series The Number of the Coil Turns

Xb

Bar Inductive Reactance

y

Winding Pitch in Slots Rotor Bars Ordinal Numbers

tst .br . tst .heal .

t st

UA, UB, UC Um w

Z  1,2... z    j Z 1 Z1 , Z 2 Zb1

The Operator of Inverse Z- Transform

Zb34

α

bev 

The Number of Teeth of IM Stator and Rotor The Impedance of Rotor Bars Rotor Bar Impedance in a Complex Form

Zb Z scr1

Arbitrary Complex Variable

Z scr 34

The Impedance of Short-Circuited Ring The Angle of Phase Zone 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 The Temperature of Heating of the Stator Windings of IM with Broken Bars The Temperature of Heating of the Rotor Bars of IM With Broken Bars The Temperature of Insulation Heating

1 2 i.h.

max max.all.

rat  

i ,  j

Maximum Operating Insulation Temperature Under Nominal Condition The Allowable Excess of Temperature 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 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



a

Motion Speed in Cartesian Coordinates for Motionless Media The Density of Electric Charges Pole Pitch Rotor Time Constant

i.d .

Insulation Durability at Temperature i.h.

s

The Time Constant of Decrease of the Rotor Rotation Frequency Magnetic Flux Scalar Magnetic Potential

 

Φ 

List of Symbols  ph

t  

Spatial Angle Between The Corresponding Phase of the Stator and the N-Th Bar of the Rotor Phi Scaling Function

 A ,  B , C

Coordinate Origin Flux Linkages for Rotor Bars in the SelfRunning-Out Mode Stator Phases Flux Linkages

 a , b ,  c

Rotor Phases Flux Linkages

 ph

The Full Flux Linkage of Stator Winding Phase Time Psi-Function, Wavelet-Function

1 ,  2 ,  Z

 t  2

 0 2 r

xix

The Mechanical Angular Speed Of Rotor Rotation The Current Electric Angular Speed Of IM The Electric Angular Speed of the Field of Stator Electric Angular Speed 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.

Damage location stator core

No.

1.

Number of failures,% 2

7.

Number of failures,% 17

2.

5

8.

11

3.

9

9.

3

4. 5. 6.

16 3 15

rotor slot wedges rotor cage bars bearings fan rotor interference with the stator

Damage location stator winding slot parts stator end windings winding joint

10. 11. 12. 13.

3 8 3 6

connections rotor winding contact rings other damages

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. 1. 2. 3. 4. 5.

PJSC «AutoKrAZ» production shop Machine assembly shop Tool production administration Experimental shop Electrotechnical shop Total

Total number of IM, pcs. 985 425

Number of broken IMs pcs. % 70 7.1 63 14.8

97 35 1542

5 3 141

5.2 8.6 9.1

4

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

1. 2. 3. 4. 5.

PJSC “AvtoKrAZ” production shop Machine assembly shop Tool production administration Experimental shop Electrotechnical shop Total

Number of broken IMs With broken With broken mechanical part stator windings pcs. % pcs. % 6 8.6 59 84

With broken rotor bars pcs. % 5 7.1

10

15.9

48

76.2

6

9.5

1

20

3

60

1

20

1

33.5

2

66.5

–

–

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

The Contemporary State of the Problem of the Diagnostics …

5

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

8.9 8,9

bars burning-out in the slot bars break beside the shortcircuited ring on two sides

13,5 13.5

bars break beside the shortcircuited ring on one side

51,4 51.4

bar break off the shortcircuited ring bar break in a slot 0

10

20

30

40

50

%

Figure 1.2. The distribution of rotor breakages for series A motors, size 11.

60

6

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov copper winding

aluminum winding

2.3 2,3

bars burning-out in the slot bars break beside the shortcircuited ring on two sides

16.7 16,7

bars break beside the shortcircuited ring on one side

36.7 36,7

bar break off the shortcircuited ring

31,9 31.9 2.3 2,3

bar break in a slot 0

5

10

15

20

25

30

35

40

%

Figure 1.3. The distribution of rotor breakages for series A motors, size 12. copper winding

aluminum winding

bars burning-out in the slot bars break beside the shortcircuited ring on two sides bars break beside the shortcircuited ring on one side

21.4 21,4

bar break off the shortcircuited ring

20 20

bar break in a slot 0

5

10

15

20

25

%

Figure 1.4. The distribution of rotor breakages for series A motors, size 13.

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.

The Contemporary State of the Problem of the Diagnostics …

7

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 squirrelcage 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 phasesequence 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].

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

The Contemporary State of the Problem of the Diagnostics …

9

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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

The Contemporary State of the Problem of the Diagnostics …

11

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

The Contemporary State of the Problem of the Diagnostics …

13

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,

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

The Contemporary State of the Problem of the Diagnostics …

15

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

The Contemporary State of the Problem of the Diagnostics …

17

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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 

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

The Contemporary State of the Problem of the Diagnostics …

19

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 mechanical and electromechanical

vibration temperature control

physical values used in induction motor diagnostics electric

current (phase, start, reverse)

electromagnetic magnetic flux and magnetic 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 with stalled rotor

motor self-running-out

Figure 1.7. IM operation modes during the diagnostics of broken rotor bars.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

The Contemporary State of the Problem of the Diagnostics …

21

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 processing

methods of making decisions

spectral analysis of the signal

comparison with the threshold value

wavelet-analysis

comparison with standard models

approximation

logic rules

determination of local extremums

artificial neural networks

obtaining integral coefficients

fuzzy logic 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:

22

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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.

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23

1.4. CONCLUSION Thus, the performed analysis of statistical data has revealed that most IM breakages occur due to broken stator windings and squirrelcage 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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 selfrunning-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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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]: k

Z2  1,  k  0, 1, 2, 3 ... , p

(2.1)

and a series of harmonics, rotating inversely with ordinal numbers: k

Z2  1,  k  1, 2, 3 ... . p

(2.2)

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

Theoretical Foundation for the Research of Induction Motor … 29 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 ; p  i      1    cos jZ 2      t   ,  j 2   2 p j   

(2.4)

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]:

30

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov  iZ     (t )   1  2  1    i cos  1   1  i  p   jZ      j cos  2 (  2t )   j  p     Z 1    i  j cos ( jZ 2  iZ1 )  j 2 2t   2 i j p p   

(2.5)

   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).

Theoretical Foundation for the Research of Induction Motor … 31 B, r.u. 0.6 0.2 -0.2 0

0.008

0.016

0.024

0.032

t, s

-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 ; t B  a H ;

rotE  

j

(2.7)

1 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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 x

2



 2 Az y

2

 

Az  J z _ out , t

(2.8)

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:

Theoretical Foundation for the Research of Induction Motor … 33 Az ; y A By   z . x

Bx 

(2.9)

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:  ph 

2l1w  Az dS , Ss S

(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: e ph (t )  

d  ph  t  dt

.

(2.12)

Thus, EMF in IM stator windings is a derivative caused by the presence of motor electromagnetic field.

34

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

-2.5

a Figure 2.2. (Continued)

0.04

0.05

t, s

Theoretical Foundation for the Research of Induction Motor … 35 E, V 1 0 0.01

0.02

0.04

0.03

0.05

0.06

t, s

-1 -2

1 broken bar

2 broken bars

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.

36

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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].

Theoretical Foundation for the Research of Induction Motor … 37 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

38

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

Theoretical Foundation for the Research of Induction Motor … 39 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-runningout 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;

40

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov   

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: s

0   . 0

(2.14)

I,A 1 0.1 0.01 10 3 10 4 0

50

100

Figure 2.4. The current spectrum of a healthy IM.

150

f, Hz

Theoretical Foundation for the Research of Induction Motor … 41 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).

42

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

Theoretical Foundation for the Research of Induction Motor … 43 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.

44

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

Theoretical Foundation for the Research of Induction Motor … 45 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

46

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 highfrequency 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).

Theoretical Foundation for the Research of Induction Motor … 47 





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 highfrequency 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 frequencytime 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

48

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 ,    fm   

(2.15)

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

Theoretical Foundation for the Research of Induction Motor … 49 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

φ

ψ

ψ wavelet function psi

1.5 1

wavelet function psi

0.5

0.5 0 1

2

3

4

5

6

-0.5

0

j

1

2

3

4

5

6

7

j

-0.5 scaling function phi

-1

a

b φ 1

ψ wavelet function psi

0.5

0 -0.5

5 scaling function phi

10

15

20

j

c 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

50

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 0 -1 100

200

300 400 500 600 700 800 Ca,b Coefficients - Coloration mode : init + by scale + abs

900

1000

900

1000

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 0 -1 100

200

300 400 500 600 700 800 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

c Figure 2.10. Sinusoid signal with a high-frequency component and its waveletspectrum with the use of Daubechies bases (а), Symlet bases (b) and Coiflets bases (c).

Theoretical Foundation for the Research of Induction Motor … 51 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 highfrequency 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:

52

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov   

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



0

Figure 2.11. The vectors of the coils testing EMF. A1, A2, A3, r. u. 1.8 1.2 A1 ( t )

0.6

A3 ( t )

0

A2 ( t )

0.008

t, s 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.

Theoretical Foundation for the Research of Induction Motor … 53 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 waveletcoefficients. 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, r. u.

e te s t 1 ( t ) e te s t 2 ( t )

1.8 e te st 3 ( t )

0.6 -0.6

t, s 0

0.008

0.016

0.024

0.032

-1.8

Figure 2.13. The test signals of stator winding coils EMF, approximate to the anticipated EMF signals.

54

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

a

b

c 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).

Theoretical Foundation for the Research of Induction Motor … 55

a

b

c Figure 2.15. The test signal of coil EMF etest2 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).

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

a

b

c Figure 2.16. The test signal of coil EMF etest3 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).

Theoretical Foundation for the Research of Induction Motor … 57 Symlet wavelet can be presented in the form [65]: ˆ     ei / 2 m0   / 2   ˆ   / 2  , 

(2.18)

where  K , ˆ    – scaling function: 





1/ 2 ˆ      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   

(2.21)

P  y  – polynomial of the type: P y 

N 1  N



k 0 

1 k  k y . k 

(2.22)

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).

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

Figure 2.17. The test wavelet-spectrum.

signal of stator winding coil group EMF etestΣ and its

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

Theoretical Foundation for the Research of Induction Motor … 59 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

a

MMF

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

Theoretical Foundation for the Research of Induction Motor … 61 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

/2



Ec 3



/2   q q 3

0 a

b

Ec 3 

Eq

Ec 2



 

0

Ec1

c 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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 , i 1

(2.24)

.

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

Theoretical Foundation for the Research of Induction Motor … 63 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.

Theoretical Foundation for the Research of Induction Motor … 65 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 selfrunning-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.

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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.

Theoretical Foundation for the Research of Induction Motor … 67 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 wavelettransform 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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

Published data IM

IM circuit mathematical models

Number of the rotor bars Value of electromagnetic parameters of IM stator and rotor windings Rotor geometric parameters

IM circuit mathematical model in a three-phase coordinates systems with presentation of the rotor as a system of short-circuited bars

Currents in the rotor bars

Number of the rotor bars Resistance of the rotor bars and ShCR IM geometric parameters Magnetic and electric properties of IM materials

Parameters of model calculation (number of rotor rotation periods, angle of rotor rotation)

For determination of distribution of currents in the rotor bars at the moment of IM disconnection from the supply mains taking into account the mode of motors operation

Circuit mathematical model on the basis of IM rotor equivalent circuit

For specification of distribution of currents in the rotor bars at the moment of IM disconnection from the supply mains taking into account the broken bars relative position

Specified values of currents in the rotor bars

Mathematical model of a twodimensional electromagnetic field in IM cross section with the use of FEM

For determination of instantaneous values of the vector magnetic potential in the stator windings by the results of calculation of magnetic field in IM cross section

Instantaneous values of vector magnetic potential in IM stator 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 ,

where L2 – rotor inductance; R2 – rotor resistance.

(3.2)

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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  dt  d C (t )  uC (t )  iC (t ) RC  dt , 

(3.3)

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

RA  RB  RC  Rs – stator phases resistances.

phases

flux

linkages;

Mathematical Models for the Research of the Method …

73

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  dt   d c  t  , 0  ic  t  Rc  dt 

(3.4)

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  d  3 (t ) 0  i (t ) R  ; 3 3 dt  ...  0  i (t ) R  d  Z 1 (t ) ; Z 1 Z 1  dt  d  ( t ) Z 0  i (t ) R  , Z Z  dt

where Z  1,2... – rotor bars ordinal numbers.

(3.5)

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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  ...  M A Z 1iZ 1  M AZ iZ ,

(3.6)

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   M 24  M 25  M 26  ...  M 2 Z 1  M 2 Z 

(3.8)

 ...  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:

Mathematical Models for the Research of the Method …

75

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   ... 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:

(3.10)

76

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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    Z  Z    2    MiZ cos     Z  1  . Z  

(3.12)

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    Z  Z    2    MiZ cos     Z  1  . Z  

(3.13)

Mathematical Models for the Research of the Method …

77

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 dt



 LA  M s

 dtA  M dt1 cos   M i1 sin   di

di

di2 2  2    cos      M i2 sin     dt Z Z     di 2  2     M 3 cos    2   M i3 sin    2   dt Z  Z   

M

...  M

diZ  1 2   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 .

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Analogously, the equations of derivatives from flux linkages of stator phases B and C: di dB 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    M

di3 2 2  2 2    cos     2   M i3 sin    2  dt 3 Z  3 Z   

...  M

diZ 1 2 2   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  

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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    

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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: di d 1 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    ...  M

diZ 1 2  2    cos     Z  2    M iZ 1 sin     Z  2    dt Z  Z   

diZ 2  2    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  

M

diC 2  2     M S iB sin    cos      MS  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:

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81

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: Me 

2p 3 3

 C   B  iA    A  C  iB    B   A  iC  .

(3.16)

Rotor motion equation: d 1  p  Me  Mc  . dt J

(3.17)

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 Ir1,Ir2...ІrZ

IA,IB,IC

STATOR

ΨA,ΨB,ΨC γ(t)

Ir1 Ir2 ROTOR

M(t) IA,IB,IC MECHANICAL ω(t) PART

... ІrZ

ΨA,ΨB,ΨC

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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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   2 2    MiZ 1 cos      Z  2   3 Z   2 2    MiZ cos      Z  1  . 3 Z  

(3.18)

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:

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83

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:

84

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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

20

0

0.501

0.15

0.53

0.559

0.3

0.588

t, s

t, s

0.45

 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).

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85

Ir,A 1 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 shortcircuited 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 1 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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

E2

Zb2

E3

Zb3

Zscr1

Zscr2

Zscr1

Zscr3

Zscr3 E34

Zscr34

Figure 3.7. IM rotor equivalent circuit.

Zscr2

Zb34

Zscr34

Mathematical Models for the Research of the Method … In Figure 3.7: E1 investigated IM; Z scr1

Zb1

87

E34 – the EMF of power supply for the Zb34

– the impedance of rotor bars;

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

5

10

15

20

25

30

Z2

35

-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

5

10

15

20

25

30

Z2

one broken bar

-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.

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89

Ib, A 15 10 5 0 -5 -10

0

5

10

15

20

25

30

Z2

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

Mathematical Models for the Research of the Method …

91

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 planeparallel; 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.

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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 LUAscript 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.

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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.

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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 selfrunning-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

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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).

96

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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.

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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 20

one broken rotor bar

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. Е, V 40

one broken rotor bar

20

0

0.01

0.02

0.03

0.04 t, s

 20  40

one broken rotor bar

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov one broken rotor bar

Е, V 100

0

0.01

100

0.02

0.03

0.04 t, s

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 with one broken rotor bar.

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

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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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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 with two broken rotor bars.

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.

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one broken rotor bar

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. one broken rotor bar

Е, V 40 20

0.01

0

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 with one broken rotor bar.

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

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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 200

Е, V 150

1 2

2 1

150

100

100

50

50 0

0.005

a

0.01 t, s

0

0.005

0.01 t, s

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).

Mathematical Models for the Research of the Method … Е, V

103

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 threephase coordinate system with the representation of the rotor as a system

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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 crosssection 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).

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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.

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107

Figure 4.4. The EMF signal of the stator winding phase of an IM with one broken rotor bar and its wavelet-spectrum.

Figure 4.5. The EMF signal of one active side of the coil of the stator winding of an IM 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.

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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.

Figure 4.9. The EMF signal of one active side of the coil of the stator winding of an IM with three broken rotor bars and its wavelet-spectrum.

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109

Figure 4.10. The EMF signal of the stator winding coil of an IM with three broken rotor bars and its wavelet-spectrum.

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 rotor bars and its wavelet-spectrum.

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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 highfrequency 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:

The Method of Induction Motor Broken Rotor Bars …

111

A

K a 

k a

a

,

l

(4.1)

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 waveletexpansion 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 one broken rotor bar three broken rotor bars

Figure 4.13. The functions of the average values of wavelet-expansion coefficients sums for the medium-frequency area.

Figure 4.14 contains function K * a 

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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov K

  a

1 – one broken rotor bar 2 – two broken rotor bars

19.2 %

,

%

2

2 8.9 %

13.7

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 waveletexpansion 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.

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113

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

E AB

EC ECA

 EB 120

EA

120

EB

 EC

E BC

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.

The Method of Induction Motor Broken Rotor Bars …

115

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: e  k   e  k t   TZ e  k t  

K

 ek z k

k 0

 E z ,

(4.2)

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,: E  z 

K

K

k 0

k 0

 e  k  zk   e  k  n zk

K

z n  e  k  n  z k n  z n E  z  . 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.

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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. 1st stage

E, V

phase EMF

100

Division of the phase

0

0.01

0.02

0.03

0.04 t, s

-100

EMF signal into signals of EMF of the coil groups

EMF of the coil group 1

E, V 100

0

nd

2 stage

0.01

0.02

0.03

0.04 t, s

group EMF signal into

EMF signal into

0

0.01

0.02

0.03

coils active sides

EMF of the coil active sides 1 broken bars

E, V

EMF of the coil 3

E, V

0.01

0.02

0.03

0.04 t, s

-20

0

0.01

0.02

0.03

0.04 t, s

-20

EMF of the coil active sides 2

10 0

-10

0.04 t, s

20 0

0.04 t, s

10

signals of EMF of the

0.03

20

-20

E, V

0.02

EMF of the coil 2

E, V

broken bars

20

3rd stage

0.01

-100

E, V

signals of the coils EMF

Division of the coil

0

-100

EMF of the coil 1

Division of the coil

EMF of the coil group 2 E, V 100

broken bars

0.01

0.02

0.03

0

0.04 t, s

0.01

0.02

0.03

0.04 t, s

-10

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 shown in Figure 4.17. To simplify the decomposition the following definitions of the signals are introduced:

The Method of Induction Motor Broken Rotor Bars … 



117

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 

Eq2  z 

z

k

E q1  z 

Figure 4.18. The block diagram of the decomposition of the signal of the winding phase EMF.

Eq2  z 

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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 wavelettransform 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.

The Method of Induction Motor Broken Rotor Bars …

119

Figure 4.21. The detached signal of the EMF of the coil group and its waveletspectrum.

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

Z1

) (Fig. 4.22).

E q1  z 

Ec2  z 

E cm  z 

z E c1  z 

k

z Ec2  z 

k

E cm  z 

Figure 4.22. The block diagram of the decomposition of the signal of the EMF of the coil group.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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

initial signal

40

detached signal

10 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 Method of Induction Motor Broken Rotor Bars …

121

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 waveletspectrum 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).

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E c1  z 

z

Et 2  z 

k

Et 2  z 

E t1  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 20

detached signal 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 wavelettransform.

The Method of Induction Motor Broken Rotor Bars …

123

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 wavelettransform 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 waveletcoefficients. 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.

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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. 1. 2.

Experiment one broken bar two broken bars

Relative position of the broken bars, degrees no

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

Figure 4.30. The comparison of the initial and detached signals of the EMF of the stator winding coil group of IM with two broken rotor bars ( br .b.  84.7 ). Е, V 50

detached signal

initial signal

25 0.04

0.08

t, s

-25 -50

Figure 4.31. The comparison of the initial and detached signals of the EMF of the

stator winding coil of IM with two broken rotor bars ( br .b.  84.7 ).

The Method of Induction Motor Broken Rotor Bars …

125

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 waveletspectrum.

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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 waveletanalysis 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 waveletspectrum.

The Method of Induction Motor Broken Rotor Bars …

127

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

The Method of Induction Motor Broken Rotor Bars …

129

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

detached signal of the coil group

10

initial signal 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

0.028

0.036

0.044

0.052

t, s

-2 -4 -6 1 – initial signal of the K  a coil 2 – detached signal of the K  a coil

Figure 4.40. The functions of the average values of the sums of the wavelet-expansion 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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 0

1 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

Figure 4.41. The functions of the average values of the sums of the wavelet-expansion 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

The Method of Induction Motor Broken Rotor Bars …

131

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 noload 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.

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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 shown in Figure 5.3.

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 Pass band Input voltage Input voltage shift, max. Nonlinearity, max Measurement accuracy Initial voltage: - minimum - medium - maximum Thermal drift Supply voltage Input resistance

Value 85 kHz ±200 mV (±300 mV max) 0.9 mV 0.3% 1% 1.18 V 2.39 V 3.61 V 4.6 mcV/°С +5V 530 kOhm

136

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov ~ 380 V

QF1

Ua PC

Uc

Ub

BVS

ADC 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.

The Experimental Verification of the Method …

137

V1 DA 1 DC/DC 5/5

XS 4 C1

1

2

4

6

R5 +

C2

C4

C3

C5 DA 3 DA 2 R1

1

1 8

2

XS 2

3

№

Circuit

1 2

U1.1in U1.2in

3

U2.1in

4 5

U2.2in U3.1in

6

U3.2in

7

U4.1in

8

U4.2in

R2

4

HCPL 7800

7

3

6

4

5

Circuit

1

U1out

2

U2out

3

U3out

4 5

U4out +12V

6

-12V

7

GND

8

+5V

8

2

R3

№

7

UD140 U1408А

6 5

R4 C6 R6

V2 DA 4 1 2

V3

DC/DC 5/± 12 V

4 5 6

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

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 noload 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.

The Experimental Verification of the Method …

139

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 shorttime 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.

140

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov 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 selfrunning-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.

The Experimental Verification of the Method …

141

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.

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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 . Pe1 



Pe1.br .

operating Pe1.heal .

under

the

nominal

condition:

, r.u.;

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:

Pe2 

Pe1.br.

Pe1.heal .

, r.u.;

The Experimental Verification of the Method … 

143

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



st .heal .

, r.u.;

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



No.

Number of broken bars

Pe1 , r.u.

Pe2 , r.u.

t st , r.u.

Ì

1. 2. 3. 4.

3% (one broken bar) 6% (two broken bars) 9% (three broken bars) 12% (four broken bars)

1.07 1.21 1.4

1.01 1.23 1.51

1.08 1.7 2.4

1 0.99 0.95

6.0



6.2





 st , r.u.

0.66



– the mathematical modeling of IM with 12% broken bars demonstrates nonoperability 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,

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 generalpurpose 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].

The Experimental Verification of the Method …

145

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



No.

Number of broken bars

1 ,

1. 2. 3. 4.

3% (one broken bar) 6% (two broken bars) 9% (three broken bars) 12% (four broken bars)

152.4 167 187 670



С

2 ,

С

106 209.2 198.6 691



– the mathematical modeling of IM with 12% broken bars demonstrates the nonoperability 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

146

M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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 No.

1. 2. 3. 4.

Number of broken bars

no 3% (one broken bar) 6% (two broken bars) 9% (three broken bars)

Pe1  Pe2 P rat 1 1.04 1.22 1.44

, r.u.

The Experimental Verification of the Method …

147

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;



for 6% of broken rotor bars:   i.d .  6.225 104  e0.07281251.22  0.94 years;



for 9% of broken rotor bars:   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.

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M. V. Zagirnyak, Zh. Iv. Romashykhina and A. P. Kalinov

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.

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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.

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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 Rated power, kW Rated rotational speed, rev/min Efficiency,% Power coefficient, r.u. Rated voltage, V Rated current, А Winding connection Stator windings resistive impedance, Ohm Rotor windings reduced resistive impedance, Ohm Stator winding inductive reactance, Ohm Rotor windings reduced inductive reactance, Ohm

Value 1.5 1395 77 0.81 220/380 6.3/3.6 Δ/Υ 5 3.69 4.35 4.01

164

Appendices Table A2. The geometric parameters of АIR80V4U2 IM

Geometric parameter Length of the stator core, mm Number of the stator slots Number of the rotor slots External diameter of the stator core, mm Internal diameter of the stator core, mm Height of the motor, mm Length of the shaft, mm Diameter of the shaft, mm Diameter of the rotor, mm Height of the rotation axis, mm Length of the rotor, mm Length of the rotor bar, mm Diameter of the bearing, mm Height of the stator slot, mm Upper width of the stator slot, mm Lower width of the rotor bar, mm Upper width of the rotor bar, mm Height of the rotor slot, mm Diameter of the rotor ring, mm Height of the rotor ring, mm Width of the rotor ring, mm

Value 104 36 34 132.6 90.5 184 370 24.9 85.7 80 124.6 106 52.1 20 3 7 4.5 1.2+2.5=3.7 82.2 2.5 9.6

Table А3. The published data of 4АN200L2U3 IM Parameter Rated power, kW Rated rotational speed, rev/min Efficiency,% Power coefficient, r.u. Rated voltage, V Rated current, А Winding connection Parameter Stator windings resistive impedance, Ohm Rotor windings reduced resistive impedance, Ohm Stator winding inductive reactance, Ohm Rotor windings reduced inductive reactance, Ohm

Value 75 2940 92 0.9 220/380 79.5/54.5 Δ/Υ Value 0.14 0.075 0.585 0.643

Appendices

165

Table A4. The geometric parameters of 4АN200L2U3 IM Geometric parameter Length of the stator core, mm Number of the stator slots Number of the rotor slots External diameter of the stator core, mm Internal diameter of the stator core, mm Diameter of the shaft, mm Diameter of the rotor, mm Height of the rotation axis, mm Length of the rotor, mm Length of the rotor bar, mm Diameter of the bearing, mm Height of the stator slot, mm Upper width of the stator slot, mm Lower width of the rotor bar, mm Upper width of the rotor bar, mm Height of the rotor slot, mm Diameter of the rotor ring, mm Height of the rotor ring, mm Width of the rotor ring, mm

Value 170 36 46 349 200 80 198 200 170 170 60 26 4 3 7.5 30 145.75 33.655 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

a

b

c 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

a

b

c Figure B2. The test signal of the EMF of coil etest2 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).

168

Appendices

a

b

c Figure B3. The test signal of the EMF of coil etest3 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

169

a

b

c 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

From 1

K 34

1

8

1 Gain 8 K 34

cos

2

[y]

2 From 18 K 34

cos

cos

2

3 3

Gain 3

K 34

9 cos

4

cos [y]

Gain 10

4 [y]

From 2

From 14 K 34

3 Gain 1 10

K 34

[y]

cos

Gain 9 From 4 cos

[y]

Figure C1. A block diagram for simulating the shift angles between the stator K 34 4 phase A winding and the rotor bars.

From 15

Gain 4 K 34

11

[y] Gain 7

From 10 cos

5

[y]

5 K 34

From 17

5 cos Gain 6

6 6

[y] From 6

12

cos

K 34 Gain 12

6

K 34

cos

Gain 5

Figure C2. A block diagram of IM mechanical part. [y]

cos

cos

[y]

7 7

From 22

8 8

K 34

13

From 7 Gain 11

K 34

7

[y] From 24

Gain 2 [y] From 9

Appendices

171

2 PsiA

34 UA

dPsiA /dt 1 s

PsiA

Integrator 1

IA 1/Ls 1 IA

Gain 3

Gain 2 Rs Gain 1 I1 1

k1

I2 2

Product 4 Gain 4

I3 3

Product 1

Lm

Product 2

Gain 5

I4 4

Product 3

I5 5

Product 5

I6 6

Product 6

I7 7

1/11

Product 8

I8 8

Product 7

I9 9

Product 9

I10 10

Product 10

I11 11

Product 11

I12 12

Product 12

I13 13

cosy

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Product 13

I14 14

I15 15

I16

Product 18

18 19

Product 15

Product 17

16 17

Product 14

Product 16

15

16 I17 17

20 21 22 23

I18 18 I19 19

24 25 26

Product 19 Product 20

Product 21

I20 20 I21 21 I22 22

Product 22

Product 23

Product 24

I24 24

25

Product 26

I26 26 I27 27

Product 27

Product 28

28 29 30 31 32

I23 23

I25 Product 25

27

I28 28 I29 29

Product 29

I30 30

Product 30

I31 31

Product 31

I32 32

Product 32

I33 33

Product 33

Figure C3. A block diagram of IM stator phase A.

33

y y

35

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) mi_addcircprop("3",I3,1) … mi_addcircprop("34",I34,1) 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) handle=openfile("AD2.txt","a"); write(handle,A2,"\n"); closefile(handle); mo_clearblock() mo_groupselectblock(3) A3=mo_blockintegral(1) handle=openfile("AD3.txt","a"); write(handle,A3,"\n"); closefile(handle); mo_clearblock() mo_groupselectblock(10) A10=mo_blockintegral(1) … mo_groupselectblock(30) A30=mo_blockintegral(1) handle=openfile("AD30...txt","a"); write(handle,A30,"\n"); 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 Е, V

177

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 with two broken rotor bars. 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 0.02

0

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 Kremenchuk Mykhailo Ostrohradskyi National University 20, Pershotravneva ul, Kremenchuk, Ukraine Email: [email protected];[email protected] Zhanna Romashykhina Senior Lecturer, PhD (Eng.) Kremenchuk Mykhailo Ostrohradskyi National University 20, Pershotravneva ul, Kremenchuk, Ukraine Email: [email protected] Andrii Kalinov Associate Professor, PhD. (Eng.). Kremenchuk Mykhailo Ostrohradskyi National University 20, Pershotravneva ul, Kremenchuk, Ukraine Email: [email protected]

INDEX A air gap, vii, xiii, 8, 13, 16, 28, 29, 30, 31, 38, 62, 94 angle of shift of stator winding coils, xviii, 51

B bar inductive reactance, 87 block of voltage sensors, 133, 134, 135, 136

C circuit model of IM rotor, 86 circuit-field mathematical model, xi, 90 coefficient of insulation durability, 146 Coiflet wavelets, 48, 49, 53 continuous wavelet transform, xi cross-section of IM, 104 current in a broken bar, 9 current spectrum, 14, 40, 41

D Daubechie wavelets, 48, 49 decomposition of the coil EMF signal, 121, 122 decomposition of the signal of winding phase EMF, 117 diagnostic signal, ix, 16, 20, 21, 25, 27, 35, 39, 42, 45, 65, 66 disconnection from the supply main, xiv, 71, 72, 84, 85, 86, 88, 89, 94, 95, 104, 139 discretization frequency, xiv, 48 discretization period, 48 distribution of currents, 85, 88, 89

E electric machine, vii, xi, 2, 10, 28, 58, 90, 159 electromagnetic field, ix, xi, 18, 31, 32, 33, 34, 35, 38, 39, 66, 71, 89, 90, 91, 92, 93, 94, 103, 104, 117, 121, 126, 130, 150, 172

182

Index

electromotive force, viii, ix, xi, 31, 58, 94, 105, 113, 114, 127, 149, 160, 162, 166, 175 electromotive force of the winding phase, 127 experimental signal, 140, 141

F finite difference method, xi, 36 finite element method, xi, 36, 66, 69, 92, 93, 104, 158, 160 finite elements grid, 38, 94 flux linkage, xix, 33, 72, 73, 74, 76, 77, 78, 79, 80, 82 Fourier transform, viii, 14, 40, 41, 42, 45, 67, 156

H high-frequency components, 43, 110

I impedance of rotor bars, xvii, 87 impedance of short-circuited ring, xvii, 87 inverse z-transform, ix, 127, 131, 149, 150

magnetic induction, xiii, 14, 15, 28, 30, 31, 32, 62, 92, 94 mathematical model, ix, 69, 70, 71, 72, 79, 81, 83, 85, 86, 89, 90, 100, 103, 104, 143, 145, 170 mutual inductance, xv, 74, 75

N nominal condition, xvi, xviii, 142, 144 number of IM poles pairs, xvi, 29 number of poles, 60 number of slots per a pole and a phase, 59, 60 number of stator slots, 60, 150 number of the coil turns, xvii, 61 number of turns in the slot, xvii, 33 number of wavelet-expansion coefficients, xv, 111

O orthogonal wavelet, 43, 44, 47, 48, 67, 105

P permeability, 31 phi scaling function, xix, 43

L R losses in IM windings, xvi, 144 losses in the rotor of IM, xvi, 142 losses in the stator windings of IM, xvi, 142

M magnetic field, vii, viii, xiv, 13, 14, 18, 28, 29, 30, 31, 34, 61, 90, 156, 157, 161 magnetic flux, 8, 27, 33, 72, 94, 95

relative permeances of IM stator, xviii rotor bar current, 84, 87 rotor current, 71, 73 rotor rotation angle, xvii, 51, 75 rotor time constant, 102

S scalar magnetic potential, xviii, 36

Index self-running-out mode, xix, 34, 35, 39, 42, 63, 65, 66, 71, 82, 83, 84, 92, 93, 94, 103, 104, 140 short-circuited ring, xi, 5, 7, 85, 86, 87, 91, 138 skew angle, xviii, 62, 63 spectral analysis, viii, 20, 22, 39, 40, 41, 42, 67 squirrel-cage rotor, vii, xi, 7, 9, 23 stator current, vii, 12, 15, 71, 72, 142, 153, 155, 158, 160 stator phase voltage, 138, 139 stator slot, xvi, 33, 34, 63, 116, 117, 164, 165, 174 Symlet wavelets, 48, 49, 53

T temperature of heating of the rotor bars, xviii, 143

183

temperature of heating of the stator windings, xviii, 143, 145 temperature of insulation heating, xviii, 146 three-phase coordinate system, 71, 72, 85, 103 time psi-function, 43

V vector magnetic potential, xi, xiii, 32, 33, 92, 94, 174

W wavelet order, 57 wavelet-basis, xi, 44, 45, 47, 53, 166 winding pitch in slots, 60 winding type, 60, 150